On March 11, 2004 a series of bombs exploded aboard four commuter trains in Madrid, Spain, killing 192 people and injuring 2050.

The Spanish police recovered a bag containing detonating devices which had a latent fingerprint that the Spanish shared with the United States Federal Bureau of Investigation (FBI).

The FBI apparently ran a check on the fingerprint using the FBI’s Automatic Fingerprint Identification System (AFIS). AFIS uses a pattern recognition algorithm to generate an ordered list of possibly matching fingerprints.

One of the fingerprints in this list matched Brandon Mayfield, a Muslim American attorney from the Portland, Oregon region. Expert latent fingerprint examiners from the FBI proceeded to positively identify the fingerprint from the Madrid train bombing as belonging to Brandon Mayfield; at least, this is what the FBI claimed at the time.

Mayfield was arrested as a material witness in the bombing and a great deal of information about him seems to have been leaked to the press. Meanwhile, the Spanish police matched the fingerprint to an Algerian man whom they arrested. The Spanish police directly challenged the FBI identification of Mayfield, leading to his eventual release.

Mayfield later successfully sued the FBI for his treatment.

A Single Fingerprint

Fingerprint identification has been in widespread use in the United States since the 1920′s where popular culture has, until recently, held that fingerprints are unique. According to some reports, people have even been executed based solely on a fingerprint identification.

This is striking since most human and automatic pattern recognition abilities and algorithms have significant false positive and false negative error rates. The author had the experience in 2002, prior to the Mayfield case, of trying to locate scientific studies confirming the accuracy of latent fingerprint identification without success. In fact, there have been a number of cases prior to the Brandon Mayfield case in which fingerprint identification was shown to have been wrong.

The suspect had an airtight alibi. DNA tests contradicted the fingerprint identification and cleared the suspect. These cases of incorrect fingerprint identification have always been blamed on fraud or error by the human latent fingerprint examiners rather than a case of identical fingerprints.

Some facts about fingerprints. Identical twins usually, perhaps always, do not have the same fingerprints. This means a fingerprint test can discriminate between the otherwise identical twins. Dramatic demonstrations of this remarkable fact helped convince juries in the 1920′s to accept fingerprint identification. However, these demonstrations do not constitute rigorous scientific statistical studies of the accuracy of latent fingerprint identification.

A small minority of people do not have fingerprints. Some people have fingers with very shallow ridges which, in practice, makes fingerprint identification more difficult. Contrary to some claims, fingerprints can be altered by scarring and wear and tear. Automatic fingerprint recognition algorithms have had substantial problems with people who work with their hands.

Discussions of the accuracy of fingerprint identification often confuse the accuracy for comparisons of all ten prints, all ten fingers, and the rates for a single or a few prints. For example, automatic fingerprint recognition algorithms were very accurate with all ten fingerprints in 2002 but much less accurate for a single print such as the forefinger or thumb. It is difficult to get all ten prints in the real world.

Latent fingerprint identification is performed by human examiners. There are automatic fingerprint recognition programs such as AFIS but these are probably not as accurate as human beings. This is not unusual. In general, human pattern recognition abilities are significantly better than automatic methods based on mathematical, statistical, or scientific methods: artificial intelligence, pattern recognition, machine learning, and other synonyms.

This is something to keep in mind when scientists, attorneys, or others denigrate eyewitness testimony. Nonetheless, human pattern recognition abilities are imperfect. There are false positive and false negative rates. Eyewitnesses do misidentify people and objects. Human fingerprint examiners almost certainly have non-zero false positive and false negative rates.

In the wake of the Mayfield case, an FBI Laboratory review committee evaluated the scientific basis of friction ridge examination (fingerprint identification) and recommended scientific research including a study of the accuracy of latent fingerprint examiners (!).

The National Research Council (NRC) also identified the need for evaluations of fingerprint examination decisions in a study in 2009. The FBI recently published a report on such a study in the Proceeding of the National Academy of Sciences (Accuracy and reliability of forensic latent fingerprint decisions, PNAS, April 25, 2011). This study found a 0.1% false positive rate and a 7.5% false negative rate.

It is worth considering this for a moment. Fingerprint identification is in widespread use in the United States. People are routinely convicted or cleared of crimes based solely or in part on fingerprint identification. Fingerprints have long been portrayed and perceived as unique.

Fingerprint identification is usually perceived as a highly scientific form of identification. Yet basic scientific studies of the accuracy of the technique appear to have been lacking until recently. This lack has only become apparent recently as unfavorable comparisons to the seemingly rigorous statistical basis of DNA profiling (formerly known as DNA fingerprinting) have been made as well as the extensive publicity received by the Mayfield case, much higher than previous misidentifications which lacked the post 9/11 terrorism angle.

The uniqueness of fingerprints seems to be one of those things that “everyone knows” that has a remarkably weak basis in fact. Indeed, this seems to happen from time to time in supposedly fact-based scientific and engineering fields. A significant number of scientific and technological breakthroughs have occurred when someone went back and questioned the underlying evidence or data behind something “everyone knew.”

In most crimes, only one or a few partial fingerprints are recovered, such as the thumb and forefinger used to hold an object, e.g. the bag in the Madrid train bombing. There are over six billion people on Earth. Suppose that one in a million people have the same or essentially the same partial prints; an examiner cannot tell the difference. This means that, in fact, there would be about six thousand (6,000) possible matches including the guilty party.

With automobiles, trains, and especially air travel, it is probable that a substantial proportion of these six thousand suspects live within traveling time of the crime and lack an alibi. There was a small possibility that Brandon Mayfield traveled secretly from Portland, Oregon in the United States to Madrid, Spain to participate in the bombings. Unlikely, but certainly possible. Even a very small false positive rate raises a reasonable doubt.

Especially since the Mayfield case, there has been more questioning of the scientific basis of fingerprint identification both by authorities such as the FBI and the National Research Council as well as in popular culture. The TV show Numb3rs, discussed in the previous post The Magical Mathematics of Numb3rs, features an episode, probably inspired by the Mayfield case, in which a man is wrongly convicted due to an error in fingerprint identification.

In their book The Numbers Behind NUMB3RS, mathematicians Gary Lorden and Keith Devlin have a chapter questioning some of the mathematical basis of fingerprint identification. DNA profiling is seemingly based on detailed rigorous scientific studies of the frequency of the various genetic markers used in the DNA tests.

Comparable studies seem to be lacking where fingerprints are concerned, hence the studies of fingerprint identification that the FBI is now performing and publishing. The comparison between DNA profiling and fingerprinting has led to questions about the accuracy of fingerprint identification.

It may also be the case that questions about the scientific basis of fingerprints may be a way of marketing DNA profiling as a more “scientific” and reliable replacement for now “old fashioned” (“legacy” in the parlance of the software industry — usually meaning it works and the market is saturated so we need to sell a new replacement technology) fingerprint identification.

The uncritical acceptance of fingerprint identification for over eighty years, without apparently performing adequate rigorous studies of the accuracy, illustrates the enormous hypnotic power of mathematics and science in our culture. The popular image of mathematics and science is that they give exact, black and white answers.

“Scientific” tests give reliable yes/no answers. The Madrid bomber was Brandon Mayfield. The bomber was not Mayfield. There are no false positive or false negative error rates. Two plus two is four, not 3.999 plus or minus 0.012. Yet, this is very rarely the case in the real world. In fact, one should almost always demand to know the error rates of numbers and be suspicious of numbers quoted without error rates or other qualifications.

Suggested Reading/References

Simon Cole, Suspect Identities: A History of Fingerprinting and Criminal Identification, Harvard University Press, Cambridge, Massachusetts, 2001

Accuracy and reliability of forensic latent fingerprint decisions

Bradford T. Ulery(a), R. Austin Hicklin (a), JoAnn Buscaglia(b),1, and Maria Antonia Roberts(c)

(a) Noblis, 3150 Fairview Park Drive, Falls Church, VA 22042;
(b) Counterterrorism and Forensic Science Research Unit, Federal Bureau of Investigation
Laboratory Division, 2501 Investigation Parkway, Quantico, VA 22135;
(c) Latent Print Support Unit, Federal Bureau of Investigation Laboratory
Division, 2501 Investigation Parkway, Quantico, VA 22135

Edited by Stephen E. Fienberg, Carnegie Mellon University, Pittsburgh, PA, and approved March 31, 2011 (received for review December 16, 2010)

Proceedings of the National Academy of Sciences (PNAS)
April 25, 2011

The Numbers Behind Numb3rs: Solving Crime with Mathematics
Keith Devlin, Ph.D. and Gary Lorden, Ph.D.
Penguin Books, New York, 2007

Credits

The fingerprint image is from Wikimedia Commons and is in the public domain. Fingerprint Image at Wikimedia Commons

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

No related posts.

At present, at any rate, very little evidence exists that great mathematicians and calculating prodigies have been endowed with an exceptional neurobiological structure.  Like the rest of us, experts in arithmetic have to struggle with long calculations and abstruse mathematical concepts.  If they succeed, it is only because they devote a considerable time to this topic and eventually invent well-tuned algorithms and clever shortcuts that any of us could learn if we tried and that are carefully devised to take advantage of our brain’s assets and get get round its limits.

Stanislas Dehaene
The Number Sense, pages 7-8

Eberhardt Rechtin was a system engineer who helped develop the Deep Space Network at the Jet Propulsion Laboratory, served as Director of the Advanced Research Projects Agency (now DARPA), Assistant Secretary of Defense for Telecommunications, chief engineer for Hewlett-Packard, President and CEO of the Aerospace Corporation, and finally a Professor at the University of Southern California before his retirement.  In 1995, he gave an interview to Frederik Nebeker of the IEEE Center for the History of Electrical Engineering on his career which is available on-line here.  This is a far ranging interview covering a lengthy and distinguished career.  In this interview, he discusses his experience with a “graduate barrier course” at Caltech (the California Institute of Technology) while getting his Ph.D. in Electrical Engineering (awarded in 1950).  This was a highly mathematical course titled Electromagnetics taught by Professor William R. Smythe.  According to Rechtin’s account, this course was designed to get rid of Ph.D. students who could not “cut it” at Caltech in the 1940s.  This article explores what the barrier course may or may not have actually been doing.

By his own account, Rechtin had been a straight A student until he took this barrier course.  He flunked the course to his surprise.  Although Caltech allowed students to retake the course, the students who flunked usually failed on a second attempt.  At least according to Rechtin’s account many years later, the odds were very much against him.  He studied the book for the course over the summer, working through problem after problem, apparently without too much success.  Then he realized that every problem in the book had two ways to work out the answer.  One was apparently the standard, brute-force answer which took a long time, too long for the short tests and exams, and was tedious to perform.  This was what he had been doing.  But in every case, there was a quick way to solve the problem by reusing mathematical solutions to other problems that had been worked out by mathematicians or engineers previously.  In his account, he mentions a problem that could be solved quickly using Bessel functions.  He knew nothing about Bessel functions when he took the course the first time.  There was always a “trick” solution to the problems that involved reusing known advanced mathematics.  Rechtin took the course a second time and passed easily according to his account.

Rechtin seems to have interpreted the Electromagnetics barrier course as a sort of intelligence test in which the smarter, better students by Caltech standards would figure out as he did that the problems were solvable by reusing various known pieces of advanced mathematics such as Bessel functions.  He also took it as a lesson for his career, to always look for quick ways to solve a problem by reusing known mathematics or previous work: don’t reinvent the wheel — certainly good advice.  But is that actually what happened; was the effect of the barrier course on other students what Rechtin thought or even what the Caltech professors thought?

The Deliberate Practice Interpretation of the Barrier Course

Deliberate practice is the central concept of K. Anders Ericsson‘s theory of expert performance, which has recently been popularized by science writer Malcolm Gladwell in his book Outliers, previously reviewed in the article Debating Deliberate Practice.  Deliberate practice is somewhat vaguely defined which is one of the major problems with the theory of expert performance.  Ericsson uses the example of the backhand in tennis, which is a relatively rare move in the game.  Tennis players who repeatedly practice rare moves such as the backhand will, in  general, defeat players who do not engage in specific deliberate practice of the backhand or other rare moves.  Someone who engages in deliberate practice of this type may well defeat players with many more years of experience playing the game, but relatively little practice of rare moves.  This is sort of the concept of deliberate practice.  In some contexts, Ericsson uses deliberate practice in a more general way to refer to a process of continuous self-improvement and conscious analysis of one’s performance and errors.

In intellectual activities such as mathematics, the notion is that, especially in a timed contest or exam, if the mathematician encounters a problem that is too complex, lengthy, and so forth to solve from first principles in the limited time available, a few hours for most exams in most college and university courses, the mathematician will fail.  On the other hand a mathematician who has specifically studied and practiced this specific type of problem, such as an electromagnetics problem that is solved with Bessel functions, will solve the problem quickly and easily.   There will be a dramatic difference between the two on many exams.

Ericsson’s theory emphasizes specific knowledge in a specific field or discipline.  Ericsson largely rejects the notion of genius or general intelligence as well as an inborn aptitude for a specific subject.  There are no born mathematicians.  It is all study and practice, and a special kind of practice — deliberate practice.  Deliberate practice is critical to Ericsson’s theory.  There are clearly many examples of mathematicians or chess players or musicians who have many, many years of experience, but do not perform at the expert or “star” level.  Why do some people with a few years of experience, often ten years, outperform people with decades of experience, especially in intellectual activities where physical aging is not as large a factor as in sports?

In fact, the barrier course that Rechtin encountered sounds like a good example of deliberate practice.  The problems apparently required detailed specific knowledge such as a knowledge of Bessel functions.  In the absence of this, the problems took too long to solve in the limited time available, a few hours usually.  Once he figured out what was going on, Rechtin probably spent many hours studying Bessel function and other specific mathematical methods, although he does not explicitly say this in his interview.

What did the barrier course actually do?

It is far from clear what the barrier course actually did or what it was actually supposed to do.  People, families, and cultures have different beliefs and attitudes toward study and practice.  In the United States “rote memorization” or “studying to the test” is generally deprecated and “thinking things through from first principles” or “thinking for yourself” is often glorified, at least in theory.  It is not unique to the United States.  The author has heard parents from India, for example, express concern that their child was not being taught to think things through in school in the United States.  The common stereotype is that Asian cultures such as China and Japan place a strong emphasis on heavy practice.  Students from a background that emphasized practice, drilling, and were already studying technical minutia like Bessel functions would have been likely to easily pass the barrier course.  On the other hand, students who were accustomed to “thinking things through,” and Eberhardt Rechtin sounds very much like this kind of student in his interview, would tend to fail.  It often would not occur to the “think it through” students to engage in deprecated “rote memorization” unless someone told them.  Rechtin is clear that no one, neither the other graduate students nor the faculty, would tell him how to pass the course; he had to figure it out on his own or already know what to do.

In his account, Eberhardt Rechtin interpreted the barrier course as an intellectual puzzle that he figured out.  That is, he thought the problem of the course through and realized that he needed to reuse existing mathematical knowledge such as Bessel functions and this general reasoning insight was the whole point of the barrier course.  Maybe it was.  Maybe it wasn’t.  He probably interpreted what he experienced from his personal and cultural background.  The barrier course could just as easily have had the effect of selecting rather unimaginative students whose high performance was a consequence of heavy drilling and who had poor abilities to think things through.  One can imagine professors eager for unimaginative drones to perform intellectual drudge work and not think things through and ask unwanted or unsettling questions: Professor Millikan, after reviewing your papers, I am pretty sure your theory that cosmic rays are caused by nuclear fusion in deep space is all wrong for reasons X, Y, and Z.

In fact, the effect and the selection of students could have been completely random.  Some students would have figured out the trick immediately without flunking the course, unlike Eberhardt Rechtin.  A few might have figured it out and passed on the second try as Rechtin did.  Many might have simply glided through the course because they were already practicing or rapidly assimilating existing specialized knowledge (maybe they could learn existing knowledge through study — reading a textbook about Bessel functions, for example — with little practice or drilling).  The barrier course could have selected several different types of students.  By his own account, Rechtin’s experience was very unusual; most students who flunked did not pass on the second attempt.

Conclusion

Obviously, one should not draw firm conclusions from a single case, let alone a verbal account of something that happened over forty-five years before.  Nonetheless, Eberhardt Rechtin’s account is similar to other selection procedures that the author has experienced or heard of in graduate programs in mathematical fields such as physics or electrical engineering.  These procedures usually have the ostensible purpose, whether stated or not, of selecting the “best and brightest” as conventionally defined.  They also often serve as a rite of passage, not perhaps unlike boot camp in the Marines or hazing in a fraternity, and this may be their true purpose and function.

For students, there are some probable lessons from this case study.  Some tests and exams can be worked out in the time available from first principles.  This often seems to be true of math and science problems in elementary, middle, and high school (K-12).  An emphasis on first principles and general reasoning methods will likely succeed with these problems, tests, and exams.  Some tests and exams have trick problems that require specific knowledge learned in advance of the test like Bessel functions in Rechtin’s account.  These require specific study and possibly heavy practice to master and overcome. These problems appear to be more common in more advanced math, science, and engineering courses at colleges or universities.  For parents and teachers, it is likely important to teach students to be aware of this and to identify the situation to the extent that this is possible.

This case study also illustrates the difficulty and perhaps impossibility of distinguishing between specific knowledge and hypothetical general intelligence or special aptitude (a born mathematician) using tests and exams.  Is there a mental horsepower and, if so, what is it?  If there is a mental horsepower, is it a single attribute or several?  Did the barrier course select “geniuses” who figured out the trick as Rechtin did or did it select intellectual “drones” who had already memorized the answers or both?  It may be that some exceptionally intelligent students were able to pass the barrier course without the specific knowledge of Bessel functions and other mathematical methods that Rechtin had to acquire through study and practice.  It may be that some students passed due to heavy practice of special methods such as Bessel functions or rapid absorption of existing knowledge through study (whether due to some innate ability to learn existing knowledge easily or studying the right, unusually clear textbook, for example, greatly reducing the need to practice).  The selection of students who could “cut it” may have been largely random.

© 2011 John F. McGowan

About the Author

John F. McGowan, Ph.D. solves problems by developing complex algorithms that embody advanced mathematical and logical concepts, including video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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Prediction is very difficult, especially about the future.

Common quotation, variously attributed to Yogi Berra, Niels Bohr, Mark Twain and others.

Introduction

In Isaac Asimov’s famous science fiction novel Foundation, a group of scientists in the distant future led by Hari Seldon discover a mathematical method to predict the course of future events, anticipating the collapse of the reigning Galactic Empire into a new Dark Age. Armed with the mathematical methods of Hari Seldon’s “psychohistory,” the scientists create a Foundation on the edge of the Galaxy that saves civilization from the prophesied Dark Age. The notion that mathematics can be used to predict human behavior in economics, finance, politics, and many other fields and activities has great appeal. With the spread of increasingly powerful computers, complex mathematical models of economics, finance, and other human activities have become more and more common. Often, the motives are much less admirable than the selfless super scientists of Asimov’s tale. Often, too, the accuracy and performance of the mathematical models has been much less impressive than Asimov’s fictional new science of psychohistory.

In the last few years, complex models of the value of mortgage backed securities have proven disastrously incorrect, a major contributing factor to the Great Recession, the present financial crisis. This is only the latest in a succession of such failures in quantitative finance. Similarly, many sophisticated econometric models of the economy have proven unreliable. To the extent that these models are shaping public policy, personal and corporate investment decisions, and so forth, the pitfalls of mathematical modeling and seemingly abstruse issues in the philosophy of science such as Karl Popper‘s doctrine of falsifiability are having a substantial impact on people and society.

In fact, applying mathematical models to economics, finance, and other human activities is especially treacherous. All mathematical modeling suffers from the deep problem that one can construct an infinite range of functions that approximate current observations and data arbitrarily well and yet make any and all possible predictions about possible new observations. In practice, human beings use various criterion to select mathematical models that are likely to be true, many of which criterion cannot be justified in any rigorous or rational way and some of which criterion are difficult to identify (intuition, “that just doesn’t feel right”, “God doesn’t play dice with the universe.”). Nonetheless, human judgment has a high error rate, though surely much more accurate than a blind guess.

In addition to this pervasive problem, many of the assumptions used in applying mathematical models to physical processes such as the motion of the planets or radioactive decay surely do not apply to economics, finance, or other forms of human behavior. Our expectation is that the motion of the planets is governed by the same “laws” today as last year or last decade or in 1605 when Johannes Kepler first recognized the elliptical orbits of the planets. Indeed, this regularity of many natural phenomenon is strongly born out by experience. On the other hand, the economy or financial markets or other human activities change and evolve. However imperfectly, human beings learn from mistakes, develop new technologies and processes. Human beings are herd animals and prone to fads and fashions that have no parallel in physical phenomena. One would not expect the market for gold to be the same today as in 1960 and it was not. The price of gold was fixed by the US federal government in 1960 and today it is not.

This article takes a look at the price of gold since 1970 when the United States ended the gold standard. Gold is currently rising sharply in price as it did in the late 1970′s and early 1980′s. This article will show how it is possible to construct many different purely symbolic mathematical models of the price of gold that make different predictions. Indeed, one can cook up whatever prediction one would like. The article will also discuss the serious problems with applying the mathematical methods of physics and other “hard” sciences to the price of gold as a specific example of a general problem.

Mathematics for Goldfinger

With the collapse of the Bretton-Woods foreign exchange system (1968-1971), the price of gold, previously fixed to the US dollar, was allowed to float free. Since 1970 the price of an ounce of gold in US dollars has risen substantially.

Nominal Gold Price

The United States government reports a consumer price index (CPI) that is purported to reflect the cost of living in US dollars for a typical US citizen. Although there are some reasons to be skeptical about the accuracy of the CPI in recent years, this article will use the CPI as a proxy for the overall price level in the United States.

United States Consumer Price Index

One can see that the CPI has generally risen at a higher rate since the United States ended the gold standard in 1970. This also corresponds to a period of overall rising energy prices and very limited progress in power and propulsion technologies. Many other areas have seen relatively limited scientific and technological progress during this period. Computers and electronics have, of course, continued their historical trend of rapid progress to the present.

Historically, the US inflation rate was high during the periods of World War I and World War II, but otherwise generally lower than the present period, since 1970. The official inflation rate, the rate of increase of the CPI, was especially high during the 1970s, a period of sharp increases in nominal and real energy prices. The inflation rate derived from the CPI in recent years does not seem to reflect the common experience of rising energy prices since the late 1990s. Nor does it show any evidence of the sharp rise in housing prices in the US from 2002 to 2005; by many estimates, housing prices in urban areas remain high compared to the historical trend.

Investors are generally interested in the real inflation-adjusted value of an investment. The value of gold in real terms fluctuated somewhat during the period of the gold standard but has fluctuated much more since 1970. Particularly notable is the large spike in the real and nominal price of gold in the early 1980′s. In real terms, the current rise in gold price has not reached the level of the spike in the early 1980′s. If inflation has been higher than the official CPI, the current real price of gold would be even lower than the spike in the early 1980s.

One can construct a simple symbolic mathematical model of the real price of gold since 1970 (the gold standard disintegrated between 1968 and August 1971 when the US government ended any attempt to tie the dollar to gold) by using a polynomial with several terms:

Where is the gold price as a function of the time . This is a simple mathematical model with little real-world justification. It will, in fact, make grossly unrealistic predictions that should call it into question. A polynomial model of the time series of real gold prices is:

Polynomial Model of Gold Price (12 Terms)

One can approximate any continuous function as accurately as one would like with a sum of a series of powers. The problem is that a power such as will grow without bound as the independent variable grows. This is usually quite unrealistic. This is a simple illustration of the difference between a symbolic mathematical model of reality and our common sense every day sense of reality.

Apocalypse or Gold Bubble?

Many other mathematical models are possible. In fact, there are an infinite number of functions or curves that agree with the data during the period from 1970 to 2009. In this way, one can predict anything. In mathematical models of physical phenomenon, it is common to try to construct the model from a set of building block functions, or more generally terms such as terms in a differential equation. This may be arbitrary or justified by arguing that the building blocks represent fundamental building blocks of the physical process in some way. In practice, one is trying to capture regularities in the data that may recur in new data. For example, the position of a pendulum is periodic; it repeats over and over again. Simple harmonic motion of this type was one of the first kinds of behavior understood mathematically by the ancients. One can construct models of the gold price data that agree with the data quite well by eye using a collection of peaks rather than periodic functions. One might, for example, represent the seeming peaks in the gold price data as the Gaussian or Normal function, commonly referred to as the “Bell Curve,”

A very simple model is the sum of two Gaussians:

Gold 2 Gaussian Model

This simple model with two Gaussians does not agree very well with the data. It predicts the following future performance:

Gold 2 Gaussian Prediction

It predicts that the real price of gold will peak in about 2025 and then drop. One can get much better agreement between the model and the gold price data by adding more Gaussians, loosely corresponding to the apparent gold peaks in about 1974, 1982, 1987, 1993, and today. The spike in 1982 is especially sharp and can better be approximated by combining two Gaussians.

Gold 7 Gaussian Model

Now the agreement by eye is much better. The curves appear essentially the same. This model makes a different prediction.

Gold 7 Gaussian Prediction

This model predicts a peak in real gold prices in just a few years, about 2012, followed by a decline to essentially zero. One can get significantly different predictions simply by using a different function to model the peaks in the gold price. For example, one can use the Cauchy-Lorentz distribution as a model for the peaks:

where is the time (year in this case), is the location of the peak, and is a measure of the width or dispersion of the peak. Initially, one can try a model with two peaks:

Gold 2 Cauchy Model

This model looks very similar to the two Gaussian model. It predicts something different however.

Gold 2 Cauchy Prediction

In this case, the price of gold trails off slowly instead of dropping to essentially zero in about a decade. This is a difference between the Gaussian and the Cauchy-Lorentz functions. Of course, the agreement between the model and data is not very good. One probably would not and should not trust it. One can achieve better agreement with more peaks, just as in the Gaussian models.

Gold 7 Cauchy Model

Now the agreement is almost exact by eye. The prediction however differs from the seven Gaussian model.

Gold 7 Cauchy Prediction

Again, the price of gold trails off slowly over a period of decades due to the difference between the Gaussian and Cauchy-Lorentz peak models. In fact, one can get essentially any prediction by choosing the appropriate mathematical model. How reasonable are these predictions? There are several theories about the present rise in gold prices. One clear contender is that the gold price rise in the last decade is yet another speculative financial bubble. In this scenario, one would expect the price of gold to drop substantially within a decade. One might also argue that the sharp run-up in gold prices will encourage overproduction of gold and the development of better gold refining, recycling technologies, alternatives to gold, and even perhaps the alchemist’s dream of converting base metals into gold (using nuclear reactors for example). At the other extreme, the rise in gold prices is tied to apocalyptic scenarios in which the US and other governments go bankrupt due to deficit spending and the dollar and other paper currencies becomes as valueless as the German mark during the infamous German hyperinflation. Thus, the real inflation-adjusted price of gold spikes as investors seek an inflation proof haven. Unfortunately, civilization collapses. Gold and other luxury items become valueless in the bitter Darwinian battle for survival in the post-apocalyptic world. The most valuable possessions are a gun, ammunition, and a large horde of dried food — all purchased from advertisers on Alex Jones web site. In either scenario, gold eventually tanks.

All kidding aside, it is usually possible to find technically sophisticated, plausible justifications for mathematical models. Usually does not mean always. The behavior of the polynomial models is grossly unrealistic. How likely is it that the price of gold would go negative, let alone extremely negative — meaning that people would be paying large sums of money to get rid of gold? The evidence that the Earth is nearly spherical with a diameter of about 8000 miles is very strong. Any argument that the Earth is really flat is extremely difficult to make in the present day. Most of us are pretty confident that the Sun will set today and rise again tomorrow and continue to do so for many, many years to come. There is some knowledge that is pretty certain. There are some mathematical models that have proven extremely accurate and reliable.

Nonetheless, it is usually possible to find plausible arguments for mathematical models and conceptual theories. One can usually explain away even grossly contradictory — “falsifying” in the language of Karl Popper — observations or experiments in a technically sophisticated, plausible way. This can create the illusion that a theory or mathematical model falls into the category of almost certain knowledge, our common sense notion of a fact. Once upon a time, many people believed that it was a fact that the Earth was flat. How could it possibly be a sphere? Political power, social status, and sizable funding tend to flow to those who claim certainty, to know hard facts rather than speculative theories. In common thinking, an expert is someone who says “I know the answer.” not someone who says “I don’t know.”

Scientists and science popularizers often seek to promote reigning scientific theories to the level of “almost certain” knowledge or “facts,” such as that the Earth is roughly spherical or the historical fact of the Holocaust. These latter two are the most common analogies cited in the popular science literature. Scientific theories are said to be supported by “overwhelming evidence.” There is a “consensus” that the theory is correct. The theory is no longer a theory, but a fact. In debates about evolution and creation, one encounters the curious claim that evolution is a falsifiable theory, that creationism or “intelligent design” is not falsifiable and thus not scientific, and that evolution is also a fact beyond any reasonable dispute or falsification. Increasingly, malcontents or die-hards who question the “consensus” are “deniers” or “denialists” in analogy to Holocaust deniers, an extremely emotional analogy indeed. There are now evolution deniers, AIDS deniers, global warming deniers, and a growing list of other deniers. Yet, certain knowledge is relatively rare especially in the frontiers of science.

In the case of gold, one can make a strong argument that the models above are unlikely to be correct. There is a certain fundamental demand for gold for industrial uses, in electronics for example. There is a long history of people buying gold for jewelry. The economic and political problems, the fears and the greed, that have historically driven the fluctuations in the price of gold are likely to continue through the 21st century. Thus, one should doubt models that show the real price of gold dropping to zero or nearly zero. Yet, both symbolic mathematics and verbal reasoning enable one to make a plausible argument that this will happen, supported by fancy symbolic mathematics and technical graphs.

A Deep Problem

It is always possible to perfectly match or fit any set of data points using a linear combination of at least linearly independent functions. Under many conditions, a function or composition of building block functions with adjustable parameters can also match or fit any set of data points. In addition, with creativity and trial and error, one can often construct mathematical models with less than parameters or building block functions that nonetheless do a good job of matching or fitting data points. These models may often fail to predict future data or data that lies outside of the region used to make the model or fit.

In general, a composition of a set of building block functions has a sort of plasticity, a certain ability to match any data to some degree. If there are building block functions or adjustable parameters and data points, this plasticity can be rigorously shown to be essentially infinite, that is the model can always match any set of data points. The philosopher of science Karl Popper called such models or theories unfalsifiable. They can never be proven wrong. They predict everything and therefore they also predict nothing. Unlike popular presentations of his doctrine of falsifiability, Popper recognized that falsifiability was often not a black and white distinction. There were degrees of falsifiability and he tried to define some logical and quantitative criterion for the degree of falsifiability. What this means is that a mathematical model or theory with adjustable parameters where is less than the number of data points may still match a broad class of possible observations or data. It can be falsified, perhaps, but only with great difficulty. There is, in the examples above, a lot of freedom to match many different sets of data with seven peaks in the model. The models have 21 adjustable parameters and there are 40 data points (40 years). Although the models cannot exactly match all sets of 40 data points, nonetheless they are likely to be good enough for most purposes. Remaining differences between the data and the models can easily be attributed to noise, measurement errors, or some similar excuse.

In physics, there are some examples of very large data sets that match very simple mathematical formulas. The motions of the planets in the Solar System conform to Kepler’s Laws and Newton’s theory of gravitation to a very high degree. This is not necessarily true for the motion of stars in the Milky Way galaxy, galaxies in galactic clusters and so forth. A deviation from Newtonian gravity is a possible explanation for the anomalies often cited as evidence for so-called “dark matter” or “dark energy.” So too, there is an enormous amount of quantitative data on the vibration of springs and strings, the motion of pendulums, simple radioactive decays, and various other physical phenomena that show precise matches to simple mathematical expressions such as Hooke’s Law for springs or exponential decay for radioactive decay. It is fair to say that the amount of data points is now in the millions or more and the models have only a few adjustable parameters such as the half-life of a radioactive isotope. Consequently, it is probably reasonable to have a high degree of confidence in these models. At the other extreme, it is clear that we should put no confidence in the predictive power of models with as many adjustable parameters or building block functions as the number of data points.

In many situations however including the frontiers of human knowledge and often areas like economics, the situation falls into a gray area. The theories, both symbolic mathematical models and conceptual models, may be complex, but not so complex as to be obviously “unfalsifiable.” The amount of data may be limited or of questionable quality, but still not obviously inadequate to draw conclusions. It is here that faulty human judgment is in practice applied, in part because human judgment, fallible though it undoubtedly is, is still the best option, more reliable than symbolic mathematics or computer programs in many cases. In constructing mathematical models in these cases, human judgment is often hidden in the definition of the symbols and the choice of building block functions or other mathematical components used in the model.

Economics, finance, politics and other human activities are especially treacherous areas for mathematical modeling. The mathematical and scientific methods used in physics and other “hard” sciences were developed to study highly repeatable phenomena that do not appear to change appreciably over time. For example, a “fair” coin will come up heads when flipped on average half the time, tails half the time. This was true in the early days of probability and statistics during the Renaissance. It is true today. Fair games of chance work the same way today as yesterday, last year, last century, and presumably thousands of years ago or in the future. One can collect vast amounts of data on these games. Similarly, physical phenomena such as the motion of the planets, radioactive decay, and so forth appear to behave the same way time after time. They do not evolve, learn from mistakes, forget lessons learned, or suddenly change for difficult to understand reasons. None of this is true of much economic or financial data.

In the example of gold, the behavior of gold changed radically from 1968 to August 1971 when the gold standard ended. Since 1970, there have been extensive political, economic, and technological changes that probably effect the price of gold. The Cold War ended. Apartheid in gold producing South Africa ended, followed by a large exodus of the white minority who dominated the mining industry. The use of electronics has increased; gold is used in electronics. Women’s fashions in clothing and jewelry have changed. The terrorist attacks of September 11, 2001 probably contributed to general unease and the rise in gold prices. Against these many changes, there is only forty years of data on gold prices. Day to day gold prices are correlated and gold seems to follow trends over several years, seemingly rising and falling in peaks. Thus, there is very limited data to analyze compared to a physical process. At the same time, one still has the freedom to construct many different mathematical models, theoretically an infinite number.

This ability to construct many different models that agree with observations or experiment but which make quite different predictions is not a problem unique to symbolic mathematical models. Rather, we encounter the same or a similar problem in everyday life, in politics, in personal relationships, where concepts, words, and pictures are the norm rather than symbolic mathematics with its illusion of certainty. It occurs when opposing attorneys are able to present radically different interpretations of the same evidence in a court case. It occurs when political activists explain the same events in radically different ways that almost always confirm their beliefs. It occurs in disputes between co-workers where each sees the same event or problem quite differently (it is all your fault). It occurs in conflicts between husbands and wives when each sees the same events differently. If verbal concepts and mental pictures are actually mathematical models maintained in the brain (but not in a symbolic way), then it may well be exactly the same problem as that encountered in mathematical modeling.

There is no doubt that human judgment is faulty and limited. In some relatively rare cases mathematics or formal logic can clearly outperform human judgment. Nonetheless, in many situations, human judgment and intuition still win out over mathematics, formal logic, or computer programs. It remains an unsolved and perhaps unsolvable problem to find a way to select the right model that, in fact, predicts new observations as well or even better than human judgment.

The nature and origin of human judgment and intuition remains an enigma. Governments have spent billions of dollars and decades on artificial intelligence in the mostly futile effort to replicate even sometimes seemingly simple aspects of human reasoning. Human beings often cannot explain either verbally or in symbolic logical or mathematical ways their successful reasoning processes. Ancient scholars and philosophers might attribute their ideas to divine inspiration or mystical insight. Indeed historical accounts of inventions and discoveries are replete with reported sudden insights or realizations such as the famous story of Archimedes in his bathtub suddenly realizing how to determine the gold content of the King’s crown without destroying the crown and then racing naked through the streets of Syracuse shouting “Heureka!” One may wonder what Archimedes feared the King would do to him if he had failed to solve the problem. The modern scientific view would probably attribute this ability of the human mind to find the right answer to an anthropic or evolutionary cause. Our brain incorporates billions of years of evolution and is thus tuned to the mysterious underlying logical or mathematical structure of the universe.

Conclusion

It is important to realize that one can construct an infinite number of mathematical models that match a set of data. In some sense, one can construct an even “larger” number of mathematical models that match a set of data “well enough,” where remaining differences can easily be attributed to noise, instrument error, or minor effects that can be ignored for practical purposes. In high school and college, one is often exposed to mathematics and geometry as a rigorous deductive system. The epitome of this is Euclid’s geometry which many people are exposed to in high school; high school math courses typically teach the first three of Euclid’s thirteen books. One starts with axioms and definitions that often seem self-evident, with the possible exception of the so-called parallel postulate. One can apply a sequence of logical steps to get a precise unambiguous answer. Similarly, high school and college science courses frequently focus primarily on extremely well-measured phenomena such as vibrating springs or radioactive decay that precisely follow simple mathematical laws. Scientists are often described as “deriving,” “figuring out,” or “deducing” mathematical laws such as Newton’s Theory of Gravitation, Maxwell’s Equations of Electromagnetism, or Schrodinger’s Equation for quantum mechanics. The implication is that these mathematical theories can be found by the application of rigorous mathematical or logical rules, much in the way that theorems in Euclidean geometry are proven. This has a great appeal compared to messy, fallible human judgment and intuition. But the reality is that the theories were found through the application of messy, fallible human judgment and intuition, perhaps assisted by some mathematics, by model fitting methods, and so forth, but in the end it was mysterious human judgment and intuition.

If we could understand what human beings are actually doing, this would be a great advance. It would be an even greater advance to find a way to improve human judgment and intuition, which is certainly quite fallible.

In the meantime, it is particularly hazardous to try to apply mathematical modeling to economics, finance, and human behavior. This does not mean that we should not try. Nor does it mean that there may not be successes in applying mathematics to human activity. Indeed, in economics, there are some mathematical rules of thumb that are often correct. It is, for example, generally observed that inflation is lower when unemployment is higher; this relation loosely follows a mathematical curve. However, we are a very long distance from a predictive mathematical method comparable to Isaac Asimov’s fictional psychohistory if this is even possible. Most people don’t think in mathematics; can human behavior really be reduced to mathematics?

The present financial crisis illustrates that these seemingly abstruse issues of mathematical models can impact the lives of many people and organizations. This is, if anything, likely to increase with growing reliance on computers and mathematical models implemented in computer software and hardware. There remains no greater wisdom than the ancient Latin saying: Caveat Emptor (Buyer Beware).

Note: An appendix with the technical details of the analysis and plots presented above follows the Suggested Reading/References section below. This includes the raw data, the models, and the Octave scripts used to fit the models to the annual gold price data. This article is primarily for informational and educational purposes. It is not investment advice.

Copyright © 2010 John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

Suggested Reading/References

Karl Popper, The Logic of Scientific Discovery, Routledge, London, England 2000 (First published 1959, English translation with new notes and appendices of Logik der Forschung, published Vienna, Austria, 1934)

Paul Feyerabend, Against Method (3rd Edition), Verso, 1993

Thomas Kuhn, The Structure of Scientific Revolutions, University Of Chicago Press; 3rd edition (December 15, 1996)

John D. Barrow and Frank J. Tipler, The Anthropic Cosmological Principle, Oxford University Press, New York, 1986

Isaac Asimov, Foundation, Spectra; Revised edition (October 1, 1991)

Roger Lowenstein, When Genius Failed: The Rise and Fall of Long Term Capital Management, Random House, New York, 2000

Emanuel Derman, My Life as a Quant: Reflections on Physics and Finance, John Wiley and Sons, Hoboken, New Jersey, 2004

Charles MacKay, Extraordinary Popular Delusions and the Madness of Crowds, Farrar, Straus, and Giroux, New York, 1932 (first published in London in 1841)

Robert J. Shiller, Irrational Exuberance, Broadway Books, New York, 2000

Dean Baker, False Profits: Recovering from the Bubble Economy, Polipoint Press (January 15, 2010)

Michael Specter, Denialism: how irrational thinking hinders scientific progress, harms the planet, and threatens our lives, Penguin Press, 2009

Seth C. Kalichman, Denying AIDS: Conspiracy Theories, Pseudoscience, and Human Tragedy, Springer, 2009

Bill McKibben, “Hot Mess: Why are conservatives so radical about climate?”, The New Republic, October 6, 2010

Appendix: Technical Details

The annual gold price data used in this article is from the World Gold Council. Here is the actual raw data.

1900    20.67    4.25        o    1900    8.14    0.037941643    544.78            1913-01-01    0.10
1901    20.67    4.25        o    1901    8.24    0.038407756    538.17            1913-01-02    0.10
1902    20.67    4.25        o    1902    8.34    0.03887387    531.72            1913-01-03    0.10
1903    20.67    4.25        o    1903    8.53    0.039759485    519.88            1913-01-04    0.10
1904    20.67    4.25        o    1904    8.63    0.040225599    513.85            1913-01-05    0.10
1905    20.67    4.25        o    1905    8.53    0.039759485    519.88            1913-01-06    0.10
1906    20.67    4.25        o    1906    8.72    0.040645101    508.55            1913-01-07    0.10
1907    20.67    4.25        o    1907    9.11    0.042462944    486.78            1913-01-08    0.10
1908    20.67    4.25        o    1908    8.92    0.041577328    497.15            1913-01-09    0.10
1909    20.67    4.25        o    1909    8.82    0.041111215    502.78            1913-01-10    0.10
1910    20.67    4.25        o    1910    9.21    0.042929058    481.49            1913-01-11    0.10
1911    20.67    4.25        o    1911    9.21    0.042929058    481.49            1913-01-12    0.10
1912    20.67    4.25        o    1912    9.4    0.043814673    471.76            1914-01-01    0.10
1913    20.67    4.25        12/31/1913    1913    9.6    0.04630723    446.37    10        1914-01-02    0.10
1914    20.67    4.25        12/31/1914    1914    9.69    0.046770302    441.95    10.1        1914-01-03    0.10
1915    20.67    4.25        12/31/1915    1915    9.74    0.047696447    433.37    10.3        1914-01-04    0.10
1916    20.67    4.25        12/31/1916    1916    10.64    0.053716387    384.80    11.6        1914-01-05    0.10
1917    20.67    4.25        12/31/1917    1917    12.82    0.063440905    325.82    13.7        1914-01-06    0.10
1918    20.67    4.25        12/31/1918    1918    15.06    0.076406929    270.53    16.5        1914-01-07    0.10
1919    20.67    4.50        12/31/1919    1919    17.3    0.087520665    236.17    18.9        1914-01-08    0.10
1920    20.67    5.60        12/31/1920    1920    20.04    0.089836026    230.09    19.4        1914-01-09    0.10
1921    20.67    5.35        12/31/1921    1921    17.9    0.080111508    258.02    17.3        1914-01-10    0.10
1922    20.67    4.69        12/31/1922    1922    16.77    0.078259219    264.12    16.9        1914-01-11    0.10
1923    20.67    4.51        12/31/1923    1923    17.07    0.080111508    258.02    17.3        1914-01-12    0.10
1924    20.67    4.68        12/31/1924    1924    17.1    0.080111508    258.02    17.3        1915-01-01    0.10
1925    20.67    4.25        12/31/1925    1925    17.53    0.082889942    249.37    17.9        1915-01-02    0.10
1926    20.67    4.25        12/31/1926    1926    17.7    0.081963797    252.18    17.7        1915-01-03    0.10
1927    20.67    4.25        12/31/1927    1927    17.37    0.080111508    258.02    17.3        1915-01-04    0.10
1928    20.67    4.25        12/31/1928    1928    17.13    0.079185363    261.03    17.1        1915-01-05    0.10
1929    20.67    4.25        12/31/1929    1929    17.13    0.079648436    259.52    17.2        1915-01-06    0.10
1930    20.67    4.25        12/31/1930    1930    16.7    0.07455464    277.25    16.1        1915-01-07    0.10
1931    20.67    4.25        12/31/1931    1931    15.23    0.067608556    305.73    14.6        1915-01-08    0.10
1932    20.67    5.90        12/31/1932    1932    13.66    0.060662471    340.74    13.1        1915-01-09    0.10
1933    20.67    6.24        12/31/1933    1933    12.96    0.061125544    338.16    13.2        1915-01-10    0.10
1934    35.00    6.88        12/31/1934    1934    13.39    0.062051688    564.05    13.4        1915-01-11    0.10
1935    35.00    7.11        12/31/1935    1935    13.73    0.063903977    547.70    13.8        1915-01-12    0.10
1936    35.00    7.02        12/31/1936    1936    13.86    0.064830122    539.87    14        1916-01-01    0.10
1937    35.00    7.04        12/31/1937    1937    14.36    0.066682411    524.88    14.4        1916-01-02    0.10
1938    35.00    7.13        12/31/1938    1938    14.09    0.064830122    539.87    14        1916-01-03    0.11
1939    35.00    7.72        12/31/1939    1939    13.89    0.064830122    539.87    14        1916-01-04    0.11
1940    35.00    8.40        12/31/1940    1940    14.03    0.065293194    536.04    14.1        1916-01-05    0.11
1941    35.00    8.40        12/31/1941    1941    14.73    0.071776206    487.63    15.5        1916-01-06    0.11
1942    35.00    8.40        12/31/1942    1942    16.3    0.078259219    447.23    16.9        1916-01-07    0.11
1943    35.00    8.40        12/31/1943    1943    17.3    0.08057458    434.38    17.4        1916-01-08    0.11
1944    35.00    8.40        12/31/1944    1944    17.6    0.082426869    424.62    17.8        1916-01-09    0.11
1945    35.00    8.61        12/31/1945    1945    18    0.084279159    415.29    18.2        1916-01-10    0.11
1946    35.00    8.61        12/31/1946    1946    19.54    0.099560544    351.54    21.5        1916-01-11    0.12
1947    35.00    8.61        12/31/1947    1947    22.34    0.108358918    323.00    23.4        1916-01-12    0.12
1948    35.00    8.68        12/31/1948    1948    24.08    0.111600424    313.62    24.1        1917-01-01    0.12
1949    35.00    9.40        12/31/1949    1949    23.85    0.109285063    320.26    23.6        1917-01-02    0.12
1950    35.00    12.50        12/31/1950    1950    24.08    0.115768075    302.33    25        1917-01-03    0.12
1951    35.00    12.50        12/31/1951    1951    25.98    0.122714159    285.22    26.5        1917-01-04    0.13
1952    35.00    12.50        12/31/1952    1952    26.55    0.123640304    283.08    26.7        1917-01-05    0.13
1953    35.00    12.50        12/31/1953    1953    26.75    0.124566449    280.97    26.9        1917-01-06    0.13
1954    35.00    12.50        12/31/1954    1954    26.88    0.123640304    283.08    26.7        1917-01-07    0.13
1955    35.00    12.50        12/31/1955    1955    26.78    0.124103376    282.02    26.8        1917-01-08    0.13
1956    35.00    12.50        12/31/1956    1956    27.18    0.127807955    273.85    27.6        1917-01-09    0.13
1957    35.00    12.50        12/31/1957    1957    28.15    0.131512533    266.13    28.4        1917-01-10    0.14
1958    35.00    12.50        12/31/1958    1958    28.92    0.133827895    261.53    28.9        1917-01-11    0.14
1959    35.00    12.50        12/31/1959    1959    29.16    0.136143256    257.08    29.4        1917-01-12    0.14
1960    35.00    12.50        12/31/1960    1960    29.62    0.137995545    253.63    29.8        1918-01-01    0.14
1961    35.00    12.50        12/31/1961    1961    29.92    0.13892169    251.94    30        1918-01-02    0.14
1962    35.00    12.50        12/31/1962    1962    30.26    0.140773979    248.63    30.4        1918-01-03    0.14
1963    35.00    12.50        12/31/1963    1963    30.62    0.143089341    244.60    30.9        1918-01-04    0.14
1964    35.00    12.50        12/31/1964    1964    31.03    0.144478557    242.25    31.2        1918-01-05    0.15
1965    35.00    12.50        12/31/1965    1965    31.56    0.147256991    237.68    31.8        1918-01-06    0.15
1966    35.00    12.50        12/31/1966    1966    32.46    0.152350787    229.73    32.9        1918-01-07    0.15
1967    35.00    12.65        12/31/1967    1967    33.4    0.15698151    222.96    33.9        1918-01-08    0.15
1968    38.94    16.23        12/31/1968    1968    34.8    0.164390666    236.87    35.5        1918-01-09    0.16
1969    40.76    16.98        12/31/1969    1969    36.67    0.174578257    233.48    37.7        1918-01-10    0.16
1970    36.07    15.03        12/31/1970    1970    38.84    0.184302775    195.71    39.8        1918-01-11    0.16
1971    41.17    16.91        12/31/1971    1971    40.51    0.190322715    216.32    41.1        1918-01-12    0.17
1972    59.00    23.58        12/31/1972    1972    41.85    0.196805727    299.79    42.5        1919-01-01    0.17
1973    97.84    39.90        12/31/1973    1973    44.45    0.213939402    457.33    46.2        1919-01-02    0.16
1974    158.96    67.96        12/31/1974    1974    49.33    0.240334523    661.41    51.9        1919-01-03    0.16
1975    160.91    72.42        12/31/1975    1975    53.84    0.257005126    626.10    55.5        1919-01-04    0.17
1976    124.71    69.05        12/31/1976    1976    56.94    0.269508078    462.73    58.2        1919-01-05    0.17
1977    147.78    84.66        12/31/1977    1977    60.61    0.287567898    513.90    62.1        1919-01-06    0.17
1978    193.39    100.75        12/31/1978    1978    65.22    0.313499947    616.87    67.7        1919-01-07    0.17
1979    304.83    143.68        12/31/1979    1979    72.57    0.355176454    858.25    76.7        1919-01-08    0.18
1980    614.61    264.20        12/31/1980    1980    82.38    0.399631394    1537.94    86.3        1919-01-09    0.18
1981    459.26    226.47        12/31/1981    1981    90.93    0.435287962    1055.07    94        1919-01-10    0.18
1982    375.28    214.38        12/31/1982    1982    96.5    0.451958564    830.34    97.6        1919-01-11    0.19
1983    423.61    279.24        12/31/1983    1983    99.6    0.469092239    903.04    101.3        1919-01-12    0.19
1984    360.50    269.77        12/31/1984    1984    103.9    0.487615131    739.31    105.3        1920-01-01    0.19
1985    317.18    244.68        12/31/1985    1985    107.6    0.506138023    626.67    109.3        1920-01-02    0.20
1986    367.72    250.66        12/31/1986    1986    109.6    0.511694891    718.63    110.5        1920-01-03    0.20
1987    446.28    272.30        12/31/1987    1987    113.6    0.534385434    835.13    115.4        1920-01-04    0.20
1988    436.79    245.20        12/31/1988    1988    118.3    0.558002121    782.77    120.5        1920-01-05    0.21
1989    380.74    232.20        12/31/1989    1989    124    0.58393417    652.03    126.1        1920-01-06    0.21
1990    383.32    214.78        12/31/1990    1990    130.7    0.619590737    618.67    133.8        1920-01-07    0.21
1991    362.10    204.65        12/31/1991    1991    136.2    0.638576701    567.04    137.9        1920-01-08    0.20
1992    343.86    194.76        12/31/1992    1992    140.3    0.657099593    523.30    141.9        1920-01-09    0.20
1993    360.00    239.68        12/31/1993    1993    144.5    0.675159413    533.21    145.8        1920-01-10    0.20
1994    384.12    250.79        12/31/1994    1994    148.2    0.693219232    554.11    149.7        1920-01-11    0.20
1995    384.05    243.31        12/31/1995    1995    152.4    0.71081598    540.29    153.5        1920-01-12    0.19
1996    387.82    248.33        12/31/1996    1996    156.9    0.734432667    528.05    158.6        1921-01-01    0.19
1997    330.98    202.10        12/31/1997    1997    160.5    0.746935619    443.12    161.3        1921-01-02    0.18
1998    294.12    177.56        12/31/1998    1998    163    0.758975499    387.52    163.9        1921-01-03    0.18
1999    278.55    172.13        12/31/1999    1999    166.6    0.77935068    357.41    168.3        1921-01-04    0.18
2000    279.10    184.09        12/31/2000    2000    172.2    0.805745801    346.39    174        1921-01-05    0.18
2001    272.67    189.36        12/31/2001    2001    177.1    0.818248753    333.24    176.7        1921-01-06    0.18
2002    309.66    206.27        12/31/2002    2002    179.9    0.83769779    369.66    180.9        1921-01-07    0.18
2003    362.91    222.20        12/31/2003    2003    184    0.853442248    425.23    184.3        1921-01-08    0.18
2004    409.17    223.36        12/31/2004    2004    188.9    0.881226586    464.32    190.3        1921-01-09    0.18
2005    444.47    244.86        12/31/2005    2005    195.3    0.911326285    487.72    196.8        1921-01-10    0.18
2006    603.95    327.68        12/31/2006    2006    201.6    0.9344799    646.30    201.8        1921-01-11    0.17
2007    695.39    347.00        12/31/2007    2007    207.34    0.972618535    714.96    210.036        1921-01-12    0.17
2008    871.65    473.17        12/31/2008    2008    215.3    0.973507634    895.37    210.228        1922-01-01    0.17
2009    972.90    621.59        12/31/2009    2009    214.54    1    972.90    215.949        1922-01-02    0.17

The models were fitted to the data using the free Octave numerical programming environment. Octave is Matlab compatible and available under the GNU Public License. Octave is available in binary as well as source code versions for Windows, Mac OS, and Linux. The standard polyfit polynomial fitting function in Octave was used to fit the polynomial models to the annual gold price data. The leasqr least squares fitting function from the optim Octave add-on package was used to fit the Gaussian and Cauchy-Lorentz peak models to the annual gold price data. Octave Forge add-on packages are available as Unix style blatz.tar.gz files. There is no need to extract the contents of these files. Octave has a command pkg install which handles installing the packages; just type pkg install blatz.tar.gz. The GNU Octave installation includes a C/C++ compiler to compile any C or C++ files included in the package. Note that the run-time errors reported by Octave, for example due to a syntax error, can be cryptic. Be patient and don’t always trust the error message.

% plot_gold.m
% Description: Octave (Matlab compatible) script to plot annual gold price data
% and fit polynomial models to the data.  Tested on Windows XP Service Pack 2
% with Octave 3.2.4 for Windows installed.
%
% Author: John F. McGowan, Ph.D.
% Copyright (C) 2010 by John F. McGowan
%
%
disp('reading gold price data...');
fflush(stdout);
data = dlmread('annual_gold_price_from_1900.txt');
disp('making plot 1');
fflush(stdout);
figure(1)
plot(data(:,1), data(:,8));
title('Inflation Adjusted Gold Price');
xlabel('Year');
ylabel('Gold Price USD (2009)');
disp('making plot 2');
fflush(stdout);
figure(2)
plot(data(:,1), data(:,2));
title('Nominal Gold Price');
xlabel('Year');
ylabel('Gold Price USD');
disp('making cpi plot');
figure(3)
plot(data(:,1), data(:,6));
title('US Consumer Price Index (CPI)');
xlabel('Year');
% annual inflation rate
disp('plotting annual inflation rate');
fflush(stdout);
figure(4)
cpi = data(:,6);
cpi_shift = shift(cpi,1);
inflation = (cpi - cpi_shift) ./ cpi_shift;
years = data(:,1);
plot(years(2:end), inflation(2:end)*100);
title('US Annual Inflation Rate');
xlabel('Year');
ylabel('Inflation (%)');
%
% fitting polynomial model to data 1970 to 2009
%
disp('making gold price prediction 12 terms');
fflush(stdout);
figure(5)
floating = years(71:end);
floating_gold_price = data(71:end, 8);
p = polyfit(floating, floating_gold_price, 12);
prediction = (1970:2020);
f = polyval(p, prediction);
plot(floating, floating_gold_price, 'o', prediction, f, '-');
title('Polynomial Model of Gold Price (12 Terms)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
disp('making gold price prediction 24 terms');
fflush(stdout);
figure(6)
floating = years(71:end);
floating_gold_price = data(71:end, 8);
p = polyfit(floating, floating_gold_price, 24);
prediction = (1970:2020);
f = polyval(p, prediction);
plot(floating, floating_gold_price, 'o', prediction, f, '-');
title('Polynomial Model of Gold Price (24 Terms)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
disp('making gold price prediction 32 terms');
fflush(stdout);
figure(7)
floating = years(71:end);
floating_gold_price = data(71:end, 8);
p = polyfit(floating, floating_gold_price, 32);
prediction = (1970:2020);
f = polyval(p, prediction);
plot(floating, floating_gold_price, 'o', prediction, f, '-');
title('Polynomial Model of Gold Price (32 Terms)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
disp('all done');
fflush(stdout);

% THE END

The script to fit the two Gaussian peak model to the data is:

% fit_gold.m
% Description: Octave (matlab compatible) script to fit two gaussian peak model
% to annual gold price data using Octave 3.2.4 and the optim package version 1.0.15
% Tested on Windows XP Service Pack 2 with Octave 3.2.4 and optim 1.0.15 installed.
%
% Author: John F. McGowan, Ph.D.
% Copyright (C) 2010 by John F. McGowan
%
disp('fitting model to gold price data');
fflush(stdout);
% fit model to gold price data
%floating_gold_price has inflation adjusted gold price since 1970
floating_years = years(71:end);

% Define functions
% model annual gold price data as two Gaussians
leasqrfunc = @(x,p) p(1) * exp(-1.0*(x - p(2)).^2/p(3)^2) + p(4) * exp(-1.0*(x - p(5)).^2/p(6)^2);
leasqrdfdp = @(x, f, p, dp, func) [exp(-1.0*(x - p(2)).^2/p(3)^2), (2*(x - p(2))/p(3)^2) * p(1) .* exp(-1.0*(x - p(2)).^2/p(3)^2), (2 * (x - p(2)).^2/p(3)^3) * p(1) .* exp(-1.0*(x - p(2)).^2/p(3)^2), exp(-1.0*(x - p(5)).^2/p(6)^2), (2*(x - p(5))/p(6).^2) * p(4) .* exp(-1.0*(x - p(5)).^2/p(6)^2), (2 * (x - p(5)).^2/p(6)^3) * p(4) .* exp(-1.0*(x - p(5)).^2/p(6)^2) ];

wt1 = ones(size(floating_gold_price));
t = floating_years;
data = floating_gold_price;

F = leasqrfunc;
dFdp = leasqrdfdp; % exact derivative
dp = [50.0; 1.0; 1.0; 50.0; 1.0; 1.0];
pin = [500.0; 1980.; 5.0; 500.0; 2010.0; 5.0 ];
stol=0.01; niter=50;
minstep = [10.0; 0.2; 0.2; 10.0; 0.2; 0.2];
maxstep = [100.0; 5.0; 5.0; 100.0; 5.0; 5.0];
options = [minstep, maxstep];

disp(size(t));
disp(size(data));
disp(size(wt1));
fflush(stdout);
figure(1);
global verbose;
verbose = 1;
[f1, p1, kvg1, iter1, corp1, covp1, covr1, stdresid1, Z1, r21] = ...
leasqr (t, data, pin, F, stol, niter, wt1, dp, dFdp, options);

%
% make a prediction
figure(2);
pred_years = [1970:2020];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
figure(3);
pred_years = [1970:2050];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
figure(4);
pred_years = [1970:2100];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');

data_2G = data;
model_2G = leasqrfunc(floating_years, p1);
diff_2G = model_2G - data_2G;
chisq_2G = diff_2G' * diff_2G;

disp('all done');
% THE END

The script to fit the seven (7) Gaussian peak model to the annual gold price data is:

% fit_gold7.m
% Description: Octave (matlab compatible) script to fit seven (7) Gaussian peak model
% to annual gold price data.
% Tested using Octave 3.2.4 with optim 1.0.15 add on package installed on Windows XP Service Pack 2
% Author: John F. McGowan, Ph.D.
% Copyright (C) John F. McGowan
%
disp('fitting 7 Gaussian model to gold price data');
fflush(stdout);
% fit model to gold price data
%floating_gold_price has inflation adjusted gold price since 1970
floating_years = years(71:end);

% Define functions

% model as seven (7) un-normalized Gaussians
leasqrfunc = @(x,p) p(1) * exp(-1.0*(x - p(2)).^2/p(3)^2) + p(4) * exp(-1.0*(x - p(5)).^2/p(6)^2) + p(7) * exp(-1.0*(x - p(8)).^2/p(9)^2) + p(10) * exp(-1.0*(x - p(11)).^2/p(12)^2) + p(13) * exp(-1.0*(x - p(14)).^2/p(15)^2) + p(16) * exp(-1.0*(x - p(17)).^2/p(18)^2) + p(19) * exp(-1.0*(x - p(20)).^2/p(21)^2);

leasqrdfdp = @(x, f, p, dp, func) [exp(-1.0*(x - p(2)).^2/p(3)^2), (2*(x - p(2))/p(3)^2) * p(1) .* exp(-1.0*(x - p(2)).^2/p(3)^2), (2 * (x - p(2)).^2/p(3)^3) * p(1) .* exp(-1.0*(x - p(2)).^2/p(3)^2), exp(-1.0*(x - p(5)).^2/p(6)^2), (2*(x - p(5))/p(6).^2) * p(4) .* exp(-1.0*(x - p(5)).^2/p(6)^2), (2 * (x - p(5)).^2/p(6)^3) * p(4) .* exp(-1.0*(x - p(5)).^2/p(6)^2), exp(-1.0*(x - p(8)).^2/p(9)^2), (2*(x - p(8))/p(9)^2) * p(7) .* exp(-1.0*(x - p(8)).^2/p(9)^2), (2 * (x - p(8)).^2/p(9)^3) * p(7) .* exp(-1.0*(x - p(8)).^2/p(9)^2) , exp(-1.0*(x - p(11)).^2/p(12)^2), (2*(x - p(11))/p(12)^2) * p(10) .* exp(-1.0*(x - p(11)).^2/p(12)^2), (2 * (x - p(11)).^2/p(12)^3) * p(10) .* exp(-1.0*(x - p(11)).^2/p(12)^2) , exp(-1.0*(x - p(14)).^2/p(15)^2), (2*(x - p(14))/p(15)^2) * p(13) .* exp(-1.0*(x - p(14)).^2/p(15)^2), (2 * (x - p(14)).^2/p(15)^3) * p(13) .* exp(-1.0*(x - p(14)).^2/p(15)^2), exp(-1.0*(x - p(17)).^2/p(18)^2), (2*(x - p(17))/p(18)^2) * p(16) .* exp(-1.0*(x - p(17)).^2/p(18)^2), (2 * (x - p(17)).^2/p(18)^3) * p(16) .* exp(-1.0*(x - p(17)).^2/p(18)^2) , exp(-1.0*(x - p(20)).^2/p(21)^2), (2*(x - p(20))/p(21)^2) * p(19) .* exp(-1.0*(x - p(20)).^2/p(21)^2), (2 * (x - p(20)).^2/p(21)^3) * p(19) .* exp(-1.0*(x - p(20)).^2/p(21)^2) ];

wt1 = ones(size(floating_gold_price));
t = floating_years;
data = floating_gold_price;

F = leasqrfunc;
dFdp = leasqrdfdp; % exact derivative
dp = [50.0; 1.0; 1.0; 50.0; 1.0; 1.0; 50.0; 1.0; 1.0; 50.0; 1.0; 1.0 ; 50.0; 1.0; 1.0; 50.0; 1.0; 1.0; 50.0; 1.0; 1.0];
pin = [500.0; 1980.; 5.0; 500.0; 2010.0; 5.0; 500.0; 1982; 2.0; 500.0; 1987; 2.0; 500.0; 1995; 5.0 ; 500.0; 1974; 5.0; 500.0; 1974; 5.0 ];
stol=0.01; niter=100;
minstep = [10.0; 0.2; 0.2; 10.0; 0.2; 0.2; 10.0; 0.2; 0.2; 10.0; 0.1; 0.1; 10.0; 0.1; 0.1; 10.0; 0.1; 0.1; 10.0; 0.1; 0.1];
maxstep = [100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0];
options = [minstep, maxstep];

disp(size(t));
disp(size(data));
disp(size(wt1));
fflush(stdout);
figure(1);
global verbose;
verbose = 1;
[f1, p1, kvg1, iter1, corp1, covp1, covr1, stdresid1, Z1, r21] = ...
leasqr (t, data, pin, F, stol, niter, wt1, dp, dFdp, options);

%
% make a prediction
figure(2);
pred_years = [1970:2020];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (7 Gaussian Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
figure(3);
pred_years = [1970:2050];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (7 Guassian Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
figure(4);
pred_years = [1970:2100];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (7 Gaussian Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');

data_7G = data;
model_7G = leasqrfunc(floating_years, p1);
diff_7G = model_7G - data_7G;
chisq_7G = diff_7G' * diff_7G;

disp('all done (7 Gaussian Fit) all done');
% THE END

The script to fit the two Cauchy-Lorentz function peak model to the annual gold price data is:

% fit_gold_cauchy.m
% Description: Octave (matlab compatible) script to fit a two Cauchy-Lorentz peak model to the annual gold price data.  Tested using Octave 3.2.4 and the optim 1.015 add-on package on Windows XP Service Pack 2.
% Author: John F. McGowan, Ph.D.
% Copyright (C) John F. McGowan
%
disp('fitting 2 Cauchy-Lorentz model to gold price data');
fflush(stdout);
% fit model to gold price data
%floating_gold_price has inflation adjusted gold price since 1970
floating_years = years(71:end);

% Define functions

% model as the linear combination of two Cauchy-Lorentz (aka Breit-Wigner) functions
leasqrfunc = @(x,p) p(1) ./(1 + (x - p(2)).^2/p(3)^2) + p(4) ./(1 + (x - p(5)).^2/p(6)^2);
leasqrdfdp = @(x, f, p, dp, func) [1.0 ./(1.0 + (x - p(2)).^2/p(3)^2), (2*p(1)*(x-p(2)))./(p(3)^2*((x-p(2)).^2/p(3)^2+1).^2), (2*p(1)*(x-p(2)).^2)./(p(3)^3*((x-p(2)).^2/p(3)^2+1).^2), 1.0 ./(1.0 + (x - p(5)).^2/p(6)^2), (2*p(4)*(x-p(5)))./(p(6)^2*((x-p(5)).^2/p(6)^2+1).^2), (2*p(4)*(x-p(5)).^2)./(p(6)^3*((x-p(5)).^2/p(6)^2+1).^2)];

wt1 = ones(size(floating_gold_price));
t = floating_years;
data = floating_gold_price;

F = leasqrfunc;
dFdp = leasqrdfdp; % exact derivative
dp = [50.0; 1.0; 1.0; 50.0; 1.0; 1.0];
pin = [500.0; 1980.; 5.0; 500.0; 2010.0; 5.0 ];
stol=0.01; niter=50;
minstep = [10.0; 0.2; 0.2; 10.0; 0.2; 0.2];
maxstep = [100.0; 5.0; 5.0; 100.0; 5.0; 5.0];
options = [minstep, maxstep];

disp(size(t));
disp(size(data));
disp(size(wt1));
fflush(stdout);
figure(1);
global verbose;
verbose = 1;
[f1, p1, kvg1, iter1, corp1, covp1, covr1, stdresid1, Z1, r21] = ...
leasqr (t, data, pin, F, stol, niter, wt1, dp, dFdp, options);

%
% make a prediction
figure(2);
pred_years = [1970:2020];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (2 Cauchy Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
figure(3);
pred_years = [1970:2050];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (2 Cauchy Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
figure(4);
pred_years = [1970:2100];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (2 Cauchy Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');

% C suffix for Cauchy Lorentz model
data_2C = data;
model_2C = leasqrfunc(floating_years, p1);
diff_2C = model_2C - data_2C;
chisq_2C = diff_2C' * diff_2C;

disp('all done');
% THE END

The Octave script to fit the seven (7) Cauchy-Lorentz peak model to the annual gold price data is:

% fit_gold7.m
% Description: Octave (matlab compatible) script to fit seven (7) Cauchy-Lorentz peak model
% to annual gold price data.
% Tested using Octave 3.2.4 with optim 1.0.15 add on package installed on Windows XP Service Pack 2
% Author: John F. McGowan, Ph.D.
% Copyright (C) John F. McGowan
%
disp('fitting seven (7) Cauchy-Lorentz model to gold price data');
fflush(stdout);
% fit model to gold price data
%floating_gold_price has inflation adjusted gold price since 1970
floating_years = years(71:end);

% Define gold price model functions

% model as linear combination of seven (7) Cauchy-Lorentz (aka Breit-Wigner) functions
leasqrfunc = @(x,p) p(1) ./(1 + (x - p(2)).^2/p(3)^2) + p(4) ./(1 + (x - p(5)).^2/p(6)^2) + p(7) ./(1 + (x - p(8)).^2/p(9)^2) + p(10) ./(1 + (x - p(11)).^2/p(12)^2) + p(13) ./(1 + (x - p(14)).^2/p(15)^2) + p(16) ./(1 + (x - p(17)).^2/p(18)^2) + p(19) ./(1 + (x - p(20)).^2/p(21)^2);

leasqrdfdp = @(x, f, p, dp, func) [1.0 ./(1.0 + (x - p(2)).^2/p(3)^2), (2*p(1)*(x-p(2)))./(p(3)^2*((x-p(2)).^2/p(3)^2+1).^2), (2*p(1)*(x-p(2)).^2)./(p(3)^3*((x-p(2)).^2/p(3)^2+1).^2),1.0 ./(1.0 + (x - p(5)).^2/p(6)^2), (2*p(4)*(x-p(5)))./(p(6)^2*((x-p(5)).^2/p(6)^2+1).^2), (2*p(4)*(x-p(5)).^2)./(p(6)^3*((x-p(5)).^2/p(6)^2+1).^2),1.0 ./(1.0 + (x - p(8)).^2/p(9)^2), (2*p(7)*(x-p(8)))./(p(9)^2*((x-p(8)).^2/p(9)^2+1).^2), (2*p(7)*(x-p(8)).^2)./(p(9)^3*((x-p(8)).^2/p(9)^2+1).^2),1.0 ./(1.0 + (x - p(11)).^2/p(12)^2), (2*p(10)*(x-p(11)))./(p(12)^2*((x-p(11)).^2/p(12)^2+1).^2), (2*p(10)*(x-p(11)).^2)./(p(12)^3*((x-p(11)).^2/p(12)^2+1).^2),1.0 ./(1.0 + (x - p(14)).^2/p(15)^2), (2*p(13)*(x-p(14)))./(p(15)^2*((x-p(14)).^2/p(15)^2+1).^2), (2*p(13)*(x-p(14)).^2)./(p(15)^3*((x-p(14)).^2/p(15)^2+1).^2),1.0 ./(1.0 + (x - p(17)).^2/p(18)^2), (2*p(16)*(x-p(17)))./(p(18)^2*((x-p(17)).^2/p(18)^2+1).^2), (2*p(16)*(x-p(17)).^2)./(p(18)^3*((x-p(17)).^2/p(18)^2+1).^2),1.0 ./(1.0 + (x - p(20)).^2/p(21)^2), (2*p(19)*(x-p(20)))./(p(21)^2*((x-p(20)).^2/p(21)^2+1).^2), (2*p(19)*(x-p(20)).^2)./(p(21)^3*((x-p(20)).^2/p(21)^2+1).^2)]

wt1 = ones(size(floating_gold_price));  % same weight for all data points
t = floating_years;    % years when gold price floats
data = floating_gold_price;   % inflation adjusted gold price

F = leasqrfunc;
dFdp = leasqrdfdp; % exact derivative
dp = [50.0; 1.0; 1.0; 50.0; 1.0; 1.0; 50.0; 1.0; 1.0; 50.0; 1.0; 1.0 ; 50.0; 1.0; 1.0; 50.0; 1.0; 1.0; 50.0; 1.0; 1.0];
pin = [500.0; 1980.; 5.0; 500.0; 2010.0; 5.0; 500.0; 1982; 2.0; 500.0; 1987; 2.0; 500.0; 1995; 5.0 ; 500.0; 1974; 5.0; 500.0; 1974; 5.0 ];
stol=0.01; niter=100;
minstep = [10.0; 0.2; 0.2; 10.0; 0.2; 0.2; 10.0; 0.2; 0.2; 10.0; 0.1; 0.1; 10.0; 0.1; 0.1; 10.0; 0.1; 0.1; 10.0; 0.1; 0.1];
maxstep = [100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0; 100.0; 5.0; 5.0];
options = [minstep, maxstep];

disp(size(t));
disp(size(data));
disp(size(wt1));
fflush(stdout);
figure(1);
global verbose;
verbose = 1;
[f1, p1, kvg1, iter1, corp1, covp1, covr1, stdresid1, Z1, r21] = ...
leasqr (t, data, pin, F, stol, niter, wt1, dp, dFdp, options);

%
% make a prediction
figure(2);
pred_years = [1970:2020];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (7 Cauchy Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
figure(3);
pred_years = [1970:2050];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (7 Cauchy Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');
%
disp('plotting figure 4');
fflush(stdout);
figure(4);
pred_years = [1970:2100];
prediction = leasqrfunc(pred_years, p1);
plot(floating_years, data, 'o', pred_years, prediction, '-');
title('Inflation Adjusted Gold Price (7 Cauchy Prediction vs Data)');
xlabel('Year');
ylabel('Gold Price USD (2009)');

disp('computing final results...');
fflush(stdout);
% C suffix for Cauchy Lorentz model
data_7C = data;
model_7C = leasqrfunc(floating_years, p1);
diff_7C = model_7C - data_7C;
chisq_7C = diff_7C' * diff_7C;

disp('all done');
% THE END

The derivatives of the models with respect to parameters were computed with the symbolic manipulation package Maxima. Maxima was also used to generate some of the Latex mathematical formulas in this article.

No related posts.

Henry Frankenstein: Look! It’s moving. It’s alive. It’s alive… It’s alive, it’s moving, it’s alive, it’s alive, it’s alive, it’s alive, IT’S ALIVE!
Victor Moritz: Henry – In the name of God!
Henry Frankenstein: Oh, in the name of God! Now I know what it feels like to be God!

Frankenstein (1931)

Introduction

In 1915, after several failed attempts, Albert Einstein promulgated the General Theory of Relativity, a mathematical theory of gravitation that reconciled gravity with his Special Theory of Relativity and explained the gravitational force as the warping of space and time by matter and energy. Amongst other things, the theory predicted a slightly different deflection of light by bodies such as the Sun than the prevailing Newtonian theory of gravitation. In 1919 Einstein became an international celebrity when the English astronomer Arthur Eddington announced results from the measurement of the deflection of starlight by the Sun during a solar eclipse confirming Einstein’s theory.  But, the story of Einstein’s triumph is more complicated than that.  Einstein patched his theory to agree with observations and the prejudices of his time in 1917.  Then, he later discarded the patch when new observations appeared to confirm his original theory of 1915.  Recently, in an astonishing about face, physicists and astronomers have resurrected Einstein’s until then embarrassing patch to force agreement between new observations and the reigning Big Bang theory/General Theory of Relativity.   This is a recent example of the construction of Frankenstein Functions in which scientists and mathematicians construct arbitrary functions out of many mathematical pieces to match observational data or simply preconceived notions.

Einstein Field Equations (General Relativity)

The General Theory of Relativity is a set of equations relating the so-called metric of space-time , loosely a measure of the curvature of space-time, to the density of mass and energy variously expressed in modern mathematical notation as:

Short version (using Einstein tensor )

is the so-called metric of space and time. is Newton’s Gravitational Constant. is the speed of light. is the so-called stress-energy tensor which loosely represents the density of mass and energy in space and time. The indices and usually run from 0 to 3 (0,1,2,3) or 1 to 4 (1,2,3,4) or over the symbols , , , and referring to the three spatial dimensions and the single time “dimension.” The author uses scare quotes for the time “dimension” because it differs radically from the three spatial dimensions in common experience, although it can be represented in a very similar symbolic mathematical way as the numerical calendar time indicated by a clock.

Using standard cosmological units , the equations are written:

The long version (using Ricci curvature tensor and scalar curvature ) is:

or

There is the symmetric” decomposition, into the scalar part:

and the traceless symmetric tensor part:

The Greatest Blunder

There was however a problem. When astronomers, physicists, and mathematicians worked out the implications of the equations of General Relativity, they found that the universe as a whole must be either expanding or contracting. The universe was not stable in the equations of General Relativity. At the time, both the evidence of observational astronomy and the philosophical bias of most scientists was that the universe was neither expanding nor contracting. The universe was static, perhaps of infinite age. Confronted with evidence apparently clearly falsifying his theory, Einstein did what scientists, philosophers, scholars, attorneys, and political activists have done since time immemorial. He patched his theory to fit the observations and prejudices of his time. Einstein added a mysterious extra term known as the “cosmological constant” that counterbalanced the predicted expansion or contraction of the universe:

where the extra term is where the constant is known as the cosmological constant. Physically the cosmological constant may correspond to a mysterious energy field filling the entire universe. With a proper choice of the cosmological constant , the universe was static, neither expanding nor contracting.

Subsequently, the redshifts of extragalactic nebulae were reinterpreted as due to the motion of the nebulae. That is, the nebulae, recognized as galaxies outside our galaxy the Milky Way, were flying away from us, causing a redshift of light. Over time, the astronomer Edwin Hubble was able to show that the dimmer the galaxy and therefore presumably the farther away (at least on average) the galaxy, the larger the redshift and thus the faster the galaxy appeared to be running away from the Earth. This rather peculiar observation could be easily explained if the universe was expanding as predicted by the original General Theory of Relativity without the cosmological term (the cosmological constant was either zero or nearly zero).

As might be imagined, Einstein did what scientists, philosophers, scholars, attorneys, and political activists have done since time immemorial when confronted by evidence apparently clearly falsifying their current theory but confirming the original un-patched theory. He dropped the cosmological term like a hot potato. In his autobiography, the physicist George Gamow recounted an alleged conversation with Einstein in which Einstein described the cosmological constant as the “greatest blunder” of his life. Whether true or not, this quotation was widely repeated in popular physics articles, textbooks, informal conversations by physicists, and so forth until the late 1990′s when observations by the Hubble Space Telescope that were apparently inconsistent with the prevailing Big Bang theory of cosmology were made. However, one could save the Big Bang theory by reintroducing the cosmological term with a non-zero cosmological constant, not strong enough to prevent the expansion of the universe but sufficient to cause an acceleration that would resolve the otherwise falsifying observations. The non-zero cosmological constant was attributed to a mysterious “dark energy” filling the universe, perhaps due to an as yet undiscovered subatomic particle/field predicted by a unified field theory or theory of everything (TOE).

The predicted expansion of the universe in the General Theory of Relativity became a theory of the origin and evolution of the universe known as the Big Bang theory. In general, the different forms of the Big Bang theory envision the universe beginning as a point, a “singularity” in the General Theory of Relativity, and expanding, exploding into the universe that we see today. The Big Bang is thought to have happened 14-25 billion years ago. The exact time has varied over the last century with different observations and theoretical calculations.

Up until the Hubble Space Telescope observations, the story of Einstein’s “greatest blunder” was widely recounted as a morality tale of the superiority of rational scientific thought over fluffy philosophy or blind prejudice. If only Einstein had stuck to his original “rational” theory rather than being influenced by fluffy (not to mention wimpy) unscientific philosophical or sociological concerns (agreeing with everybody else), all would have been well. Thus, the “greatest blunder” became a tale about the primacy of science over other “ways of knowing,” to use an appropriately fluffy New Age cliche.

The Big Bang theory and modern cosmology has a curious history. There have been repeated observations that appear to falsify the theory or major components of the theory such as the theory of gravitation, whether Einstein’s or Newton’s theory of gravity. This has led to the introduction of several new concepts and components expressed in symbolic mathematics like the cosmological term. These include “inflation” to account for the puzzling uniformity of the universe in certain measurements (for example, the cosmological microwave background radiation is extremely smooth which is difficult to explain in many versions of the Big Bang theory) and several types of as yet undetected “dark matter” to account for gross anomalies in the measured angular momentum distribution of galaxies, clusters of galaxies, and so forth. There is an entire industry seeking to detect the mysterious particles that comprise the as yet hypothetical “dark matter” and “dark energy.”

The saga of the cosmological term and the other patches to the General Theory of Relativity and the Big Bang theory illustrates a deep mathematical problem that has bedeviled science and human society at least since the ancient Greeks (and probably Babylonians or Sumerians as well) constructed detailed mathematical models of the motion of the planets. It is a problem that appears both in symbolic mathematics, in the effort to construct predictive symbolic mathematical theories of the world, as well as in conceptual, verbal reasoning and discourse. It seems likely that the human mind, unlike present-day symbolic mathematics or computer programs, has a limited, imperfect ability to resolve this problem.

What is the problem? Given a set of observations — the positions and motions of the planets, the waveform of speech, the time and intensity of earthquakes, the values of stock prices, blood pressure, any quantitative measurement — it is possible to choose many different sets of building block functions that can be combined (added, multiplied, etc.) to match (or “fit”) the observations as accurately as desired (simply add more building block functions). One can construct not one, but many, in fact an infinite number, of “Frankenstein Functions” that match the data. While these sets of building block functions can be chosen to match the observational data, the “training” set in artificial intelligence terminology, they often will fail to predict new observations. It is often necessary to add new building block functions to save or patch the theory as new observations are made. However, if the building block functions share some characteristics in common with the unknown “true” theory or mathematics, then the theory may give somewhat correct predictions but still be wrong or incomplete.

At a conceptual, verbal level, it usually proves possible to devise a plausible, technically sophisticated concept to explain away the apparently falsifying observations and to justify the “patch” expressed in purely symbolic mathematical terms. For example, unified field theories or theories of everything (TOE) usually predict new particles that are otherwise unknown. These new, unknown particles might in turn provide the dark matter or dark energy needed to explain the contrary observations. Certain commonalities among known particles and forces suggest an underlying unity.

The ancient Greeks constructed (or inherited from the even older Babylonian civilization) a mathematical theory of the universe, basically our modern solar system, in which the planets orbited the Earth, a sphere about eight-thousand miles in diameter. However, from the very beginning, this theory had a serious problem. Certain planets, notably Mars, actually backed up during their journey through the Zodiac. This was grossly inconsistent with a simple orbit. Hence, the Greeks (and possibly the Babylonians before them) introduced the now infamous epicycles. The planets, envisioned as the Gods themselves, executed a complex dance in which they performed a circular motion around the simple circular orbit around the Earth. This could produce a period when the outer planets — Mars, Jupiter, and Saturn — would appear to back up in the Zodiac, exactly as observed. Yet, the theory never quite worked. Over the centuries and ultimately almost two millenia, astronomers and astrologers (mostly the same thing) added more and more epicycles to create the “Ptolemaic” theory that existed at the time of Copernicus, Galileo, and Kepler.

The Ptolemaic theory had hundreds of epicycles, epicycles on top of epicycles on top of epicycles. It was very complex and required extensive time and effort to make predictions using the pen, paper, and printed mathematical tables of the time. There were no computers. It could predict the motion of Mars to around one percent accuracy. This was actually much better than the original heliocentric theory proposed by Copernicus. In fact, Copernicus also used epicycles. A hard headed comparison of the geocentric and heliocentric theories based solely on quantitative goodness of fit measures would have selected the traditional geocentric Ptolemaic theory. It was not until at least 1609 when Kepler published his discovery of the elliptical orbits and possibly even later (Kepler made mistakes) that the heliocentric theories clearly outperformed the Ptolemaic theory.

The orbits of the planets around the Sun are almost periodic. The motion of the planets as seen from Earth is quasi-periodic. Thus, if one uses periodic functions such as the uniform circular motion of the Ptolemaic epicycles, one can reproduce much of the observed motion of the planets. The Ptolemaic models had some predictive power.

The lesson of Copernicus, Galileo, and Kepler as well as subsequent successes in science seemed to be to prefer “simple” theories with few building block functions, few terms in the mathematical expressions, and so forth. This led Einstein to select his original General Theory of Relativity as the simplest or one of the simplest sets of differential equations consistent with the Special Theory of Relativity as well as known observations (the theory had to largely reproduce Newtons’ theory of gravitation). For many years, popular science and popular physics accounts such as the “greatest blunder” stories embraced this preference for “simplicity” under the banner of Occam’s Razor as the obviously scientific, rational way to do things. It is actually difficult to justify this preference. Today, however, the popular science orthodoxy has changed as otherwise falsifying observations have accumulated. For example,

But Einstein did not exclude terms with higher derivatives for this or for any other practical reason, but for an aesthetic reason: They were not needed, so why include them? And it was just this aesthetic judgment that led him to regret that he had ever introduced the cosmological constant.

Since Einstein’s time, we have learned to distrust this sort of aesthetic criterion. Our experience in elementary particle physics has taught us that any term in the field equations of physics that is allowed by fundamental physics is likely to be there in the equations.

(…several paragraphs of explication in a similar vein…)

Regarding his introduction of the cosmological constant in 1917, Einstein’s real mistake was that he thought it was a mistake.

Steven Weinberg (Nobel Prize in Physics, 1979)
“Einstein’s Mistakes”
Physics Today, November 2005, pp. 31-35

Incidentally, Professor Weinberg’s article has the ironic subtitle “Science sets itself apart from other paths to truth by recognizing that even the greatest practitioners sometimes err.”   This is probably a veiled jab at traditional religion.  It is probably doubly ironic in that some forms of traditional religion clearly recognize the fallibility of their prophets.   For example, in his letters to his fellow feuding Christians of the first century, the Apostle Paul makes a clear distinction between his personal opinions, which he considers fallible, and divine revelation.

With respect to Einstein’s aesthetic judgment, essentially any continuous function can be approximated to arbitrary accuracy by a polynomial of sufficiently high degree — or indeed any of any infinite number of compositions of arbitrarily chosen building block functions.  A polynomial is the sum of powers of . For example,

or

In general,

What does this mean? Let consider, for example, an arbitrary function such as the trigonometric sine function . Here is the sine function plotted by Octave, a free Matlab compatible numerical programming environment:

x = (0.0 : 0.1 : 20*pi)';
y = sin(x);
plot(x,y,'-');

Sine Function

Octave, like many similar tools such as Mathematica, has a built in function, polyfit in this case, to fit a polynomial to data:

function [] = plot_sinfit(x,y, n, m)
  %  plot_sinfit(x, y, n)  fits polynomial of degree n to data (x,y) in range 0.0 to 2*pi
  %
  if nargin > 3
    span = m;
  else
    span = 3;
  end

  myx = x(1:63);
  myy = y(1:63);
  p = polyfit(x,y,n)
  f = polyval(p,x);
  x = (0: 0.1: span*2*pi)';
  y = sin(x);
  f = polyval(p,x);
  plot(x,y,'o',x,f,'-')
  axis([0  span*2*pi  -1  1])
end

A sixth degree polynomial is fitted to the data in the range 0.0 to 6.28   (the “training” set), but the fitted function is displayed in the range 0.0 to 18.42 .  This gives:

Polynomial Fit with 6 Terms

One can see the agreement with six terms is poor.  However, one can always add more terms:

Polynomial Fit with 12 Terms

The agreement is somewhat better but still poor.  One can still add more terms:

Polynomial Fit with 24 Terms

Now the fit is getting better, but there is still room for improvement.  Both examination by eye and a rigorous goodness of fit test would show the mathematical model and the observational data disagree.  One can still add more terms:

Polynomial Fit with 48 Terms

The agreement is even better, although not perfect.  One can see the disagreement by looking at a larger range of data (recall the model is fitted to the range 0.0 to 6.28 only):

Polynomial Fit with 48 Terms (5 Cycles)

As one moves farther away from the region used for the fit (0.0 to 6.28), the training set in the language of artificial intelligence, the agreement will generally worsen.  However, one can make the agreement as good as one wants by adding more and more terms, more and more powers of .  It is important to realize that a sequence of powers of can never really work.  It will never predict the long term behavior of the data.  The data in this illustrative example is periodic.  In contrast, powers of grow without bound.  Eventually, as grows without bound, the largest power of will dominate the mathematical model of the data and the model will blow up, growing without bound, and failing at some point to agree with the observations, the data.

If one used a mathematical model constructed from periodic functions, other than the sine, one could patch together a “Frankenstein Function” that would have some predictive power and share some of the gross characteristics of the actual data.  This is what happened with the Ptolemaic epicycles centuries ago.  In fact, one can construct a Frankenstein Function out of randomly chosen functions,  Gaussians, polynomials, trig functions, pieces of other functions, and so forth that will agree with observational data to any desired level of agreement.

Many techniques in pattern recognition and artificial intelligence, such as the Hidden Markov Model (HMM) speech recognition engines and artificial neural networks, are attempts to construct extremely complex mathematical models composed of, in some cases, hundreds of thousands of building block functions, to replicate the ability of human beings to classify sounds or images or other types of data.  In these attempts, many of the same problems that have occurred in mathematical models such as the Ptolemaic model of the solar system have recurred.  In particular, it has been found that neural networks and similar models can often exactly agree with a training set.  In fact, this seeming agreement is often bad.  The training or fitting process is often intentionally stopped before completion because while the mathematical model of classification will agree with the training set, it will often fail to classify new data such as a so-called “validation” data set.  Even extremely complex models such as those used in speech recognition today continue to fail to reach the human level of performance, possibly for many of the same reasons the epicycles of the Ptolemaic theory failed.

Falsifiability and Occam’s Razor

These difficulties with mathematical modeling lead one directly to two pillars of popular and sometimes scholarly science:  the doctrine of falsifiability, usually attributed to the philosopher of science Karl Popper, and Occam’s Razor.  The doctrine of falsifiability holds that science proceeds by the falsification of theories by new evidence.  In mathematical terms, one can compare the mathematical theory’s predictions with experimental data, apply a goodness of fit test, and conclusively rule out the theory.  This is supposed to differentiate science decisively from fluffy philosophical, religious, mystical, and political “knowledge.”  Science may not be able to tell us what is true, but it can tell us conclusively what is false.  The doctrine of falsifiability is often touted in discussions of so-called pseudoscience, especially in the context of debates about the theory of evolution and “creation science,” or, more recently, so-called “intelligent design.”  Usually, the argument is that falsifiability allows us to distinguish between true science which should be taught in schools and generally accepted and dubious pseudoscientific “knowledge.”

The problem with this, as the saga of the cosmological constant now clearly shows, is that theories can be patched and are frequently patched, sometimes obviously and sometimes more subtly.  Adding more and more terms to a polynomial approximation is pretty obvious.  The epicycles in the Ptolemaic theories were pretty obvious.  Actually, add-ons such as the cosmological constant are pretty obvious.  On the other hand, quantum field theory and the various unified field theories/theories of everything such as superstrings are so complex and have such a long learning curve for most people that it is often difficult to evaluate what might be a patch (e.g. the concept of renormalization) and what might not be a “patch.”

At this point, Occam’s Razor is usually invoked.  William of Occam was an English Franciscan friar and scholastic philosopher who lived from about 1288 to about 1348.  He was involved in a range of theological and political conflicts during which he formulated his so-called Razor, quite possibly for political and theological reasons quite alien to the modern use (or misuse) of Occam’s Razor.  In its modern form, Occam’s Razor is usually expressed as a need not to make ad hoc assumptions or to keep a theory as simple as possible while still agreeing with observations.   Of course, it is hard to define an ad hoc assumption or simplicity in practice.  In disputes about evolution and creation, Occam’s Razor is often used to attack creationist explanations of the radioactive (and other) dating of the Earth and fossils to million or billions of years of age.  This evidence of the great age of the Earth is by far the most difficult observational evidence for creationists to explain.  In his criticism of the teaching of evolution, William Jennings Bryan, who was not a fundamentalist (biblical literalist) as many believe, simply accepted the great age of the Earth as did many religious leaders of his time.

Here is Steven Weinberg again on the new revised Occam’s Razor:

Indeed, as far as we have been able to do the calculations, quantum fluctuations by themselves would produce an infinite effective cosmological constant, so that to cancel the infinity there would have to be an infinite “bare” cosmological constant of the opposite sign in the field equations themselves.  Occam’s razor is a fine tool, but it should be applied to principles, not equations.

Of course, in what might be called the strong AI theory of symbolic mathematics, the symbols in the equations must correspond either directly or in some indirect but precise, rigorous way to “principles.”  We just don’t understand the correspondence yet.  Unless the strong AI theory of symbolic math is wrong and some concepts cannot be expressed in symbolic mathematical form.

Where does this leave us?  We know rigorously that it is possible to construct many arbitrarily complex functions or differential equations that can be essentially forced to fit current observational data.  How do we choose which ones are likely to be true?  Human beings seem ultimately to apply some sort of judgment or intuition.  Often they are wrong, but still they are right much more often than random chance would suggest.  Historically, it seems that simplicity at both the level of verbal concepts and at the level of precise symbolic mathematics has been a good criterion.  We can’t really justify this experience “scientifically,” at least as yet.

Frankenstein Functions in the Computer Age

With modern computers, mathematical tools such as Mathematica, Matlab, Octave, and so forth, and modern mathematics with its myriad special functions, differential equations, and other exotica, it is now possible to construct Frankenstein Functions on a scale that dwarfs the Ptolemaic epicycles.  Artificial intelligence techniques such as Hidden Markov Model based speech recognition, genetic programming, artificial neural networks and other methods in fact explicitly or implicitly incorporate mathematical models with, in some cases, hundreds of thousands of tunable parameters.  These models can match training sets of data exactly and yet they fail significantly, sometimes totally when confronted with new data.

In fundamental physics, the theories have grown increasingly complex.  Even the full Lagrangian for the reigning standard model of particle physics (for which Steven Weinberg shared the Nobel Prize in 1979) is quite complex and features such still unobserved ad hoc entities as the Higgs particle.  Attempts at grand unified theories or theories of everything are generally more complex and elaborate.  The reigning Big Bang theory has grown increasingly baroque with the introduction of inflation, numerous types of dark matter, and now dark energy — the rebirth of the cosmological constant.  Computers, mathematical software, advanced modern mathematics, and legions of graduate students and post doctoral research associates all combine to make it possible to construct extremely elaborate models far beyond the capacity of the Renaissance astronomers.   The very complexity and long learning curve of the present day models may become a status symbol and protect the theories from meaningful criticism.

Conclusion

Aesop’s Fables include the humorous tale of The Astronomer who spends all his time gazing up at the heavens in deep contemplation.  He is so mesmerized by his star gazing that he falls into a well at his feet.  The moral of the tale is “My good man, while you are trying to pry into the mysteries of heaven, you overlook the common objects that are at your feet.”    This may have a double meaning in the current age of Frankenstein Functions.  On the one hand, scientists and engineers may well have become enamored of extremely complex models and forgotten the lesson of past experience that extreme complexity is often a warning sign of deep problems, the lesson that Einstein initially heeded.  It also raises the question of whether ordinary people, business leaders, policy makers, and others in the “real world” need be concerned about these complex mathematical models, usually implemented in computer software, and the difficulties associated with them.

Extremely complex mathematical models, some apparently successful, some probably less so, are increasingly a part of life.  Complex models incorporating General Relativity are used by the Global Positioning System to provide precise navigation information — to guide everything from hikers to ships to deadly missiles.  Widely quoted economic figures such as the unemployment rate and inflation rate are actually the product of increasingly complex mathematical models that bear a less than clear relationship to common sense definitions of unemployment and inflation.  Is a “discouraged worker” really not unemployed?  Are the models that extrapolate from the household surveys to the nationally reported “unemployment rate” really correct?   What should one make of hedonic corrections to the inflation rate in which alleged improvements in quality are used to adjust the price of an item downward?  Should the price of houses used in the consumer price index (CPI) be the actual price of purchase or the mysterious “owner equivalent rent.”  What is the average person to make of the complex computer models said to demonstrate global warming beyond a reasonable doubt?  Should we limit the production and consumption of coal, oil, or natural gas based on these models?  How do oil companies and governments like Saudi Arabia calculate the “proven reserves” of oil that they report each year?  Are we experiencing “Peak Oil” as some claim or is there more oil than commonly reported?

In the lead up to the present financial crisis and recession, the handful of economists and financial practitioners (Dean Baker, Nouriel Roubini, Robert Shiller, Paul Krugman, Peter Schiff, and some others) who clearly recognized and anticipated the housing bubble and associated problems used very simple, back of the envelope calculations and arguments to detect the bubble.  Notably, housing prices in regions with significant zoning restrictions on home construction rose far ahead of inflation, something rarely seen in the past and then usually during previous housing bubbles.  Home prices in regions with significant zoning restrictions became much higher than would be expected based on apartment rental rates in the same areas.  In other words, it was much cheaper to rent than to own a home of the same size, something with little historical precedent.  In contrast, the large financial firms peddling mortgage backed securities used extremely complex mathematical models, not infrequently cooked up by former physicists and other scientists, that proved grossly inaccurate.

It is likely that simplicity and Occam’s Razor as commonly understood have some truth in them, even though we do not truly understand why this is the case.  They are not perfect.  Sometimes the complex theory or the ad hoc assumption wins.  Nonetheless, Frankenstein Functions and extreme complexity, both in principles (verbal concepts) and precise symbolic mathematics should be viewed as a warning sign of trouble.  By this criterion, the Big Bang theory, General Relativity, and quantum field theory may all be in need of significant revision.

Suggested Reading/References

“Einstein’s Mistakes (PDF),” Steven Weinberg, Physics Today, November 2005, pp. 31-35

Aesop’s Fables, Selected and Adapted by Jack Zipes, Penguin Books, New York, 1992

George Gamow, My World Line — An Informal Autobiography, Viking Press, New York, 1970

Copyright © 2010 John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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The 2004 movie/documentary What the Bleep Do We Know? has a simple answer to life’s problems: Quantum Mechanics! Can’t get a date? The answer: Quantum Mechanics. Hate your job? The answer: Quantum Mechanics. Worried about the long term health effects of your prescription anxiety pills? Have no fear. Quantum Mechanics is the answer. The movie intermixes a maelstrom of flashy computer generated special effects, the ultimately uplifting story of Amanda, a divorced photographer hooked on anxiety pills, who finds enlightenment and a good man through the power of quantum mechanics, and interviews with fourteen experts including physicist Amit Goswami, physicist Fred Alan Wolf, physicist John Hagelin, physicist David Albert, neuroscientist and pharmacologist Candace Pert, anesthesiologist Stuart Hameroff, a number of others, and, oh yes, New Age spiritual guru JZ Knight channeling Ramtha, the 35,000 year old spirit of a warrior from the lost continent of Lemuria. As it happens, the filmmakers and some of the experts are adherents of JZ Knight/Ramtha.

What the Bleep Do We Know? implicitly advances the theory that not only does human consciousness cause the mysterious collapse of the wave function in quantum mechanics but in fact can control or influence the outcome of the collapse, thus allowing human beings to influence or control events through the power of their mind and quantum mechanics. What the Bleep Do We Know? is vague about the actual textbook theory of quantum mechanics, not surprisingly since this theory gives no clear answer as to why, when, or how the quantum wave function collapses. The movie/documentary is one of the more prominent recent examples of the linking of quantum mechanics and mysticism, sometimes by experts with impressive resumes.

Like most popular science, the movie makes very limited use of mathematics: rendering the movie title in mathematical symbols on the DVD case and web site and a brief sequence of computer generated flying equations that are never explained. This is a very common technique in video, both science “fact” and science fiction where the audience is briefly shown a mathematical formula or equation in cryptic, often Greek, symbols with either no explanation or a vague explanation. Brian Greene’s The Elegant Universe PBS/Nova special, which manages to promote the highly mathematical field of superstrings with almost no mathematics, features a couple of scenes with cryptic equations on a blackboard or notebook page, for example.

What is Quantum Mechanics Actually?

The textbook theory of non-relativistic quantum mechanics is actually quite simple. Quantum Field Theory, sometimes abbreviated QFT, which purports to reconcile quantum mechanics and the theory of special relativity is extremely complex, abstract, and difficult to learn. The author once attended a lecture by the physicist Leonard Susskind in which he stated that it took ten years to learn Quantum Field Theory. This seems to be true for most people. In contrast, basic quantum mechanics is straightforward and could probably be explained in a movie/documentary like What the Bleep Do We Know?

The basic quantum mechanics formulated by Niels Bohr, Max Born, Werner Heisenberg, and their colleagues over the objections of Albert Einstein, Erwin Schrodinger, and Prince Louis deBroglie asserts that sub-atomic particles (and presumably macroscopic objects like tables and chairs) are characterized by a quantum state or quantum wave function usually represented by the Greek letter . This wave function is a complex number representing a wave with an amplitude and phase. For example:

The quantum wave function is governed by a wave equation, a partial differential equation. There are several different wave equations depending on the type of particle: the original Schrodinger Equation, the Dirac Equation, and the Klein-Gordon equation. As far as the author can determine, these equations were found empirically — that is they were educated guesses that turned out to match experimental data on certain types of particles. Subatomic particles are thought to have a mysterious instrinsic “spin” that is somewhat analogous to the spin/angular momentum of a macroscopic body. Different wave equations govern the quantum wave functions of particles of different spin. The basic Schrodinger Equation which describes the electron in the hydrogen atom approximately (the electron is a spin 1/2 particle which is exactly described by the Dirac Equation) is:

where is Planck’s constant divided by , is the mysterious “wave function”, is the potential such as the electrical potential of the hydrogen atom, and is the position.

Qualitatively, one can think of the quantum wave function as a physical wave like a water wave, a sound wave, or a radio wave. These familiar phenomema are governed by wave equations as well, similar to the Schrodinger equation in many ways. This similarity to known phenomena and mathematics made Schrodinger’s wave picture and his wave equation appealing to the physicists of his time (the 1920′s) and easy to use.

So far so good. The problem is that when one conducts experiments with electrons, one sees something consistent with point particles, not waves. Electrons are detected as points on photographic film or a modern electronic imaging device, physical tracks in a cloud chamber or bubble chamber. Waves interfere. Interference effects appear with electrons in accumulated patterns of many electrons. If one fires properly prepared electrons through a crystal, dots will appear on a sheet of photographic film. Each electron appears at a specific point on the film, unlike a conventional wave which would be spread out over the film. As many dots appear on the film, unexpectedly an interference pattern will appear as if the electrons are somehow also a wave diffracting through the crystal. Similar effects are observed with other subatomic particles such as neutrons.

At this point, the formal mathematical theory of quantum mechanics expressed in symbols gets fuzzy. Bohr, Born, Heisenberg, and their colleagues argued that there was something called a “measurement” or “observation,” which they did not define clearly. In most practical experiments, the measurement is the determination of the position of a subatomic particle such as an electron (a dot on a sheet of photographic film for example). The square of the amplitude of the quantum wave function was the probability of observing a particle at the position .

In general, the “measurement” was represented in symbolic mathematics as an operator acting on the quantum wave function . The measurement supposedly caused the quantum wave function to instantaneously jump to an “eigenfuction” or “eigenstate” of the operator. An “eigenfunction” is a simple concept with a fancy German name. An operator is something, such as taking the derivative or simply multiplying by two, that transforms one function into another function.

An eigenfunction of an operator is a function such that the operator transforms the function into a factor, known as an “eigenvalue,” times the original untransformed function:

With a position measurement, this meant the quantum wave function collapsed to a point, often represented by a so-called Dirac function where the Dirac function is zero everywhere except at and infinite at . This infinity is such that the Dirac function integrated to unity (1):

A measurement could also be a measurement of the momentum of the particle. In this case, the appropriate operator is the momentum operator:

In this case, the so-called eigenfunction or eigenstate of momentum is:

where the eigenvalue is the measured momentum . In this case, the probability of measuring a momentum is the so-called inner product of the original quantum wave function with the complex conjugate of the so-called eigenstate:

There is something remarkable and unexplained here. The quantum wave function jumps discontinuously and instantaneously from one form to another. If the measurement is a position measurement, the quantum wave function, which may be spread out over quite a distance, collapses instantaneously (faster than light) to a single point. What causes the collapse? What is a measurement? Schrodinger’s Equation and the other equations of basic quantum mechanics do not tell us. Representing the measurement as a symbolic operator does not tell us; it obscures the problem. Textbook quantum theory does not tell us what physical process actually causes the collapse, how the collapse occurs (it is presumably instantaneous), or why. In the textbook theory of quantum mechanics, the collapse of the wave function to a point is completely random . A conscious observer has no power to influence or control the outcome of the collapse (the measurement or observation).

The Quantum Measurement Problem

From the very beginning of modern quantum mechanics in 1927, some physicists, notably Albert Einstein, Erwin Schrodinger, and Prince Louis deBroglie, were dissatisfied with this hazy, hand waving explanation of what was going on. This has led to a number of attemps to clarify quantum mechanics including the Many World Interpretation of Hugh Everett, the pilot wave theory of Louis deBroglie and David Bohm, and a number of less well known theories and interpretations. Significantly, a few physicists including Eugene Wigner and John Wheeler speculated that consciousness might be directly involved in the collapse. Perhaps a conscious observer was needed to collapse the wave function. This in turn opened the door to a number of mystical possibilities.

The quantum measurement problem is a significant problem. It is often said that quantum mechanics makes accurate predictions for a wide range of physical phenomena and that it has never been wrong. There is supposedly no experiment that contradicts quantum mechanics in over ninety years of experiments in atomic, nuclear, and particle physics. Yet, quantum mechanics is a plastic theory. Because the theory does not provide a clear rigorous theory of when the measurement occurs, the predictions are ambiguous. If an experiment unexpectedly shows that a particle behaves as a wave, the measurement process is sufficiently ambiguous that one can probably argue that this is in fact what should have been seen. One simply didn’t understand quantum mechanics prior to the experiment. Conversely, if one observes particle-like behavior where one naively expected wave-like behavior, once again one only has to redefine what constitutes a “measurement” that collapses the wave function.

One of the problems that quantum mechanics was developed to solve is the stability of atoms. If matter is comprised of atoms and if an atom consists of a positively charged nucleus with negatively charged electrons somehow “orbiting” the nucleus, there is a serious problem. In the classical electromagentic theory of Maxwell, a charged particle that is accelerating, which can mean either changing speed or direction, will radiate electromagnetic waves. The atom should quickly decay. Yet, this was not observed. There was clear evidence that atoms contained a positively charged nucleus and seemingly electrons in something like an orbit around the nucleus. Nonetheless, the atoms did not decay.

In orthodox quantum mechanics, the electrons in an atom are uncollapsed wave functions. The phase of the wave function oscillates in time but the amplitude of the wave function is constant. This means somehow that the spatial distribution of the electric charge around the nucleus is constant — unchanging. Thus, there is no acceleration and the atom is stable. Amongst other things, the observed stability of atoms means that the wave function must refer to a physical reality, not to the observer’s state of knowledge about the electrons in the atom. If the electrons had a definite position unknown to the observer, e.g. a scientist performing an experiment, the atom would radiate electromagnetic waves and decay. The wave function cannot collapse for the atom to be stable.

These electrons in uncollapsed wave functions nonetheless appear to interact with other particles and objects, apparently without collapsing. For example, a mirror reflects light. This is thought to be due to the electric charge in the electrons in the atoms. Yet, seemingly, the electron wave functions in the atoms of the mirror do not collapse. The atoms in the surface of the mirror remain stable and do not decay as one might naively expect. So, too, if one pushes on the mirror, the firm resistance of the mirror to one’s hand is attributed to electrostatic repulsion between the electrons in the mirror and the electrons in the atoms of one’s hand. Yet, again, it does not seem that pushing on the surface of a mirror causes the electron wave functions to collapse and the atoms on the surface of the mirror to decay as one might naively expect. However, if one blasts the mirror with energetic x-rays, the x-rays will knock loose electrons which can be observed as tracks in detectors — seemingly point particles. The electron wave functions collapse and the mirror will physically degrade when bombarded with x-rays.

Why does one interaction or “observation” appear to collapse the wave function and another does not? To be sure, there is probably some hand-waving explanation of this seeming problem with quantum mechanics in the advanced physics research literature. But really, is it understood? Probably not.

These observations with mirrors and other macroscopic objects arguably bracket the conditions under which the wave function collapses. There are many practical and mathematical difficulties in determining what quantum mechanics predicts for a macroscopic object comprised of trilions of trilions of trillions of atoms and molecules. Presumably, scientists can study this behavior at the level of atoms, molecules, or small numbers of atoms with atomic force microscopes and other modern instruments and determine experimentally when and under what conditions the wave function is collapsing (if this is even a correct interpretation of the physical phenomenon). It is probably more accurate to say that experiments may be able to determine when and under what conditions the wave function collapses assuming that the wave function collapse really exists as a meaningful physical concept (for example, the pilot wave theory of quantum mechanics does not require the wave function to collapse to explain the puzzling observations).

Symbols and Concepts

The expression of quantum mechanics as a formal mathematical system of symbols (that could be programmed on a modern computer for example) does not contain a predictive mathematical formula for when and under what conditions the wave function will or will not collapse. The expression of the “measurement” or “observation” as a symbolic operation such as the operator does not answer this question. It actually sweeps the problem under a symbolic rug as Einstein, Schrodinger, and deBroglie realized. Rather physicists are using fuzzy verbal concepts such as Niels Bohr’s “complementarity” or unstated theories or speculations to specify when the wave function collapses and when it does not.

This is not unusual. When physical theories are expressed in purely symbolic forms such as Maxwell’s Equations or Schrodinger’s Equation, a great deal is often hidden in the definition of the symbols. The quantum mechanical is a good example. But, even in Maxwell’s Equations, exactly what is the definition of the electric field and the magnetic field . How are these symbols in the formal mathematics connected to real physical measurements in the laboratory or in the field? In physics and engineering, ultimately the abstract symbols must correspond to physical macroscopic objects and systems that form sense perceptions in the human mind, e.g. dots on a photograph, the deflection of a meter, and so forth. Somewhere in this process shapes and more generally concepts that the human mind can handle, but which we cannot (yet) express in purely symbolic terms usually appear.

Quantum Mysticism

There is a long history of physicists and other scientists wandering off into mystical or spiritual areas. During the Renaissance, the sharp separation and alleged conflict between science, religion, and mysticism largely did not exist. Johannes Kepler was an astrologer and mystic. Tycho Brahe was an astrologer and alchemist. Isaac Newton practiced alchemy and had a range of religous and mystical views. Even in the early nineteenth century, Michael Faraday was deeply religious although he tried to keep his personal religious views out of his scientific publications. Faraday and many of the early electrical experimenters and theorists were deeply influenced by mystical ideas about the unity of nature which led them to actively seek the evidence of a strong connection between electricity, magnetism, light, and chemistry which they found. What the Bleep Do We Know? features an entertaining and probably false story in which the Indians (Native Americans) were supposedly unable to see Christopher Columbus’s ships because they were not mentally prepared to see the ships — until a wise shaman somehow opened their eyes. Astonishingly, early electrical experimenters had been studying electricity extensively for at least twenty years without noticing that there were magnetic effects (fields in modern parlance) around wires carrying electrical currents. The invention of the electric battery in about 1800 made it possible to conduct extensive, detailed, repeatable experiments with electricity. The mystically inclined Hans Christian Oersted easily found the magnetic field around a current-carrying wire with a magnetic compass, something that anyone could have done for…twenty years. He saw what he was mentally prepared to see; it does happen (all the time).

The evolution and growth of modern science has been accompanied by the adoption of a militant, materialistic, reductionist, and atheistic creed by many leading scientists and popularizers of science (see, for example, works by Steven Weinberg and Carl Sagan). Nonetheless, there have been a number of modern physicists, including some with impressive resumes, who have bucked the trend and become enamored of various mystical, parapsychological, and spiritual ideas. Arthur Eddington developed a complex mathematical theory of everything with strong mystical elements with which he apparently became obsessed. This may have contributed to the notorious bitter feud between Eddington and Subrahmanyan Chandrasekhar in which Chandrasekhar’s discoveries threatened to shatter Eddington’s theory. The Nobel-prize winning physicist Brian Josephson is probably one of the best known, most prominent recent examples. Josephson became fascinated with extrasensory perception (ESP) and other exotic ideas. Although many of his ideas are rather materialistic, the physicist David Bohm became associated with various mystical and spiritual individuals and groups such as the spiritual guru Krishnamurti. Bohm’s ideas, in particular, have become intertwined wih many mystical ideas about quantum mechanics. John Hagelin, who appears in What the Bleep Do We Know?, features an impressive resume before leaving mainstream (government funded) theoretical particle physics to join the Maharishi Mahesh Yogi‘s transcendental meditation movement. Relevant to the theme of What the Bleep Do We Know?, the physicist and engineer Robert Jahn conducted experiments for years at Princeton claiming to detect small effects where people could slightly alter the outcome of random, presumably quantum mechanical processes such as radioactive decays or electronic noise.

The organized, mostly government-funded physics “community” has taken various measures to distance itself from these mystical, spiritual, and parapsychological theories and experiments. It is almost impossible to publish research papers on these sorts of things in major physics publications such as Physical Review. A few years ago, the prominent and influential arxiv.org preprint server shifted from a policy of being open to all to requiring an institutional affiliation, a sponsor to endorse the preprint, and some other “precautions,” generally to block publications by real or alleged “crackpots.” The mainstream physicist David Albert who appeared in What the Bleep Do We Know? rapidly distanced himself from the movie, claiming he was quoted out of context and did not realize the subject matter of the film; this may well be true.

It is difficult to know what to make of this sort of thing. Some important aspects of quantum mechanics are almost certainly not understood. Thus, it is not possible to rule out a direct role for consciousness or other mystical extensions to quantum mechanics. It is also clear the textbook theory does not require a role for consciousness as What the Bleep Do We Know? and similar materials claim — quite the opposite. Most physicists appear to be attracted to the modern field by the discovery that they are good at physics in school, which often means good at specific calculations or derivations. Modern physics is little more than a glory-driven professional sport like football or basketball — with plenty of egos and mindless competition and precious little “enlightenment”. Intrinsically, fundamental physics such as theoretical particle physics, cosmology, and so forth deals with fundamental questions about the nature and origin of everything. Some people come into physics primarily driven by curiosity, spiritual, or philosophical feelings about these basic questions. There is obviously an appeal to finding evidence of God or some comforting mystical reality in science and in empirically measurable phenomena such as quantum mechanical phenomena. More cynically, career prospects and salaries in physics are relatively poor; becoming a spiritual guru with a Ph.D. in Physics may be a career option, even a lucrative one in rare cases. Also, despite the general lack of mainstream government funding, over the years, a number of wealthy philanthropists such as the late Laurance Rockefeller and Robert Bigelow have funded a plethora of far out research topics such as parapsychology, crop circles, and UFOs.

Crypto-Mysticism in Mainstream Physics

Probably most readers of this article will find the story of a New Age guru allegedly channelling the 35,000 year old spirit of a warrior from the lost continent of Lemuria who promotes a questionable extension of quantum mechanics amusing, if not ridiculous, and may enjoy a smug sense of superiority over the poor benighted folks who believe this sort of thing. Yet, labels like cult, pseudoscience, bad science, junk science, and pseudomathematics are generally applied only to relatively small fringe groups with little money, power, or prestige. In fact, it is common to encounter religious and mystical language and imagery in popular physics articles. Sometimes it is deadly serious. Often the words are used in a way that lends itself to a double meaning. That is, a statement has one meaning if the words are given their common meaning in English, yet a different meaning if the “special meaning” used by physicists is used. In many cases, the religious or mystical language or imagery is ostensibly tongue in cheek, not really serious.

A fairly extreme example of this is the title of Leon Lederman’s book The God Particle, about the Higgs particle, the “Holy Grail” (another religious image) of particle physics. The title says it all. Is the Higgs particle really equivalent to God? Albert Einstein referred to God frequently in various public appearances. There is a famous statement by Einstein “Raffiniert ist der Herrgott aber boshaft ist Er nicht.” which is commonly translated as “The Lord is subtle, but not malicious.” Abraham Pais used this translation as the title of his biography of Einstein. What Einstein meant by “God” is not clear and the subject of some controversy. The famous American rabbi and Jewish leader Herbert Goldstein specifically asked Einstein in a telegram if he believed in God. Einstein allegedly responded with:

I believe in Spinoza’s God who reveals Himself in the orderly harmony of what exists, not in a God who concerns himself with fates and actions of human beings.

It is worth understanding that the philosopher Spinoza’s views were very far from either traditional rabbinical Judaism or Christianity. He was, in fact, excommunicated by his Jewish religious community. This custom of using the term “God” in a way that differs dramatically from common usage is frequent in the popular physics literature. Here is a fairly clear statement of this practice from Leonard Susskind’s popular physics book The Cosmic Landscape: String Theory and the Illusion of Intelligent Design (page 8):

I don’t know the religous beliefs of Davies or Greenstein, but I would be wary of too literal an interpretation. Physicists often use terms like design, agency, and even God as metaphors for what is not known — period. I have used the term agent in print and have been sorry every since. Einstein often spoke of God. “God is cunning but He is not malicious.” “God does not play dice.” “I want to know how God created the world.” Most commentators believe Einstein was using the term God as a metaphor for an orderly set of laws of nature.

Well, maybe. Why not use “what is not known” instead of “God,” which in common usage means something totally different?

It is also common to encounter language about “knowing God,” “glimpsing God,” “knowing the mind of God,”, or “glimpsing the mind of God” in the popular physics literature. There is an example of this in Brian Greene’s PBS/Nova special The Elegant Universe where a physicist talks about knowing the mind of God.

It is also common to encounter mystical or religious references that are supposedly tongue in cheek. For example, the PBS/Nova special The Elegant Universe contains a seemingly tongue in cheek story about the original supposed breakthrough in superstrings. The two physicists are struggling to perform a complex symbolic derivation. There is however a raging storm outside which threatens to distract them from sucess. Are the Gods themselves seeking to prevent the discovery? The physicists, of course, succeed despite the raging storm. It is all reminiscent of the jealous Creator of some Gnostic Christian sects who seeks to keep men ignorant, but knowledge gnosis of the true supreme God sets men free. Murray Gell-Mann used the mystical term “The Eightfold Way” to humorously label his theory of quarks. Particle physicists initially named two of the quarks “truth” and “beauty,” but later renamed them “top” and “bottom.”

The central idea of most mystical systems is the absolute unity of all things, an absolute One identified with God and often the universe as well. In the western Neoplatonic tradition, this is closely associated with the mathematical mysticism of Pythagoras and Plato, in which mathematics plays a central role in the nature and architecture of the universe. The quest for a unified field theory or theory of everything (TOE) in the popular physics literature bears a striking resemblance to this ancient mystical belief. Even the “landscape” of a myriad alternate universes, the subject of Susskind’s book, bears a suspicious resemblance to the myriad failed worlds that God creates before our universe in some versions of the Jewish Kabbalah, the Jewish mystical tradition. In the Kabbalah, one of the supreme goals is to learn and master the most secret name of God, which confers infinite power on whoever speaks it; this bears a marked resemblance to the quest for a secret equation, a theory of everything, that explains the entire of creation and which might confer great power as well (antigravity, warp drives, quantum teleportation, time travel, who knows?).

Scientism is the idea that natural science is the most authoritative worldview or aspect of human education, and that it is superior to all other interpretations of life. Its’ appeal to many scientists is easy to see. Science can take the place of traditional religion. With quantum mysticism, the mysticism and spirituality is out in the open, clearly recognizable — proudly touted, in fact, whatever its’ actual merits. The disguised (crypto) mysticism of the popular physics, and other popular science, literature is less clear, easy to deny as just words used in an odd way or as light hearted humor. No one is really claiming the Higgs particle is God…or are they? But the dangers from heavily funded and prestigious mainstream science gone off the rails would dwarf the dangers from even the largest New Age cult to insignificance.

Conclusion

Returning to What the Bleep Do We Know? and similar materials, caution is well advised. Consciousness is not a part of regular textbook quantum mechanics nor is it clearly necessary to explain the logical and philosophical problems with quantum mechanics. It may be that we will one day find that mysticism, parapsychology, or some similar exotic idea is involved in quantum mechanics, but we do not know this today nor are we close to an answer.

Speaking mathematically, it may be possible to find the explanation for the quantum measurement problem by simply adding a non-linear term to the Schrodinger Equation that causes the collapse of the quantum wave function under certain physical conditions, most probably some sort of interaction between the wave functions of different particles. It is very hard to know what the precise form of this additional term might be. There have been a few attempts in the physics literature. Most likely it is necessary to think carefully about the conceptual issues swept under the rug in the quantum measurement operators to make educated guesses about the nature of this term if it exists. It remains quite possible that solving these problems requires a radical change, discarding the Schrodinger Equation and other known mathematics almost entirely.

Suggested Reading/References

	The Quantum Ten: A Story of Passion, Tragedy, Ambition, and Science Sheila Jones Oxford University Press, USA 2008 336 pages
	Quantum Dialogue: The Making of a Revolution Mara Beller University Of Chicago Press 2001 365 pages
	Infinite Potential: The Life and Times of David Bohm F. David Peat Basic Books 1997 380 pages
	The Undivided Universe: An Ontological Interpretation of Quantum Theory David Bohm, Basil J. Hiley Routledge 1993 397 pages
	Opticks: Or a Treatise of the Reflections, Refractions, Inflections & Colours of Light-Based on the Fourth Edition London, 1730 Isaac Newton, Albert Einstein (Foreword) Dover Publications 1952
	Quantum Theory At The Crossroads – Reconsidering The 1927 Solvay Conference Guido Bacciagaluppi, Antony Valentini Cambridge University Press 2009 556 pages
	Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy J. S. Bell Cambridge University Press 2004 288 pages
	The Cosmic Landscape: String Theory and the Illusion of Intelligent Design Leonard Susskind Little Brown and Company New York 2005

Copyright © 2010, John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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Hobbes: “A new decade is coming up.”

Calvin: “Yeah, big deal! Hmph. Where are the flying cars? Where are the moon colonies? Where are the personal robots and the zero-gravity boots, huh? You call this a new decade?! You call this the future?? HA! Where are the rocket packs? Where are the disintegration rays? Where are the floating cities?”

Calvin and Hobbes
by Bill Watterson
December 30, 1989

Indeed, where are the flying cars, the intelligent robots, the bases on the Moon, the atomic powered lawnmowers? Forty years ago futurists confidently predicted all of these things in our time. Some of these predictions were undoubtedly unjustified hype. Nonetheless, the predictions were more reasonable than may now seem the case. Power and propulsion technologies experienced dramatic progress for almost two centuries from the introduction of the separate condenser steam engine in the 1770′s until the manned landing on the Moon in 1969. This progress included both steady evolutionary progress and several technological revolutions: the separate condenser steam engines, the high pressure expansive steam engines, internal combustion engines, the electric battery, electric generators and motors, the diesel engine, jet engines, rocket engines, and atomic power. Progress in power and propulsion technology has been very limited over the last forty years.

So too, progress in many other areas ranging from artificial intelligence to fundamental physics has been quite limited. This has made electronics and computers highly visible with society pinning its hopes for a better future on these fields, in a way that was not the case with comparable advances in the past — radio and television for example — that were overshadowed by dramatic advances in power, propulsion, and other technologies. Probably a number of factors, including just plain bad luck in some cases, have contributed to the limited scientific and technical progress of the last forty years. This article will focus on excessive reliance on symbolic manipulation in place of the conceptual reasoning and conceptual leaps usually involved in major scientific or technological breakthroughs.

What is symbolic manipulation?

The ancient Greeks knew mathematics as arithmetic and geometry. Arithmetic consisted of addition, subtraction, multiplication, and division of numbers. The Greeks had a crude number system similar to the Roman Number system familiar to most students today (I = 1, V = 5, X = 10, L = 50, C = 100, etc.). Geometry consisted of words and pictures, verbal proofs, derivation, or calculations, and diagrams constructed with a compass and ruler. When it is said the Greeks could find the roots of certain quadratic equations, this meant performing visual geometric operations with a compass and ruler. When the Greeks referred to the square of a number, they meant a literal square drawn on a piece of paper. This style of mathematics is found in the works of Euclid, Archimedes, and Apollonius of Perga. It became the standard in Europe for over fifteen hundred years. Johannes Kepler wrote his book New Astronomy, published in 1609, detailing the discovery of the elliptical orbit of Mars and the precise mathematical law of planetary motion along the ellipse in this style. Isaac Newton used a similar style in his famous Principia published in 1687.

In about 1600, the French mathematician Francois Viete introduced the concept of using letters to represent arbitrary numbers, what we now call variables. Viete’s system was crude by modern standards. He used only vowels to represent variables and consonants for constants. It took several generations for symbolic mathematics to mature and replace the traditional Greek style of words and pictures. Mathematics and physics have moved in the direction of increasing reliance on symbolic mathematics and abstraction since then.

Most people encounter symbolic mathematics in high school algebra. For example, what is in:

This can be solved easily by subtracting the number three (3) from both sides of the equation.

With symbols, this solution can be generalized to an entire class of problems:

Then, subtract from both sides of the equation.

This is a general solution for all equations of the form . Up to a point, this general solution will be true independent of the interpretation or meaning of the symbols , , and . This is part of the power and appeal of symbolic mathematics. It is also one of the pitfalls of symbolic mathematics. In physics or engineering, what if the phenomema represented by , , and don’t actually have the properties of numbers? In this computer age, another important aspect of symbolic formulas, equations, and solutions is that they can be programmed on a computer. Essentially all computer programming languages have been designed to incorporate variables and at least simple symbolic mathematics involving addition, subtraction, multiplication, and division. Some programming languages and environments such as Matlab and Mathematica can handle quite advanced symbolic mathematics. Some programming languages and environments such as Mathematica and Maxima (formerly known as MACSYMA) can perform certain types of symbolic manipulations, automatic theorem proving, and other exotic functions.

The example above is very simple. Symbolic mathematics can be much more complex and solve much more difficult problems. A more difficult problem that took many centuries to find a solution is the general solution for the quadratic equation. This is usually taught in high school algebra. It is actually rare in everyday life, but occasionally surfaces as a real problem in computer graphics. This is an example of a quadratic equation:

This equation has a simple solution. It can be factored into two terms:

The solution is . In general, the square of is:

With some thought, this actually gives us a way to solve the general quadratic equation:

where , , and can be any three numbers. In a nutshell, we want to reorganize the equation so it is in the form:

This is known as “completing the square”. First, divide both sides of the general quadratic equation by :

To turn the left hand side into , must be . So, subtract from both sides of the equation:

Then, add to both sides of the equation.

Now, the equation is in the form where and . Therefore:

Then, subtract from both sides of the equation and remove the factor outside the square root sign to get the formula for .

This is the general solution for the quadratic equation in symbolic form. Once again, up to a point, this formula is correct regardless of whether , , , and refer to distances on a surveyor’s map, distances in a blueprint of a building, a machine, or an electromagnetic field. The answer seems to be found in a precise, step-by-step logical procedure seemingly devoid of unpredictable and sometimes wrong intuition.

Most high school algebra textbooks stop at this point and mention the existence of general solutions to the cubic and quartic equations:

(Cubic Equation)

and

(Quartic Equation)

This practice in high school algebra textbooks is somewhat misleading and probably contributes to magical thinking about the power of symbolic manipulation. The solution to the quadratic equation by symbolic manipulation involves only about a dozen steps which are typically spelled out step by step (as above) in a high school algebra textbook. As mathematical problems become more complex, the possible symbolic manipulations grow, often exponentially, in number. A successful solution to a problem in higher mathematics may involve hundreds, thousands, or even more steps. There are many possible rearrangements of the symbolic formulas that lead nowhere, much as there are many choices in a game of chess that lead only to defeat. The number of possible sequences of substitutions and other symbolic manipulations grows beyond the limited ability of human beings to exhaustively try each sequence and often beyond the capacity of a computer program, as powerful as computers have become, to exhaustively try each possible sequence. Thus, the mysterious human “intuition” reappears. Mathematicians develop a sense of the sequence of steps that is likely to result in a useful answer. Even so, they are often wrong and have to make repeated attempts — trial and error — to find a useful result, but many less trials than an exhaustive brute force search.

The Status of Symbolic Manipulation

Symbolic mathematics and symbolic manipulation have acquired great status and a reputation bordering on magic both in general society and much of the scientific, engineering, and mathematical community. Not without good reason. Symbolic manipulation has solved many problems and undoubtedly will continue to do so. Nonetheless, it has substantial limitations relative to human abilities and judgement, our conceptual reasoning abilities as will be discussed further below.

Probably the epitome of the mystique of symbolic mathematics and manipulation in general culture is the famous equation:

This cryptic equation became closely associated with Albert Einstein and the atomic bomb. In the equation, represents energy, represents mass, and represents the speed of light. In the Special Theory of Relativity one can derive this equation to show an equivalence between mass and energy. The atomic bomb derives its enormous destructive power from the conversion of a fraction of a percent of its mass into energy. A bomb that converted all of its mass into energy would be hundreds of times more powerful than the most powerful thermonuclear weapons in the arsenals of the leading nuclear powers.

The general press, popular science, textbooks and many other sources frequently display pictures of scientists, engineers, and mathematicians standing in front of blackboards covered with cryptic symbols, reinforcing the message that the symbolic mathematics and manipulations is the essence of science, engineering, and mathematics.

The Scientist at the Blackboard (An Iconic Image of Popular Science)

Richard Tolman and Albert Einstein	Paul Dirac
Theodor Von Karman	Ed Witten

The nineteenth and twentieth century saw the steady “professionalization” of science and engineering. This accelerated during and after World War II which saw a huge expansion in the size, the scope, and the level of public funding of scientific research and technological invention, much of it associated with military research and development. Hundreds of years ago, science and engineering, invention and discovery, were conducted by a hodge podge of often wealthy amateurs, craftsmen such as clockmakers, blacksmiths, bicycle makers (the Wright Brothers), and university professors who often relied on teaching rather than research grants for their income. Prizes for inventions and discoveries, in principle (if not always in practice) open to anyone, rather than proposal-driven grants and R&D contracts to specific researchers or research groups were much more common. This changed gradually over the nineteenth and early twentieth century. Germany, in particular, witnessed the rise of a professional civil-service of university professors conducting basic and applied scientific research. Many of the German customs were deliberately copied in the United States or imported with the huge wave of refugees from Nazi Germany.

The professionalization of science and engineering was accompanied by an increasing emphasis on the importance of mathematics and symbolic mathematics. In the ideal, the professional scientist or engineer would conduct sober logical scientific research, perform sophisticated calculations or derivations correctly the first time through rigorous professional training, and produce scientific and technical advances as needed, avoiding the messy intuition and time consuming, unpredictable trial and error of the mere “inventor” or “tinkerer.” The reality, of course, never matched this ideal, but the enormous success of wartime projects such as the Manhattan Project, the German V-2 rocket program, and lesser known successes such as the Jet Assisted Take Off (JATO) rockets in the United States (all these projects led by German or German-trained scientists and engineers — J. Robert Oppenheimer with a Ph.D. from the University of Gottingen, Werner Von Braun, and Theodor Von Karman) contributed to the credibility of this picture of the central role of symbolic mathematics, and symbolic manipulation in particular, in the post-World War II professional science and engineering. According to the semi-official history of the Manhattan Project, the atomic bomb succeeded on the first test, a remarkable achievement in the history of invention where repeated failures and repeated trials have been the norm, based in part on purely theoretical calculations by Hans Bethe, J. Robert Oppenheimer, and other theoretical physicists, giving great credibility to mathematically driven science and engineering.

The primacy of symbolic manipulation offered a comforting world in which the scientist, engineer, or mathematician would be an intellectual plumber, assigned to solve a specific narrow problem in a specific time without trial and error and without having to deal with any “big picture” issues, “above his pay grade” in government-speak. Policy makers, business leaders, and the generals who dominated the vast military R&D apparatus of the post World War II world could deal with the important issues while assigning the solution of a specific problem to a narrow technical expert. Conceptual analysis is much more likely than symbolic manipulation to raise uncomfortable and sometimes explosive issues. In his famous book New Astronomy, Johannes Kepler, a devout Lutheran, devoted much of the introduction to arguing that his discovery of the elliptical orbit of Mars did not in fact conflict with the Bible, which seems to state that the Earth is stationary and the Sun moves around the Earth in a few places (such as the Book of Joshua when God miraculously stops the Sun in its path). Indeed, the wiser contemporaries of Galileo and Kepler advised both men to present their discoveries as mere mathematical conveniences for computing the positions of the planets and moons. In this way, the big picture and potentially explosive issues (the Bible is wrong) could be avoided. By its nature conceptual reasoning can raise uncomfortable issues and raise those issues in ordinary words and pictures that are easily understood by non-experts, unlike abstract symbols. Galileo, the epitome of the vain and arrogant scientist, blindly ignored the advice and got into serious trouble. Kepler, not without his faults but far more conciliatory in style than Galileo, avoided a similar fiasco (he was excommunicated by the Lutheran Church — a much less serious matter than a trial by the Inquisition — over an unrelated matter of religious doctrine).

Nor is the sensitive nature of conceptual analysis a problem restricted to Galileo, Kepler, and the Renaissance. The logical and philosophical problems with quantum mechanics that puzzled Albert Einstein, Erwin Schrödinger, and Prince Louis de Broglie ultimately led Einstein to deduce the non-local nature of quantum mechanics, something that appears to have been confirmed in numerous experiments over the last several decades. This quantum entanglement in which widely separated objects maintain some sort of mysterious instantaneous connection — certainly faster than light — raises many curious issues and has been embraced by mystics, parapsychologists and others as empirical evidence of the oneness of all things claimed in most mystical systems. Quantum mechanics as an abstract algorithm expressed in symbolic mathematics raises none of these issues — or, at least, they are well disguised.

Conceptual Reasoning and Analysis

Curiously, most people never use symbolic mathematics and symbolic manipulation. They encounter it in high school algebra and often never again. For the most part, human beings reason with concepts, with mental pictures, and only very rarely with explicit numbers, mathematical symbols, symbolic formulas, or symbolic manipulation. After decades of mostly failed research into artificial intelligence, we have some idea why this is. For certain classes of geometric objects such as circles, spheres, cubes, and so forth, there are exact symbolic formulas.

(The Circle)

This is the formula for a circle with arbitrary location and size. The radius of the circle is . The location of the center of the circle is . We can do the same with spheres:

(The Sphere)

This is a sphere with radius centered at in three dimensions.

So far, so good. What is the symbolic formula for the shape of a cat? A tree? A car? A chair? The letter “A”? A door handle? The issue is not the symbolic formula for a single instance of a cat, a tree, and so forth. Movies are now full of specific computer generated cats, trees, cars, and so forth. But what is the symbolic formula for the class of cats, the class of trees, the class of cars, etc.? How do human beings look at cats — big cats, small cats, thin cats, fat cats, brown cats, orange cats, cats dyed purple — and recognize that each specific cat is an instance of the class of cats, the concept of a cat? We have no idea. Object oriented programming such as found in Java, C#, C++ and many other programming languages was inspired in part by this problem, but object oriented programs still cannot think conceptually like human beings. Various highly mathematical artificial intelligence methods such as artificial neural networks were developed to solve the same problem, but so far they have not succeeded.

For most everyday shapes, objects, and phenomena, we do not know a symbolic mathematical formula, unlike simple geometrical shapes like the circle or sphere. Somehow, through conceptual reasoning skills and “intuition” human beings can reason successfully about the many shapes, objects, and phenomena they encounter every day without anything clearly identifiable as symbolic or numerical mathematics. This can be very precise. Human beings have little difficulty picking up and manipulating quite complex shapes. Computers and robots, even using the most advanced mathematics that we know, often cannot do the same. This is why many common tasks such as clerks at convenience stores or assembly tasks in factories are still performed by human beings. This is a very precise ability, as if the human mind was subconsciously computing the precise coordinates of surfaces. There is little conscious awareness of this if some sort of mathematics is in use as it probably is.

Most major advances in power and propulsion have involved at least one conceptual leap, usually a new architecture — a new component or a radically new system. In many cases, these are new shapes and/or combinations of materials that work in a substantially different way from past systems. Since in most cases, the new shapes or structures cannot be represented as a symbolic formula, it is difficult to see how any known symbolic manipulation could find the improvement. In most cases, it was a mental leap — so-called intuition. One can often simulate a specific instance of a new architecture or design with modern mathematics and computers. Some critical equations such as the Navier-Stokes equations remain difficult or impossible to simulate numerically or solve exactly through symbolic methods. Neglecting the problems with solving or numerically simulating equations like the Navier-Stokes equation, it remains the case that there is no way to express the general shapes or concepts as symbolic formulas, although one can often express a specific instance as a computer or symbolic model. Consequently, both symbolic manipulation and computer simulations are simply unable to find a new shape, structure, or system for power or propulsion or many other forms of mechanical invention. They can be helpful in verifying a concept once found, determining that it is likely to succeed.

In fundamental physics, most major advances consisted of a conceptual leap, often expressed in words or pictures, followed by the development of a mathematical formula that implemented the new concept. Kepler realized that the orbit of Mars was something like an ellipse. He was fortunate that the ellipse was a simple geometric form that had been studied successfully by Apollonius of Perga over a thousand years before. He was basically able to look up the mathematics that matched his mental concept. In probably the most extreme example of the separation of concepts and mathematics, Michael Faraday constructed a mental picture of the electrical and magnetic fields (as pressures and circulating vortices in the hypothetical luminiferous ether) entirely without mathematics. William Thomson (later Lord Kelvin) and then James Clerk Maxwell, Thomson’s student, were able to translate the pictures into mathematical language, eventually Maxwell’s Equations. In the case of Faraday and Maxwell, it was not a simple matter of looking up the mathematics in an old book. Rather, Maxwell had to construct mechanical models of the hypothetical ether, assign values to special or representative points in the mechanical model, and then infer the differential equations that comprise Maxwell’s Equations. Einstein developed the concept that gravity was due to mass and energy warping space-time well before he was able, after several failures, to find a specific equation that implemented his concept and appeared to match certain observational measurements (notably the precession of Mercury). Schrödinger started from Louis de Broglie’s concept that the electron had a wave associated with it and tried for some time to find a wave equation that matched the spectrum of hydrogen. There are many possible wave equations and Schrödinger, like Kepler before him, was fortunate that a simple, easy to guess formula could be found.

If one tries to perform fundamental physics purely by symbolic manipulation without any mental visualization or conceptual analysis, one has very little to work with. One must basically guess symbolic formulas, often by randomly adding terms to known formulas and expressions. This is very easy to do with the Lagrangian formalism. The literature of theoretical physics in the last forty or fifty years is filled with guesses at Lagrangians for various forces and phenomena and especially the theory of everything (TOE). Historically, physicists used concepts to narrow down the number and range of symbolic formulas to try. There was still a lot of trial and error but often the conceptual analysis, the “intuition,” seems to have been effective.

Conclusion

Symbolic mathematics and symbolic manipulation are very powerful in some cases. They are not a panacea. In fact, given the current state of mathematical knowledge, they have great limitations. Most people have, use, and rely for their survival on conceptual reasoning abilities every day that we don’t know how to express in symbolic mathematical form or program on a computer, if this is even possible. To make major progress, even in highly mathematical areas like pure mathematics or theoretical physics or power systems, we probably need to make heavy use of conceptual analysis and visualization just as the successful scientists and engineers of the past did.

Copyright © 2010 John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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“Our advice: Beware of geeks bearing formulas.”
— Warren Buffett, Letter to Shareholders of Berkshire Hathaway, Reporting Huge Losses

In the classic 1961 Disney science fiction comedy movie The Absent Minded Professor, Professor Ned Brainard of Medfield College of Technology has been struggling for three months to make a breakthrough in his garage, home to a low budget chemistry lab and his antique Model “T” car. A blackboard shows a collection of cryptic formulas including:

Professor Brainard has a sudden realization that the formula is in error and changes it to:

The Professor proceeds to make a series of frantic changes to his bubbling chemistry experiment, forgetting about his wedding, and there is an explosion. When he wakes up (after missing his wedding and losing his fiance), he has discovered “flying rubber” or “flubber,” a miraculous material that seems to violate every law of physics and provides the driver for the rest of the comedy. The movie ends with the Professor (and his fiance in a true Disney happy ending) flying his flubber-powered Model “T” to the United States capital where he outflies jet fighters sent to intercept him. Incidentally, the corrected formula is actually the formula for the enthalpy, a common thermodynamic concept.

This portrayal of the near magical power of mathematics is common in science fiction, especially popular movies and video, serious as well as comedic. The classic science fiction movie The Day The Earth Stood Still (1951) features a famous scene in which the alien visitor Klaatu solves a problem in celestial mechanics on the blackboard of Professor Jacob Barnhardt, supposedly revealing the secret of interplanetary (perhaps interstellar) travel, a somewhat curious act for the representative of an interplanetary civilization supposedly fearing atomic attack from the barbaric Earth men. The original Star Trek television series featured several episodes in which either the science officer Mr. Spock or the ship’s super-intelligent computer solved some new, never before encountered problem by performing some mysterious calculations, all by the end of each less than one hour episode. The 2003 disaster movie The Core, which has the dubious distinction of some of the worst and most inaccurate physics in any major movie, features several scenes where the scientists perform some complex calculation, sometimes in their heads in seconds, and solve an otherwise fatal problem (e.g. the end of the world). The television series Numb3rs features mathematicians who help the FBI solve otherwise unsolvable criminal cases through the magic power of mathematics.

In general, it is common to encounter a scene in science fiction movies and video with a scientist, engineer, or mathematician at a blackboard covered with cryptic symbols. The blackboard usually faces the camera and often dominates the scene. Sometimes a modern whiteboard or a notebook page may be used, but the blackboard is still the most common icon. Frequently, these scenes depict the scientist performing some sort of mysterious symbolic manipulation, such as a substitution of one expression for another or the sign change in The Absent Minded Professor. This symbolic manipulation is often followed explicitly or implicitly by miraculous results. There are some obvious reasons for these scenes in movies and video. A blackboard covered with cryptic symbols is a highly visual image; it tells the story very quickly and clearly. The use of cryptic symbols and mysterious symbolic manipulations saves the script writer from explaining the invention or discovery: how would a real flubber work? Finally, these scenes are very similar to traditional portrayals of ceremonial magic where the magician performs marvels through mysterious symbols, hand gestures, and incantations.

How realistic are these portrayals? Based on the history of mathematics, they are far from realistic. For example, in The Absent Minded Professor, Professor Brainard is stumped for three months — only three months — before making his breakthrough. Most major breakthroughs similar to the fictional flubber have taken years, usually at least five years. Very often, the inventor or discoverer was stumped, on the wrong track, for most of that period; in this respect, the portrayal in The Absent Minded Professor is somewhat accurate.

How important have mathematical calculations been in breakthroughs in power and propulsion technologies, such as the fictional flubber? Not as much as one would think. Major breakthroughs in power and propulsion technologies usually involve a conceptual leap in which the architecture of the system is changed. This usually means either the introduction of a new component or a radical redesign of the system. It has often been difficult or impossible to calculate what would happen with the new design. Even modern computer simulations are typically calibrated on current designs, components, and so forth. A new component or, even worse, a radical new architecture often falls outside the realm of validation of current simulation models. Imagine trying to design an anti-gravity drive using the simulation software for the Space Shuttle. In a number of engineering fields such as aviation the Navier-Stokes equation is thought to describe fluid dynamics, air and water for example. The Navier-Stokes equation has proven impossible to accurately simulate numerically on a computer in many cases for deep mathematical reasons that remain unresolved. This is one of the reasons that the Navier-Stokes equation is one of the Millennium Problems for which the Clay Institute has offered a one million dollar prize. Consequently, engineers and designers must still rely on empirical methods in many cases: wind tunnels, flight tests, and so forth.

Professor Brainard bears a suspicious resemblance to the rocket pioneer Robert Goddard (1882-1945), who was very much in the news and public consciousness in 1961, the height of the post-Sputnik space race (Hubert Alyea (1903-1996), a Professor of Chemistry at Princeton who gave entertaining chemistry demonstrations somewhat similar to Brainard’s antics in the movie often claimed to be the model for the character). Goddard was a professor at a small technical university, Clark University. He even married the secretary of the University President, as Professor Brainard does in the movie. Goddard was a physicist who certainly used mathematical calculations in his work. Yet, remarkably, most of the rocket components were developed empirically by inspired trial and error. The mathematics of Goddard’s time was simply too primitive to simulate the operation of complex shapes like rocket motors, turbo-pumps and so forth. Goddard made all the same key advances in rocketry also made by the German rocketeers including switching from unreliable powder explosives to liquid propellants, the torpedo shaped design with the payload on top, the tank in the middle, and the rocket engine trailing behind (yes, early liquid propellant rockets had the engine in front), and the use of the extremely complex turbo-pump to achieve the high thrust needed for long range and eventually orbit. These were all conceptual leaps that could not be found by symbolic manipulation of mathematical formulas. Note that it took Goddard over five years just to realize to discard unreliable powdered explosives and switch to liquid propellant rockets. Each major conceptual leap in his rocket design took years!

Goddard’s arch-nemesis, Professor Theodor von Karman (1881-1963) of Caltech, became a key adviser to the Air Force and United States Federal Government in the years after World War II. A graduate of the highly mathematical University of Gottingen in Germany, Von Karman heavily promoted the concept of using detailed theoretical calculations to perform aviation and rocketry research and development in a highly “scientific” way, contrasted to the intuition and trial and error of traditional “inventors.” The irony of this is that the development of the Jet Assisted Take Off (JATO) rockets during World War II that made Von Karman’s reputation is about as far from the ideal of mathematically driven research and development as one could imagine. The actual inventor “Jack” Parsons (1914-1952) was a truly bizarre character with negligible mathematical skills. He did have however many years of hands-on experience in building rockets, which mostly blew up unsuccessfully, and explosives. One day in 1942, he somehow realized that one could make stable, reliable solid fuel rockets by replacing the unreliable explosive powders of the day with a mixture of asphalt and potassium perchlorate (a widely used oxidizer in explosives). By some accounts, he was inspired by his reading about the mysterious Greek Fire weapon of the medieval Byzantine Empire which may have been a mixture of oil and a powerful oxidizing agent. Again, this was a conceptual leap with negligible math. With the right concept, it took only a few weeks in 1942 for Parsons and his associates, known as the “Suicide Squad,” to build some working prototypes, mostly by trial and error. Note, however, that Parsons had been working on rockets, mostly failing, for almost ten years before his breakthrough.

Of course, as an applied mathematician, the author does not want to give the impression that mathematics and calculations are useless or unimportant. This is certainly not the case. There are cases where machines were built according to theoretical calculations and worked (mostly) right the first time. The first atomic bomb, built based in part on theoretical calculations, worked the first time. However, even with the atomic bomb, there was an enormous amount of intuition and trial and error at the component level. The Space Shuttle was designed based on wind tunnel data, experimental flight data from the X-15, some flight tests of partial mockups, and theoretical calculations. Nonetheless, the first flight of the Space Shuttle was a success, although the mission came close to catastrophic failure during reentry when the control system began to oscillate unexpectedly; the theoretical calculations were not quite right. Machines that work right the first time based on theoretical calculations appear to be rare, even with modern computer simulations. In a recent example, the highly computer oriented space startup SpaceX had to try four times before successfully launching a satellite into orbit. Three computer designed rockets in a row blew up or crashed during launch.

The magical mathematics of science fiction has considerable influence. This is how many people, including many scientists and engineers who should probably know better, view mathematics. Theodor von Karman and others like him promoted a very similar picture of the power of mathematics in public policy circles after World War II and to this day. Popular science, which presumably is not fiction, such as the PBS Nova specials, Scientific American articles, and other sources often portrays essentially the same picture of mathematics. Policy makers, business leaders, and the general public often have a picture of mathematics and mathematicians, especially the power of symbolic manipulation of mathematical formulas, that borders on magic. Many people are blissfully unaware of the large amount of trial and error and calendar time frequently involved in developing applied mathematical formulas (or computer programs based on them) that solve real problems. The role of conceptual leaps in invention and discovery has been deprecated in favor of the power of symbolic manipulations like Professor Brainard’s switch from to in his magic formula. The overused cliche “thinking out of the box” is often a synonym for trying a slightly different term in a mathematical formula rather than, for example, realizing that combining asphalt and potassium perchlorate avoids the problems of explosive powders in rocketry. The author has actually heard someone use “out of the box” to refer to using a different compiler flag on the GNU C Compiler to compile a mathematical program implemented in the C programming language.

There are several reasons why sci-fi math seems believable even to scientists, engineers, and mathematicians who should presumably know better. First, some people are able to achieve phenomenal performance on standardized math tests (e.g. the SAT, GRE), academic exams, qualifying exams in graduate school, formal contests such as the International Math Olympiad and the Putnam Exam, and so forth. Some people can answer these tests very accurately, even perfectly, in a short time period. This seems to match the performance of mathematicians in science fiction. It is likely this is because most tests and exams test the knowledge of known facts, calculations, and methods. The very top graduate schools are populated almost exclusively by people of this type. When prodigies of this type encounter new, unknown problems their error rate often rises dramatically. Further, the academic tests rarely measure the conceptual reasoning skills often used in breakthroughs; we don’t even know exactly what those skills are.

It is not clear why some people can perform so well on various tests, exams, competitions, academic homework, and so forth. This high performance is often attributed to extremely high innate intelligence. The psychologist K. Anders Ericsson has argued that expert performance is due to what he calls “deliberate practice.” These ideas have been popularized by Malcolm Gladwell in his recent book Outliers. Ericsson argues that extremely high performers in games such as chess, sports, performing arts, and mathematics engage in “deliberate practice” where they spend many, many hours practicing relatively rare problems, methods, and techniques. This differs from ordinary experience or practice. The example that Ericsson often uses is the backhand in tennis. The backhand is relatively rare in tennis. Usually a tennis player does not need to perform the backhand tennis swing. Consequently, even highly experienced casual tennis players may be weak on the backhand. In contrast, champion tennis players practice the backhand heavily. Thus, when they encounter the backhand, they can easily defeat players who have not practiced this rare move. Although Ericsson often uses the tennis analogy, he started from studies of champion chess players such as Bobby Fischer and generalized the results to most fields of expertise, arguing similar patterns occur in most fields of expertise.

It is likely that someone who engaged in large amounts of deliberate practice in mathematics could perform extremely well on mathematical tests, exams, and other competitive measures so long as these tests involved calculations or derivations that had been practiced. The problem is that by their nature inventions and discoveries involve problems that have never been solved by anyone. There is no way to practice in this way. Deliberate practice is very time consuming. Ericsson argues that most experts engage in 10,000 hours of deliberate practice, typically over ten years. In part this is derived from original studies of chess where champion chess players have almost always spent at least ten years of intensive study and practice before reaching the International Grand Master level. Ericsson has found similar patterns in many other fields. It is quite possible that deliberate practice at this level can substantially reduce the time available to study basic concepts and to develop the conceptual reasoning skills frequently used in invention and discovery.

Both Ericsson and Malcolm Gladwell have a strong environmental bias, that is attributing expertise and extreme levels of success to external environmental factors rather than innate, presumably genetic, characteristics of the individual. In his research papers, Ericsson assumes that expert performance translates into invention and discovery, developing a new method or technique in one’s field of expertise. One gets better and better through years of deliberate practice and finally reaches the rareified level where one starts to make original contributions to one’s field, inventions and discoveries. One problem with Ericsson’s theory is that some inventors and discoverers (like the physicist Albert Einstein) actually are not that technically proficient (expert) as conventionally defined in their field. In extreme cases, they have had to seek out technical experts to implement their ideas and concepts. Nonetheless, there is probably some truth in Ericsson’s theory of “deliberate practice.”

Second, popular science accounts, textbooks, and even many scholarly studies of scientific and technological breakthroughs frequently downplay or omit the long period of calendar time and large amounts of trial and error often involved. Many accounts focus on the “flash of insight”, the “Eureka moment” which can happen very rapidly according to many accounts, but almost always after a long period of study, preparation, and often failure. Even more remarkable, the “flash of insight” often occurs when the inventor or discoverer is taking a break — not thinking consciously about the problem. There are many accounts of breakthroughs during recreational walks, vacations, and other leisure activities. In his study of mathematical invention, the great mathematician Jacques Hadamard concluded this was typical of mathematical inventions. Taken out of context, these dramatic leaps forward sound like instances of extreme intelligence, extremely rapid solutions of problems by exceptional people, just like the superhuman feats in science fiction. Trial and error, especially the many failures, is extremely boring and tedious to describe; it is also not the way to inspire students to pursue scientific, engineering, or mathematical careers.

Third, many working scientists, engineers, and mathematicians are unaware of the actual history of the inventions or discoveries in their field. To the extent that they have studied this, they are usually relying consciously or not on stories in textbooks or popular science accounts or even word of mouth that are often quite inaccurate on close examination. Working scientists, engineers, and mathematicians are busy with their work and don’t have time to study the past. A number of accounts of major inventions in US history textbooks (such as the Wright brothers and the airplane) are highly misleading. They are probably intended to inspire students and promote patriotism.

Astonishingly, the costs of the widespread belief in science fiction mathematics may run into the trillions of dollars. Questionable mathematical models for the valuation of mortgage-backed securities played a central role in the financial crisis of 2008 and the on-going economic and financial problems. Economist Dean Baker estimates that the housing bubble in the United States amounted to $8 trillion in excess valuation, much of this in the form of loans bundled into mortgage-backed securities, most of which remain to be written off. Similar mathematical models were implicated in the failure of the Long Term Capital Management (LTCM) hedge fund, several derivative securities failures in 1994, and the stock market crash of 1987.

One can find numerous examples of start-up companies such as Lernout and Hauspie and Kurzweill Applied Intelligence whose exaggerated stock valuations depended on unrealistic ideas about the prospects for quickly solving artificial intelligence problems such as speech recognition, handwriting recognition (the pen computing fad of the early 1990s), and even reproducing human thought entirely. Stagnant research programs like tokamaks for nuclear fusion and superstrings in theoretical physics often seem based on exaggerated ideas about the power of mathematics, especially symbolic manipulation, and computer simulations to take the place of the conceptual thinking and conceptual leaps usually involved in major breakthroughs.

Mathematics, especially cryptic symbolic formulas, is often both intimidating and awe-inspiring, especially since mathematics has sometimes resulted in or contributed to astonishing accomplishments such as the atomic bomb, the manned landing on the Moon, GPS navigation, DVD video, and many other marvels of the modern world. Mastering modern mathematics is a major undertaking. Even top mathematicians like Andrew Wiles are often specialists in one area of mathematics. How are non-mathematicians to navigate the growing hazards of our increasingly mathematical world? Policy makers, business leaders, and the general public can verify the long calendar time and large amounts of trial and error usually reported in mathematical as well as other inventions and discoveries with careful research. The enormous mostly failed efforts by highly qualified scientists and engineers in numerous areas such as artificial intelligence can be verified from publicly available federal budgets. In many cases, the performance of the mathematics can be independently evaluated (the accuracy of speech recognition, power output of tokamaks, etc.) without any understanding of the abstruse mathematics.

Business leaders and policy makers became enamored, sincerely or not, with complex financial engineering in the housing market that seemed to offer a quick fix to the recession that followed the collapse of the Internet Bubble and the September 11 terrorist attacks, all on the time scale of the next election or even the next quarterly earnings report. In the present severe economic downturn, the appeal of another seeming scientific-technical quick fix — whether in the form of exaggerated expectations for machine learning algorithms, miraculous green energy technologies, or other purported “breakthroughs” — is easy to anticipate. In the current financial and energy crises, everyone should become familiar with the historical record of mathematical research and development: relying on primary historical sources where possible and not on science fiction portrayals or popular science that is really science fiction. Otherwise trillions more — personal savings, pension funds, public funds — may be squandered on ineffective high-tech panaceas offered, sincerely or not, as quick fixes to current economic problems and rising energy prices.

Notes

It is difficult to define exactly when an inventor or discoverer began working on an invention or discovery as well as when they succeeded. For example, the famous 1903 Kitty Hawk flight of the supposed first airplane took place in an extremely high wind; this is why the experiments were done at Kitty Hawk which had the highest winds in the United States according to US Weather Service data. The Wright Flyer of 1903 almost certainly could not have flown in still air. It was not until years later that the Wright Brothers (and Octave Chanute) achieved a flyer that could take off and land in still air.

The role of mathematics in mechanical inventions is often hard to determine. During the nineteenth century, an elaborate mathematical theory of thermodynamics usually attributed to Sadi Carnot was developed to explain the operation of steam engines. On close examination, early steam pioneers like John Fitch and Henry Voigt, Oliver Evans, and others developed high performance steam engines well before this theory, mostly by intuition and trial and error. James Watt, with assistance from John Robison and Joseph Black, clearly used mathematics to understand the efficiency of the steam engine, but he exaggerated the importance of the mathematics later in life, in part to lay claim to being a true “scientist” rather than a mere “tinkerer.” Similarly, more recently, Theodor Von Karman made a big deal out of mathematics and theoretical calculations by his graduate student Frank Malina in the development of the JATO rockets, but on close examination this invention was mostly “intuition” and empirical trial and error, especially by Jack Parsons.

Inventions and discoveries vary a lot. While many involve the “flash of insight” or “Heureka moment,” there are also many where there is no clearly identifiable single leap forward of this type. The “flash of insight” is a very dramatic story, ideal for popular science articles or a movie.

Accounts of inventions and inventors on Wikipedia tend to significantly understate the amount of trial and error involved in many inventions and discoveries. This is a common feature of popular and even scholarly accounts. It is often necessary to track down original notebooks, first hand accounts and so forth to fully realize the amount of trial and error usually involved in breakthrough inventions and discoveries.

Suggested Reading/References

The Papers of Robert H. Goddard: Volumes I-III
Esther C. Goddard (Editor) et al.
McGraw Hill Book Company
New York, 1970

Strange Angel: The Otherwordly Life of Rocket Scientist John Whiteside Parsons
George Pendle
Harcourt Inc.
Oakland, 2005

Sex and Rockets: The Occult World of Jack Parsons
John Carter, Robert Anton Wilson (Introduction)
Feral House
Los Angeles, 2004

The Wind and Beyond
Theodor Von Karman
Little and Brown Company
Boston, 1967

The Rocket Team
Frederick C. Ordway III, Mitchell R. Sharpe
Foreward by Werner Von Braun
Apogee Books, 2003 (original copyright 1979)

From Runway to Orbit: Reflections of a NASA Engineer
Kenneth W. Iliff and Curtis L. Peebles
National Aeronautics and Space Administration
NASA History Office
Washington, D.C.
2004

The Making of the Atomic Bomb
Richard Rhodes
Simon and Schuster, New York, 1986

The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field
Jacques Hadamard
One of Princeton University Press’s Notable Centenary Titles.
With a new preface by P. N. Johnson-Laird
1996

Outliers: The Story of Success
Malcolm Gladwell
Little, Brown and Company
2008

False Profits: Recovering from the Bubble Economy
Dean Baker
Polipoint Press
2010

When Genius Failed: The Rise and Fall of Long Term Capital Markets
Roger Lowenstein
Random House
New York, 2000

Startup: A Silicon Valley Adventure
Jerry Kaplan
Houghton Mifflin Co, Boston, 1995

“How High-Tech Dream Shattered in Scandal at Lernout & Hauspie”, by Mark Maremont, Jesse Eisinger, and John Carreyrou, Wall Street Journal, December 7, 2000

“ANATOMY OF THE KURZWEIL FRAUD
How Kurzweil’s straight-arrow CEO went awry”
By Mark Maremont in Waltham, Mass.
Business Week,
September 16, 1996

Copyright © 2010 John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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The word “mathematics” comes from the Greek μάθημα (máthema) meaning lesson and the verb μαθαίνω (mathéno) meaning “to learn”. It could be argued that math anxiety is anxiety about learning in general but that’s probably stretching things a bit. Although math is often perceived as the subject of proofs and absolute truth, there are lots of math myths and math legends. Perhaps enough for a dedicated blog.

Many of the legends have become part of history, others are history in the making. Pappus (“old man”) of Alexandria is the nameless giant and last of the ancient geometricians. Charles Babbage was the last of the Industrial Revolution mathematics machine-makers. Who, after watching “A beautiful mind” will ever forget the story of John Forbes Nash, Jr. battling his paranoid schizophrenia, not with pills but with logic and his winning a Nobel prize in Economic Science? And then there are the truly monumental life sagas of the largely unknown: Srinivasa Ramanujan, Jakow Trachtenberg, Andrew Wiles and Grigori Perelman.

A young, self-taught Ramanujan, set sail for England in a merchant ship in 1914 just before the outbreak of WWI and became the first Indian to be elected a Fellow of Trinity College, Cambridge just before the war’s end in 1918. Trachtenberg, imprisoned for 7 years in Nazi concentration camps, without pen or paper, worked mentally and developed speed arithmetic. The mathematics professor Wiles kept a 7 year-secret, working in recluse day and night and finally proved the 17th Century Fermat’s Last Theorem in 1994. Perhaps most mysterious of them all is Perelman, the Russian genius who posted a series of eprints to the free arXiv server in 2002, that proved the 1904 Poincaré Conjecture. He continues to live in poverty refusing the $1,000,000 Clay Millenium Prize awarded to him. Each of their stories is so impressive that each one should be made into a movie. Then there are the legends we ourselves create. We all have our personal heroes and heroines. Each of them a true legend.

And then there are the math myths. We are surrounded by many inspiring minds but the myths seem to linger on. Many have tried to dispel them. Perhaps no one more that Paul Halmos, a Hungarian-born American mathematician. At the University of Connecticut’s Gallery of Mathematicians, his portrait shares a wall with other great wizards such as Archimedes, Descartes, Euclid, Galileo and Newton. Not bad, at all. But I can hear you saying, “who the hell was he”? Heard of Q.E.D. (“quod erat demonstrandum”)? It’s what mathematicians write at the end of proof. The more common end-of-proof mark is “∎” which is Unicode symbol U+220E also known as “the Halmos”. He made fundamental advances in the areas of probability theory, statistics, Hilbert spaces and algebraic structure of mathematics and won medals for his ability to communicate mathematics. His “I Want to Be a Mathematician: An Automathography” is well worth a read.

Five of the most common math myths are:

1. The Genius Myth (that good mathematicians are born with special math talent and enormous left brains);
2. The Good Memory Myth (that good mathematicians have a phenomenal memory for formulas);
3. The Using-Tools-Is-Cheating Myth (that good mathematicians don’t use fingers, toes and calculators);
4. The Gender Myth (that good mathematicians are all men despite the abundance of female bio-statisticians);
5. The Who Needs it Anyway Myth (that math is useful only to mathematicians).

But the biggest myth of them all is the “I-Cant-Do-Math Myth”. I recently taught multivariate calculus to a class of non-mathematicians and social scientists. It wasn’t just them asking the question “why are we here”? But an open-mind is a powerful adversary. They soon dusted-off this myth in a matter of months. Yes, for some, math is like sorcery. We all have our superstitions to overcome. The good news is that we can. Arthur C Clarke who once said that, “any sufficiently advanced technology is indistinguishable from magic”. If that technology is born of math then however miraculous or foreboding it appears, we will learn to embrace it. Mathematics – to learn. Let’s face our fears, dispel the myths and advance. There are legends to be made.

Dr Michael Taylor (PhD, CPhys)
National Observatory of Athens &
American University of Athens
http://patternizer.wordpress.com/

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During the seventeenth century, the physicist Isaac Newton conducted a series of experiments studying the properties of light. Newton encountered a puzzling enigma. He found that light behaved both like a particle and like a wave. Some of his experiments made the most sense if he interpreted light as a stream of tiny particles or “corpuscles”. On the other hand, he observed clear interference effects indicative of waves. Further, he examined the surface of mirrors using the early microscopes of his time. He found that while the surface appeared smooth to the naked eye, the surface of a mirror under a microscope was rough with hills and valleys. If light was tiny particles like billiard balls bouncing off the surface of the mirror, the light particles should scatter randomly in all directions, rather than forming a reflected image. Reflection in a mirror might naively seem like evidence for light particles, but microscopic investigation showed that it was not that simple. Newton published his results in his book Opticks, a semi-popular book written in more accessible English rather than the Latin preferred by the scholars of his time. Opticks helped contribute to Newton’s broad reputation with the public in England. Newton presented his results with limited commentary, focusing on the empirical facts rather than theory. Nonetheless, he had some theories.

In particular, Newton studied a phenomenon that has come to be called Newton’s rings, although Newton did not actually discover the phenomenon. The phenomenon of Newton’s rings is an interference pattern caused by the reflection of light between two surfaces – a spherical surface and an adjacent flat surface. When viewed with monochromatic light such as a modern laser the interference pattern is a series of concentric, alternating light and dark rings centered at the point of contact between the two surfaces. When viewed with white light (Newton used white light for many of his experiments) Newton’s rings is a concentric ring pattern of rainbow colors.

Newton’s rings were essentially impossible to explain either by picturing light as a simple particle like a billiard ball or a particle that interacted with matter through a simple repulsive force that grew weaker with distance similar to the modern repulsive electrical force between two like charges. One could, for example, explain the reflection of light from the microscopically rough surface of the mirror by introducing a repulsive force between the light particle and the body of the mirror. In this case, the repulsive force would be averaged over the surface of the mirror. The light particle would never actually strike the rough surface, but rather slow down and bounce backward in the air above the surface of the mirror, feeling an average repulsive force as if the mirror were perfectly smooth. It was necessary to introduce some sort of wave or vibration to explain Newton’s rings.

Newton speculated on the existence of an “Aetherial Medium” permeating the universe to account for the interference effects, electricity, magnetism, and gravity. Here is Newton’s Query 21 from the Project Gutenberg version of Newton’s Opticks. When Newton refers to “alternate Fits of easy Transmission and easy Reflexion” he is referring to interference effects such as Newton’s rings that he observed. It is also significant, for reasons to be explained below, that Newton realized that the vibrations in his hypothetical Aether must travel faster than light to account for his observations (the light particles travel at the experimentally measured speed of light but the waves in the Aether must travel faster to produce some of the observed interference effects).

(Note: Qu. is an abbreviation for Query or Question)

Qu. 21. Is not this Medium much rarer within the dense Bodies of the Sun, Stars, Planets and Comets, than in the empty celestial Spaces between them? And in passing from them to great distances, doth it not grow denser and denser perpetually, and thereby cause the gravity of those great Bodies towards one another, and of their parts towards the Bodies; every Body endeavouring to go from the denser parts of the Medium towards the rarer? For if this Medium be rarer within the Sun’s Body than at its Surface, and rarer there than at the hundredth part of an Inch from its Body, and rarer there than at the fiftieth part of an Inch from its Body, and rarer there than at the Orb of Saturn; I see no reason why the Increase of density should stop any where, and not rather be continued through all distances from the Sun to Saturn, and beyond. And though this Increase of density may at great distances be exceeding slow, yet if the elastick force of this Medium be exceeding great, it may suffice to impel Bodies from the denser parts of the Medium towards the rarer, with all that power which we call Gravity. And that the elastick force of this Medium is exceeding great, may be gather’d from the swiftness of its Vibrations. Sounds move about 1140 English Feet in a second Minute of Time, and in seven or eight Minutes of Time they move about one hundred English Miles. Light moves from the Sun to us in about seven or eight Minutes of Time, which distance is about 70,000,000 English Miles, supposing the horizontal Parallax of the Sun to be about 12”. And the Vibrations or Pulses of this Medium, that they may cause the alternate Fits of easy Transmission and easy Reflexion, must be swifter than Light, and by consequence above 700,000 times swifter than Sounds. And therefore the elastick force of this Medium, in proportion to its density, must be above 700000 x 700000 (that is, above 490,000,000,000) times greater than the elastick force of the Air is in proportion to its density. For the Velocities of the Pulses of elastick Mediums are in a subduplicate Ratio of the Elasticities and the Rarities of the Mediums taken together.


As Attraction is stronger in small Magnets than in great ones in proportion to their Bulk, and Gravity is greater in the Surfaces of small Planets than in those of great ones in proportion to their bulk, and small Bodies are agitated much more by electric attraction than great ones; so the smallness of the Rays of Light may contribute very much to the power of the Agent by which they are refracted. And so if any one should suppose that AEther (like our Air) may contain Particles which endeavour to recede from one another (for I do not know what this AEther is) and that its Particles are exceedingly smaller than those of Air, or even than those of Light: The exceeding smallness of its Particles may contribute to the greatness of the force by which those Particles may recede from one another, and thereby make that Medium exceedingly more rare and elastick than Air, and by consequence exceedingly less able to resist the motions of Projectiles, and exceedingly more able to press upon gross Bodies, by endeavouring to expand it self.

Subsequent physicists emphasized Newton’s theory of light particles alone. Physics students are often taught incorrectly that Newton’s theory of light was a simple theory of tiny light particles, implying that Newton was unaware of wave-like phenomena such as Newton’s rings. In the early nineteenth century new observations of interference effects by Thomas Young and others led to the abandonment of the particle theory of light. A wave theory of light became dominant. The electromagnetic theory of Michael Faraday and James Clerk Maxwell seemed to definitively demonstrate that light was an electromagnetic wave, possibly in the mysterious luminiferous ether, with no particle characteristics at all. Maxwell’s Equations seems to preclude particles of light.

The late nineteenth and early twentieth century saw new electrical equipment and experiments: light bulbs, vacuum tubes, and so forth. Experimental physicists and other experimenters encountered a plethora of bewildering results. Both light and the recently discovered electron seemed to behave sometimes like a wave and sometimes like a particle. In one experiment, electrons seemed to diffract through a crystal. Each individual electron appeared randomly as a discrete localized point on a sheet of photographic film, but these points accumulated to form an interference pattern as many electrons passed through the crystal. The results were baffling and no good predictive theory existed. Several partial theories competed for the explanation of the strange results, notably the so-called “matrix mechanics” of Werner Heisenberg, a highly abstract theory associated with the University of Gottingen and Niels Bohr’s Institute for Theoretical Physics in Copenhagen.

Prince Louis de Broglie, a graduate student and perhaps significantly a student of the history of physics, proposed that the electron was accompanied by a mysterious wave to account for some of the baffling results. de Broglie’s suggestion led the physicist Erwin Schrödinger to seek a “wave equation”, a partial differential equation that would govern this mysterious wave and enable precise quantitative predictions. After an unknown number of attempts (certainly more than one as it is known Schrödinger tried at least one equation that did not work), Schrödinger found the famous partial differential equation that bears his name:

where is Planck’s constant divided by , is the mysterious “wave function”, is the potential such as the electrical potential of the hydrogen atom, and is the position. When Schrödinger applied this equation to the hydrogen atom, thought to consist of a single electron somehow “orbiting” a positively charged nucleus, he was able to compute both the spectrum of hydrogen and the intensity of the spectral lines of hydrogen. He was also able to quickly demonstrate that his equation was mathematically equivalent to the abstruse “matrix mechanics” of Werner Heisenberg.

Schrödinger’s equation took the small world of atomic physics by storm. It worked and it was much easier to use than “matrix mechanics”. Most physicists were familiar with partial differential equations and wave equations, mainstays of nineteenth century “classical” physics. It was quite easy to learn Schrödinger’s equation, compute, and publish new results. Schrödinger was something of an outsider: an established professor at a lesser university with a colorful and questionable personal life. His success threatened to upend the dominant position of both Niels Bohr and the Institute of Theoretical Physics in Copenhagen and the University of Gottingen in the small world of atomic physics.

There was the peculiarity that no one understood what Schrödinger’s Equation meant. What was exactly? How could an electron in a hydrogen atom also be a wave or have a wave associated with it? What was waving and why were electrons detected as points on photographic film or as tracks in detectors that seemed to imply a billiard ball like particle flying from point A to point B. The same problem that had puzzled Newton and led to his theories of light particles interacting with waves in the “aether” had reappeared, now for both light particles (photons) and electrons. Schrödinger, de Broglie, Albert Einstein, and a minority of physicists would remain puzzled about the meaning of the wave function and quantum mechanics. This may be the true state of our knowledge to this day.

A bitter dispute developed between Prince Louis de Broglie, Albert Einstein, and Erwin Schrödinger on the one hand and Niels Bohr and his school, the department of physics at the University of Gottingen, and others over quantum mechanics. Confronted with Schrödinger’s sudden leap forward which threatened to render matrix mechanics and their associated theories irrelevant, the other physicists managed to turn the tables on Schrödinger by arguing that they had figured out the meaning of the wave function ! was the probability density to observe an electron or other particle at x. Nature was inherently probabilistic. Schrödinger may have been able to calculate the spectrum of hydrogen precisely but he didn’t know what he was doing and they did. Schrödinger and Max Born, who conceived of the probabily wave interpretation of , would argue bitterly about the issue for the rest of their lives.

This heated dispute may seem rather silly to outsiders. There is an old saying that “academic disputes are the worst because the stakes are so small.” It is worth remembering that most physicists in the story were German. Germany had been defeated in World War I and suffered enormous economic and political problems. The 1920′s were a difficult time in Germany and to a lesser extent the other European nations still recovering from arguably the worst war in recorded history. In Germany, a tenured full professor in physics was a civil servant with a small salary and more job security than most. Even so, many of the atomic physicists would lose their jobs, and often much more, when Adolf Hitler rose to power a few years later. Max Born would flee to England and lose many of his relatives to the Holocaust. Even today when quantum mechanics underlies many electronic devices in common use like computer chips, lasers, and so forth, there are few jobs for experts in quantum mechanics.

By many accounts, the dispute came to a head at the Fifth Solvay Congress at the Institut de Physiologie in Parc Leopold, Brussels which began on October 24, 1927. Ernst Solvay was a wealthy Belgian industrialist who had organized the Solvay Congresses in his name as an invitation only meeting of the top physicists in the world. The Solvay Congresses continue to this day. The winners came to describe the 1927 Solvay Congress as a decisive victory in which the victors routed de Broglie, Schrödinger, Einstein and their ill conceived ideas and objections to the “revolutionary” new quantum mechanics of Bohr, Born, Heisenberg, and their associates. In fact, de Broglie may have been cowed by attacks on his ideas for a time, but Erwin Schrödinger and Albert Einstein remained openly defiant. The losers were painted with much success as reactionary die-hards clinging childishly to the comforting but outmoded world of “classical” physics.

What then was the problem? de Broglie, Schrödinger, and Einstein had a range of logical and philosophical objections to the so-called Copenhagen theory or interpretation of quantum mechanics of Niels Bohr, Max Born, and their associates. In a nutshell, most objections concerned how, why, and when the supposed probability wave function “collapsed” into a point particle or a measurement indistinguishable from a point particle. Schrödinger expressed some of his objections in a famous “thought experiment” known as Schrödinger’s Cat. Briefly, Schrödinger envisioned a hapless cat in a sealed box with a radioactive sample that randomly triggers the breaking of a vial of cyanide depending on quantum mechanics. In the Copenhagen version of quantum mechanics, the cat will exist as a superposition of a live cat wave function and a dead cat wave function until a measurement is made. Then, somehow the wave function collapses to either a live cat or a dead cat when a measurement is made. Well…does this happen when the box is opened? Does it happen before the box is opened? What if the scientists who opens the box is in a sealed room? Does the wave function collapse when someone else opens the sealed room to observe the scientist? Schrödinger and Einstein would have none of it.

Einstein eventually summarized his objections to quantum mechanics in a short, elegant, and now famous paper published in Physical Review in 1935 now known as EPR after its authors Einstein, Podolsky, and Rosen. Niels Bohr immediately responded with his own extremely confusing paper published shortly thereafter in Physical Review in 1935 which was widely taken as a decisive rebuttal of Einstein. EPR was largely ignored for many years, but eventually, in part due to the work of the late physicist John Bell, spawned a series of astonishing experiments usually referred to as EPR experiments.

In a typical EPR experiment, an atom undergoes a quantum mechanical transition and radiates two photons in opposite directions. The spin of these photons is related by conservation of angular momentum. The photons travel some distance; some experiments use fiber optics to transport the photons over a considerable distance before a measurement is made. In Copenhagen quantum mechanics, some sort of wave function describing both photons is expanding outward from the atom until a measurement is made. Then, the spin of each photon is measured “simultaneously,” so there is not enough time for even a light signal, the fastest known signal, to travel from one measuring device to another. Nonetheless, quantum mechanics predicts a correlation between the measurements that depends on what measurement one chooses to make. The probability of measuring the spin of photon A with a particular value depends on how the experimenter chooses to measure the spin of photon B. The experimenter cannot communicate actual information between photon B and photon A by doing this, but he can cause a difference that can be detected when the results of measuring the spins of A and B are compared. This correlation has been repeatedly observed in EPR experiments. So, presumably the probability density wave where x is the position of one photon and y is the position of the other photon, is collapsing almost instantaneously and faster than light, something that would seem inconsistent with Einstein’s Special Theory of Relativity which seemingly precludes travel faster than light. The photons in EPR experiments are said to be entangled until a measurement is made. Over time, the EPR theory and experiments have given rise to Quantum Computation and Quantum Cryptography, seeking to exploit the strange properties of quantum entanglement for ostensibly practical purposes.

For the most part, physicists have avoided seeking a mechanism to explain quantum mechanics. There is almost a taboo on doing this, probably partly a consequence of the bitter dispute over the foundations of quantum mechanics. Probably the best known attempt to “explain” quantum mechanics and the meaning of the wave function is the pilot wave theory of Louis de Broglie and David Bohm. In the 1920′s, de Broglie theorized that the electron was guided or propelled by an extended wave that was the wave function, not unlike Newton’s aether waves and the light particles. Thus, there was no collapse of the wave function. Rather, the electron followed a definite path in response to the wave function, the pilot wave. The pilot wave would exhibit interference patterns. The statistical behavior came from uncertainties in the starting position of the electron. Unmeasurable tiny variations in the starting position of the electron would be exponentially magnified by the non-linear behavior of the quantum wave function (the pilot wave). de Broglie speculated that the electron was actually a non-linear excitation of the wave itself. de Broglie’s idea was reportedly bitterly attacked at the Fifth Solvay Congress (1927) and de Broglie fell silent until the physicist David Bohm, one of Einstein’s small number of research assistants, rediscovered the pilot wave theory in the 1950s and resolved some of the technical problems in de Broglie’s original attempt. David Bohm would spend much of his life researching and promoting the pilot wave theory. Bohm had many unorthodox ideas and interests and his pilot wave theory, intrinsically rather mechanistic, became associated with mystical viewpoints.

The astute reader will notice that while the pilot wave picture is intuitively appealing for a single particle, a single photon or a single electron, it becomes complicated and problematic for multiple particles including the entangled multi-particle systems. Each particle seems to need a separate wave that guides only it, except for the entangled states of multiple particles in which case a complex multidimensional wave is needed to jointly guide the system of particles until a measurement occurs. David Bohm was able to find a way to do this, but it is complex and cumbersome and lacks the immediate intuitive appeal of the pilot wave for a single particle.

The quantum catfight left a powerful imprint on physics which lasts to this day. A high proportion of theoretical and mathematical physicists trace their academic lineage to the winners. The University of Gottingen and Bohr’s Institute for Theoretical Physics had many alumni. In contrast, de Broglie, Einstein, and Schrödinger had few students. Significantly, one of the major challenges to the reigning Copenhagen theory came from Einstein’s erstwhile research assistant David Bohm. Einstein, de Broglie, and Bohm were all involved in left wing politics which may also have contributed to the unpopularity of their views, especially during the Cold War (Bohm fled the United States under a cloud and eventually settled in England). Schrödinger’s unorthodox and rather questionable personal life may also have been a factor as well. Bohm’s views became associated with mystical and paranormal viewpoints, in part due to Bohm’s association with the spiritual guru Krishnamurti. More generally, quantum entanglement has been embraced by mystics, parapsychologists, and others as evidence of the mystical oneness implicit in most mystical and magical systems and alleged psychic phenomena (see, for example, the 2004 movie What the Bleep Do We Know).

In conclusion, the logical and philosophical problems with quantum mechanics that bothered de Broglie, Einstein, and Schrödinger remain, probably unresolved: the so-called quantum measurement problem. Schrödinger’s Equation and the other equations of quantum mechanics such as the Dirac Equation appear to make accurate predictions, but the meaning of the wave function probably remains unknown. It seems likely that quantum mechanics is like Kepler’s laws of planetary motion, correct, predictive, but incomplete. It took later work to discover that Kepler’s laws could be derived from Newton’s Theory of Gravitation with its inverse square law:

where is the attractive gravitational force, is the universal gravitational constant, is the mass of body 1 (e.g. the Sun), is the mass of body 2 (e.g. the planet Mars), and is the distance between the two bodies.

Newton sought a deeper understanding of gravity in the concept of an Aetherial Medium with faster than light waves as illustrated in the quote from Opticks above. So too, the explanation for quantum mechanics may lie in some sort of faster than light waves that transmit signals between entangled particles. Another possibility is a “hyperspace” that connects all points in space-time together, bypassing normal space-time. Even more exotic possibilities may exist. Mathematically speaking, one is looking for a deeper, more fundamental equation or equations from which Schrödinger’s Equation can be derived.

Suggested Reading/References

	The Quantum Ten: A Story of Passion, Tragedy, Ambition, and Science Sheila Jones Oxford University Press, USA 2008 336 pages
	Quantum Dialogue: The Making of a Revolution Mara Beller University Of Chicago Press 2001 365 pages
	Infinite Potential: The Life and Times of David Bohm F. David Peat Basic Books 1997 380 pages
	The Undivided Universe: An Ontological Interpretation of Quantum Theory David Bohm, Basil J. Hiley Routledge 1993 397 pages
	Opticks: Or a Treatise of the Reflections, Refractions, Inflections & Colours of Light-Based on the Fourth Edition London, 1730 Isaac Newton, Albert Einstein (Foreword) Dover Publications 1952
	Quantum Theory At The Crossroads – Reconsidering The 1927 Solvay Conference Guido Bacciagaluppi, Antony Valentini Cambridge University Press 2009 556 pages
	Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy J. S. Bell Cambridge University Press 2004 288 pages

Copyright © 2010, John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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1. Einstein: Superman or Super Stubborn?

“It is not the critic who counts: not the man who points out how the strong man stumbles or where the doer of deeds could have done better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood, who strives valiantly, who errs and comes up short again and again, because there is no effort without error or shortcoming, but who knows the great enthusiasms, the great devotions, who spends himself for a worthy cause; who, at the best, knows, in the end, the triumph of high achievement, and who, at the worst, if he fails, at least he fails while daring greatly, so that his place shall never be with those cold and timid souls who knew neither victory nor defeat.”

Theodore Roosevelt
“Citizenship in a Republic,”
Speech at the Sorbonne, Paris, April 23, 1910

Einstein’s Mistakes: The Human Failings of Genius Hans C. Ohanian W.W. Norton and Company New York 2008 394 pages

On November 6, 1919, the British astronomer Arthur Eddington announced his results confirming the deflection of light by the Sun during an eclipse as predicted by the General Theory of Relativity of the German/Swiss physicist Albert Einstein, then little known outside of his field. The announcement took place at a special joint meeting of the Royal Astronomical Society and the Royal Philosophical Society in what appears to have been a carefully staged media event shortly before the first anniversary of the armistice that ended World War I (November 11, 1918). In what may not have been a coincidence, the next day was the second anniversary of the Bolshevik Revolution (November 7, 1917). Significantly, Eddington was a Quaker and a pacifist who had been a conscientious objector during the war. His colleague and correspondent Einstein was also a pacifist who had avoided military service during the war. The story of a British astronomer and a German physicist working together on a revolutionary breakthrough offered an inspiring example to a war weary world buffeted by revolution and severe economic problems. On November 7, 1919, Albert Einstein became an international celebrity with adulatory articles on the front pages of the London Times and the New York Times. That celebrity continues to the present day.

A substantial mythology grew up around Einstein, aided and abetted by Einstein himself, his secretary Helen Dukas who inherited much of his estate, many fellow physicists, well meaning admirers, the German government of the Weimar Republic, avaricious book publishers, and others. Einstein became a superman and a saint. In many accounts, he launched the Manhattan Project to build the atomic bomb by writing a letter to President Franklin Roosevelt at the urging of his friend and onetime business partner the Hungarian physicist Leo Szilard. He was an erstwhile pacifist reluctantly forced to advocate the atomic bomb by the menace of Nazism.

Einstein became an icon of popular culture and counter-culture. Einstein was probably the model for Professor Jacob Barnhart in the famous sci-fi movie The Day the Earth Stood Still (1951). A myth even grew up that Einstein succeeded (partially) in his quest for a theory unifying electromagnetism and gravity, creating an electromagnetic invisibility device that transported a US Navy ship at the Philadelphia naval yard into an alternate dimension, peddled by William Moore and Charles Berlitz in their book The Philadelphia Experiment (1979) and other sources. Einstein actually did work as a consultant for the Navy in Philadelphia during World War II. Posters of Einstein decorate dorm rooms at MIT, Caltech, and other technical universities. Every year sees new Einstein books, most ranging from positive to very positive.

Within the rarefied world of theoretical physics, there was another, less flattering mythology about Einstein, one rarely exposed to the general public until recently. Einstein, along with Erwin Schrödinger and Prince Louis de Broglie, was on the losing side of a bitter dispute over the “completeness” of quantum mechanics. In a nutshell, Einstein, Schrödinger, and de Broglie argued that quantum mechanics was logically inconsistent or incomplete. The theory could not be correct. Primarily they focused on the problem with how and when the wave function “collapsed” in the winning Copenhagen interpretation of quantum mechanics. Some further fundamental insight was needed. Despite Einstein’s public celebrity, most leading physicists trace their “lineage” to the winners: Niels Bohr and his school at the Institute for Theoretical Physics in Copenhagen and the physics department at the prestigious University of Gottingen in Germany. In this mythology, Einstein was a mediocre mathematician, a has been, a loopy old man who couldn’t keep up with the revolutions in quantum and nuclear physics, a useful mascot for awing the unwashed masses but certainly not to be taken seriously by the real physicists of today. Generations of physics graduate students have been taught this alternate mythology quietly behind closed doors, not infrequently by the winners or the students of the winners or the students of the students of the winners.

Over the years, only a few books have been highly critical of Einstein. A well known example is The Private Lives of Albert Einstein by Roger Highfield and Paul Carter which purports to detail Einstein’s numerous affairs, failed marriage, and other personal failings. Hans Ohanian’s recent book Einstein’s Mistakes takes aim at Einstein the scientist, arguing that Einstein made numerous sometimes serious and sometimes obvious errors in many of his papers and even some famous results.

On the whole, Einstein’s Mistakes is well written, an enjoyable read, and probably a worthwhile antidote to the more extreme adulation of Einstein. It also suffers from several minor factual errors, a glaring flaw in a book about Einstein’s Mistakes. More seriously, one can question the standards and philosophy used to measure and critique Einstein.

The author Hans Ohanian is a physicist. Physics is a highly competitive field, especially since the atomic bomb elevated its status, making physicists advisers to Presidents, Prime Ministers, and Premiers. Through about the middle of graduate school, advancement in physics is determined largely by performance on competitive exams, standardized tests like the Scholastic Aptitude Test (SAT) and Graduate Record Exams (GRE) in the United States, qualifying exams in graduate school and so forth. The winners, especially in theoretical or mathematical physics, are those who can perform various derivations and calculations quickly and accurately in a few hours. The winners in modern physics are extremely good, some able to perform difficult calculations perfectly under tight time constraints. Einstein was not competitive in this way, ending up with a job at a patent office while working toward a Ph.D. part time at the less prestigious University of Zurich. As Einstein’s Mistakes notes, many of Einstein’s contemporaries were astounded by his discoveries. His former professor Minkowski was flabbergasted, having reputedly referred to Einstein as that “lazy dog.” In the even more competitive world of physics today, the ideal is to be one hundred percent right fast. Ohanian’s book reflects this prejudice.

The problem is that most inventors and discovers, including in mathematical fields such as theoretical physics, make lots of mistakes. Ohanian notes with scorn grievous errors by Galileo, Kepler, and Newton amongst others. In fact, this is not unusual. Almost every individual or team remembered for a great discovery or invention failed, failed again, kept failing, screwed up some more, made even more mistakes, and finally got it right. Not infrequently, subsequent researchers had to repair gaping holes. The theory or technology taught in a textbook usually turns out to be much improved from the actual theory or invention as originally discovered. Einstein’s work is no exception to this common pattern.

It seems likely that some people can perform extremely well or perfectly on academic exams because they are applying a known method or formula for solving a problem (e.g. addition using place value arithmetic on an SAT mathematics exam). This ability, whether inborn or acquired, appears only somewhat correlated with the discovery of new methods and new concepts. Regardless of academic performance, discovering new methods and concepts often involves a large amount of trial and error as well as conceptual reasoning and visualization that is difficult to measure in a typical exam. Thus Einstein made numerous discoveries that his more technically proficient colleagues missed.

Einstein’s Mistakes largely accepts the winner’s narrative of the dispute over the foundations of quantum mechanics. Einstein was and is often criticized for the “mistake” of refusing to accept quantum mechanics — more accurately arguing that quantum mechanics is “incomplete,” a somewhat arcane term of physics jargon. Yet the logical problems and paradoxes that Einstein, Schrödinger, and de Broglie noted remain, largely unexplained. How and when does Max Born’s probability density wavefunction collapse to a single point or a measurement indistinguishable from a point particle? Was Einstein foolishly stubborn or foresightful not to accept Bohr and Born’s confusing explanations of their theory?

In quantum mechanics, the state of a “particle” such as an electron in the atom is represented by a state or wavefunction often represented by the Greek letter . In basic quantum mechanics, the wavefunction is governed by a partial differential equation, Schrödinger’s Equation, discovered by the physicist Erwin Schrödinger. In the quantum mechanics of Niels Bohr and Max Born, the wavefunction is interpreted as the probability density of making an observation or measurement at the point x. The wavefunction is often said to collapse to a point when an observation or measurement is made, for example when an electron strikes a sheet of photographic film or a modern electronic imaging device. Each electron appears as a distinct point on the film, yet an interference pattern is seen as many electrons strike the film over time. But what exactly is a measurement and how and when does the wavefunction “collapse”? When electrons diffract through a crystal creating an interference pattern on film, does the wavefunction collapse at the film or earlier (or later when the scientist examines the film)… how and why? Such questions have bedeviled quantum mechanics since its establishment in 1927 and often been swept under the rug by confusing rhetoric.

Einstein spent much of his life in the apparently futile pursuit of a unified field theory that would combine electromagnetism, gravity, and perhaps all other forces, also hoping that this unified field theory would explain the paradoxes of quantum mechanics to his satisfaction. Einstein was then and is now mocked for this pursuit. Yet, Einstein’s position at the Institute for Advanced Study in Princeton was secure. He had nothing to lose and everything to gain. Better to try and fail than to never try. Physics and many other fields are full of papers of no consequence, computing some parameter that no one should care about to another decimal place because it can be done, safely and securely. Rather, both taxpayers and donors should surely want the Einsteins of the world to try to do something significant, rather than settling for safe mediocrity.

Ohanian compares Einstein’s quest for the unified field theory to the obsessive pursuit of string theory by most of the current generation of theoretical physicists. It is a fair comparison and a cautionary tale. But, it was not wrong to pursue string theory given the promising results published in 1984. Rather the mistake has surely been to pursue string theory almost exclusively and continuously in the face of poor results. More likely if the theoretical physicists of the last thirty years had diversified their efforts, working on the problems of quantum mechanics that Einstein noted as well as many other ideas including string theory, instead of focusing single-mindedly on string theory, more substantial discoveries would have occurred. So, too, if Einstein failed with his unified field theory while others succeeded with quantum electrodynamics and possibly quantum field theory, so much the better. If everyone had followed Einstein’s lead, as arguably happened with the prominent theoretical physicist Ed Witten and string theory in 1984, then everyone would have failed in Einstein’s time as well.

In conclusion, Einstein’s Mistakes is well worth reading. It is unhealthy to deify scientists like Einstein, Richard Feynman, Ed Witten, or many others. However, readers should keep in mind that Einstein’s tale of mistakes and errors, far from unusual or odd, is often the story of actual invention and discovery. It is not inappropriate to recall the Aesop’s Fable of the Tortoise and the Hare. The Tortoise won the race by stubbornly and doggedly advancing forward while the faster easily distracted Hare ultimately lost. Einstein, like many inventors and discovers, spent many years stubbornly pursuing his great discoveries — about seven years for special relativity and the other discoveries published in 1905 and seven years for General Relativity. His accomplishments did not come easily or quickly or without error.

Copyright © 2010, John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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