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A Big List of Interesting Mathematics Books Released in 2011

Antonio Cangiano — Fri, 16 Dec 2011 19:11:17 +0000

The following is a list of 46 interesting math books that were released in 2011.

As some of you know, I run a service called Any New Books?, which emails you a list of new books that are related to the categories of your choice each week. For the most part I pulled this list from the weekly staff picks there throughout this past year, just in time for your Christmas shopping.

The books are ordered by their current sale rank on Amazon (from the most popular to the least popular at the moment, with hardcovers first). I hope this page will help you discover a few titles you may have not have noticed yet.

[Programming book list]

	The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Sharon Bertsch McGrayne ISBN: 0300169698 Publisher: Yale University Press Publication date: May 17, 2011 Binding: Hardcover Estimated price: $17.60 Share this book on Twitter \|Facebook
	The Man of Numbers: Fibonacci’s Arithmetic Revolution Stores: USA \| Canada \| Italy \| Kindle \| UK Kindle By Keith Devlin ISBN: 0802778127 Publisher: Walker & Company Publication date: July 5, 2011 Binding: Hardcover Estimated price: $14.50 Share this book on Twitter \|Facebook
	The Mathematics of Life Stores: USA \| Canada \| Italy \| Kindle By Ian Stewart ISBN: 0465022383 Publisher: Basic Books Publication date: June 7, 2011 Binding: Hardcover Estimated price: $12.13 Share this book on Twitter \|Facebook
	The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1 Stores: USA \| UK \| Canada \| Italy By Donald E. Knuth ISBN: 0201038048 Publisher: Addison-Wesley Professional Publication date: January 22, 2011 Binding: Hardcover Estimated price: $53.15 Share this book on Twitter \|Facebook
	SuperCooperators: Altruism, Evolution, and Why We Need Each Other to Succeed Stores: USA \| Canada \| Italy \| Kindle By Martin Nowak, Roger Highfield ISBN: 1439100187 Publisher: Free Press Publication date: March 22, 2011 Binding: Hardcover Estimated price: $8.33 Share this book on Twitter \|Facebook
	Undocumented Secrets of MATLAB-Java Programming Stores: USA \| UK \| Canada By Yair M. Altman ISBN: 1439869030 Publisher: Chapman and Hall/CRC Publication date: December 12, 2011 Binding: Hardcover Estimated price: $60.57 Share this book on Twitter \|Facebook
	Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Paul J. Nahin ISBN: 0691144257 Publisher: Princeton University Press Publication date: August 28, 2011 Binding: Hardcover Estimated price: $18.74 Share this book on Twitter \|Facebook
	Stats: Data and Models (3rd Edition) Stores: USA \| UK \| Canada \| Italy \| Kindle By Richard D. De Veaux, Paul F. Velleman, David E. Bock ISBN: 0321692551 Publisher: Addison Wesley Publication date: January 8, 2011 Binding: Hardcover Estimated price: $98.99 Share this book on Twitter \|Facebook
	Functional Analysis: Introduction to Further Topics in Analysis Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Elias M. Stein, Rami Shakarchi ISBN: 0691113874 Publisher: Princeton University Press Publication date: September 11, 2011 Binding: Hardcover Estimated price: $60.00 Share this book on Twitter \|Facebook
	Cluster Analysis Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Brian S. Everitt, Dr Sabine Landau, Dr Morven Leese, Dr Daniel Stahl ISBN: 0470749911 Publisher: Wiley Publication date: March 8, 2011 Binding: Hardcover Estimated price: $59.99 Share this book on Twitter \|Facebook
	Viewpoints: Mathematical Perspective and Fractal Geometry in Art Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Marc Frantz, Annalisa Crannell ISBN: 0691125929 Publisher: Princeton University Press Publication date: July 25, 2011 Binding: Hardcover Estimated price: $35.31 Share this book on Twitter \|Facebook
	The Glorious Golden Ratio Stores: USA By Alfred S. Posamentier, Ingmar Lehmann ISBN: 1616144238 Publisher: Prometheus Books Publication date: November 22, 2011 Binding: Hardcover Estimated price: $14.46 Share this book on Twitter \|Facebook
	One, Two, Three: Absolutely Elementary Mathematics Stores: USA \| UK \| Canada \| Italy \| Kindle By David Berlinski ISBN: 0375423338 Publisher: Pantheon Publication date: May 10, 2011 Binding: Hardcover Estimated price: $9.89 Share this book on Twitter \|Facebook
	Game Theory and the Humanities: Bridging Two Worlds Stores: USA \| UK \| Canada \| Italy By Steven J. Brams ISBN: 0262015226 Publisher: The MIT Press Publication date: March 4, 2011 Binding: Hardcover Estimated price: $28.81 Share this book on Twitter \|Facebook
	Majority Judgment: Measuring, Ranking, and Electing Stores: USA \| UK \| Canada \| Italy By Michel Balinski, Rida Laraki ISBN: 0262015137 Publisher: The MIT Press Publication date: January 28, 2011 Binding: Hardcover Estimated price: $31.89 Share this book on Twitter \|Facebook
	The Oxford Handbook of Random Matrix Theory Stores: USA \| UK \| Canada \| Italy By Gernot Akemann, Jinho Baik, Philippe Di Francesco ISBN: 0199574006 Publisher: Oxford University Press, USA Publication date: September 25, 2011 Binding: Hardcover Estimated price: $173.05 Share this book on Twitter \|Facebook
	An Elementary Introduction to Statistical Learning Theory Stores: USA \| UK \| Canada \| Italy By Sanjeev Kulkarni, Gilbert Harman ISBN: 0470641835 Publisher: Wiley Publication date: August 2, 2011 Binding: Hardcover Estimated price: $73.99 Share this book on Twitter \|Facebook
	The Ultimate Challenge: The 3x+1 Problem Stores: USA \| UK By Jeffrey C. Lagarias ISBN: 0821849409 Publisher: American Mathematical Society Publication date: January 14, 2011 Binding: Hardcover Estimated price: $59.00 Share this book on Twitter \|Facebook
	Connections in Combinatorial Optimization Stores: USA \| UK \| Canada \| Italy By Andras Frank ISBN: 0199205272 Publisher: Oxford University Press, USA Publication date: June 1, 2011 Binding: Hardcover Estimated price: $94.04 Share this book on Twitter \|Facebook
	The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Burt C. Hopkins ISBN: 0253356717 Publisher: Indiana University Press Publication date: September 7, 2011 Binding: Hardcover Estimated price: $30.00 Share this book on Twitter \|Facebook
	The Evolution of Principia Mathematica: Bertrand Russell’s Manuscripts and Notes for the Second Edition Stores: USA \| UK \| Canada \| Italy By Bernard Linsky ISBN: 110700327X Publisher: Cambridge University Press Publication date: July 11, 2011 Binding: Hardcover Estimated price: $138.09 Share this book on Twitter \|Facebook
	Comparing Groups: Randomization and Bootstrap Methods Using R Stores: USA \| UK \| Canada \| Italy By Andrew S. Zieffler, Jeffrey R. Harring, Jeffrey D. Long ISBN: 0470621699 Publisher: Wiley Publication date: June 15, 2011 Binding: Hardcover Estimated price: $64.99 Share this book on Twitter \|Facebook
	Lectures in Game Theory for Computer Scientists Stores: USA \| UK \| Canada \| Italy By Editors at Cambridge University Press ISBN: 0521198666 Publisher: Cambridge University Press Publication date: February 14, 2011 Binding: Hardcover Estimated price: $50.00 Share this book on Twitter \|Facebook
	Probability Concepts and Theory for Engineers Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Harry Schwarzlander ISBN: 0470748559 Publisher: Wiley Publication date: March 1, 2011 Binding: Hardcover Estimated price: $67.99 Share this book on Twitter \|Facebook
	Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form Stores: USA \| UK \| Canada \| Italy By Kevin O’Meara, John Clark, Charles Vinsonhaler ISBN: 0199793735 Publisher: Oxford University Press, USA Publication date: September 16, 2011 Binding: Hardcover Estimated price: $71.55 Share this book on Twitter \|Facebook
	The Art of R Programming: A Tour of Statistical Software Design Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Norman Matloff ISBN: 1593273843 Publisher: No Starch Press Publication date: October 12, 2011 Binding: Paperback Estimated price: $21.99 Share this book on Twitter \|Facebook
	Here’s Looking at Euclid: From Counting Ants to Games of Chance – An Awe-Inspiring Journey Through the World of Numbers Stores: USA \| Canada \| Italy \| Kindle By Alex Bellos ISBN: 1416588280 Publisher: Free Press Publication date: April 19, 2011 Binding: Paperback Estimated price: $4.50 Share this book on Twitter \|Facebook
	Matlab, Second Edition: A Practical Introduction to Programming and Problem Solving Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Stormy Attaway ISBN: 0123850819 Publisher: Butterworth-Heinemann Publication date: August 11, 2011 Binding: Paperback Estimated price: $35.44 Share this book on Twitter \|Facebook
	The Best Writing on Mathematics 2011 Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Editors at Princeton University Press ISBN: 0691153159 Publisher: Princeton University Press Publication date: November 27, 2011 Binding: Paperback Estimated price: $12.14 Share this book on Twitter \|Facebook
	Algorithmic Puzzles Stores: USA \| UK \| Canada \| Kindle By Anany Levitin, Maria Levitin ISBN: 0199740445 Publisher: Oxford University Press, USA Publication date: October 14, 2011 Binding: Paperback Estimated price: $21.13 Share this book on Twitter \|Facebook
	Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 Stores: USA \| Canada By Jessica F. Shumway ISBN: 1571107908 Publisher: Stenhouse Pub Publication date: April 10, 2011 Binding: Paperback Estimated price: $17.88 Share this book on Twitter \|Facebook
	Agent-Based and Individual-Based Modeling: A Practical Introduction Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Steven F. Railsback, Volker Grimm ISBN: 0691136742 Publisher: Princeton University Press Publication date: November 6, 2011 Binding: Paperback Estimated price: $41.39 Share this book on Twitter \|Facebook
	Foundations of Geometry (2nd Edition) Stores: USA \| UK \| Canada By Gerard Venema ISBN: 0136020585 Publisher: Addison Wesley Publication date: July 16, 2011 Binding: Paperback Estimated price: $60.35 Share this book on Twitter \|Facebook
	Taming the Infinite: The Story of Mathematics from the First Numbers to Chaos Theory Stores: USA \| UK \| Canada \| Italy By Ian Stewart ISBN: 1847247687 Publisher: Quercus Publication date: November 1, 2011 Binding: Paperback Estimated price: $6.03 Share this book on Twitter \|Facebook
	The Number Mysteries: A Mathematical Odyssey through Everyday Life Stores: USA \| Canada \| Italy \| Kindle By Marcus du Sautoy ISBN: 0230113842 Publisher: Palgrave Macmillan Publication date: May 24, 2011 Binding: Paperback Estimated price: $10.66 Share this book on Twitter \|Facebook
	The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition Stores: USA \| UK \| Canada \| Italy \| Kindle By Stanislas Dehaene ISBN: 0199753873 Publisher: Oxford University Press, USA Publication date: April 29, 2011 Binding: Paperback Estimated price: $20.04 Share this book on Twitter \|Facebook
	Introduction to Proof in Abstract Mathematics Stores: USA \| UK \| Canada \| Italy By Andrew Wohlgemuth, Mathematics ISBN: 0486478548 Publisher: Dover Publications Publication date: February 17, 2011 Binding: Paperback Estimated price: $12.89 Share this book on Twitter \|Facebook
	Wizardry: Baseball’s All-Time Greatest Fielders Revealed Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Michael Humphreys ISBN: 0195397762 Publisher: Oxford University Press, USA Publication date: March 30, 2011 Binding: Paperback Estimated price: $11.00 Share this book on Twitter \|Facebook
	Differential Geometry: Bundles, Connections, Metrics and Curvature Stores: USA \| UK \| Canada \| Kindle By Clifford Henry Taubes ISBN: 0199605874 Publisher: Oxford University Press, USA Publication date: December 1, 2011 Binding: Paperback Estimated price: $36.20 Share this book on Twitter \|Facebook
	Fatal Numbers: Why Count on Chance Stores: USA \| UK \| Kindle \| UK Kindle By Hans Magnus Enzensberger ISBN: 1935830015 Publisher: Upper West Side Philosophers, Inc. Publication date: February 2, 2011 Binding: Paperback Estimated price: $10.05 Share this book on Twitter \|Facebook
	Matroid Theory Stores: USA \| UK \| Canada \| Italy By James Oxley ISBN: 0199603391 Publisher: Oxford University Press, USA Publication date: April 22, 2011 Binding: Paperback Estimated price: $53.16 Share this book on Twitter \|Facebook
	Visual Thinking in Mathematics Stores: USA \| UK \| Italy \| Kindle \| UK Kindle By Marcus Giaquinto ISBN: 0199575533 Publisher: Oxford University Press, USA Publication date: November 14, 2011 Binding: Paperback Estimated price: $30.57 Share this book on Twitter \|Facebook
	Mathematics, Ideas and the Physical Real Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Albert Lautman, Simon Duffy ISBN: 144112344X Publisher: Continuum Publication date: August 4, 2011 Binding: Paperback Estimated price: $14.50 Share this book on Twitter \|Facebook
	The Algebra Solution to the Mathematics Reform: Completing the Equation Stores: USA \| UK \| Canada \| Italy By Frances R. Spielhagen ISBN: 0807752312 Publisher: Teachers College Press Publication date: June 17, 2011 Binding: Paperback Estimated price: $17.43 Share this book on Twitter \|Facebook
	A Career in Statistics: Beyond the Numbers Stores: USA \| UK \| Canada \| Italy By Gerald J. Hahn, Necip Doganaksoy ISBN: 0470404418 Publisher: Wiley Publication date: June 28, 2011 Binding: Paperback Estimated price: $48.55 Share this book on Twitter \|Facebook
	The Star and the Whole: Gian-Carlo Rota on Mathematics and Phenomenology Stores: USA \| UK \| Canada By Fabrizio Palombi ISBN: 1568815832 Publisher: A K Peters/CRC Press Publication date: August 24, 2011 Binding: Paperback Estimated price: $20.07 Share this book on Twitter \|Facebook

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Start Your Own Blog About Mathematics

Antonio Cangiano — Wed, 21 Sep 2011 14:30:29 +0000

We don’t usually include off-topic posts here, but I feel this may interest some of our readers, plus it’s a shameless plug as well.

If you’ve always wanted to start your own technical blog, perhaps about a mathematical or scientific topic, but never got around to do it or failed to attract a following, read on.

My book about technical blogging is now available in Beta from The Pragmatic Bookshelf:
http://pragprog.com/book/actb/technical-blogging

It’s the kind of book that will teach you everything you need to realistically know to succeed at blogging. In it I provide a complete road map that can be applied whether you blog about math, science, technology, programming, or just about any other professional area.

Check it out and let me know what you think. If you’re just hearing about this book for the first time now, be sure to read the free introduction and excerpts to get an idea of how practical and useful this book is.

Happy technical and scientific blogging!

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The Magical Mathematics of Numb3rs

John F. McGowan, Ph.D. — Mon, 29 Aug 2011 11:00:51 +0000

Numb3rs is a television show that ran for six seasons on CBS from 2005 to 2010 about FBI agent Don Eppes and his brother Charles, a child math prodigy turned math professor at CalSci (a thinly disguised Caltech), who fight crime with mathematics in a sunny, smog-free TV version of Los Angeles filled with an astonishing number of extremely attractive young women. All six seasons are now available on DVD. Numb3rs features some real mathematics used in solving some real crimes as well as pure science fiction. In many respects, Numb3rs is a techno-thriller that features a mix of real present-day technology, advanced technology that may exist, and technology and mathematics that might plausibly exist in the near future. Mathematics and science is especially well integrated into many episodes in the first and second seasons of the show. Unlike over-the-top science fiction shows like Eureka, a viewer could believe that Numb3rs is a realistic presentation of mathematics and science used in crime fighting and other applications today.

Actor David Krumholtz (Charlie Eppes in Numb3rs)

Real life mathematicians including Caltech statistics professor Gary Lorden consulted for the show. Gary Lorden is listed in the credits for each episode as “Math Consultant”. In later seasons, Stephen Wolfram’s Wolfram Research also provided consulting advice to the series. There are scenes with references to Mathematica, Wolfram’s flagship product, and close-up shots of Wolfram’s magnum opus A New Kind of Science on Charlie Eppes desk. Wolfram also has a Caltech connection; he received his Ph.D. in Physics from Caltech in 1979. Gary Lorden and fellow mathematician Keith Devlin published a popular book The Numbers Behind Numb3rs: Solving Crime with Mathematics in 2007: “A companion to the hit CBS crime series Numb3rs presents the fascinating ways mathematics is used to fight real-life crime.” The shadowy National Security Agency (NSA), probably the largest patron of mathematics and mathematicians in the United States and the world, makes several appearances in Numb3rs.

Millikan Library at Caltech (appears often in Numb3rs)

CBS, Texas Instruments (a leading maker of digital signal processor or DSP chips), and the National Council of Teachers of Mathematics (NCTM) developed the “We All Use Math Every Day” initiative, sometimes abbreviated WAUMED, to inspire students to achieve more in math by showing how the subject is relevant to their lives:

Using the hit CBS television show, Numb3rs, the “We All Use Math Every Day” initiative provides free classroom activities online at cbs.com/Numb3rs that help students understand how the math they are learning in the classroom applies to the real world. The activities explore the math derived from the concepts used to solve cases in the FBI crime-solving show.

The show used the “We All Use Math Every Day” tagline in the opening introduction to the show in the first and second seasons.

How Realistic is Numb3rs?

Although explicitly fiction, in many respects Numb3rs paints a picture of mathematics and science that is similar to ostensibly factual popular science such as Scientific American articles, PBS/Nova video programs, Congressional testimony by leading scientists, and informal discussions at fundraising cocktail parties — unless the scientists or mathematicians are in the rare and unusual position of having to explain an obvious failure to a lay audience:

“Well, Senator, as everyone knows, science is a risky enterprise. Eighty to ninety percent of our research projects fail. Surely your staff briefed you on that; the proposal committee mentioned this clearly in italics in footnote 83 in Appendix C of the Proposal for the New Manhattan Project that Will Produce Miraculous Results by the Next Election.”

In several important respects, Numb3rs is very unrealistic. It is also the case that many people ranging from Silicon Valley executives trying to use mathematical methods for their businesses — for example, the current fad trying to use machine learning for recommendation engines in social networking and search businesses — to practicing scientists and engineers who one might think would know better often have expectations similar to what is portrayed in Numb3rs. These misconceptions almost certainly contributed in a major way to the multi-trillion dollar global housing bubble and crash through the widespread use of invalid mathematical models for the valuation of mortgage-backed securities. With business and political leaders seemingly floundering in the current economic difficulties, these misconceptions may wreak even greater havoc.

Before launching into a critique of Numb3rs, it is important to realize that there have been many successes in applied mathematics and mathematical software including impressive advances in video compression such as used by YouTube and Skype, audio compression such as the widely used MP3 standard, still image compression such as JPEG images, computer generated imagery in movies and video games, the Global Positioning System (GPS) that tells people where they are, and even speech recognition which is finally finding some practical use. Modern computers are extremely powerful, comparable to the supercomputers of previous decades; this power is mostly unused because we do not have the mathematics to put this power to practical use. Today’s powerful computers and new mathematics probably can solve or help solve many pressing problems, even trillion dollar problems such as energy shortages or major diseases such as cancer. Success in solving problems with mathematics requires realistic expectations, realistic planning, and adequate time and resources.

In The Numbers Behind Numb3rs (page 208), the mathematicians Keith Devlin and Gary Lorden, a full professor at Caltech, write:

One thing that is completely unrealistic is the time frame. In a fast-paced 41-minute episode, Charlie has to help his brother solve the case in one or two “television days.” In real life, the use of mathematics in crime detection is a long and slow process. (A similar observation is equally true for the use of laboratory-based criminal forensics as depicted in television series such as the hugely popular CSI franchise.)

Also unrealistic is that one mathematician would be familiar with so wide a range of mathematical and scientific techniques as Charlie. He is, of course, a television superhero — but that’s what makes him watchable. Observing a real mathematician in action would be no more exciting than watching a real FBI agent at work! (All that sitting in cars waiting for someone to exit a building, all those hours sifting through records or staring at computer screens… boring.)

It’s also true that Charlie seems able to gather masses of data in a remarkably short time. In real-life applications of mathematics, getting hold of the required data, and putting it into the right form for the computer to digest, can involve weeks or months of labor-intensive effort. And often the data one would need are simple not available.

(Emphasis Added)

In their discussion of the episode “Manhunt” (Airdate: May 13, 2005,The Numbers Behind Numb3rs, page. 78), in which Charlie Eppes uses Bayesian statistics to predict the actions and location of an escaped killer, Devlin and Lorden also write:

As is often the case with dramatic portrayals of mathematics or science at work, the length of time available to Charlie to produce his ranking of the reported sightings [of the escaped killer] is significantly shortened, but the idea of using the mathematically based technique of Bayesian analysis is sound.

(Emphasis Added)

Real-life mathematics and mathematical software development involves much more time, much more trial and error, much more debugging, and much more risk than depicted in Numb3rs. Scientists often claim an eighty to ninety percent failure rate in their research projects, frequently when explaining an obvious failure to disappointed graduate students, donors, policy makers, and others who expected more. Charlie Eppes almost never fails! There is historical evidence that the failure rate in genuine “breakthroughs” is higher, quite possibly ninety-nine percent or worse. Some of the mathematics that Charlie whips up in a few “television days” in the show would actually qualify as breakthroughs in real-life, notably some mathematics and algorithms for artificial intelligence and pattern recognition (see below). Historically, genuine breakthroughs have usually involved at least five years of effort when successful. To give a recent example, Grigoriy Perelman’s proof of the Poincare Conjecture took him at least seven years. There appear to have been about one hundred failed published attempts to prove the conjecture by mathematicians prior to Perelman’s success.

The reality is, in fact, worse than Devlin and Lorden concede in their book. Numb3rs has several episodes that portray artificial intelligence (AI), pattern recognition, machine learning, and similar technologies far superior to reality at the time the show aired (2005-2010) or even today (2011). In one episode, Charlie whips up an image/object recognition algorithm in a few hours to enable the NSA to track a yellow truck carrying a contraband missile guidance system through their satellite images of LA to a terrorist (“Finders Keepers,” Original Air Date: January 12, 2007). Similarly, remarkably effective face recognition algorithms play a role in several episodes. Many of Charlie’s AI and pattern recognition algorithms and the other pattern recognition technology shown in Numb3rs works much better than the real algorithms and math.

The Specter of 9/11

Numb3rs is a fast-paced entertaining show with sexy, idealistic, highly effective heroes and heroines. Although it is sometimes critical of security agencies like the CIA and powerful institutions like pharmaceutical companies, in many respects it is Hollywood product placement for the post 9/11 world of massive, expensive high-tech surveillance and security measures both overseas and at home — in which mathematics plays an important and growing role. It reminds one of President Eisenhower’s speeches during the 1950′s:

The worst to be feared and the best to be expected can be simply stated.

The worst is atomic war.

The best would be this: a life of perpetual fear and tension; a burden of arms draining the wealth and the labor of all peoples; a wasting of strength that defies the American system or the Soviet system or any system to achieve true abundance and happiness for the peoples of this earth.

Every gun that is made, every warship launched, every rocket fired signifies, in the final sense, a theft from those who hunger and are not fed, those who are cold and are not clothed. This world in arms is not spending money alone.

It is spending the sweat of its laborers, the genius of its scientists, the hopes of its children.

The cost of one modern heavy bomber is this: a modern brick school in more than 30 cities.

It is two electric power plants, each serving a town of 60,000 population.

It is two fine, fully equipped hospitals. It is some 50 miles of concrete highway.

We pay for a single fighter plane with a half million bushels of wheat.

We pay for a single destroyer with new homes that could have housed more than 8,000 people.

This, I repeat, is the best way of life to be found on the road the world has been taking.

This is not a way of life at all, in any true sense. Under the cloud of threatening war, it is humanity hanging from a cross of iron.

Chance for Peace (April 16, 1953)
President Dwight David Eisenhower (shortly after the death of Joseph Stalin)

President Dwight D. Eisenhower

Eisenhower and his advisers were no shrinking violets. They were well aware the world can be a nasty, dangerous place. They presided over a massive military buildup and controversial covert operations in Guatemala, Iran, Vietnam, and other countries. By the end of his Presidency Eisenhower and his advisers found that it was never enough. Even thousands of nuclear weapons, ships, tanks, spies, and what we now know was a massive lead over the Soviet Union was not enough to satisfy what he famously labeled the “military industrial complex” in his Farewell Address. Eisenhower found himself attacked by Republicans and Democrats alike for not spending even more money on guns and preparations for war!

Osama Bin Laden: The Trillion Dollar Man

Following the reported death of Osama Bin Laden, Andrea Millen Rich, writing in the libertarian Reason magazine, estimated the direct cost of getting Bin Laden at $1.1 trillion. Tim Fernholz and Jim Tankersley, writing in The Atlantic estimated the total cost at $3 trillion over fifteen years. Sam Stein of the Huffington Post, citing a Congressional Research Service report of March 29, 2011, put the cost at at least 1.283 trillion.

According to the United States Centers for Disease Control, the leading causes of death in the United States in the calendar year 2007 were:

Number of deaths for leading causes of death

* Heart disease: 616,067
* Cancer: 562,875
* Stroke (cerebrovascular diseases): 135,952
* Chronic lower respiratory diseases: 127,924
* Accidents (unintentional injuries): 123,706
* Alzheimer’s disease: 74,632
* Diabetes: 71,382
* Influenza and Pneumonia: 52,717
* Nephritis, nephrotic syndrome, and nephrosis: 46,448
* Septicemia: 34,828

All homicides, of which terrorist attacks are a small fraction even in 2001, do not make the top ten. In 2007, the Centers for Disease Control listed all homicides as the 15th leading cause of death:

All homicides

* Number of deaths: 18,361
* Deaths per 100,000 population: 6.1
* Cause of death rank: 15

It is worth noting that the US invasion of Iraq in 2003 resulted in a dramatic drop in Iraqi oil production, undoubtedly contributing substantially to the large increases in oil and energy prices in the last decade. So too the US invasion of Afghanistan in 2001 seems to have scuttled any chance of constructing a pipeline for natural gas from Turkmenistan to the Indian Ocean, also undoubtedly contributing to high energy prices.

It is difficult to improve on President Eisenhower’s words today. Bayesian statistical analyses that predict terrorist attacks, even if they work, don’t make up for dwindling supplies of inexpensive oil and natural gas. They don’t feed people. They don’t cure diseases like cancer or prevent heart attacks. How much more could have been and could still be accomplished if today’s powerful computers and new mathematics were applied to substantive problems such as energy, food, and health instead of the will-o’-the-wisp of perfect security or the pseudo-scientific financial engineering that helped cause the current Great Recession? Mathematicians, scientists, business leaders, and policy makers can do better than we have done.

Conclusion

Numb3rs is a fun, entertaining show. If you are a mathematician, it will probably make you feel great about your profession unless you are in the unfortunate position of dealing with an employer, client, investor, or funding agency that expects you to do what Charlie Eppes does in every episode of Numb3rs. Some of the math and science in Numb3rs is completely realistic. Some of the math is somewhat exaggerated. Some of the math is pure science fiction even though it generally seems very real and believable. As Devlin and Lorden admit in their book, the time frame is, in most cases, completely unrealistic.

The world is presently confronted with serious and worsening problems, possibly due to a dwindling supply of inexpensive oil and natural gas. The political and economic leadership of the world appears paralyzed and unable to deal with the problems, bickering over debt ceilings and other silliness. We do have vast unused resources in the computational power of hundreds of millions of computers and other devices. With the proper mathematics and creative thinking, we may be able to harness this power to resolve many of the current problems, without waiting for paralyzed governments or blundering Too Big To Fail banks to act wisely.

Most mathematics and mathematical software has been developed by individuals and small teams working over periods of several months to several years with total costs of tens of thousands to a few million dollars per project. Success requires realistic expectations about the size, scope, difficulty level, and risks of developing and implementing mathematics and mathematical software. In these difficult times, mathematicians and scientists must gain support for realistic projects that can find real solutions to our pressing problems, and honestly reject the fantasy elements of Numb3rs.

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

The Shadow Factory: The Ultra-Secret NSA from 9/11 to the Eavesdropping on America
James Bamford
Doubleday, New York, 2008

The Cost of Iraq, Afghanistan, and Other Global War on Terror Operations Since 9/11
Amy Belasco, Congressional Research Service, Washington, D.C, March 29, 2011

Credits
The picture of actor David Krumholtz at the Serenity Premiere is from Wikimedia Commons, licensed under the Creative Commons Attribution 2.0 Generic license.

This image was originally posted to Flickr by RavenU at http://flickr.com/photos/36330825119@N01/45967991. It was reviewed on 10:00, 30 April 2007 (UTC) by the FlickreviewR robot and confirmed to be licensed under the terms of the cc-by-2.0.

Millikan Library at Caltech Image from Wikimedia Commons.

Official Portrait of President Dwight D. Eisenhower, May 29, 1959

(from Wikipedia) This image is a work of an employee of the Executive Office of the President of the United States, taken or made during the course of the person’s official duties. As a work of the U.S. federal government, the image is in the public domain.

The image of Usama Bin Laden (Osama Bin Laden) is from the FBI Ten Most Wanted Poster.

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.

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Teaching Prime Numbers to Children

Antonio Cangiano — Wed, 16 Mar 2011 18:00:16 +0000

The other day I received a math book that’s geared towards children. It’s called “You Can Count On Monsters” (USA | UK | Canada) by Professor Richard Evan Schwartz. This mathematics book is unconventional in the sense that it’s mostly comprised of drawings, with relatively few words or formulas. It’s square shaped, enjoyably colorful, and printed on stark black paper which really makes the illustrations pop.

Though it might surprise some to know, this is one of the most popular math books in print today – it’s even an Amazon best seller (when looking at overall Amazon sales, not just in the math category). Even more surprising, the book isn’t brand new and therefore just riding a wave of immediate (but ultimately short-lived) success. It was released over a year ago and has brought in some incredible sales figures over that time. On Amazon, You Can Count On Monsters mostly has five star reviews, and it’s been well reviewed by several outlets, including NPR.

So what is this book all about? I like Publishers Weekly’s factual and succinct description of the book:

This compact, innovative book counts to 100 using prime numbers represented as monsters, each with identifying characteristics (two resembles a bee with two buggy eyes, and three is an angry-looking triangular creature). The book opens with explanations of multiplication, prime and composite numbers, and factor trees, then moves on to a list of numbers. Each prime number looks unique, while composite numbers are represented by scenes involving their prime monsters (eight is illustrated as three of the beelike twos, i.e., two times two times two. Readers may have difficulty deciphering the pictures, which come to resemble little works of abstract geometric art. But especially for creative learners, visualizing the roles each monster plays may lead to deeper number sense. Ages 4 to 8.

This book is charming, interesting, fun, and clearly comes at the topic in a child-friendly manner, yet it’s still fascinating enough to engage older readers, too. It builds on the basic concept of integer multiplications and moves on to intuitively (and visually) illustrate the concept of prime numbers, composite numbers, factorization trees, and even more advanced topics such as the Sieve of Eratosthenes, as well as an illustration of Euclid’s proof for the existence of an infinite number of prime numbers.

While the illustrations may be targeted at children, the math in this book is rigorous and non-trivial considering the age group it’s geared towards. The approach taken is remarkable as it doesn’t fall into the trap of presenting mathematics to children as a dry, mechanical series of steps in an algorithm that will eventually solve a given problem. It takes an exploratory approach where experimentation and imagination are associated with the general idea of doing math.

The most important lesson this book presents is how to reason when it comes to viewing numbers like mathematician (and by the same token, how fun it can be to think in terms of numerical patterns).

2 x 7 = 14

You Can Count On Monsters may help children overcome their trepidations about mathematics, and even aid in the development of a real penchant for math from an early age, which will obviously be of great benefit to any child as they go through their school years and on to adulthood.

In short, we highly recommend picking up this excellent (and delightfully illustrated) book for your children, and spending some quality time together with them as you explore the colourful pages of You Can Count On Monsters. Your youngsters will enjoy the experience – and it may even help you take a new approach to how you help your child learn math from here on out.

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30 Great Math Books as Recommended by Our Readers

Antonio Cangiano — Thu, 10 Feb 2011 18:00:50 +0000

A couple of days ago we asked you to tell us about your favorite math book. Now that the results are in, we must say that while some titles were expected, there are quite a few surprises as well. Quite frankly we were blown away by the great list of math books we compiled with your input.

Listed in the order received, along with quotes from those who submitted them:

1. A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form by Paul Lockhart

Good light book — Folex

2. The Mathematical Experience by Phillip J. Davis, Reuben Hersh

When I was about to finish my degree in math, I had the opportunity to enjoy this new (to me) point of view. The historical and philosophical aspects make it really worth reading. — Jose Rodriguez

3. Winning Ways for Your Mathematical Plays: Volumes 1 and 2 by Elwyn R. Berlekamp, John H. Conway, Richard K. Guy

It expands on the concept of numbers tremendously. — Jerry Anning

4. Fourier Analysis by T. W. Körner

Lots of interesting math, lots of historical information and stories, some interesting applications. What’s not to like? — Peter

5. Tools of the Trade by Paul J. Sally, Jr.

Paul F***in Sally, nuff said QED, bitch. — John Rodgers

6. The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger, Rotraut Susanne Berner, and Michael Henry Heim

Great introduction to many mathematical topics for tweens. — Mark Bell

7. Logicomix: An Epic Search for Truth by Apostolos Doxiadis, Christos H. Papadimitriou, Alecos Papadatos, and Annie Di Donna

This comic is structured on the life of Bertrand Russel and his search for truth. It’s concise, easy to understand and most of all beautifully illustrated. It truly emerges you in one of the most important mathematical stories like no other normal book could. Certainly a must read for anyone with an interest in Mathematics and Logic. — Matthew Edwards

8. Geometry and the Imagination by David Hilbert and S. Cohn-Vossen

A book compiled by one of David Hilbert’s students covering a series of lectures about geometry. It is a great look into modern geometry and I think many undergraduates in math could follow along nicely. Very expansive. — Michael Music

9. Mathematics: Form and Function by Saunders MacLane

A beautiful and clear exposition of the branches of mathematics and their interconnections. — Nick Kirby

10. How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics by William Byers

I like the treatment/discussion of many foundational aspects of mathematics: logic, paradox, ambiguity, infinity, uncertainty, “truth,” that people too often take for granted or overlook. — Shecky R.

11. The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century by David Salsburg

This book introduces the history of statistics without getting too technical. Having studied statistics in school, it is amazing to hear the stories of the people who invented the statistical techniques. — Shigeru Sasao

12. Unknown Quantity: A Real and Imagined History of Algebra by John Derbyshire

Exposed me to whole areas of mathematics I’d never heard of, such as topology, abstract algebra, Reimann surfaces, and so on. — Scott McWilliams

13. Here’s Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math by Alex Bellos

I thought this was a great look into some origins of numbers and math concepts in a fascinating way. I didn’t use it in the classroom, but several of the examples and interesting facts easily made their way into my classroom discussions. — Dan Olexio

14. Real Analysis by N. L. Carothers

It’s a simple, intuitive, down to earth look at real analysis on metric space without losing on the rigour typically found in an upper year undergraduate course. — Nikolay Hristov

15. Professor Stewart’s Cabinet of Mathematical Curiosities by Ian Stewart

The book is fun to read and easy to understand. The mathematics is laid out beautifully within entertaining puzzles. — Danielle Stewart

16. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth by Paul Hoffman

Very entertaining biography by Paul Hoffman. Erdos is portrayed as a very quirky yet endearing genius. — Janet Dee

It is a wonderful combination of biography and mathematics. It is a very well-written biography, intriguing and inspiring. In illustrating the life of the great mathematician Paul Erdos, Hoffman slips in snippets of interesting mathematical ideas and discoveries, allowing readers like me who are not yet familiar with more advanced mathematics to feel its magic. — Cissy Chan

17. The Visual Display of Quantitative Information, 2nd edition by Edward R. Tufte

It’s a wonderful book that can both function as a introductory textbook to visualization or a coffee table book. It is filled with fantastic diagrams and visuals both original and from a wide variety of sources, including international examples. It manages to have great breath and skim through topics without feeling deficient. — Joshua Niederriter

18. Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory by Carl M. Bender and Steven A. Orszag

Deeply insightful and utterly fascinating.

I had the privilege to explore this guide to the universe of asymptotics under Prof. Bender. I used to think, and still do (about 2 decades on), that this book is like the Rosetta Stone, and enables us to solve anything in applied mathematics.

A must have for anyone looking to understand the incredible universe we find ourselves in! — Monish Verma

19. Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter

The book is a marvel on many levels. — Hemanth Kapila

20. What Is Mathematics? An Elementary Approach to Ideas and Methods by Richard Courant, Herbert Robbins, and Ian Stewart

After reading this:

“It doesn’t matter what mathematical things are: it’s what they do that counts. Thus mathematics hovers uneasily between the real and the not real; its meaning does not reside in formal abstractions, but neither is it tangible. This may cause problems with philosophers who like tidy categories, but it is the great strength of mathematics — what I have elsewhere called its ‘unreal reality.’ Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either.” – Ian Stewart, What is Mathematics?

…I was hooked and after reading it several times, I still can’t put it down! Great job to all who contributed to creating this masterpiece — Bill Zimmerly

21. A Mathematician’s Apology by G. H. Hardy

Wonderful description of why being a mathematician and mathematics matters which calls upon our higher selves rather than utilitarian notions. — Robert Janes

Beautiful stuff. — A.S.

22. The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele

Inspiring and beautiful proofs. — Sung guan Yun

23. Calculus Basic Concepts for High School by L.V. Tarasov, V Kisin, and A Zilberman

The entire book is written in the form of a dialogue between a teacher and a student. Calculus is introduced through sequences and inequalities. The discussion proceeds as naturally and intuitively as a first learner of the subject always wishes for. I was introduced to this book by a friend of mine when I was in high school. Today, as a teacher and a student of mathematics, I thank him wholeheartedly for that. This Soviet era mathematical gem is a must read for any one interested in pre-college mathematics. — Ashani Dasgupta

24. Visual Complex Analysis by Tristan Needham

Visual Complex Analysis is filled with deep insights about what’s going on in the complex plane. Tristan Needham writes in an informal style and it is difficult not to be drawn in by his enthusiasm and excitement. In addition to the traditional calculus topics, the book includes fascinating chapters on Mobius transformations, non-Euclidean geometry, and topology. Every chapter ends with several pages of clever and engaging problems, many of which will grab you like a good mystery novel and won’t let go until you solve them (well, at least if you’re the kind of person who reads math blogs). Who knew complex analysis could be so much fun? Needham did, and fortunately he’s shared his ideas and discoveries with the rest of us. — Joel Schwartz

25. Geometry: a High School Course by Serge Lang and G. Murrow

I was awful at geometry for years due to being taught it piecemeal, as a random assortment of processes and formulas to memorize without any unifying principle or foundation. As a result, I avoided or struggled though any bit of math that hinted at it – algebra because of the trigonometry, calculus because of the conic sections – until I finally decided to tackle it with everything I had. I pulled the first book off the shelf of the library’s geometry section and resolved that I would work through it cover to cover. Thank God it was Lang’s book. He starts off with the very basics, lines and the concept of distance, but in such a way as to revolutionize your conception of them and all other similar, presumably dull axioms. The book serves doubly as a text on problem-solving – after only a few chapters of thoughtfully constructing proofs my confidence in my mathematic ability was increased tenfold. What had previously seemed a veiled framework whose mysteries were penetrable only through mastery of abstruse tomes now seemed open and accessible to me through my inherent attribute of reason. I now know that this is the result of working through any systematic geometry text starting with Euclid, but Lang’s remains my favorite for the continually fascinating proofs and presentation. He had a wonderful gift for explaining things to the uninitiated without sacrificing rigor or intellectual strength. — Lauren Tubbs

26. The Princeton Companion to Mathematics by Timothy Gowers, June Barrow-Green, and Imre Leader

Coherent overview of all of pure mathematics. — Nicholas Urfe

27. Principles of Mathematical Analysis by Walter Rudin

The amount of mathematical rigor applied to the fundamental concepts of calculus. — Zachary Gilmartin

28. Mathematics for the Million by Lancelot Thomas Hogben

First read this book in 1946 when I was 14. It was, at the time, the most exciting book I’d ever read and created in me a lifelong interest in mathematics. As a result I ended up as an electrical engineer, one son as a professor of Physics, and another as a fixed income portfolio manager (heavy on Math). I ascribe all of this to to Lancelot Hogben’s book. It may be out-of-date now but I can think of no better introduction to the subject for the young mind. — Ronald Francis

29. Number Theory: An Elementary Introduction Through Diophantine Problems by Daniel Duverney

The exhibition of the beauty of the number theory, achieved on the basis of its own simplicity and clarity, is one of the best features of this book. The book is both well written and pragmatic, while its material is pedagogical and comprehensive. This book illustrates various elementary topics in number theory through very clear explanations, concise definitions, and plenty of exercises along with their solutions. — Ana B. Momidik-Reyna

30. Mathematical Thought from Ancient to Modern Times (Vol. 1) by Morris Kline

Traces the origin of mathematics all the way to Babylonian times. A very comprehensive book. — Matt Swass

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What’s your favorite math book?

Antonio Cangiano — Tue, 08 Feb 2011 18:17:20 +0000

Let’s do a small experiment. I’m not sure what the outcome will be, but definitely think it’s worth a shot no matter what. Using this link you can fill out a form and let us know about one great math book you’ve read and loved. It can be a textbook, a popular science book, or even a mathematical novel. It doesn’t matter.

What we’ll do is collect the titles you share with us and turn them into a list, then we’ll show those books in a blog entry (in the near future). This should be a fun way to get more people exposed to books they may not have heard of. If your favorite book is a classic like Flatland or GEB, feel free to submit those, too – any math related book is welcome!

If you provide your name and a link in the form, we’ll attribute the submission and link it to your non-commercial, non-spammy blog or site (we reserve the right to decide which sites are included).

Submit a book

Please don’t use the comment section to suggest the book, but rather use the link above. We are genuinely curious to see what titles we’ll end up with on this cool list.

Sponsor’s message: Check out Math Better Explained, an insightful ebook and screencast series that will help you see math in a new light and experience more of those awesome “aha!” moments when ideas suddenly click.

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Developing Math Intuition

Antonio Cangiano — Thu, 13 Jan 2011 20:00:22 +0000

Today Kalid Azad released a premium edition of his well-received Math Better Explained ebook.

This brand new edition (available for a limited time for just $47) boasts a screencast for each of the 12 chapters in the book, as well as a 3-part interview about developing your mathematical intuition with Scott H Young, who has written extensively about study techniques. The package also includes a PDF version of the book, a worksheet, and slides/image as a zip file.

We were provided early access to the material, and have found the ebook + screencasts combo to be highly effective in conveying fundamental mathematical concepts that will serve you well for years to come.

To celebrate the announcement, and as a Math Blog exclusive, today we are publishing an excerpt from the book here. It covers how to develop math intuition and the great importance this skill plays in the field of mathematics. Enjoy!

Developing Math Intuition

Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean?
Suppose we want to define a “cat”:

Caveman definition: A furry animal with claws, teeth, a tail, 4 legs, that purrs when happy and hisses when angry. . .
Evolutionary definition: Mammalian descendants of a certain species (F. catus), sharing certain characteristics. . .
Modern definition: You call those definitions? Cats are animals sharing the following DNA: ACATACATACATACAT. . .

The modern definition is precise, sure. But is it the best? Is it what you’d teach a child learning the word? Does it give better insight into the “catness” of the animal? Not really. The modern definition is useful, but after getting an understanding of what a cat is. It shouldn’t be our starting point.

Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is.

Let’s approach ideas from a different angle. I imagine a circle: the center is the idea you’re studying, and along the outside are the facts describing it. We start in one corner, with one fact or insight, and work our way around to develop our understanding. Cats have common physical traits leads to Cats have a common ancestor leads to A species can be identified by certain portions of DNA. Aha! I can see how the modern definition evolved from the caveman one.

But not all starting points are equal. The right perspective makes math click — and the mathematical “cavemen” who first found an idea often had an enlightening viewpoint. Let’s learn how to build our intuition.

1.1 What is a Circle?

Time for a math example: How do you define a circle?

There are seemingly countless definitions. Here’s a few:

The most symmetric 2-d shape possible
The shape that gets the most area for the least perimeter (the isoperimeter property)
All points in a plane the same distance from a given point (drawn with a compass, or a pencil on a string)
The points (x,y) in the equation x² + y² = r² (analytic version of the geometric definition above)
The points in the equation r · sin(t),r · cos(t), for all t (really analytic version)
The shape whose tangent line is always perpendicular to the position vector (physical interpretation)

The list goes on, but here’s the key: the facts all describe the same idea! It’s like saying 1, one, uno, eins, “the solution to 2x + 3 = 5” or “the number of noses on your face” — just different names for the idea of unity.

But these initial descriptions are important — they shape our intuition. Because we see circles in the real world before the classroom, we understand their “roundness”. No matter what fancy equation we see (x² + y² = r²), we know deep inside that a circle is round. If we graphed that equation and it appeared square, or lopsided, we’d know there was a mistake.

As children, we learn the caveman definition of a circle (a really round thing), which gives us a comfortable intuition. We can see that every point on our “round thing” is the same distance from the center. x² + y² = r² is the analytic way of expressing that fact (using the Pythagorean theorem for distance). We started in one corner, with our intuition, and worked our way around to the formal definition.

Other ideas aren’t so lucky. Do we instinctively see the growth of e, or is it an abstract definition? Do we realize the rotation of i, or is it an artificial, useless idea?

1.2 A Strategy For Developing Insight

I still have to remind myself about the deeper meaning of e and i — which seems as absurd as “remembering” that a circle is round or what a cat looks like! It should be the natural insight we start with.

Missing the big picture drives me crazy: math is about ideas — formulas are just a way to express them. Once the central concept is clear, the equations snap into place. Here’s a strategy that has helped me:

Step 1: Find the central theme of a math concept. This can be difficult, but try starting with its history. Where was the idea first used? What was the discoverer doing? This use may be different from our modern interpretation and application.
Step 2: Explain a property/fact using the theme. Use the theme to make an analogy to the formal definition. If you’re lucky, you can translate the math equation (x² + y² = r²) into a plain-english statement (“All points the same distance from the center”).
Step 3: Explore related properties using the same theme. Once you have an analogy or interpretation that works, see if it applies to other properties. Sometimes it will, sometimes it won’t (and you’ll need a new insight), but you’d be surprised what you can discover.

Let’s try it out.

1.3 A Real Example: Understanding e

Understanding the number e has been a major battle. e appears everywhere in science, and has numerous definitions, yet rarely clicks in a natural way. Let’s build some insight around this idea. The following section has several equations, which are simply ways to describe ideas. Even if the equation is gibberish, there’s a plain-english concept behind it. Here’s a few common definitions of e:

The first step is to find a theme. Looking at e’s history, it seems it has some- thing to do with growth or interest rates. e was discovered when performing business calculations (not abstract mathematical conjectures) so “interest” (growth) is a possible theme.

Let’s look at the first definition, in the upper left. The key jump, for me, was to realize how much this looked like the formula for compound interest. In fact, it is the interest formula when you compound 100% interest for 1 unit of time, compounding as fast as possible. The chapter on e describes this interpretation.

Definition 1: Define e as 100% compound growth at the smallest increment possible.

Let’s look at the second definition: an infinite series of terms, getting smaller and smaller. What could this be?

After noodling this over using the theme of “interest” we see this definition shows the components of compound interest. Now, insights don’t come instantly — this insight might strike after brainstorming “What could 1 + 1 + 1/2 + 1/6 + . . . ” represent when talking about growth?”

Well, the first term (1 = 1/0!, remembering that 0! is 1) is your principal, the original amount. The next term (1 = 1/1!) is the “direct” interest you earned — 100% of 1. The next term (0.5 = 1/2!) is the amount of money your interest made (“2nd-level interest”). The following term (.1666 = 1/3!) is your “3rd-level interest” — how much money your interest’s interest earned!

Money earns money, which earns money, which earns money, and so on — the sequence separates out these contributions (the chapter on e describes how Mr. Blue, Mr. Green & Mr. Red grow independently). There’s much more to say, but that’s the “growth-focused” understanding of that idea.

Definition 2: Define e by the contributions each piece of interest makes.

Neato. Now to the third, shortest definition. What does it mean? Instead of thinking “derivative” (which turns your brain into equation-crunching mode), think about what it means. The feeling of the equation. Make it your friend.

It’s the calculus way of saying “Your rate of growth is equal to your current amount”. Well, growing at your current amount would be a 100% interest rate, right? And by always growing it means you are always calculating interest – it’s another way of describing continuously compound interest!

Definition 3: Define e as always growing by 100% of your current value.

Nice — e is the number where you’re always growing by exactly your current amount (100%), not 1% or 200%.

Time for the last definition — it’s a tricky one. Here’s my interpretation: Instead of describing how much you grew, why not say how long it took?

If you’re at 1 and growing at 100%, it takes 1 unit of time to get from 1 to 2. But once you’re at 2, and growing 100%, it means you’re growing at 2 units per unit time! So it only takes 1/2 unit of time to go from 2 to 3. Going from 3 to 4 only takes 1/3 unit of time, and so on.

The time needed to grom from 1 to A is the time from 1 to 2, 2 to 3, 3 to 4. . . and so on, until you get to A. The first definition defines the natural log (ln) as shorthand for this “time to grow” computation.

ln(a) is simply the time to grow from 1 to a. We then say that e is the number that takes exactly 1 unit of time to grow to. Said another way, e is is the amount of growth after waiting exactly 1 unit of time!

Definition 4: Define the time needed to grow continuously from 1 to as ln(a). e is the amount of growth you have after 1 unit of time.

Whablamo! These are four different ways to describe the mysterious e. Once we have the core idea (“e is about 100% continuous growth”), the crazy equations snap into place — it’s possible to translate calculus into English. Math is about ideas!

1.4 What’s the Moral?

In math class, we often start with the last, most complex idea. It’s no wonder we’re confused: we’re showing students DNA and expecting them to see a cat. I’ve learned a few lessons from this approach, and it underlies how I understand and explain math:

Search for insights and apply them. That first intuitive insight can help everything else snap into place. Start with a definition that makes sense and “walk around the circle” to find others.
Be resourceful. Banging your head against an idea is no fun. If it doesn’t click, come at it from different angles. There’s another book, another article, another person who explains it in a way that makes sense to you.
It’s ok to be visual. We think of math as rigid and analytic — but visual interpretations are ok! Do what develops your understanding. Imaginary numbers were puzzling until their geometric interpretation came to light, decades after their initial discovery. Looking at equations all day didn’t help mathematicians “get” what they were about.

Math becomes difficult and discouraging when we focus on definitions over understanding. Remember that the modern definition is the most advanced step of thought, not necessarily the starting point. Don’t be afraid to approach a concept from a funny angle — figure out the plain-English sentence behind the equation. Happy math.

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The Best Writing on Mathematics 2010

Antonio Cangiano — Tue, 04 Jan 2011 18:00:00 +0000

While compiling the new math releases for my site that helps you “discover new books“, I came across a brand new book by Princeton University Press, called “The Best Writing on Mathematics 2010“. While I haven’t had the opportunity to read this book yet, (it was just released yesterday), I think it looks extremely promising.

Here is a description of it directly from the publisher:

This anthology brings together the year’s finest writing on mathematics from around the world. Featuring promising new voices alongside some of the foremost names in mathematics, The Best Writing on Mathematics makes available to a wide audience many articles not easily found anywhere else–and you don’t need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today’s hottest mathematical debates. Here readers will discover why Freeman Dyson thinks some mathematicians are birds while others are frogs; why Keith Devlin believes there’s more to mathematics than proof; what Nick Paumgarten has to say about the timing patterns of New York City’s traffic lights (and why jaywalking is the most mathematically efficient way to cross Sixty-sixth Street); what Samuel Arbesman can tell us about the epidemiology of the undead in zombie flicks; and much, much more.

In addition to presenting the year’s most memorable writing on mathematics, this must-have anthology also includes a foreword by esteemed mathematician William Thurston and an informative introduction by Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us–and where it’s headed.

I’m eagerly waiting for my copy to arrive, as it sounds like the kind of book that can provide readers with an overview of the direction in which some of the most exciting and cutting edge mathematical research is currently headed.

If you’d like to pick up your own copy to start the year off with some advanced math reading, this title is now available and in stock on Amazon.

Sponsor’s message: Check out Math Better Explained, an elegant and insightful ebook that will help you see math in a new light and experience more of those awesome “aha!” moments when ideas suddenly click.

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Free Kindle Math Books

Antonio Cangiano — Fri, 31 Dec 2010 00:03:57 +0000

The CK-12 Foundation is a wonderful non-profit organization. Recently they released several free Kindle ebooks about Mathematics. Currently in the Kindle store you can find the following CK-12 Foundation math titles, all priced at $0.00:

Advanced Probability and Statistics

From a cursory look it would seem that the content of these ebooks is pretty solid and certainly worth checking out. While providing free math books isn’t a new concept, having said books formatted for your Kindle is a major plus (though there are a few concerns about their legibility in the reviews; these points are being addressed by the foundation).

More textbooks are available on the CK-12 Foundation site, which enables you to customize the content of a textbook through their intriguing “FlexBook” system. In their own words:

Traditional textbooks are both expensive and rigid. FlexBooks conform to national and state textbook standards. They are free, easy to update and easy to customize. With FlexBooks, you can customize your textbooks to support your innovative work in the classroom. The CK-12 Foundation provides FlexBooks free to anyone who wants to use them.

While it’s safe to assume that many of our readers already own a Kindle device at this point, it warrants mentioning that you don’t actually need one to read Kindle ebooks (there is Kindle software that’s available for Windows, Mac, iPad, and other mobile platforms that enables you to read Kindle ebooks sans owning an actual ebook reader).

However, on a related side note, if you’re buying a Kindle with the aim of reading technical material and textbook, I highly suggest that you opt for the DX version, because the screen on the regular sized one is too small to comfortably read such detailed material (particularly if you have free math PDFs you’d like to read).

Happy reading and all the very best to everyone in 2011, from the Math Blog team.

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Bad Mathematics: A Trillion Dollar Problem

John F. McGowan, Ph.D. — Tue, 21 Sep 2010 19:00:51 +0000

“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.

*A Comparison of Sci-Fi Math and Real Math*
Sci-Fi Math	Real Math
Fast. Difficult calculations in seconds. Even breakthroughs in moments, hours, a few months.	Slower. Breakthroughs usually take several years, often five or more.
Little or no trial and error. Few or no errors.	Usually higher error rates. Breakthroughs usually involve large amounts of trial and error: often hundreds to tens of thousands of trials.
Symbolic manipulation often results in answers or breakthroughs.	Conceptual analysis and conceptual leaps. Precise math is often used to verify that a new concept will work or is likely to work.

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.

**Appendix: Some Mathematically Oriented Inventions and Discoveries**
Invention or Discovery	Inventor/Discoverer (s)	Duration	Heureka Moment
Kepler’s First Two Laws of Planetary Motion	Johannes Kepler	1600-1605 (5 years)	Sudden realization of answer over Easter Season 1605; probably on vacation.
Kepler’s Third Law of Planetary Motion	Johannes Kepler	1600-1619 (19 years)	Sudden realization of answer in a few weeks in 1619.
Separate Condenser Steam Engine	James Watt (some assistance from John Robison and Joseph Black)	1759-1765 (5 years)	Sudden realization during a walk in the park at the University of Glasgow in April 1765
High Pressure Steam Engine/Steam Boat	John Fitch, Henry Voigt	c. 1782-1787 (about 5 years)	None known.
Precision Nautical Chronometer for Measuring Longitude	John Harrison	c. 1730-1761 (31 years)	None known.
Maxwell’s Equations (Mathematical Theory of Electromagnetism)	James Clerk Maxwell, Michael Faraday	c. 1860-1865 (about 5 years)	None known.
Powered Aircraft	Wilbur Wright, Orville Wright, Octave Chanute	c. 1897-1906 (9 years)	Sudden realization of wing warping steering mechanism while working at bike store.
Special Theory of Relativity	Albert Einstein	c. 1898 – 1905 (7 years)	Sudden leap forward in 1905: “a storm in my mind”
Theory of Photoelectric Effect	Albert Einstein	c. 1898 – 1905 (7 years)	Sudden leap forward in 1905: “a storm in my mind”
Theory of Brownian Motion (“proof” of atoms)	Albert Einstein	c. 1898 – 1905 (7 years)	Sudden leap forward in 1905: “a storm in my mind”
General Theory of Relativity	Albert Einstein	c. 1908 – 1915 (7 years)	None known. Publishes several failed attempts before 1915.
Schrodinger’s Equation	Erwin Schrodinger	c. 1921 – 1926 (about 5 years)	Sudden realization during ski vacation in the alps with girlfriend.
EPR (Non-Local Nature of Quantum Mechanics)	Albert Einstein, Boris Podolsky, Nathan Rosen	c. 1927 – 1935 (7 years)	None known.
Atomic Bomb	Manhattan Project (huge team)	1939-1945 (6 years)	None known.
Sounding Rocket (USA)	Robert Goddard (team of about six technicians, engineers, physicists)	1914-1941 (27 years) (	Several leaps forward — see text above.
Jet Assisted Take Off (JATO) solid fuel rocket	Marvel Whiteside “Jack” Parsons	c. 1932 – 1942 (about 10 years)	Sudden realization in early 1942.
Orbital Rocket (USA)	Werner Von Braun and “Rocket Team” (about 200 people)	1927-1958 (31 years)	Several leaps forward; parallels Goddard.
Manned Landing on Moon (USA)	Apollo Project	1962-1969 (7 years)	None known. Largely scaling up proven orbital rocket design.
Proof of Fermat’s Last Theorem	Andrew Wiles, Richard Taylor	1986-1995 (9 years)	Not known. Incorrect proof published in 1993.
Proof of Poincare Conjecture	Grigoriy Perelman	1995-2002 (7 years)	None known.

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

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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	Probability Concepts and Theory for Engineers Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Harry Schwarzlander ISBN: 0470748559 Publisher: Wiley Publication date: March 1, 2011 Binding: Hardcover Estimated price: $67.99 Share this book on Twitter \|Facebook
	Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form Stores: USA \| UK \| Canada \| Italy By Kevin O’Meara, John Clark, Charles Vinsonhaler ISBN: 0199793735 Publisher: Oxford University Press, USA Publication date: September 16, 2011 Binding: Hardcover Estimated price: $71.55 Share this book on Twitter \|Facebook
	The Art of R Programming: A Tour of Statistical Software Design Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Norman Matloff ISBN: 1593273843 Publisher: No Starch Press Publication date: October 12, 2011 Binding: Paperback Estimated price: $21.99 Share this book on Twitter \|Facebook
	Here’s Looking at Euclid: From Counting Ants to Games of Chance – An Awe-Inspiring Journey Through the World of Numbers Stores: USA \| Canada \| Italy \| Kindle By Alex Bellos ISBN: 1416588280 Publisher: Free Press Publication date: April 19, 2011 Binding: Paperback Estimated price: $4.50 Share this book on Twitter \|Facebook
	Matlab, Second Edition: A Practical Introduction to Programming and Problem Solving Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Stormy Attaway ISBN: 0123850819 Publisher: Butterworth-Heinemann Publication date: August 11, 2011 Binding: Paperback Estimated price: $35.44 Share this book on Twitter \|Facebook
	The Best Writing on Mathematics 2011 Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Editors at Princeton University Press ISBN: 0691153159 Publisher: Princeton University Press Publication date: November 27, 2011 Binding: Paperback Estimated price: $12.14 Share this book on Twitter \|Facebook
	Algorithmic Puzzles Stores: USA \| UK \| Canada \| Kindle By Anany Levitin, Maria Levitin ISBN: 0199740445 Publisher: Oxford University Press, USA Publication date: October 14, 2011 Binding: Paperback Estimated price: $21.13 Share this book on Twitter \|Facebook
	Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 Stores: USA \| Canada By Jessica F. Shumway ISBN: 1571107908 Publisher: Stenhouse Pub Publication date: April 10, 2011 Binding: Paperback Estimated price: $17.88 Share this book on Twitter \|Facebook
	Agent-Based and Individual-Based Modeling: A Practical Introduction Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Steven F. Railsback, Volker Grimm ISBN: 0691136742 Publisher: Princeton University Press Publication date: November 6, 2011 Binding: Paperback Estimated price: $41.39 Share this book on Twitter \|Facebook
	Foundations of Geometry (2nd Edition) Stores: USA \| UK \| Canada By Gerard Venema ISBN: 0136020585 Publisher: Addison Wesley Publication date: July 16, 2011 Binding: Paperback Estimated price: $60.35 Share this book on Twitter \|Facebook
	Taming the Infinite: The Story of Mathematics from the First Numbers to Chaos Theory Stores: USA \| UK \| Canada \| Italy By Ian Stewart ISBN: 1847247687 Publisher: Quercus Publication date: November 1, 2011 Binding: Paperback Estimated price: $6.03 Share this book on Twitter \|Facebook
	The Number Mysteries: A Mathematical Odyssey through Everyday Life Stores: USA \| Canada \| Italy \| Kindle By Marcus du Sautoy ISBN: 0230113842 Publisher: Palgrave Macmillan Publication date: May 24, 2011 Binding: Paperback Estimated price: $10.66 Share this book on Twitter \|Facebook
	The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition Stores: USA \| UK \| Canada \| Italy \| Kindle By Stanislas Dehaene ISBN: 0199753873 Publisher: Oxford University Press, USA Publication date: April 29, 2011 Binding: Paperback Estimated price: $20.04 Share this book on Twitter \|Facebook
	Introduction to Proof in Abstract Mathematics Stores: USA \| UK \| Canada \| Italy By Andrew Wohlgemuth, Mathematics ISBN: 0486478548 Publisher: Dover Publications Publication date: February 17, 2011 Binding: Paperback Estimated price: $12.89 Share this book on Twitter \|Facebook
	Wizardry: Baseball’s All-Time Greatest Fielders Revealed Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Michael Humphreys ISBN: 0195397762 Publisher: Oxford University Press, USA Publication date: March 30, 2011 Binding: Paperback Estimated price: $11.00 Share this book on Twitter \|Facebook
	Differential Geometry: Bundles, Connections, Metrics and Curvature Stores: USA \| UK \| Canada \| Kindle By Clifford Henry Taubes ISBN: 0199605874 Publisher: Oxford University Press, USA Publication date: December 1, 2011 Binding: Paperback Estimated price: $36.20 Share this book on Twitter \|Facebook
	Fatal Numbers: Why Count on Chance Stores: USA \| UK \| Kindle \| UK Kindle By Hans Magnus Enzensberger ISBN: 1935830015 Publisher: Upper West Side Philosophers, Inc. Publication date: February 2, 2011 Binding: Paperback Estimated price: $10.05 Share this book on Twitter \|Facebook
	Matroid Theory Stores: USA \| UK \| Canada \| Italy By James Oxley ISBN: 0199603391 Publisher: Oxford University Press, USA Publication date: April 22, 2011 Binding: Paperback Estimated price: $53.16 Share this book on Twitter \|Facebook
	Visual Thinking in Mathematics Stores: USA \| UK \| Italy \| Kindle \| UK Kindle By Marcus Giaquinto ISBN: 0199575533 Publisher: Oxford University Press, USA Publication date: November 14, 2011 Binding: Paperback Estimated price: $30.57 Share this book on Twitter \|Facebook
	Mathematics, Ideas and the Physical Real Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Albert Lautman, Simon Duffy ISBN: 144112344X Publisher: Continuum Publication date: August 4, 2011 Binding: Paperback Estimated price: $14.50 Share this book on Twitter \|Facebook
	The Algebra Solution to the Mathematics Reform: Completing the Equation Stores: USA \| UK \| Canada \| Italy By Frances R. Spielhagen ISBN: 0807752312 Publisher: Teachers College Press Publication date: June 17, 2011 Binding: Paperback Estimated price: $17.43 Share this book on Twitter \|Facebook
	A Career in Statistics: Beyond the Numbers Stores: USA \| UK \| Canada \| Italy By Gerald J. Hahn, Necip Doganaksoy ISBN: 0470404418 Publisher: Wiley Publication date: June 28, 2011 Binding: Paperback Estimated price: $48.55 Share this book on Twitter \|Facebook
	The Star and the Whole: Gian-Carlo Rota on Mathematics and Phenomenology Stores: USA \| UK \| Canada By Fabrizio Palombi ISBN: 1568815832 Publisher: A K Peters/CRC Press Publication date: August 24, 2011 Binding: Paperback Estimated price: $20.07 Share this book on Twitter \|Facebook