Hobbes: “A new decade is coming up.”

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

Calvin and Hobbes
by Bill Watterson
December 30, 1989

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

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

What is symbolic manipulation?

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

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

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

 x $+$ 3 = 5

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

 x = 5 $-$ 3 = 2

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

x $+$ a = b

Then, subtract  a from both sides of the equation.

x = b $-$ a

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

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

 x^2 $-$ 2x + 1 = 0

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

 (x $-$ 1)(x $-$ 1) = 0

The solution is  x = 1 . In general, the square of  x + a is:

 (x + a)(x + a) = x^2 + 2a + a^2

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

 ax^2 + bx + c = 0

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

 (x+e)^2 = f

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

 x^2 + \frac{b}{a}x + \frac{c}{a} = 0

To turn the left hand side into (x+e)^2, \frac{b}{a} must be 2e. So, subtract \frac{c}{a} from both sides of the equation:

 x^2 + \frac{b}{a}x = -\frac{c}{a}

Then, add  (\frac{b}{2a})^2 to both sides of the equation.

 x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 = -\frac{c}{a} + (\frac{b}{2a})^2

Now, the equation is in the form  (x+e)^2 = f where  e = \frac{b}{2a} and  f = -\frac{c}{a} + (\frac{b}{2a})^2 . Therefore:

 x + \frac{b}{2a} = \sqrt{ -\frac{c}{a} + (\frac{b}{2a})^2 }

Then, subtract  \frac{b}{2a} from both sides of the equation and remove the factor  \frac{1}{2a} outside the square root sign to get the formula for  x .

x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\hspace{2cm}a\neq0

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

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

 ax^3 + bx^2 + cx + d = 0 (Cubic Equation)


 ax^4 + bx^3 + cx^2 + dx + e = 0 (Quartic Equation)

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

The Status of Symbolic Manipulation

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

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

 e = m c^2

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

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

Richard Tolman and Albert Einstein

Richard Tolman and Albert Einstein

Paul Dirac

Paul Dirac

Theodor Von Karman

Theodor Von Karman

Ed Witten

Ed Witten

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

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

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

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

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

Conceptual Reasoning and Analysis

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

 (x-a)^2 + (y-b)^2 = r^2 (The Circle)

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

 (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 (The Sphere)

This is a sphere with radius r centered at (a,b,c) in three dimensions.

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

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

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

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

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


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

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

About the Author

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

Sponsor’s message: Receive free weekly updates about new math books. Don’t miss great new titles in the genres you love (such as Mathematics, Science, Programming, and Sci-Fi):

Get more stuff like this

Get interesting math updates directly in your inbox.


  1. H September 27, 2010
    • Antonio Cangiano September 27, 2010
      • H September 27, 2010
        • H September 27, 2010
          • Cider October 20, 2010
    • Zafar September 29, 2010
    • Jan October 20, 2010
  2. Matt J. September 27, 2010
    • Michael November 3, 2012
  3. Peter O. Sagay September 27, 2010
  4. Vincent Johns September 27, 2010
  5. Seven Star Hand September 28, 2010
  6. Phil Brass September 28, 2010
  7. Archie Z. September 30, 2010
  8. John Faben November 3, 2010
  9. Priscilla Fierge January 28, 2013

Leave a Reply

Join thousands of
math enthusiasts

Get free math updates directly in your inbox.