Bad Mathematics: A Trillion Dollar Problem

“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:

[tex] H = E $-$ PV [/tex]

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

[tex] H = E + PV [/tex]

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 [tex]$-$[/tex] to [tex]$+$[/tex] in his magic formula. The overused cliche “thinking out of the box” is often a synonym for trying a slightly different term in a mathematical formula rather than, for example, realizing that combining asphalt and potassium perchlorate avoids the problems of explosive powders in rocketry. The author has actually heard someone use “out of the box” to refer to using a different compiler flag on the GNU C Compiler to compile a mathematical program implemented in the C programming language.

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

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

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

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

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

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

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

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

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

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

Notes

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

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

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

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

Suggested Reading/References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

About the Author

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

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