MATHEMATICS
AND THE EMPEROR'S NEW CLOTHES.
© 2001, G. William Moore, MD, PhD.

     One of the luxuries of being an amateur mathematician, who does not earn his living at mathematics, is that one may ask questions that might get a salaried mathematician fired from his job. That is, the questions are so outrageous or apparently simple-minded, that one is fired either for blasphemy or for gross incompetence.

     In the child's fairy tale, The Emperor's New Clothes, the child asks, ingenuously, why the emperor is naked, while the adults are admiring his (non-existent) new clothes. The most obvious meaning of this story is that children will blurt out an obvious truth that adults are either too stupid or too timid to say, out of a fear of being considered stupid by others or being arrested by the emperor's police. The next layer of meaning, which I didn't appreciate until I was an adult, is that the adults are perfectly aware of the emperor's nakedness, but need to know two things: (1) REALITY, namely that the emperor is actually naked, which the child blurts out; and (2) CONVENTIONAL WISDOM, namely that the emperor is wearing beautiful garments. Some social settings call for perception (1); other social settings call for perception (2). For example, a citizen of Baghdad will have a much better life if he repeats what Saddam Hussein SAYS is true, rather than if he makes an independent judgment. These are simple survival skills in a totalitarian country.

     So what does all this have to do with mathematics? Somewhere late in my graduate school training in biomathematics, it dawned on me that there are about a dozen central ideas in mathematics, all of them basically fairly simple once understood, from which one may derive all the important theories of mathematics. The amazing thing is that such simple things took such a long time to internalize in our culture.

     For example, why was the Greco-Latin culture so resistant to the idea of ZERO, discovered one thousand years B.C. by the Babylonians (as a place-holder on the abacus)? The idea was banished from ancient Greece, and not really embraced in Europe until the sixteenth century, by merchants not mathematicians, who could do their accounting far more easily with Arabic numerals (with zero) than with Roman numerals (without zero).

     A few more examples: Pythagoras's proof [Singh]; infinity; infinitesimals (Calculus; Newton/Leibniz; Weierstrass; Robinson's calculus); open/closed sets (Heine-Borel theorem); computational complexity (NP complete problem; why the Sieve of Eratosthenes takes so long to solve); symbolic logic [Boole; Lewis and Langford]; Goedel's proof [Casti and DePauli]; Hilbert's Tenth Problem [Davis]; Fermat's Last Theorem [Singh]; Riemann hypothesis [Davis]; fractals [Lauwerier], etc.

     These ideas, once learned, are not very difficult to understand and internalize; but from them derive many of the fundamental theorems of mathematics. So what's the big deal? Why did it take so long to grasp such fundamentally simple ideas? Why do the supposedly smartest men and women of our society devote their lives to cultivating and teaching these ideas? Have mathematicians purposely obfuscated the ideas from the rest of us, to keep their own employment stable? Or, as a prominent mathematician recently stated, is it true that: MATHEMATICS IS HARD, EVEN FOR MATHEMATICIANS.

     First, let's resist the temptation to belittle mathematicians: their ideas are basically simple, but they are also devilishly unintuitive at certain junctures. This is why it takes years of training to understand them, and why it took thousands of years of history to ensconce/stabilize them in Western mathematical culture. I remember that it took me four years between first blush and becoming thoroughly comfortable with the main ideas of calculus. Now I can hardly remember what all the fuss was about.

     Second, let's resist the temptation to belittle the ancients. Some of those guys were pretty clever. Archimedes (The Sand Reckoner) was basically aware of calculus. First century Chinese mathematician Sun Tse [Schneier] discovered the Chinese Remainder Theorem, a core concept in modern cryptography algorithms. Somewhere between the stabilization/canonization of the Hebrew Old Testament (the T'nakh) in the second century B.C. and the mediaeval Jewish Kabbalists, the idea of Goedel enumeration was born [Casti and DePauli]. The actual construction method for Goedel numbers from prime numbers was known to Euclid [Schneier]. The seventh century Arabs invented algebra, and with it, the early rumblings of group theory, matrix algebra, etc. Connecting with my adult interpretation of the Emperor's New Clothes, the ancients probably understood the above issues quite well, but they did not have the resources to carry their questions to completion. If ancient mathematicians had been more numerous, if they had had longer life spans, Pentium computers, and the Internet, they might well have been our equals.


APPENDIX A.
SOME CENTRAL IDEAS OF
MATHEMATICS.


PROOF OF PYTHAGORAS'S THEOREM.
     What could be easier than Pythagoras's Theorem about right triangles? If a, b, are legs of a right triangle, and c is the hypoteneuse, then:
a2 + b2 = c2 ,
Apparently the ancient Egyptians knew Pythagoras's Theorem on a heuristic basis for hundreds of years, but each new right triangle was a separate calculation. There was no predictability from one right triangle to the next. The idea that Pythagoras's Theorem ALWAYS WORKED, and that there was a proof of this, was left to the ancient Greeks. So were the ancient Egyptians stupid, or what?

     The basic idea underlying proof of the Pythagorean Theorem is summarized in this simple diagram:
If the ancient Egyptians had only understood this diagram, they would not have had to work out the Pythagorean Theorem for each new triangle. The larger square has an area of (a+b)2, and the smaller square has an area of c2. The four triangles lying outside the smaller square each have an area of (1/2)×a×b, or a total of 2×a×b for all four triangles. The expression (a+b)2 expands to:
a×a + a×b + a×b + b×b.
If you subtract away 2×a×b, which accounts for the four triangles, then the following:
a×a + b×b.
is what is left behind. That is:
c2 = a×a + b×b.
or:
c2 = a2 + b2.
Presto! All this manipulation was within the grasp of ancient Egyptian mathematicians. Remember, these guys weren't dummies. They're the ones who designed and built the pyramids.


INFINITY.
     Infinity is the companion of zero. The Hindu mathematicians (esp. Brahmagupta) loved it [Singh]. Pythagoras was suspicious of it. The proof of the irrationality of the square root of two depends upon an INFINITE REGRESS. Hippasus of Metapontum was EXECUTED for revealing the irrationality of the square root of two outside Pythagoras's secret cult [Seife]. Aristotle, and by extension, Aristotle's Christian apologist, St. Thomas Aquinas, hated it [Seife], since it challenged the (finite) elephant-tortoise cosmology of ancient Greece, and the finite origins of man (Adam and Eve). Infinity doesn't behave like all other numbers; but you need the concept in mathematics for all sorts of things, starting with division.

     Contrary to popular belief, infinity is NOT a vague idea. You can build up a concept of infinity methodically from ordinary whole numbers, and from the usual operators of symbolic logic, most notably, NOT. Even with all his flaws, Aristotle was a good friend of NOT [Aristotle]. Every whole number is finite; but you can always make a larger whole number by multiplying by two or greater. So, the thing that is greater than any whole number is INFINITY. Infinity has special rules of arithmetic, but they are manageable. You just have to pay attention, and do your bookkeeping carefully.

     It wasn't until the late nineteenth century that Georg Cantor, who was eventually driven crazy by David Hilbert's relentless badgering [Bell], clearly understood that there is an infinite hierarchy of DIFFERENT INFINITIES. Not until the late twentieth century did Cohen prove the relationship between Cantor's infinities and the Axiom of Choice. So what's the big deal? Why did it take so long?


INFINITESIMALS.
     The fundamental building blocks of integral calculus and differential calculus were known to the contemporaries of Sir Isaac Newton and Gottlob Leibniz. DIFFERENTIAL CALCULUS was known as FLUXIONS; INTEGRAL CALCULUS was known as QUADRATURE [Courant]. It fell to these two men to understand the fundamental relationship between fluxions and quadrature. Alas, they devoted much of their lives to fighting one another over priority [Hawking]. Another battle, fought with less rancor: Weierstrass; Robinson's calculus.


OPEN/CLOSED SETS.
     Heine-Borel theorem.


COMPUTATIONAL COMPLEXITY.
     Computational complexity is the idea that some calculations take more effort to perform than others [Tarjan; Davis]. Many calculations can be broken down into n units of calculation, such as: sorting a list of n items; finding a particular item in an unsorted list of n items; or traversing a decision-tree with n nodes [Knuth]. The problems are typically simple for small n, but require lots of mathematical horsepower for larger n, such as n=quarter-million patients in a 1000-bed hospital census, or n=quarter-billion eventual social security recipients in the USA.

     Some problems can be solved in n steps, such as finding a particular item in an unsorted list of n items. The n-sorting problem can require n2 steps, or even n3 steps, for the novice computer programmer. But there are well-understood methods for sorts requiring only n×log2(n) steps. The expression log2 denotes log-to-the-base 2. Therefore,
log2(4)=2,
log2(8)=3,
log2(16)=4,
log2(32)=5
log2(64)=6
log2(128)=7...
log2(1,048,576)=20....
For n=1,048,576 (a little over a million), the difference between n2 = 1,098,304,000,000 steps and n×log2(n) = 20,971,520 steps (over 52,000-fold) may mean the difference between finding (and/or billing) a patient, or not.

     The n-sized problems may be classified as:
Linear: n steps.
Polynomial: nk steps, for a fixed k.
Exponential: 10n steps.
Pressburger Algebra: finite but beyond exponential.
Undecidable: Gödel's theorem.
For all intents and purposes, all beyond-polynomial problems are effectively insoluble for large n in the foreseeable future. Finally, according to Gödel's theorem, every mathematical system of any complexity (including ordinary arithmetic, ordinary geometry, ordinary set theory, etc.) contains some true but undecidable statements, i.e., statements with infinite computational steps.

     There is a very large class of mathematical problems, known as non-polynomial (NP) complete problems, which are known to be at most exponential, for which no polynomial solution is known, but for which there is no proof that a polynomial solution exists. This problem-class includes various traveling problems, airline scheduling problems, decision-tree problems, computer wiring problems, etc. If a polynomial solution is found for one, it will apply (with some modifications) to all. This problem has been an active research area for decades. (I gave up on it thirty years ago [Moore, 1971].) Sir Andrew Wiles, fresh from his triumph on Fermat's Last Theorem, has set his considerable mathematical gifts to work on this problem. Trouble is, if Sir Andrew finds that a polynomial solution is NOT possible (which is what I anticipate), then computer programmers and their clients will be no better off.

     Why the Sieve of Eratosthenes takes so long to solve.
SYMBOLIC LOGIC.
     George Boole [18xx-18xx], Irish mathematician and philosopher, usually gets the credit for inventing SYMBOLIC LOGIC [Boole]. However, Gottlob Leibniz [16xx-16xx] conceived of a RATIOCINATOR UNIVERSALIS, or universal language of discourse, but got stuck on the wrong definition of LOGICAL-OR. (Leibniz used EXCLUSIVE-OR, i.e., x OR y, but NOT both; he should have used INCLUSIVE-OR, i.e., x OR y OR BOTH.)

     The hook for symbolic logic is that you can set up equations like addition and multiplication, using true-false statements instead of whole numbers. Using 1=true and 0=false, LOGICAL-OR behaves almost like addition, and LOGICAL-AND behaves almost like multiplication, as follows: 
 true AND true = true.
   1  × 1  =  1.

 
 true AND false = false.
   1  × 0  =  1.

 
 false AND true = false.
   0  × 1  =  1.

 
 false AND false = false.
   0  × 0  =  0.

 
 true OR true = true.
   1  +   1  =  1.

 
 true OR false = true.
   1  +   0  =  1.

 
 false OR true = true.
   0  +   1  =  1.

 
 false OR false = false.
   0  +   0  =  0.

GOEDEL'S PROOF.
TURING MACHINE.
VON NEUMANN COMPUTER.
    
FRACTALS.
     Everybody thinks that Benoit Mandelbrot invented FRACTALS. He certainly deserves credit for popularizing the idea; unifying the literature from diverse sources; and contributing massively to the field himself. However, nineteenth century mathematicians (and, eventually, Euler [17xx-17xx]) clearly understood the underlying ideas of fractals [Lauwerier].

    
HILBERT'S TENTH PROBLEM.
     Matiyasevic's demonstration that Hilbert's Tenth Problem is insoluble [Davis].
FERMAT'S LAST THEOREM.
     Pierre de Fermat was a seventeenth century French attorney and civil servant, who occupied his leisure time with mathematics. He was very secretive with his proofs, and shared them with colleagues only when he could use them to demonstrate his own superiority as a mathematician. Fermat's Last Theorem is the name given to Diophantus's conjecture, with Fermat's infamous inscription:
Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.
The conjecture is:
ak + bk = ck ,
where a, b, c, and k, are all integers. Fermat's Last Theorem is the assertion that there are no solutions where k > 2.
RIEMANN HYPOTHESIS.
     The RIEMANN HYPOTHESIS is an unsolved conjecture of mathematics which asserts a fundamental, almost mysterious relationship between imaginary numbers and prime numbers. If you come up with a proof (or a counterexample) for this hypothesis, then you can apply for a million-dollar prize from the Courant Institute of Mathematics in New York.

REFERENCES.

     1. Courant R, Robbins H, Stewart I.
What is Mathematics? An Elementary Approach to Ideas and Methods. Second Edition.
Oxford: Oxford University Press. 1996.

     2. Lauwerier H.
Fractals. Endlessly Repeated Geometric Figures.
Princeton, NJ: Princeton University Press. Princeton Science Library. 1991.

      3. Schneier B.
Applied Cryptography, Second Edition. Protocols, Algorithms, and Source Code in C.
New York: John Wiley & Sons, 1996.

     4. Polanyi M. Personal Knowledge.

     5. Archimedes. The Sand Reckoner. In: Loeb Classical Library. Greek Mathematical Works: Thales to Euclid. Translated by Ivor Thomas.

     6. Bell ET. Men of Mathematics.

     7. Seife C.
Zero. The Biography of a Dangerous Idea.
London: Penguin Books. 2000.
ISBN: 0-670-88457-X, 248 pages.
This book includes an account of the execution of Hippasus of Metapontum, a member of the Pythagorean cult, who had dared to reveal the existence of irrational numbers to persons outside the cult.

     8. Abbott E. Flatland. 1884.

     9. Stewart I.
Flatterland. Like Flatland. Only More So.
Cambridge, MA: Perseus Publishing. 2001.
ISBN 0-7382-0442-0, 301 pages.
This book is a sequel of Edwin A. Abbott's FLATLAND, published in 1884, and cited in Stephen Hawking's A BRIEF HISTORY OF TIME.

     10. Casti JL, DePauli W.
Gödel. A Life of Logic.
Cambridge, MA: Perseus Publishing. 2000.
ISBN 0-7382-0274-6, 210 pages.

     11. Aleksandr I, Morton H.
An Introduction to Neural Computing. Second Edition.
London: International Thomson Computer Press. 1995.
ISBN 1-85032-167-1, 284 pages.

     12. Scarborough D, Sternberg S.
Methods, Models, and Conceptual Issues. An Invitation to Cognitive Science. Volume 4.
Cambridge, MA: MIT Press. 1998.
ISBN 0-262-65946-0, 950 pages.

     13. Changeux J-P, Connes A.
Conversations on Mind, Matter, and Mathematics
Ed & Transl: DeBevoise MB. Princeton, NJ: Princeton University Press. 1995.
ISBN 0-691-08759-8, 260 pages.

     14. Hockey S.
A Guide to Computer Applications in the Humanities.
Chapter 8. Sound Patterns. pp.168-188.
Baltimore: The Johns Hopkins Univ Press. 1980.
ISBN 0-8018-2891-0, 248 pages.
Cited: Ott W. Metrical Analysis of Latin Hexameter by Computer. Revue 4:7-24, 1966.
Cited: Greenberg NA. Scansion Purement Automatique de l'Hexamère Dactylique. Revue 1967;3:1-25.
A computer program successfully scanned 95% of hexameters of Virgil's Aeneid, by recognizing the usual conventions for long and short vowels, as well as elisions, such as "-que" before a vowel. In 3-4% of sentences, more than one scansion was proposed by the computer program, and in 1-2% of sentences, the scansion was abandoned by the computer program, and it was determined that Virgil had not obeyed the rules. In about 5% of lines with an equivocal scansion, it was determined that the equivocal vowel-weight (such as a first-declension nominative (short) versus ablative (long)) had to be determined from the semantic context.
Analyses of Homer's Odyssey and Aristotle's Nicomachean and Eudemean Ethics and Plato's Seventh Letter and Apology are also discussed in this book. An early analysis of 440 lines from Homer's Odyssey found eight false scansions, but manual analysis of these lines revealed, as you noted for Aeschylus, that in these lines there were special semantic circumstances that allowed the usual scansion rules to be "relaxed.

     15. Woodger JH.
The Axiomatic Method in Biology.
Out of Print.

     16. Woodger JH.
Techniques of Theory Construction.
Out of Print.

     17. Cate FH.
Privacy in the Information Age.
Washington: Brookings Institution Press. 1997.
ISBN 0-8157-1316-9

     18. Davis M.
Computability and Unsolvability.
New York: Dover Publications, Inc. 1958.
ISBN 0-486-61471-9, 248 pages.
A short description of the major issues in the field of Computability and Unsolvability. A nice appendix, with a review of the major theorems of Number theory, and Matiyasevic's demonstration that Hilbert's Tenth Problem is insoluble. Good reference section.

     19. Farmer R, Miller D, Lawrenson R.
Lecture Notes on Epidemiology and Public Health Medicine. Fourth Edition.
Oxford: Blackwell Science. 1996.
ISBN 0-86542-611-2, 288 pages.

     20. Orwant J, Hietaniemi J, Macdonald J.
Mastering Algorithms with Perl.
Cambridge: O'Reilly. 1999.
ISBN 1-56592-398-7, 684 pages.


     21. Greek Mathematics.
Goold GP, ed. Thomas I transl. Loeb Classical Library. #335. Cambridge, MA: Harvard University Press. 1939.
ISBN 0-674-99369-1, 511 pages.
The Loeb formula of facing page translations, Greek and English. Includes: Pythagoras, Thales, Plato, Aristotle, Euclid.

     22. Hippocrates.
Hippocrates. Volume I.
Jones WHS, transl. Loeb Classical Library. Cambridge, MA: Harvard University Press. 1923.
ISBN 0-674-99162-1, 361 pages.
Includes Hippocrates' Oath, with explanatory notes.

     23. Singh S.
Fermat's Enigma. The Epic Quest to Solve the World's Greatest Mathematical Problem.
New York: Anchor Books. A Division of Random House, Inc. 1997.
ISBN 0-385-49362-2, 315 pages.


     24. Giere W.
Dokumentation und Informationsaufbereitung fuer den Arzt. Beitraege zur Medizischen Informatik.
Kirsten K, ed. Darmstadt: Epilog Verlag. 1996.
ISBN 3-9803214-7-9, 437 pages.
Collected works of Germany's foremost medical informatician.

     25. Hofstadter DR.
Gödel, Escher, Bach. An Eternal Golden Braid.
New York: Basic Books. A member of the Perseus Books Group. 1979.
ISBN 0-465-02656-7, 777 pages.
Pulitzer Prize-Winning book, that introduced the general educated public to Gödel, Time Magazine's mathematician of the twentieth century. A really dynamite set of references, including notes, in the field of mathematical logic.

     26. Watson F.
India. A Concise History.
New York: Thames and Hudson. 1974.
ISBN 0-500-027164-X, 192 pages.
A quick coverage of Indian history, that shows the early communications between the Mediterranean culture and Indian culture through the excursions of Alexander the Great. The cultivation of zero in India. The achievements of Islamic mathematicians. The British raj began slowly, with the Brits first using Farsi as the language of discourse! The Brits taught the Indians about their own history, and in doing so, awakened the Indians to their own heritage, and sowed the seeds of revolt.

     27. Andrews GL.
Number Theory.
New York: Dover Publications, Inc. 1971.
ISBN 0-486-68252-8, 259 pages.


     28. Croxton FE.
Elementary Statistics. With Applications in Medicine and the Biological Sciences.
New York: Dover Publications, Inc. 1953.
376 pages.


     29. Eco U.
A Theory of Semiotics.
Bloomington, IN: A Midland Book. Indiana University Press. 1979.
ISBN 0-253-35955-4, 354 pages.
Serious academic work by the beloved author of The Name of the Rose.

     30. Ulam SM.
Adventures of a Mathematician.
Berkley, CA: University of California Press. 1991.
ISBN 0-52007154-9, 329 pages.
My favorite chapter is chapter 15, random reflections, in which much of the lore of mathematics is discussed.

     31. Bernstein PL.
Against the Gods. The Remarkable Story of Risk.
New York: John Wiley & Sons, Inc. 1996.
ISBN 0-471-29563-9, 383 pages.
A fantastic excursion through the history of probability and chance, starting with the ancient Egyptians and ending with modern worldwide business practices. Probability was originally studied in order to INCREASE BENEFITS, as in winning at gambling or staying alive longer. Now, probability has its most important applications

     32. DeCew JW.
In Pursuit of Privacy. Law, Ethics, and the Rise of Technology.
Ithaca, NY: Cornell University Press. 1997.
ISBN 0-8014-3380-0, 199 pages.


     33. Singh S.
Fermat's Enigma. The Epic Quest to Solve the World's Greatest Mathematical Problem.
New York: Anchor Books. A Division of Random House, Inc. 1997.
ISBN 0-385-49362-2, 315 pages.
The on-again, off-again proof by Sir Andrew Wiles of Fermat's Last Theorem. A brief run-through of the history of Diophantus's famous conjecture, with Fermat's infamous inscription:
Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.
The conjecture is:
ak + bk = ck ,
where a, b, c, and k, are all integers. There are no solutions where k > 2.

     34. Chametzky RA.
A Theory of Phrase Markers and the Extended Base.
Albany: State University of New York Press. 1996.
ISBN 0-7914-2972-5, 206 pages.

     35. Scarborough D, Sternberg S.
An Invitation to Cognitive Science. Second Edition. Methods, Models, and Conceptual Issues. Volume 4.
Cambridge, MA: A Bradford Book. The MIT Press. 1998.
ISBN 0-262-65046-0, 950 pages.

     36. Croxton FE.
Elementary Statistics with Applications. in Medicine and the Biological Sciences.
New York: Dover Publications, Inc. 1953.

     37. Murphy EA.
A Companion to Medical Statistics.
Baltimore: The Johns Hopkins University Press. 1985.

     38. Hardy GH, Wright EM.
An Introduction to the Theory of Numbers. Fourth Edition.
Oxford: Oxford University Press. 1960.

     39. Tarjan.
Computational Complexity.
Murray Hill, NJ: Bell Laboratories. 19xx.

     40. Hawking SF.
A brief history of time.
Cambridge: Cambridge University Press.

     41. Knuth D.
The Art of Computer Programming. Volume 1.
Reading, MA: Addison-Wesley Publishing Co. 1973.

     42. Knuth D.
The Art of Computer Programming. Volume 2.
Reading, MA: Addison-Wesley Publishing Co. 1973.

     43. Knuth D.
The Art of Computer Programming. Volume 3. Searching and Sorting.
Reading, MA: Addison-Wesley Publishing Co. 1973.

     44. Boole G.
An Investigation of the Laws of Thought. On which are founded the Mathematical Theories of Logic and Probabilities.
New York: Dover Publications, Inc. 1954.

     45. Lewis CI, Langford CH.
Symbolic Logic. Second Edition.
New York: Dover Publications, Inc. 1932.

     46. Aristotle.
Nichomachaean Ethics. Book One. Chapter Six.
"Amicus Plato, sed magis amica veritas." Quoted in Tymoczko.

     47. Tymoczko T.
New Directions in the Philosophy of Mathematics.
Princeton, NJ: Princeton University Press. 1998.
Foundations. Plato's Ideals. Hilbert's Formalism. Brouwer's Intuitionism. Quasi-emperical mathematics. "Gödel was the last, great Platonist."

     48. The Holy Bible. Containing the Old and New Testaments. Set forth in 1611 and commonly known as the King James Version.
New York: American Bible Society.

     49. Aquinas T.
Summa Theologica.
Kreeft P, ed. A Summa of the Summa. The Essential Philosophical Passages of St. Thomas Aquinas' Summa Theologica. Edited and Explained for Beginners. San Francisco: Ignatius Press. 1925.


Last modified: 8/4/2001, by G. William Moore, MD, PhD.