Magazine article Phi Kappa Phi Forum

Proof and Truth in Mathematics

Magazine article Phi Kappa Phi Forum

Proof and Truth in Mathematics

Article excerpt

The idea of "proof" is the guiding light of mathematics. No matter how many examples you can give for the reality of your theorem, if you cannot offer a valid proof, then your theorem is merely a conjecture. After confidently mastering elementary algebra, most students are taken aback by the difference between algebra and geometry. Elementary algebra is taught as a tautology. Expressions are set equal to each other, and these equations are manipulated to obtain the desired reduction in complexity, such as finding the value of a variable (equating the variable to a number). Geometry, however, is an exercise in logic. In a geometric proof, each step follows logically from the others, and there is a chain of truth that extends from beginning to end.

Euclidian geometry as truth was an underpinning of mathematics until the nineteenth century, when mathematicians found that it was based on a flawed axiom. That axiom, the "Parallel Postulate," states that if you have a line and a point not on the line, only one line can be drawn through the point parallel to the first line. A trio of mathematicians--Lobachevsky, Bolyai, and Riemann--showed that there exists, in one case, more than one line; or in another, no such line. The truth of Euclidian geometry was either destroyed or transformed into three new truths, depending on your mood. Mathematicians preferred the three truths to no truth, and mathematical life went on.

Bertrand Russell continued the assault on mathematical truth in the twentieth century. Russell is famous for the nearly 2,000-page tome, Principia Mathematica, coauthored with Alfred North Whitehead, in which he attempted to reduce all mathematics to a form of logic. Russell used logic in the form of a paradox as his weapon against truth. Russell's Paradox, outlined in a letter to fellow mathematician Gottlob Frege, has an analogy in the statement by Epimenides, a Cretan, that "All Cretans are liars." Russell's mathematical statement of this paradox implied that there could be no truth in mathematics, since mathematical logic was flawed at a basic level.

This logical assault on mathematical truth continued in the work of Kurt Godel, an esteemed associate of Albert Einstein. Godel's Incompleteness Theorem, popularized by Douglas R. Hofstadter in his book, Godel, Escher, Bach: An Eternal Golden Braid (Basic Books, New York, 1979), caused quite a stir at its publication. This theorem states that certain statements in mathematics exist in a shadow world in which they are neither true nor false; they are "undecidable." The consequence of this is that there is a fundamental uncertainty in mathematics.

Godel's Incompleteness Theorem is to mathematics what the Heisenberg Uncertainty Principle is to physics. The Heisenberg Uncertainty Principle, published at about the same time as Godel's Incompleteness Theorem, states that some things in the physical world cannot, in principle, be known. Many physicists, Einstein included, were not convinced that, in effect, some things Man was not meant to know. Today, the Heisenberg Uncertainty Principle is taken as fact, and it is even a useful tool. Likewise, mathematicians have chosen to live with incompleteness. Gregory J. Chaitin of IBM has stated that Godel's Theorem "has had no lasting impact on the daily lives of mathematicians or on their working habits; no one loses sleep over it any more."

In 1950, Alan Turing, a founding father of computer science, proposed a test of machine intelligence that he called an "imitation game." This game is now called the "Turing Test," and the modern form has a person conversing with a computer program and guessing whether he or she is chatting with a machine or a real person. Turing thought it was necessary for the computer program to occasionally answer some questions wrong, lest its perfection prove it was not human. Mathematicians, of course, are human, and Andrew Hodges of Wadham College, Oxford, UK, remarks in the Stanford Encyclopedia of Philosophy, "Turing's post-war view was that mathematicians make mistakes, and so do not in fact see the truth infallibly. …

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