Truth | Research & Encyclopedia Articles

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Truth Summary

Truth

In their search for truth, many of the great philosophers have looked to mathematics as a paradigm for how such a search should be undertaken. The reason is that mathematicians have a very precise notion of what truth is within their discipline. A proposition in mathematics is true if and only if it has been rigorously proved according to the very stringent laws of logic. Not everything can be proved, of course; we must start somewhere by making a few definitions and assumptions, but logicians and mathematicians keep these to a bare minimum. Moreover, they assume only those things that any reasonable person would agree to. Beyond that, deductive mathematics is essentially about seeking the truth, or falsity, of propositions by proof. In no other discipline is the standard of truth so high. No proposition, no matter how obvious, can be accepted as true until it has been proved. A perfect example of this is Fermat's Last Theorem. The French mathematician Pierre de Fermat (1601-1665) had written in the margin of a book that he had a marvelous proof that for all integers greater than 2, the equation aⁿ+bⁿ=cⁿ has no integer solutions for a, b, and c. Unfortunately, Fermat said that his proof would not fit in the margin of the book. No one ever discovered Fermat's alleged proof and for the next 350 years, some of the greatest mathematicians of all time tried and failed to prove Fermat's Last Theorem. Nevertheless, some did show that the theorem was true in specific cases. By 1839, it had been proved for n=3,4,5, and 7, so the evidence for the truth of the theorem was starting to build. By 1970, Fermat's conjecture had been established for all n < 4003 by brute force with computers. In a court of law, such evidence would be considered proof beyond a reasonable doubt. If a scientist tested a hypothesis 4002 times and had it always confirmed, she would proclaim it a law of nature. But, unlike in the courtroom, "evidence" is not good enough in mathematics. Unlike in the experimental sciences, a finite number, however large, of confirmed hypotheses is not sufficient to make a claim for truth in mathematics. There must be a proof. The 1970s and 1980s saw the advent of the supercomputer and the confirmation of the Fermat hypothesis for n up to 4,000,000, but there was still no proof of the general case. Fermat's Last Theorem remained only a conjecture. In the late 80s, unbeknownst to anyone else, a quiet, reserved Princeton University mathematics professor named Andrew Wiles began a journey to immortality. Using only pen, paper, and his mind, Wiles spent the next seven years of his life working on a problem that had vexed him since childhood--Fermat's Last Theorem. Using existing mathematics and mathematics that he created especially for this problem, Wiles completed the proof of Fermat's Last Theorem in 1994. His proof could not have been Fermat's proof, although Wiles could also claim that his proof would not fit in the margin--it was 200 pages long and had to be checked and rechecked by other mathematicians before it was proclaimed a valid proof. Fermat's Last Theorem is now considered to be one of the truths of mathematics. The point here is that by 1990, there were most likely no mathematicians who did not believe that Fermat's Last Theorem was true; yet not a single one would proclaim it to be true until a correct proof had been given. This is the standard for truth in mathematics.

Given that mathematicians know clearly what truth is for their discipline, does mathematical truth have anything to do with truth outside of mathematics? Does the fact that Fermat's Last Theorem is true, have any relevance to anything in the physical world? Does mathematical truth imply truth about events in the real world? Physicists would most likely answer the last question in the affirmative, because it is difficult, if not impossible, to do physics without mathematics. All of the laws of physics are stated in the language of mathematics. The spectacular success of the calculus as a tool to study the physical world, suggests the possibility that truth in mathematics is related to truth in nature. On the other hand, many pure mathematicians claim that their work is only about mathematics, that the theorems they prove are true as a result of careful adherence to the laws of logic which have nothing to do with objects or events in nature. One school of mathematicians, the formalists, claim that the objects of mathematical discourse are nothing more than meaningless symbols which are moved about and strung together according to accepted logical inference patterns. They see mathematics as a "game" to be played by the rules of symbolic logic. In this view, mathematics is not "about" anything outside the game itself and the great usefulness of mathematics in physics is merely a fortuitous coincidence.

Another group of mathematicians, called intuitionists, say that the objects of mathematical study are ideas or "intuitions" in the minds of individual mathematicians. Intuitionists are notable for denying the so-called "Law of Excluded Middle", a fundamental rule of standard logic, which states that any statement is either true or false--there is no middle ground. Intuitionists say that there is a middle ground, namely, the set of propositions which are either as yet undecided or perhaps undecidable. Under this view, Fermat's Last Theorem is neither true nor false until a correct proof is presented, and true intuitionists cannot accept Andrew Wiles' proof because it is an indirect proof. Wiles' assumed that Fermat's Last Theorem was false and showed that this led to a contradiction of a theorem which had already been proved. Indirect proof, a staple of traditional mathematics, is based upon the Law of Contraposition, which states that the propositions "If p, then q" and "If not q, then not p" are equivalent. Thus under standard logic, if either of these propositions is true, so is the other one. The problem for the intuitionists is that the Law of Contraposition is equivalent to the Law of Excluded Middle, hence indirect proof is outlawed under intuitionist logic.

It is probably the case that most mathematicians are neither formalists nor intuitionists. Most tend to regard themselves as realists. Realism in mathematics is derived from Platonism, named for the Greek philosopher Plato (c.428-c.347 BC), who believed that the objects of the physical world were mere "shadows" of the real objects, which he called "Forms." So, according to this view, an earthly man is only a shadow, a poor copy, of the ideal man. For the realist mathematician, the objects of mathematics are much like Plato's Forms--they have an independent existence, and physical entities that are modeled after them are mere approximations. Thus, what we usually call the real world, the world studied by the physicist, is only a shadow of the true reality, which is mathematics. This allows the realist mathematician to say that whenever the physicist's experiments do not agree with the mathematics, it is because the physical universe can only be an approximation to the real universe of mathematics. The world studied by physics is rampant with experimental error, but no such error can occur in the realm of mathematics. There are no experiments in mathematics, only pure forms following the dictates of pure logic, which for the realist is the only realm in which it is possible to find truth with complete certainty.

In this discussion of truth in mathematics, it is only fitting to give the last word to the great mathematician and logician, Bertrand Russell (1872-1970). Russell was a member of yet another school of mathematicians called logicists because they claim that mathematics is actually derived from pure logic. Like the formalists, they see mathematics as the correct manipulation of symbols according to logical principles; but, unlike the formalists, they believe that these symbols and rules of inference have an independent existence that is part of the fabric of the universe. Mathematics is more than just a game for the logicists. Nevertheless, logicists say that within the realm of mathematics, neither the symbols nor the rules of inference have an interpretation related to the natural world. Any interpretation or application of mathematics is not mathematics. Russell and Alfred North Whitehead wrote a massive three-volume work called Principia Mathematica claiming to derive all of mathematics from pure logic. They and most of the mathematical community believed that they had succeeded until 1931, when the brilliant Austria logician Kurt Gödel (1906-1978) published a paper entitled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." In this paper, one of the most important of the 20th Century, Gödel showed that in any axiomatic system based on arithmetic, there will always be propositions that cannot be proved within the system. This came to be known as "Gödel's Undecidability Theorem" or "Gödel's Incompleteness Theorem." The theorem showed that such systems can be either consistent or complete, but not both. This shattered Russell's attempts to show that Principia Mathematica was both consistent and complete. Furthermore, it called into question the very concept of mathematical truth, leaving Russell to say that "mathematics is that subject in which we never know what we are talking about nor whether what we are saying is true."

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