MATHEMATICS: DISCOVERED OR CONSTRUCTED? Klaus Truemper Erik Jonsson School of Engineering and Computer Science University of Texas at Dallas Aussois, January 7–11, 2013 The talk was given during the 17th Combinatorial Workshop 2013 in Aussois, France. Since the slides by themselves are not sufficient to convey the content of the presentation, we have added the verbal comments from memory, using a smaller font as done here. We also have revised some slides to clarify points and correct errors, and have included some material that could not be discussed due to the one-hour time limit for the talk.
Mathematics: Discovered or Constructed? In 1978, J. Edmonds arranged that we could visit the University of Waterloo for one year. It was a terrific opportunity. During the day, we would work on mathematical problems at the university. At night when we would be too tired to continue, we started to study the writings of the philosopher L. Wittgen- stein. Our uncle F. H¨ ulster had done so years earlier and had written up his understanding of the books Tractatus Logico-Philosophicus , 1921, and Philosophische Untersuchungen , 1953, in two enlightening manuscripts. In 1980, back at our home institution, the University of Texas at Dallas, the topic of this talk came up in an informal discussion with one of our Ph.D. students. He argued that mathematics was discovered, with a reasoning that did not seem convincing. We made up our mind that we should look into the matter and study what others had said about the question. Aussois — January 7–11, 2013 2
Mathematics: Discovered or Constructed? We use “discovered” in the sense that an existing thing is detected, and “constructed” in the sense that a previously not existing thing is produced. Here are some examples. Discovered: Antarctica DNA double helix Constructed: Water pump of Archimedes Constitution of the USA Aussois — January 7–11, 2013 3
Mathematics: Discovered or Constructed? Let us look at some questions concerning discovery and construction. Specific questions: Are mathematical theorems discovered or constructed? 1 Are the complex numbers discovered or constructed? 2 Are the natural numbers discovered or constructed? 3 If we want to argue for discovery, then the task becomes progressively easier as we proceed from case 1 to case 3. Indeed, the items claimed to be discovered become simpler as we proceed down the list. On the other hand, proving construction becomes progressively harder. Aussois — January 7–11, 2013 4
Mathematics: Discovered or Constructed? General question: Are at least some parts of mathematics discovered, or is all of mathematics constructed? Compared with the previous questions, discovery is easiest to argue, and construction is hardest. No matter what the view, there is general agreement: Counting and the natural numbers are the start of arithmetic. Indeed, they can be viewed as the beginning of mathematics. There is a subtle point, though. Before we can count things such as apples, we must implicitly have the notion of equivalence classes. It is safe to assume that mankind learned that notion by evolution. For example, the people who didn’t recognize an approaching lion as dangerous if they had not seen that particular animal before, were simply eaten. Counting with the fingers and toes also implicitly requires the notion of equivalence classes. Aussois — January 7–11, 2013 5
General agreement: Two examples R. Dedekind ( Stetigkeit und irrationale Zahlen , 1872), a representative of the Axiomatic Method: “I view all of Arithmetic to be a necessary or at least natural result of the most elementary act, the counting. And counting is nothing but the successive creation of an infinite sequence of the positive whole numbers, in which each individual is defined from its immediate predecessor.” H. Weyl ( ¨ Uber die neue Grundlagenkrise der Mathematik , 1921), a representative of Intuitionism: “Mathematics starts with the series of natural numbers, that is, the law that from nothing creates the number 1 and from each already created number the successor.” The Axiomatic Method and Intuitionism are two opposing philosophies about mathematics. Over- simplifying, one might say that Intuitionism does not allow use of an element of a set based solely on a proof that the set is nonempty. The Axiomatic Method has no compunction about such use. C. Thiel of the University of Erlangen/N¨ urnberg pointed out to us that both statements contain some vagueness. Dedekind does not specify who does the counting, and Weyl does not say where the cited law comes from. Thiel says that both are “hedging their bets.” Aussois — January 7–11, 2013 6
A wide range of opinions about discovery vs. construction Over at least 2,400 years, a wide variety of opinions have been offered about discovery versus construction. Plato (424–348 BC, quoted by Euthydemos 290 BC, translation by C. Thiel): “Geometers, arithmeticians, and astronomers are in a sense seekers since they do not produce their figures and other symbols at will, but only explore what is already there.” Clearly Plato is in favor of discovery. The ideas of Plato dominated philosophical thought for centuries. Instead of examining examples so influenced, we jump right up to the 19th century. C.F. Gauss (letter to Bessel, 1830): “ . . . number is purely a product of our minds . . . ” Now here is a contrast. Since Gauss declares numbers to be constructed, he evidently believes that all of mathematics is obtained that way. Gauss was a deeply religious person, so his viewpoint about mathematics may seem surprising. But he felt that he could not draw any conclusion about the mysteries of his belief, and carefully separated facts and arguments about mathematics and the world from his religious beliefs. Aussois — January 7–11, 2013 7
A wide range of opinions about discovery vs. construction (cont’d) G. Cantor ( ¨ Uber unendliche, lineare Punktmannigfaltigkeiten , 1883): “ . . . the essence of mathematics just consists of its freedom.” Cantor caused an upheaval in mathematics by his investigation into infinite sets and the intro- duction of transfinite numbers. His friend J. P. G. L. Dirichlet reviewed Cantor’s papers prior to submission to a journal and advised him that publication would be detrimental to his career. Cantor went ahead anyway. Dirichlet’s prediction turned out to be correct, since Cantor never attained a position at a major university, something he surely deserved. L. Kronecker (quoted by Weber, 1893): “God made the integers; all else is the work of man.” Kronecker was a finitist, that is, he only accepted an existence proof if it involved a finite construc- tion starting with the natural numbers. For consistency, he could not advocate the construction of the entire set of integers by humans. Instead, he invoked a religious entity for the task. Aussois — January 7–11, 2013 8
A wide range of opinions about discovery vs. construction (cont’d) R. Dedekind ( Stetigkeit und irrationale Zahlen , 1872): “ . . . the negative and rational numbers are created by man . . . ” We have already seen another citation of the paper, where counting creates the natural numbers and it is not stated who does the counting. In the present case, creation by man is clearly asserted. The section of the paper that defines the (Dedekind) cut is titled “Creation of the irrational numbers.” Dedekind emphasizes that this creation process is an axiom. The (Dedekind) cut involves splitting the ordered rational numbers into two disjoint sets where all numbers in one set are smaller than the numbers in the other one. Dedekind then postulates that a number lies between the two sets. Carrying out this operation in all possible ways, he creates the real numbers. This process is unacceptable to an intuitionist since the mere fact that the two sets are disjoint is used to claim existence of a number between them. Worse yet, this is done in one fell swoop for all ways of splitting up the ordered rationals, with no prescription how this is to be accomplished. Aussois — January 7–11, 2013 9
A wide range of opinions about discovery vs. construction (cont’d) Here is a statement by B. Russell we read many years ago. We have not been able to find the reference, despite Google. If anybody has that information, please send it to us. B. Russell: “I cannot imagine a world where 1+1 is not equal to 2.” Russell’s statement baffled us for many years. It seemed impossible to find a contradiction to the implied claim that the first counting step is part of every world. We will return to the statement later. Next are two books. The first one implicitly votes for construction of the complex numbers, and the second one for discovery not just of the natural numbers, but also of the zero. The latter number is essential for implementation of the idea of place value, a powerful idea. A. Hodges (book One to Nine , 2007): “If complex numbers are so intrinsic to reality, why are we unaware of them?” C. Reid (book From Zero to Infinity , 1956): The Zero is the “first of the numbers, was the last to be discovered.” Aussois — January 7–11, 2013 10
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