Get Real

1. Get Real the challenges of mathematical epistemology Jason Douma University of Sioux Falls November 18, 2003 presented to the SDSU Senior Seminar in Mathematics

2. What distinguishes mathematics from the usual “natural sciences?” Mathematics is not fundamentally empirical —it does not rely on sensory observation or instrumental measurement to determine what is true. Indeed, mathematical objects themselves cannot be observed at all!

3. Does this mean that mathematical objects are not real? Does this mean that mathematical knowledge is arbitrary? Good questions! These are the things that keep mathematical epistemologists awake at night.

4. The Question of Epistemology:an unreasonably concise history Through the 18th Century, an understanding that mathematics was in some way part of “natural philosophy” was widely accepted. In the 19th Century, several developments (non-Euclidean geometry, Cantor’s set theory, and—a little later—Russell’s paradox, to name a few) triggered a foundational crisis.

5. The Question of Epistemology:an unreasonably concise history The final decades of the 19th Century and first half of the 20th Century were marked by a heroic effort to make the body of mathematics axiomatically rigorous. During this time, competing epistemologies emerged, each with their own champions. After lying relatively dormant for half a century, these philosophical matters are now receiving renewed, as reflected by the Philosophy of Mathematics SIGMAA unveiled in January, 2003.

6. In the modern mathematical community, there is very little controversy over what it takes to show that something is “true”…this is what mathematical proof is all about. Most disagreements over this matter are questions of degree, not kind. (Exceptions: proofs by machine, probabilistic proof, and arguments from a few extreme fallibilists) However, when discussion turns to the meaning of such “truths” (that is, the nature of mathematical knowledge), genuine and substantial distinctions emerge.

7. Gabriel’s Horn Gabriel’s Horn can be gener-ated by rotating the curve over [1,∞) around the x-axis. As a solid of revolution, it has finite volume. As a surface of revolution, it has infinite area. Picture and equations generated by Mathematica.

8. The Peano-Hilbert Curve (from analysis) There exists a closed curve that completely fills a two-dimensional region. Image produced by Axel-Tobias Schreiner, Image produced by John Salmon Rochester Institute of Technology, and Michael Warren, Caltech “Programming Language Concepts,” “Parallel, Out-of-core methods for http://www.cs.rit.edu/~ats/plc-2002-2/html/skript.html N-body Simulation,” http://www.cacr.caltech.edu/~johns/pubs/siam97/html/online.html

9. A Theorem of J.P. Serre (from homotopy theory) If n is even, then is a finitely generated abelian group of rank 1.

10. The Platonist View Mathematical objects are real (albeit intangible) and independent of the mind that perceives them. Mathematical truth is timeless, waiting to be “discovered.” Pictures courtesy of the MacTutor History of Mathematics Archive, http://www-gap.dcs.st-and.ac.uk/~history/

11. The Formalist View Mathematical objects have no external meaning; they are structures that are formally postulated or formally defined within an axiomatic system. Mathematical truth refers only to consistency within the axiomatic system. Picture courtesy of the MacTutor History of Mathematics Archive, http://www-gap.dcs.st-and.ac.uk/~history/

12. The Intuitionist/Constructivist View Mathematical objects finitely derived from the integers have real meaning; the rest is mathematical fantasy. Appeal to the law of the excluded middle is not a valid step in a mathematical proof. Picture courtesy of the MacTutor History of Mathematics Archive, http://www-gap.dcs.st-and.ac.uk/~history/

13. The Empiricist and Pragmatist Views Mathematical objects have a necessary existence and meaning inasmuch as they are the underpinnings of the empirical sciences. The nature of a mathematical object is constrained by what we are able to observe (or comprehend). Picture courtesy of the Harvard University Department of Philosophy, http://www.fas.harvard.edu/~phildept/html/emereti.html

14. The Logicist View Mathematical objects are values taken on by logical variables. Mathematical truth is logical tautology. Picture courtesy of the MacTutor History of Mathematics Archive, http://www-gap.dcs.st-and.ac.uk/~history/

15. The Humanist View Mathematical objects are mental objects with reproducible properties. These objects and their properties (truths) are confirmed and understood through intuition, which itself is cultivated and normed by the practitioners of mathematics. Picture courtesy of the MacTutor History of Mathematics Archive, http://www-gap.dcs.st-and.ac.uk/~history/

16. Name that Epistemology: “I would say that mathematics is the science of skillful operations with concepts and rules invented for just this purpose. The principal emphasis is on the invention of concepts. ... The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible.” Eugene Wigner

17. Name that Epistemology: “Certain things we want to say in science may compel us to admit into the range of values of the variables of quantification not only physical objects but also classes and relations of them; also numbers, functions, and other objects of pure mathematics.” “To be is to be the value of a variable.” W.V. Quine

18. Name that Epistemology: “Mathematical knowledge isn’t infallible. Like science, mathematics can advance by making mistakes, correcting and recorrecting them. A proof is a conclusive argument that a proposed result follows from accepted theory. ‘Follows’ means the argument convinces qualified, skeptical mathematicians.” Reuben Hersh

19. Name that Epistemology: “Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science, which have been by slow degrees vouchsafed to man…all of which must have existed in that sublimely omniscient Mind from eternity.” Mary Somerville

20. Name that Epistemology: “Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true.” Kurt Gődel

21. Every Rose has its Thorn:a perfect epistemology is hard to find A Critique of Platonism: The Platonistic appeal to a separate realm of “pure ideas” sounds a lot like good ‘ol Cartesian dualism, and is apt to pay the same price for being unable to account for the integration of the two realms.

22. Every Rose has its Thorn:a perfect epistemology is hard to find A Critique of Formalism: Three words: Gődel’s Incompleteness Theorem. In any system rich enough to support the axioms of arithmetic, there will exist statements that bear a truth value, but can never be proved or disproved. Mathematics cannot prove its own consistency.

23. Every Rose has its Thorn:a perfect epistemology is hard to find A Critique of Intuitionism/Constructivism: Some notion of the continuum—such as our real number line—seems both plausible and almost universal, even among those not educated in modern mathematics. What’s more, the mathematics of the real numbers works in practical application.

24. Every Rose has its Thorn:a perfect epistemology is hard to find A Critique of Empiricism/Pragmatism: This doctrine tends to lead inexorably to the conclusion that “inconceivable implies impossible.” Yet history is filled with examples that were for centuries inconceivable but are now common knowledge. What’s more, mathematics provides us with objects that yet seem inconceivable, but are established to be mathematically possible.

25. Every Rose has its Thorn:a perfect epistemology is hard to find A Critique of Logicism: Attempts to reduce modern mathematics to logical tautologies have failed miserably in practice and may have been doomed from the start in principle. Common notion, local convention, and intuitive allusion all appear to obscure actual mathematics from strictly logical deduction.

26. Every Rose has its Thorn:a perfect epistemology is hard to find A Critique of Humanism: This view is pressed to explain the universality of mathematics. What about individuals, such as Ramanujan, who produced sophisticated results that were consistent with the systems used elsewhere, yet did not have the opportunity to “norm” their intuition against teachers or colleagues?

27. When assessing metaphysical or epistemological paradigms, it’s often helpful to compare the various paradigms against the “sticky wickets” to see which view is best able to make sense out of the puzzling case at hand. Let’s give it a whirl…

28. Gabriel’s Horn Gabriel’s Horn can be gener-ated by rotating the curve over [1,∞) around the x-axis. As a solid of revolution, it has finite volume. As a surface of revolution, it has infinite area. Picture and equations generated by Mathematica.

29. The Peano-Hilbert Curve (from analysis) There exists a closed curve that completely fills a two-dimensional region. Image produced by Axel-Tobias Schreiner, Image produced by John Salmon Rochester Institute of Technology, and Michael Warren, Caltech “Programming Language Concepts,” “Parallel, Out-of-core methods for http://www.cs.rit.edu/~ats/plc-2002-2/html/skript.html N-body Simulation,” http://www.cacr.caltech.edu/~johns/pubs/siam97/html/online.html

30. A Theorem of J.P. Serre (from homotopy theory) If n is even, then is a finitely generated abelian group of rank 1.

31. Whaddaya think?

32. A Brief Bibliography for the (amateur) Philosopher of Mathematics Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics, Prentice-Hall, 1964. Philip Davis and Reuben Hersh, The Mathematical Experience, Houghton Mifflin, 1981. Judith Grabiner, “Is Mathematical Truth Time-Dependent?”, American Mathematical Monthly 81: 354-365, 1974. Reuben Hersh, What is Mathematics, Really?, Oxford Press, 1997. George Lakoff and Rafael Nuñez, Where Mathematics Comes From, Basic Books, 2000. Edward Rothstein, Emblems of Mind, Avon Books, 1995.