Monica Clark

HON 213

Prof. Bryan Smith

27 April 2001

Truth and Proof in Mathematics and Philosophy

What in us really wants "truth"? Indeed we came to a long halt at the question about the cause of this will—until we finally came to a complete stop before a still more basic question. We asked about the value of this will. Suppose we want truth: why not rather unturth? and uncertainty? even ignorance?¹

One major difference between the disciplines of mathematics and philosophy on truth, proof and meaning, is mathematics’ assumption that there is a truth to study through rational abstractions. In philosophy, assumptions about the nature of truth itself are questioned, as German philosopher Friedrich Nietzsche artfully exclaimed in the above quotation. While most mathematicians strive to understand, explain and validate mathematical truths, philosophers question even the most basic assumptions in understanding truth (or untruth). However, philosophers, too, must return to the realm of explanation and validation to prove their arguments about the nature of understanding. In this paper I will compare mathematical and philosophical approaches to truth, proof and meaning. While mathematicians and philosophers share similar interests in developing and proving truth, they utilize different approaches developed from their distinct traditions.

Generally speaking, mathematics and philosophy pursue truth. "Truth" itself is an ambiguous term, which implies an actual and real conformity or fact.² As humans, we possess an ability to ask questions, and through affirmations and negations we recognize that there can, and thus arguably should, be a truth to explain all the unexplainable. This desire for the explainable creates a distinction between practical and theoretical truth. There are abstract theories about what "truth" means as a form of logical reason, evidence, or form and then a practical application of taking what is said to be "true" as the foundation of different tasks. Mathematics tends generally toward the practical form of truth, assuming the validity of truth, while philosophy is more concerned about the very nature of "truth" as a theory. The development of mathematics and philosophy, therefore, reflect their different motivations for interpreting "truth."

Mathematical truth is often built out of the assumptions of set theory and logic, which assume the idea of an "objective" or "detached" truth, i.e. a truth that transcends the human mind. For example, logic assumes that form and reason can account for valid expressions of what is true.³ So, logical arguments in set theory can be used to prove the existence of numbers and many mathematical assumptions, which are supposedly valid in logic alone and not culturally, socially, or subjectively constructed.⁴ Philosophy, on the other hand, debates and questions the very idea of "objective" truth. In contrast to Nietzsche’s scoff toward the blind search for "truth," Ben-Ami Scharfstein declares, "the point I want to emphasize here is that I believe whole-heartedly that there is an objective truth, and that an argument such as mine is a contribution, however modest, towards its attainment."⁵ The history of philosophy, therefore, has many approaches to truth including subjectivism, relativism and perspectivism each carrying different connotations about how to access truth. Subjectivism, initially started with the Sophists, is the contrast to objectivism, and brings into consideration the situation, perspective, and context toward defining truth.⁶ Relativism affirms "that in some areas at least, truth itself is relative to the standpoint of the judging subject."⁷ Perspectivism is Nietzsche’s term for truth determined by certain perspectives, and ultimately defined by the most accurate or elite perspective.⁸ So, while most mathematicians quickly turn toward debates about proving and validating the truths they discover, most philosophers continue to debate the very value of the claim of "truth."

Seeking truth requires a method, which in both mathematics and philosophy is found in arguments and proofs. Proofs are usually used to validate one’s assumed truth.⁹ In mathematics, formal proofs are often logical statements working in a preset theory, such as an axiomatic system, to validate the theorems and propositions as true or false. The use of a "proof" was originally developed by the Greeks Thales and Pythagoras to systematize mathematics and geometry.¹⁰ However, proofs can also take on a more flexible definition in the development of mathematical ideas. Imre Lakatos suggests a more flexible definition in his essay describing the history of mathematical ideas, "I propose to retain the time-honored technical term ‘proof’ for a thought-experiment – or ‘quasi-experiment’..."¹¹ Thus, Lakatos challenges the traditional use of proofs to validate truths by maintaining that "one does not prove what one has set out to prove," but instead builds an interpretation of truth along the way.¹² Technically, a mathematical proof is used to validate the system, although a variety of approaches to proofs have been developed. Thus a mathematical "proof" is used to validate mathematical arguments, but also reflects the historical development of mathematical attitudes toward proofs.

In philosophy, outside the strict logical tradition, arguments are used as more informal proofs to reflect the rational underpinnings in one’s conclusion. This more informal and lucid formation of proofs, then, allow for rhetoric, persuasion and psychology. In addition, most philosophers continue to stress the "reason" and "clarity" found in formal proofs.¹³ While reason and clarity may mirror the structure of a formal proof in mathematics, a philosophical argument is not focused solely on a theorem or proposition to be concluded, but is itself questionable. After all, philosophy does not maintain the clear-cut validity built into an axiomatic system. As Scharfstein indicates, a philosopher "lives in an intellectually competitive life, and to ask him to accept arguments without real resistance is like asking a chess player both to forego the pleasure of the game and to declare himself, all the same, to have been mated."¹⁴ Thus appealing to persuasion, philosophical arguments also maintain an essentially psychological element identified with each individual philosopher: "All the emotions, love, pride, possessiveness, jealously, shame, guilt, anger, hate, can come into play, all the attitudes that humans take toward one another can also be taken toward ideas."¹⁵ In philosophy, proof is created through a method of argument and debate that is very dependent on the human element, contrary to the formal systemization of a proof in mathematics.

Definitions of truth and proof, though, have varied throughout history in both mathematics and philosophy. As already suggested in Lakatos, the history of mathematics has reflected a less formal and objective truth than originally appears. Suggesting a similar idea, Ernst Snapper argues that the mathematical schools of Logicism, Intuitionism and Formalism have philosophical bases that have been developed "to give a firm foundation to mathematics."¹⁶ Hence one must necessarily recognize the historical background to modern definitions of truth and proof in mathematics.¹⁷

Similarly, philosophy has an intricate history of schools and ideas that has shaped the discipline that we know today: "In the sense of solutions arrived at, not only is its future unknown, but its past, too, is unknown."¹⁸ Each argument has developed along a path that is shrouded in psychology, history and circumstances and is continually challenged by future ideas. The development of different traditions including analytic and continental philosophy and the range of schools and branches such as metaphysics, logic, philosophy of language and ethics, reflect the vast range of ideas on truth and argument within philosophy.

Finally, there is also interplay between mathematics and philosophy. As Snapper and Lakatos argued, philosophy creates a necessary foundation for the systems developed in mathematics. Both mathematicians argue that mathematics is inextricably linked to philosophical foundations, which are necessary to validate proofs. Thus the philosophical arguments about truth inevitably affect the foundation of mathematics as well. If Nietzsche were to dissuade philosophical assumptions about truth, the foundation of mathematics would be arguably weakened. However, the history of philosophy and mathematics makes such questions about truth interesting, but not very threatening. Similarly, philosophical arguments reflect the rigor of mathematical proofs, especially seen in the 17^th and 18^th centuries when Descartes, Spinoza, Leibniz and other philosopher-mathematicians developed philosophical theories from mathematical foundations. Descartes claims that to put philosophy on a firm and stable foundation, "the only order which I could follow was that normally employed by geometers, namely to set out all the premises on which a desired proposition depends, before drawing any conclusions about it."¹⁹ Spinoza actually employs this axiomatic method by outlining Definitions, Axioms and Propositions in his Ethics.²⁰ However, as Keith Devlin summarizes, while mathematics is essential for developments in thought, "the existing techniques of logic and mathematics—indeed of the traditional scientific method in general—are inadequate for understanding the human mind."²¹ Philosophy, therefore, also requires the history of mathematics for insight into philosophy of mind, epistemology and logic. So, while each discipline remains distinct in its purpose, many of the ideas and methods are overlapped in their historical development.

In all, mathematics and philosophy share a similar interest in developing, proving, and maintaining truth, but through different approaches and traditions. "Truth" in philosophy is an idea to be debated, while in mathematics it is the foundation of proofs and theorems. The method of seeking truth in both philosophy and mathematics is through proof and argument. In mathematics, though, the axiomatic system has tightly bound proofs to logic, while philosophy leans toward psychological and rhetorical methods of proof and argument. The background of both disciplines strongly influences their general attitudes toward truth, proof and meaning, but each discipline also maintains diverse perspectives and a continued debate about these elements. I believe that both disciplines have much to learn from each other as philosophy requires the clarity and systemization of mathematics in order to apply reason to its arguments, while mathematics must constantly consult philosophy as a grounds to its assumptions. Without the dialogue between these disciplines, both share a similar potential of losing touch with reality and becoming too embroiled in their own abstract ideas of truth, proof and meaning.

Sources:

1 F. Nietzsche, Beyond Good and Evil in W. Kaufmann, ed., Basic Writings of Nietzsche (New York: The Modern Library, 1992), p. 199.

2 The American Heritage Dictionary of the English Language, Third Edition, www.dictionary.com (Houghton Miffin Company, 1996).

3 D. Cannon, "Formal Logic," (PHIL 261, Fall 2000).

4 A. Whitehead and B. Russell, Principia Mathematica (Cambridge: Cambridge University Press, 1962) and J. Derrida, Of Grammatology, trans. Gayatri Chakravorty Spivak (Baltimore: Johns Hopkins University Press, 1976).

5 B. Scharfstein, The Philosophers: Their Lives and the Nature of Their Thought (New York: Oxford University Press, 1980), p. 17.

6 S. Blackburn The Oxford Dictionary of Philosophy (Oxford: Oxford University Press, 1996), p. 365.

7 Ibid., p. 326.

8 Nietzsche, Beyond Good and Evil, 236.

9 I. Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery (Cambridge: Cambridge University Press, 1976). Lakatos gives a brief description of proofs in mathematics and logic, p. 106n.

10 M. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History (New York: W.H. Freeman and Company, 1993), p. 7.

11 Lakatos, p. 9.

12 Ibid., p. 41.

13 R. Watson, Writing Philosophy: A Guide to Professional Writing and Publishing (Carbondale: Southern Illinois University Press, 1992), p. 3-4.

14 Scharfstein, p. 5.

15 Ibid., p. 97.

16 E. Snapper, "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism," Mathematics Magazine 52 (Sept. 1979), 183.

17 See for example R. Lawlor, Sacred Geometry: Philosophy and Practice (New York: The Crossroad Publishing Company, 1982).

18 Scharfstein, p. 31.

19 R. Descartes, Meditations of First Philosophy in Central Readings in the History of Modern Philosophy: Descartes to Kant, ed. R. Cummins and D. Owen (Belmont: Wadsworth Publishing Company, 1999), p. 3.

20 B. Spinoza, Ethics in Central Readings in the History of Modern Philosophy: Descartes to Kant, ed. R. Cummins and D. Owen (Belmont: Wadsworth Publishing Company, 1999).

21 K. Devlin, Goodbye Descartes: The end of Logic and the Search for a New Cosmology of the Mind (New York: John Wiley & Sons, Inc., 1997), p. viii.