Philosophy of Mathematics

Philosophy of Mathematics is an active branch of Philosophy addressing questions about the character of Mathematics, the conduct of mathematical inquiry, and the role of mathematical objects in describing empirical phenomena. As a form of philosophical inquiry, it examines the record of mathematical inquiry and poses questions regarding its aims, its conduct, and its results. Although the questions are diverse and never-ending, a number of recurrent themes can be recognized:

1. What are the sources of mathematical subject matter?
2. What does it mean to refer to a mathematical object?
3. What is the character of a mathematical proposition?
4. What kinds of inquiry play a role in mathematics?
5. What are the objectives of mathematical inquiry?
6. What gives mathematics its grip on experience?
7. What is the bearing of beauty on mathematics?

The terms “philosophy of mathematics’” and ’“mathematical philosophy’” are not synonyms. The former refers to the study of the role and meaning of numbers and mathematical concepts, while the latter refers to the “doing” of actual theorems and various proofs in Mathematics and Meta-Logic.




This is a process of abstraction that produces empirically bound concepts and formally free concepts in tandem, and that brings about a threefold relation among contingent experiences, concepts of physical space, and concepts of mathematical space. Achieving a more thorough understanding of this process, by which mathematical patterns are abstracted from concrete experience, developed as quasi-autonomous forms, and then applied back to experience in far-reaching and surprising ways, is one of the essential services that philosophical examination can perform for the benefit of mathematical thought.

Mathematical propositions, at least at first sight, appear to differ from other sorts of propositions, but in ways that have, historically speaking, been difficult to define precisely. One distinctive feature of mathematical propositions is, as Hilary Putnam sketched a common view of it, “the very wide variety of equivalent formulations that they possess”, by which he does not mean the sheer number of ways of saying the same thing but “rather that in mathematics the number of ways of expressing what is in some sense the same fact (if the proposition is true) while apparently not talking about the same objects is especially striking” (Putnam, 170).

Related Approaches

Realism

Mathematical Realism, like Realism in general, holds that mathematical entities exist independently of the human Mind. Thus humans do not invent Mathematics, but rather discover it, and any other intelligent beings in the Universe would presumably do the same. In this point of view, there is really one one sort of Mathematics that can be discovered: Triangles, for example, are real entities, not our creations.

Platonism

Platonism is the form of Realism suggesting mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the naive view most people have of numbers. The term Platonism is used because such a view is seen to parallel Plato’s belief in a “World of Ideas”, an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato’s view partly came from Pythagoras, and his followers the Pythagoreans, who believed that the world was, quite literally, built up by the numbers.

Logicism

Logicism is the provable thesis that Mathematics is reducible to Logic, and hence a branch of Logic (Carnap 1931/1883, 41). Logicists hold that Mathematics can be known a priori, but suggest that our knowledge of Mathematics is just part of our knowledge of Logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, Logic is the proper foundation of Mathematics, and all mathematical statements are necessary logical truths.

Empiricism

Empiricism is a form of Realism that denies Mathematics can be known a priori at all. It says we discover mathematical facts by empirical research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th Century, but primarily arose in the middle of the century. This argues statements like “2 + 2 = 4” come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.

Formalism

Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the “game” of Euclidean Geometry (which is seen as consisting of some strings called “axioms”, and some “rules of inference” to generate new strings from given ones), one can prove that the Pythagorean Theorem holds (that is, you can generate the string corresponding to the Pythagorean Theorem). Mathematical truths are not about numbers and sets and triangles and the like ? in fact, they aren’t “about” anything at all. Thus, Formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold.

Intuitionism

Intuitionism is a program of methodological reform whose motto is that “there are no non-experienced mathematical truths” (L.E.J._Brouwer). From this springboard, intuitionists seek to reconstruct what they consider to be the “corrigible” portion of Mathematics in accordance with concepts of constructibility, provability, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects. (CDP, 542)

Constructivism

Like Intuitionism, Constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, Mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction.

Fictionalism

Fictionalism was introduced in 1980 when Hartry Field rejected and in fact reversed Quine’s indispensability argument, that Mathematics was indispensable for our best scientific theories, and therefore should be accepted as true. Field suggested that Mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian Mechanics that didn’t reference numbers or functions at all. He started with the “betweenness” axioms of Hilbert’s Geometry to characterize Space without coordinatizing it, and then added extra relations between points to do the work formerly done with Vector Fields.

References

• Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181?531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
• Audi,_Robert (ed., 1999), The Cambridge Dictionary of Philosophy, Cambridge University Press, Cambridge, UK, 1995. 2nd edition, 1999. Cited as CDP.
• Benacerraf,_Paul, and Putnam,_Hilary (eds., 1983), Philosophy of Mathematics, Selected Readings, 1st edition, Prentice?Hall, Englewood Cliffs, NJ, 1964. 2nd edition, Cambridge University Press, Cambridge, UK, 1983.
• Berkeley,_George (1734), The Analyst; or, a Discourse Addressed to an Infidel Mathematician. Wherein It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith, London & Dublin. Online text, David R. Wilkins (ed.), Eprint.
• Bourbaki,_N. (1994), Elements of the History of Mathematics, John Meldrum (trans.), Springer-Verlag, Berlin, Germany.
• Boyer,_Carl_B. (1991), A History of Mathematics, 1st edition, 1968. 2nd edition, Uta C. Merzbach (ed.), Isaac Asimov (foreword), John Wiley and Sons, New York, NY.
• Carnap,_Rudolf (1931), “Die logizistische Grundlegung der Mathematik”, Erkenntnis 2, 91?121. Republished, “The Logicist Foundations of Mathematics”, E. Putnam and G.J. Massey (trans.), in Benacerraf and Putnam (1964). Reprinted, pp. 41?52 in Benacerraf and Putnam (1983).
• Chandrasekhar,_Subrahmanyan (1987), Truth and Beauty. Aesthetics and Motivations in Science, University of Chicago Press, Chicago, IL.
• Hadamard,_Jacques (1949), The Psychology of Invention in the Mathematical Field, 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954.
• Hardy,_G.H. (1940), A Mathematician’s Apology, 1st published, 1940. Reprinted, C.P. Snow (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.
• Hart,_W.D. (ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
• Huntley,_H.E. (1970), The Divine Proportion: A Study in Mathematical Beauty, Dover Publications, New York, NY.
• Kleene,_S.C. (1971), Introduction to Metamathematics, North?Holland Publishing Company, Amsterdam, Netherlands.
• Klein,_Jacob (1968), Greek Mathematical Thought and the Origin of Algebra, Eva Brann (trans.), MIT Press, Cambridge, MA, 1968. Reprinted, Dover Publications, Mineola, NY, 1992.
• Kline,_Morris (1959), Mathematics and the Physical World, Thomas Y. Crowell Company, New York, NY, 1959. Reprinted, Dover Publications, Mineola, NY, 1981.
• Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, NY.
• K?,_Julius_(Gyula) (1905), “&1826;er die Grundlagen der Mengenlehre und das Kontinuumproblem”, Mathematische Annalen 61, 156?160. Reprinted, “On the Foundations of Set Theory and the Continuum Problem”, Stefan Bauer-Mengelberg (trans.), pp. 145?149 in Jean van Heijenoort (ed., 1967).
• Leibniz,_G.W., Logical Papers (1666?1690), G.H.R. Parkinson (ed., trans.), Oxford University Press, London, UK, 1966.
• Mac_Lane,_Saunders (1998), Categories for the Working Mathematician, 1st edition, Springer-Verlag, New York, NY, 1971. 2nd edition, Springer-Verlag, New York, NY.
• Maddy,_Penelope (1990), Realism in Mathematics, Oxford University Press, Oxford, UK.
• Maddy, Penelope (1997), Naturalism in Mathematics, , Oxford University Press, Oxford, UK.
• Maziarz, Edward A., and Greenwood, Thomas (1995), Greek Mathematical Philosophy, Barnes and Noble Books.
• Peirce,_Benjamin (1870), “Linear Associative Algebra”, ? 1. See American Journal of Mathematics 4 (1881).
• Peirce,_C.S., Collected Papers of Charles Sanders Peirce, vols. 1?6, Charles Hartshorne and Paul_Weiss (eds.), vols. 7?8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931?1935, 1958. Cited as CP (volume).(paragraph).
• Peirce, C.S., Writings of Charles S. Peirce, A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN, 1981?. Cited as CE (volume), (page).
• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9 (1870), 317?378. Reprinted (CP 3.45?149), (CE 2, 359?429).
• Peirce, C.S. (c. 1896), “The Logic of Mathematics; An Attempt to Develop My Categories from Within”, 1st published as CP 1.417?519 in Collected Papers.
• Peirce, C.S. (1902), “The Simplest Mathematics”, MS dated January?February 1902, intended as Chapter 3 of the “Minute Logic”, CP 4.227?323 in Collected Papers.
• Plato, “The Republic, Volume 1”, Paul Shorey (trans.), pp. 1?535 in Plato, Volume 5, Loeb Classical Library, William Heinemann, London, UK, 1930.
• Plato, “The Republic, Volume 2”, Paul Shorey (trans.), pp. 1?521 in Plato, Volume 6, Loeb Classical Library, William Heinemann, London, UK, 1935.
• Putnam,_Hilary (1967), “Mathematics Without Foundations”, Journal of Philosophy 64/1, 5?22. Reprinted, pp. 168?184 in W.D. Hart (ed., 1996).
• Robinson,_Gilbert_de_B. (1959), The Foundations of Geometry, University of Toronto Press, Toronto, Canada, 1940, 1946, 1952, 4th edition 1959.
• Russell,_Bertrand (1919), Introduction to Mathematical Philosophy, George Allen and Unwin, London, UK. Reprinted, John G. Slater (intro.), Routledge, London, UK, 1993.
• Smullyan,_Raymond_M. (1993), Recursion Theory for Metamathematics, Oxford University Press, Oxford, UK.
• Steiner,_Mark (1998), The Applicability of Mathematics as a Philosophical Problem, Harvard University Press, Cambridge, MA.
• Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras, Berkeley Hills Books, Berkeley, CA.
• Styazhkin,_N.I. (1969), History of Mathematical Logic from Leibniz to Peano, MIT Press, Cambridge, MA.
• Tait,_W.W. (1986), “Truth and Proof: The Platonism of Mathematics”, Synthese 69 (1986), 341?370. Reprinted, pp. 142?167 in W.D. Hart (ed., 1996).
• Tarski,_A. (1983), Logic, Semantics, Metamathematics: Papers from 1923 to 1938, J.H. Woodger (trans.), Oxford University Press, Oxford, UK, 1956. 2nd edition, John Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.
• Tymoczko,_Thomas (1998), New Directions in the Philosophy of Mathematics, Catalog entry?
• Ulam,_S.M. (1990), Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators, A.R. Bednarek and Fran&31721;se Ulam (eds.), University of California Press, Berkeley, CA.
• van_Heijenoort,_Jean (ed. 1967), From Frege To G?: A Source Book in Mathematical Logic, 1879?1931, Harvard University Press, Cambridge, MA.
• Wigner,_Eugene (1960), “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Communications in Pure and Applied Mathematics 13. Eprint
• Wilder,_Raymond_L. (1952), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY.

Further Reading

• Colyvan, Mark (2004), “Indispensability Arguments in the Philosophy of Mathematics”, Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
• Davis,_Philip_J. and Hersh,_Reuben (1981), The Mathematical Experience, Mariner Books, New York, NY.
• Devlin,_Keith (2005), The Math Instinct: Why You’re a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs), Thunder’s Mouth Press, New York, NY.
• Dummett,_Michael (1991 a), Frege, Philosophy of Mathematics, Harvard University Press, Cambridge, MA.
• Dummett, Michael (1991 b), Frege and Other Philosophers, Oxford University Press, Oxford, UK.
• Dummett, Michael (1993), Origins of Analytical Philosophy, Harvard University Press, Cambridge, MA.
• Ernest,_Paul (1998), Social Constructivism as a Philosophy of Mathematics, State University of New York Press, Albany, NY.
• George,_Alexandre (ed., 1994), Mathematics and Mind, Oxford University Press, Oxford, UK.
• Kline,_Morris (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, NY.
• Lakoff,_George, and Rafael E. N?N? Rafael E. (2000), Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Basic Books, New York, NY.
• Peirce,_C.S.,_Bibliography.
• Raymond, Eric S. (1993), “The Utility of Mathematics”, Eprint.
• Shapiro,_Stewart (2000), Thinking About Mathematics: The Philosophy of Mathematics, Oxford University Press, Oxford, UK.

Journals

• Philosophia Mathematica journal
• The Philosophy of Mathematics Education Journal homepage

See Also

• Logic
• Philosophy of Language
• Philosophy of Science
• Scientific Method
• Set Theory
• Truth
• Truth Theory
• History of Philosophy

External Links

• The Philosophy of Real Mathematics (Weblog).
• Jones, R.B., Philosophy of Mathematics (Webpage).
• Tymoczko,_Thomas (1998), New Directions in the Philosophy of Mathematics, Princeton University Press, Princeton, NJ. Link.
• Philosophia Mathematica (Journal)
• Google Directory Category : Philosophy of Mathematics.
• Open Directory Project : Philosophy of Mathematics.

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