The Philosophy of Mathematics is a specialization of Logic closely related to the Philosophy of Logic, which are all a part of Philosophy. Addressing questions about the character of Mathematics and “Number”, the conduct of mathematical inquiry, the Philosophy of Mathematics is focused on the role of mathematical objects in describing empirical phenomena. As a form of philosophical inquiry aligned with the Philosophy of Logic, the Philosophy of Mathematics examines the record of mathematical inquiry and poses questions regarding its aims, conduct, results. The terms “Philosophy of Mathematics'” and '“Mathematical Logic'” are not exactly synonyms, and not exactly not: 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 Logic, or “Meta-Logic”. These are all academic specializations which look at different aspects of the same fundamental things, and it is important to note that Mathematics is a Language of Logic.

Involved is a process of Abstraction that produces Concepts in a relation to Experiences, including Concepts of physical Space and mathematical Space. Achieving understanding of the process in which mathematical patterns are abstracted from concrete Experiences, developed as quasi-autonomous Forms, and then applied back to Experience in far-reaching and surprising ways, is one of the essential services philosophical examination can perform for the benefit of mathematical thought. In short, this defines the Philosophy of Mathematics.

Background

Realism

Mathematical Realism, like Realism in general, holds that mathematical entities exist independently of the human Mind. 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, and not just our creations.

Platonism

Platonism is a form of Realism suggesting mathematical entities are abstract, they have no spatiotemporal or causal properties, and they are eternal and unchanging. This is often claimed to be the naïve view most people have of Numbers. This is called Platonism because this view is parallel to Plato's belief in a “World of Ideas”, an unchanging Ultimate Reality which 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 Numbers.

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 discovered in any of the other Sciences. Statements like “2 + 2 = 4” are uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet. On this view, the logic of the mathematical proposition is irrelevant.

Logicism

Logicism refers to a provable thesis that Mathematics is actually reducible to Logic, and hence, Math is a branch of Logic, which comports with common sense if Numbers are not Ideal or Realistic Forms. Logicists hold that Mathematics can be known a priori, but our knowledge of Mathematics is just part of our knowledge of Logic in general, even if informal, and so it is “Analytic”, not requiring any special faculty of mathematical intuition. Logic, and thus, Philosophy, is the proper foundation of Mathematics, because no one has ever “seen” a “Number” in a literal sense. So, mathematical statements are necessary logical statements, and not vice-versa.

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 (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 to say, you can generate the string corresponding to the Pythagorean Theorem). Mathematical truths are not about Numbers and Sets and Triangles and the like, and in fact they aren't “about” anything at all. Thus, Formalism need not mean that Mathematics is a meaningless symbolic game. It just means there exists some Interpretation on 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”, as L.E.J. Brouwer argued as founder of the school of thought. 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 held that mathematical objects arise from the a priori Forms of Volitions that inform the Perception of Empirical Objects.

Constructivism

Like Intuitionism, Constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed should be admitted to any mathematical discourse. In other words, Mathematics is an exercise of the Human Intuition, not merely a Game played with meaningless Symbols. Math is about Entities we can create directly through mental activity.

Fictionalism

Fictionalism was introduced in 1980 when Hartry Field rejected and reversed W.V.O. Quine's “Indispensability Argument”, that Mathematics was indispensable for our best scientific theories, and therefore should be accepted as true. Field suggested Mathematics was indeed dispensable, and could be rejected as false. He did this by giving a complete axiomatization of Newtonian Mechanics that referenced no Numbers or mathematical Functions. So, Math could be little more than a modeling Language, and if not based in Logic, what does it describe?

Scholarship by M.R.M. Parrott

Synthetic A Priori: Philosophical Interviews
Interviews, Discussion

©1998-1999 M.R.M. Parrott
First Published: 99,00,02,08,11

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The Generation of 'X': Philosophical Essays 1991-1995
Academic Papers

©1991-1995 M.R.M. Parrott
First Published: Oct 2002

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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), “ܢer 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篩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.

Journals

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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