Philosophy of mathematics

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Philosophy of mathematics is a branch of philosophy that studies the assumptions, foundations, and implications of mathematics, and purports to provide a viewpoint of the nature and methodology of mathematics, and to understand the place of mathematics in relation to people.


  • According to the dominant view, the reflection on mathematics is the task of a specialized discipline, the philosophy of mathematics, starting with Frege, characterized by its own problems and methods, and in a sense “the easiest part of philosophy”. In this view, the philosophy of mathematics “is a specialized area of philosophy... Many of the questions that arise within it... occur within the philosophy of mathematics in an especially pure, or especially simplified, form”. ...[However,] like applied mathematics, pure mathematics draws its concepts from experience, observation, scientific theories and even economics. The questions considered by the reflection on mathematics have, therefore, all the impurity and complexity of which philosophical problems are capable.
    • Carlo Cellucci, "Introduction" to Filosofia e matematica, 18 Unconventional Essays on the Nature of Mathematics (2006) ed., Reuben Hersh.
  • Philosophy of mathematics has been slow to draw the analogy from the Kuhnian sea-change in philosophy of science, but during the last decade, a growing number of younger philosophers of mathematics have turned their attention to the history of mathematics and tried to make use of it in their investigations. The most exciting of these concern how mathematical discovery takes place, how new discoveries are structured and integrated into existing knowledge, and what light these processes shed on the existence and applicability of mathematical objects.
    • Emily Grosholz, The Growth of Mathematical Knowledge (2000) Introduction, p. xii.
  • We are still in the aftermath of the great foundationist controversies of the early twentieth century. Formalism, intuitionism and logicism, each left its trace in the form of a certain mathematical research program that ultimately made its own contribution to the corpus of mathematics... As philosophical programs, as attempts to establish a secure foundation for mathematical knowledge, all have run their course and petered out or dried up. Yet there remains, as a residue, an unstated consensus that the philosophy of mathematics is research on the foundations of mathematics. If I find [that] uninteresting or irrelevant, I conclude that I'm simply not interested in philosophy (thereby depriving myself of any chance of confronting my own uncertainties about the meaning, nature, purpose or significance of mathematical research).
  • The doctrine that mathematical knowledge is a priori—mathematical apriorism...—has been articulated in many different ways... To name only the most prominent defenders... since the seventeenth century, Descartes, Locke, Berkeley, Kant, Frege, Hilbert, Brouwer, and Carnap... Most of the disputes... conducted in our century represent internal differences... among apriorists. ...I shall offer a picture of mathematical knowledge which rejects mathematical apriorism.
    • Philip Kitcher, The Nature of Mathematical Knowledge (1984) Introduction, p. 3.
  • [M]athematical apriorism... has not gone completely unquestioned. J. S. Mill attempted to argue that mathematics is an empirical science, thereby making himself the subject of Frege's biting criticism. More recently, W. V. Quine, Hillary Putnam, and Imre Lakatos have... challenged the... thesis. However, none of these have offered a systematic account of our mathematical knowledge. ...[T]he alternative...—mathematical empiricism—has never been given a detailed articulation. I shall try... I have gained much from insights of Quine and Putnam... [and] learned from Mill... My quarrel with earlier empiricists is, for the most part, that they have been incomplete rather than mistaken.
    • Philip Kitcher, The Nature of Mathematical Knowledge (1984) Introduction, p. 4.
  • Philosophy of mathematics appears to become a microcosm for the most general and central issues in philosophy—issues in epistemology, metaphysics, and philosophy of language—and the study of those parts of mathematics to which philosophers... most often attend (logic, set theory, aritmetic) seems designed to test the merits of large philosophical views about the existence of abstract entities of the tenability of a certain picture of human knowledge. ...[A]re [there] not other tasks ...that arise either from the current practice of mathematics or the history of the subject... the kinds of issues that occupy those who study the other branches of human knowledge... as: How does mathematical knowledge grow? What is mathematical progress? What makes some mathematical ideas (or theories) better than others? What is mathematical explanation?
    • Philip Kitcher, History and Philosophy of Modern Mathematics (1988) p. 17.
  • The dialogue takes place... The class gets interested in a Problem: Is there a relation between the number of vertices V, the number of edges E and the number of faces F of a polyhedra—particularly of regular polyhedra—analogous to the trivial relation between the number of vertices and edges of polygons, namely, that there are as many edges as vertices: V = E? ...After much tiral and error they notice that for all regular polyhedra V - E + F = 2. Somebody guesses that this may apply for any polyhedron whatever. Others try to falsify [test] this conjecture... it holds good. The results corroborate the conjecture, and suggest that it could be proved. It is at this point—after the stages problem and conjecture—that we... offer a proof.
    • Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery (1976)
  • A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed... were made explicit when logic was formalized early in the this century... These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a... conjecture. ...Heuristic arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. ...Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof.
  • By "philosophy of mathematics" I mean the specific set of concepts, categories, and theories employed, implicitly or explicitly, by philosophers and mathematicians in their discourse about mathematics. Understood in this way, philosophy of mathematics would include, among other things, some rather ethereal discussions on the nature of numbers by several hermetic philosophers, and the status of various notions, including number, space, infinity, according to the philosophers and mathematicians operating in the seventeenth century, as well as several other areas of investigation. Therefore I must introduce a qualification contained in the concept of "mathematical practice." ...I use this term as it is used today in mathematical logic and philosophy of mathematics, simply to indicate mathematics as it is done, not as it should be done according to some preconceived philosophical viewpoint. ...[F]ar from eliminating the philosophical questions, an interest in mathematical practice has actually extended their range. Addressing the issue of mathematical practice requires a detailed knowledge of the mathematical literature of the period.
    • Paolo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (1996) Introduction, pp. 3-4.
  • Philosophers and logicians have been so busy trying to provide mathematics with a "foundation" in the past half-century that only rarely have a few timid voices dared to voice the suggestion that it does not need one. I wish here to urge with some seriousness the view of the timid voices. I don't think mathematics is unclear; I don't think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs "foundations." The much touted problems in the philosophy of mathematics seem to me, without exception, to be problems internal to the thought of various system builders. The systems are doubtless interesting as intelIectual exercises; debate between the systems and research within the systems doubtless will and should continue; but I would like to convince you (of course I won't, but one can always hope) that the various systems of mathematicaI philosophy, without exception, need not be taken seriously.
    • Hillary Putnam, "Mathematics Without Foundations", Journal of Philosophy (1967) Vol. 64, No. 1, pp. 5-22.

The Philosophy of Mathematical Practice (2008)

by Paolo Mancosu
  • There is an interesting analogy... with the philosophy of the natural sciences, which has flourished under the combined influence of both general methodology and classical metaphysical questions (realism vs. antirealism, space, time, causation, etc.) interacting with detailed case studies in... (physics, biology, chemistry, etc.)... [C]ase studies both historical (studies of Einstein's relativity, Maxwell's electromagnetic theory, statistical mechanics, etc.). By contrast, with few exceptions, philosophy of mathematics has developed without the corresponding detailed case studies.
  • Already in the 1960s, first with Lakatos and later through a group of 'maverick' philosophers of mathematics (Kitcher, Tymoczko, and others), a strong reaction set in against the philosophy of mathematics conceived as foundation of mathematics. ...What these philosophers called for was an analysis of mathematics that was more faithful to its historical development.
  • A characterization... of the main features of the maverick tradition could be..; a. antifoudationalism, i.e. there is no certain foundation... mathematics is... fallible... b. anti-logicism, i.e. mathematical logic cannot provide the tools for an adequate analysis of mathematics and its development; c. attention to mathematical practice: only detailed analysis and reconstruction of large and significant parts of mathematical practice can provide a philosophy of mathematics...

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