Philosophy of mathematics Quotes

“Mathematics”

“Mathematical Platonism”

“The Philosophy of Mathematics (Comte)”

“Wikipedia: Philosophy of mathematics”

“The Philosophy of Mathematics*BJones.com”

“Philosophy of Mathematics*hilPapers.org”

“Philosophy of Mathematics*he London Philosophy Study Guide”

“Philosophy of Mathematics*em>Stanford Encyclopedia of Philosophy”

“Category Archives: Philosophy of Mathematics*em>Internet Encyclopedia of Philosophy”

“Philosophy of Mathematics Education Journal*chool of Education, University of Exeter”

“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.”

“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...”

“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.”

“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.”

“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.”

“[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.”

“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.”

“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.”

“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?”

“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.”