The branch of philosophy that studies the nature of mathematical entities such as numbers and sets, and how, if at all, we can have knowledge of them. The Stanford Encyclopedia of Philosophy frames its central task as determining what the nature of mathematical entities consists in and how we can have knowledge of mathematical entities, a question that sets mathematics apart from the empirical sciences because mathematical truths seem to be established by proof rather than observation. The field is closely related to, but distinct from, the foundations of mathematics, the more technical project of grounding mathematical practice in a formal system.
Facts
Central QuestionWhat is the nature of mathematical entities such as numbers and sets, and how, if at all, can we have knowledge of them? 1 Key DebateMathematical platonism against nominalism and fictionalism: whether numbers, sets and other mathematical objects exist independently of the mind and of any physical instantiation, as platonists hold, or whether mathematical language is a useful fiction with no such independent objects behind it. A further, more recent debate concerns mathematical pluralism, the view that any consistent mathematical theory describes a free-standing mathematical universe and that no such theory is more true than any other, which unsettles the very idea of a single fact of the matter about which axioms are correct. 1 Disputed
Key DebateWhether a proof by exhaustive computer calculation, as in Appel and Haken's 1976 proof of the four color theorem, counts as genuine mathematical proof given that no human being can survey it, or marks a break from the traditional requirement that a proof be humanly checkable. 2 Kenneth Appel and Wolfgang Haken's 1976 proof of the four color theorem, verified by computer case-checking no mathematician could survey by hand, forced this question onto philosophers of mathematics. Cross-Tradition Connections
Sources
1. Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)
Stanford Encyclopedia of PhilosophySection 1, opening paragraphQuote, Section 1, opening paragraph
what the nature of mathematical entities consists in and how we can have knowledge of mathematical entities
1. Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)
Stanford Encyclopedia of PhilosophySection 5.2Quote, Section 5.2
any consistent mathematical theory describes a free-standing mathematical universe, and that no such theory is more true than any other
1. Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)
Stanford Encyclopedia of PhilosophyAssociated With: Gottlob FregeView the Source 1. Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)
Stanford Encyclopedia of PhilosophyAssociated With: Bertrand RussellView the Source 1. Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)
Stanford Encyclopedia of PhilosophyIncludes: Platonism, entry: Philosophy of Mathematics, Logic, and the Foundations of Mathematics, Section 3Quote, Includes: Platonism, entry: Philosophy of Mathematics, Logic, and the Foundations of Mathematics, Section 3
On the platonistic conception, the subject matter of mathematics consists of abstract entities.
2. Stanford Encyclopedia of Philosophy
Stanford University, Center for the Study of Language and Informations.v. Philosophy of MathematicsQuote, s.v. Philosophy of Mathematics
If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology.
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