40 thinkers, books, and ideas connected to this field.
British philosopher of mathematics who has worked on logicism, the nature of mathematical knowledge, and the philosophy of mathematics.
The view in philosophy of mathematics that abstract objects (numbers, sets) are obtained by abstracting from concrete instances — identifying equivalence classes under abstraction principles like Hume's Principle.
Dutch logician who was the leading expert on intuitionistic logic and constructive mathematics — writing the definitive metamathematical studies of intuitionistic systems.
Canadian philosopher working at the intersection of philosophy of mathematics, logic, and feminist epistemology.
The Brouwer-Heyting-Kolmogorov (BHK) interpretation of intuitionistic logic — a proposition is true if and only if there is a proof of it; the meaning of each logical connective is defined in terms of what counts as a proof.
New Zealand philosopher who has argued against platonism in mathematics by developing the epistemic challenge: if mathematical objects are causally inert abstract entities, how can we have knowledge of them?
A fundamental theorem of first-order logic: a set of sentences has a model if and only if every finite subset of it has a model.
The mathematical program requiring that proofs of existence be constructive — providing a method to exhibit or compute the object, rather than merely demonstrating that its non-existence leads to contradiction.
The axioms characterizing the natural numbers — zero is a natural number, every natural number has a successor, zero is not the successor of any number, different numbers have different successors, and induction — providing the standard foundation for arithmetic.
Dutch logician and philosopher of mathematics who wrote the definitive biography of Brouwer and co-authored the reference work on constructive mathematics.
German philosopher who founded phenomenology — the rigorous description of the structures of consciousness — and whose epoché (bracketing of the natural attitude) opened a new domain of philosophical investigation.
Chinese philosopher of mathematics who has developed strict finitism — only finite mathematical objects exist — and applied it to the philosophy of applied mathematics.
American logician and philosopher of mathematics who developed plural quantification, rehabilitated Frege's logicism via neo-logicism, and wrote on provability logic.
German mathematician who created set theory and the theory of transfinite numbers — proving that infinite sets come in different sizes and that the continuum cannot be put into bijection with the natural numbers.
German mathematician and philosopher who invented modern predicate logic and introduced the sense/reference distinction, founding analytic philosophy of language.
American philosopher who transformed philosophy of language with semantic externalism and the twin earth thought experiment, and later evolved from scientific realism to pragmatism.
Hungarian philosopher of science who developed the methodology of scientific research programmes — a more sophisticated account of scientific rationality than Popper's falsificationism — and Proofs and Refutations, a revolutionary philosophy of mathematics.
Canadian philosopher working on the relationship between numerical cognition research and philosophical questions about mathematical ontology.
American philosopher who revived the notion of essence as real definition — distinct from modal necessity — developed the theory of grounding, and contributed to neo-logicism.
Austrian-American logician who proved the incompleteness theorems — any consistent formal system strong enough to express arithmetic contains true statements it cannot prove — transforming logic, philosophy of mathematics, and epistemology.
American logician who provided a simplified and generalized proof of Gödel's completeness theorem using the Henkin construction — adding witness constants to extend consistent theories to complete consistent theories.
Same entity as Leon A. Henkin (ID 252) — American logician who proved completeness of first-order logic via the Henkin construction.
Israeli-American philosopher who extended Wigner's puzzle into a systematic philosophical analysis of the applicability of mathematics to physics.
Finnish philosopher working on the empirical and cognitive foundations of mathematics — arguing that mathematical knowledge has roots in embodied and evolved cognitive capacities.
British philosopher working on embodied approaches to the philosophy of mathematics and mathematical cognition.
Canadian philosopher and historian of mathematics who wrote the definitive philosophical study of Cantor's set theory and its foundational motivations.
Italian-American philosopher of mathematics who has worked on mathematical explanation, abstraction principles in neo-logicism, and the history of 20th century logic.
American philosopher who posed Benacerraf's dilemma — if mathematical objects are abstract entities, how can we have knowledge of them? — and argued that numbers cannot be sets because there is no principled choice among competing set-theoretic reductions.
American philosopher who developed a naturalistic account of mathematics — defending mathematical realism on empiricist grounds — and then moved to a second philosophy naturalizing the study of mathematical practice.
Scottish philosopher who has worked on neo-logicism, a priori knowledge, and the foundations of mathematics.
Three-volume collaborative work by Bertrand Russell and Alfred North Whitehead attempting to derive all of mathematics from purely logical axioms using type theory.
Swiss-American cognitive scientist who collaborated with George Lakoff on Where Mathematics Comes From — arguing that mathematics is a human cognitive achievement grounded in embodied experience.
American philosopher who has made major contributions to neo-logicism (proving and analyzing Frege's Theorem) and to the philosophy of language.
American philosopher of mathematics who has worked on neo-logicism, the paradoxes of self-reference, and the logic of comics.
Australian philosopher who has worked on mathematical explanation, the metaphysics of time, and time travel.
American philosopher of mathematics who developed ante rem structuralism — mathematical objects are positions in structures that exist independently of their instantiations — and has written on philosophy of logic.
Kit Fine's systematic investigation of neo-Fregean abstractionism, examining which abstraction principles are acceptable and what the limits of the neo-logicist program are.
The standard axiomatic foundation for mathematics — a list of axioms about sets (extensionality, pairing, union, power set, infinity, replacement, foundation) that avoid the Russell and Burali-Forti paradoxes and suffice to develop virtually all mathematics.
Norwegian philosopher who has contributed to plural logic, neo-logicism, and a potentialist interpretation of set theory — sets are not fully formed objects but potential collections.