Postulates, Peano

Peano postulates (Peano arithmetic)

Mathematics, foundations of

By Detlefsen, Michael

Conceived of philosophically, the foundations of mathematics concern various metaphysical and epistemological problems raised by mathematical practice, its results and applications. Most of these problems are of ancient ...

Logic in the early 20th century

By Moore, Gregory H.

The creation of modern logic is one of the most stunning achievements of mathematics and philosophy in the twentieth century. Modern logic – sometimes called logistic, symbolic logic ...

Successor

Induction, mathematical

Russell, Bertrand Arthur William (1872–1970)

By Griffin, Nicholas

Bertrand Russell divided his efforts between philosophy and political advocacy on behalf of a variety of radical causes. He did his most important philosophical work in logic and ...

Completeness (of a theory)

Categorical theory

Soundness (of a theory)

Proper names

By Forbes, Graeme

The Roman general Julius Caesar was assassinated on 14 March 44 bc by conspirators led by Brutus and Cassius. It is a remarkable fact that, in so ...

Beth’s theorem and Craig’s theorem

By Swijtink, Zeno

Beth’s theorem is a central result about definability of non-logical symbols in classical first-order theories. It states that a symbol P is implicitly defined by a theory T ...

Brouwer, L.E.J. (1881–1966)

By van Atten, Mark

L.E.J. Brouwer was a mathematician and philosopher. He graduated from the University of Amsterdam in 1907 and remained there, from 1913 to 1951, as full professor. Brouwer was ...

Logic in the 19th century

By Dipert, Randall R.

The nineteenth century was one of the most active periods for logic in Western philosophy. It is regarded foremost as being the first time logic became ‘symbolic’ and ...

Logic machines and diagrams

By Dipert, Randall R.

By ‘logical diagrams’ we generally mean any two-dimensional representations of logical relationships, such as of class inclusion or consequence. One usually also means representations using non-typographical symbols or ...

Arithmetic, philosophical issues in

By Potter, Michael

The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point ...

Poincaré, Jules Henri (1854–1912)

By Stump, David J.

Although primarily a mathematician, Henri Poincaré wrote and lectured extensively on astronomy, theoretical physics, philosophy of science and philosophy of mathematics at the turn of the century. In ...

Reference

By Devitt, Michael

It is usual to think that referential relations hold between language and thoughts on one hand, and the world on the other. The most striking example of such ...

Numbers

By Priest, Graham

Numbers are, in general, mathematical entities whose function is to express the size, order or magnitude of something or other. Historically, starting from the most basic kind of ...

Skolem, Thoralf (1887–1963)

By Brady, Geraldine

The twentieth-century mathematician Thoralf Skolem is known principally for two achievements. The first is the statement and proof of the Löwenheim-Skolem theorem. The second is his construction of ...

Twentieth-century philosophy

By Livingston, Paul M.

Although it is difficult to generalize, twentieth-century philosophy has a number of broadly characteristic and widely shared concerns. These include the ambition to clarify the nature and foundations ...

Brunschvicg, Léon (1869–1944)

By Loi, Maurice

Brunschvicg occupied a central place in French philosophy during the first part of the twentieth century. In 1909 he became a professor at the Sorbonne, teaching there and ...

Dedekind, Julius Wilhelm Richard (1831–1916)

By Stein, Howard

Dedekind is known chiefly, among philosophers, for contributions to the foundations of the arithmetic of the real and the natural numbers. These made available for the first time ...

Intuitionism

By McCarty, David Charles

Ultimately, mathematical intuitionism gets its name and its epistemological parentage from a conviction of Kant: that intuition reveals basic mathematical principles as true a priori. Intuitionism’s mathematical lineage ...

Non-constructive rules of inference

By Hazen, A.P.

For some theoretical purposes, generalized deductive systems (or, ‘semi-formal’ systems) are considered, having rules with an infinite number of premises. The best-known of these rules is the ‘ω-rule’, ...