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Search Results 1 - 25 of 73. Results contain 488 matches


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Overview

Mathematics, foundations of

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 ...

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Thematic

Pythagoreanism

Pythagoreanism refers to a Greek religious-philosophical movement that originated with Pythagoras in the sixth century bc. Although Pythagoreanism in its historical development embraced a wide range of ...

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Thematic

Intuitionistic logic and antirealism

The law of excluded middle (LEM) says that every sentence of the form A∨¬A (‘A or not A’) is logically true. This law is accepted in classical logic, ...

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Second-order logic, philosophical issues in

Typically, a formal language has variables that range over a collection of objects, or domain of discourse. A language is ‘second-order’ if it has, in addition, variables that ...

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Thematic

Inductive definitions and proofs

An inductive definition of a predicate R characterizes the Rs as the smallest class which satisfies a basis clause of the form (β(x)→Rx), telling us that ...

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Thematic

Mereology

Mereology is the theory of the part–whole relation and of derived operations such as the mereological sum. (The sum of several things is the smallest thing of which ...

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Logicism

The term ‘logicism’ refers to the doctrine that mathematics is a part of (deductive) logic. It is often said that Gottlob Frege and Bertrand Russell were the first ...

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Intuitionism

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 ...

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Constructivism in mathematics

Constructivism is not a matter of principles: there are no specifically constructive mathematical axioms which all constructivists accept. Even so, it is traditional to view constructivists as insisting, ...

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Thematic

Hilbert’s programme and formalism

In the first, geometric stage of Hilbert’s formalism, his view was that a system of axioms does not express truths particular to a given subject matter but rather ...

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Antirealism in the philosophy of mathematics

Realism in the philosophy of mathematics is the position that takes mathematics at face value. According to realists, mathematics is the science of mathematical objects (numbers, sets, lines ...

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Realism in the philosophy of mathematics

Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities. ...

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Arithmetic, philosophical issues in

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 ...

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Geometry, philosophical issues in

The least abstract form of mathematics, geometry has, from the earliest Hellenic times, been accorded a curious position straddling empirical and exact science. Its standing as an empirical ...

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Analysis, philosophical issues in

The term ‘mathematical analysis’ refers to the major branch of mathematics which is concerned with the theory of functions and includes the differential and integral calculus. Analysis and ...

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Analysis, nonstandard

Nonstandard analysis is an important application of mathematical logic to the rest of mathematics. Invented in 1960, it provided a long-sought-for rigorous justification for the use of infinitely ...

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Category theory, introduction to

A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For ...

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Recursion-theoretic hierarchies

In mathematics, a hierarchy is a ‘bottom up’ system classifying entities of some particular sort, a system defined inductively, starting with a ‘basic’ class of such entities, with ...

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Polish logic

The term ‘Polish logic’ was coined by McCall to signal the important contributions to modern logic by logicians from Poland between the wars. There were several centres of ...

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Forcing

The method of forcing was introduced by Paul J. Cohen in order to prove the independence of the axiom of choice (AC) from the basic (ZF) axioms of ...

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Thematic

Fuzzy logic

The term ‘fuzzy’ refers to concepts without precise borders. Membership in a ‘fuzzy’ set – the set of things to which a ‘fuzzy’ concept (fuzzily) applies – is ...

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Thematic

Logic in the 19th century

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 ...

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Logic in the early 20th century

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 ...

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Thematic

Modal logic

Modal logic, narrowly conceived, is the study of principles of reasoning involving necessity and possibility. More broadly, it encompasses a number of structurally similar inferential systems. In this ...

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