Philosophy of mathematics
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"2 + 2 = 5"--Nineteen Eighty-Four (1949) by George Orwell |
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The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics, and to provide a viewpoint of the nature and methodology of mathematics and to understand the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts.
The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably. The latter, however, may be used to refer to several other areas of study. One refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term "mathematical philosophy" to be an allusion to the approach to the foundations of mathematics taken by Bertrand Russell in his books The Principles of Mathematics and Introduction to Mathematical Philosophy.
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Major themes
Mathematical realism
Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered; triangles, for example, are real entities, not the creations of the human mind.
Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but the continuum hypothesis conjecture might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.
Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them. Major forms of mathematical realism include Platonism.
Mathematical anti-realism
Mathematical anti-realism generally holds that mathematical statements have truth-values, but that they do not do so by corresponding to a special realm of immaterial or non-empirical entities. Major forms of mathematical anti-realism include Formalism and Fictionalism.
See also
- Axiomatic set theory
- Axiomatic system
- Category theory
- Definitions of mathematics
- Formal language
- Formal system
- Foundations of mathematics
- Golden ratio
- History of mathematics
- Intuitionistic logic
- Logic
- Mathematical beauty
- Mathematical constructivism
- Mathematical logic
- Mathematical proof
- Metamathematics
- Model theory
- Naive set theory
- Non-standard analysis
- Philosophy of language
- Philosophy of science
- Philosophy of probability
- Proof theory
- Rule of inference
- Science studies
- Scientific method
- Set theory
- The Unreasonable Effectiveness of Mathematics in the Natural Sciences
- Truth
- Ultimate ensemble
Related works
- The Analyst
- Euclid's Elements
- Gödel's completeness theorem
- Introduction to Mathematical Philosophy
- New Foundations
- Principia Mathematica
- The Simplest Mathematics
Historical topics