Probability & Statistics Article
Mathematics has no generally accepted definition.[6][7] Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart.[42]
In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[43]
A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[6] There is not even consensus on whether mathematics is an art or a science.[7] Some just say, "Mathematics is what mathematicians do."[6]
Intuitionist definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."[44] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct. Intuitionists also reject the law of excluded middle (i.e., P ∨ ¬ P {\displaystyle P\vee \neg P} {\displaystyle P\vee \neg P}). While this stance does force them to reject one common version of proof by contradiction as a viable proof method, namely the inference of P {\displaystyle P} P from ¬ P → ⊥ {\displaystyle \neg P\to \bot } {\displaystyle \neg P\to \bot }, they are still able to infer ¬ P {\displaystyle \neg P} \neg P from P → ⊥ {\displaystyle P\to \bot } P\to \bot . For them, ¬ ( ¬ P ) {\displaystyle \neg (\neg P)} {\displaystyle \neg (\neg P)} is a strictly weaker statement than P {\displaystyle P} P.[47]
In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[43]
A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[6] There is not even consensus on whether mathematics is an art or a science.[7] Some just say, "Mathematics is what mathematicians do."[6]
Intuitionist definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."[44] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct. Intuitionists also reject the law of excluded middle (i.e., P ∨ ¬ P {\displaystyle P\vee \neg P} {\displaystyle P\vee \neg P}). While this stance does force them to reject one common version of proof by contradiction as a viable proof method, namely the inference of P {\displaystyle P} P from ¬ P → ⊥ {\displaystyle \neg P\to \bot } {\displaystyle \neg P\to \bot }, they are still able to infer ¬ P {\displaystyle \neg P} \neg P from P → ⊥ {\displaystyle P\to \bot } P\to \bot . For them, ¬ ( ¬ P ) {\displaystyle \neg (\neg P)} {\displaystyle \neg (\neg P)} is a strictly weaker statement than P {\displaystyle P} P.[47]