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Single Idea 18788

[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism ]

Full Idea

For the intuitionist, talk of mathematical objects is rather misleading. For them, there really isn't anything that we should call the natural numbers, but instead there is counting. What intuitionists study are processes, such as counting and collecting.

Gist of Idea

For intuitionists there are not numbers and sets, but processes of counting and collecting

Source

Edwin D. Mares (Negation [2014], 5.1)

Book Ref

'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.196

A Reaction

That is the first time I have seen mathematical intuitionism described in a way that made it seem attractive. One might compare it to a metaphysics based on processes. Apparently intuitionists struggle with infinite sets and real numbers.

The 14 ideas from 'Negation'

18781 Inconsistency doesn't prevent us reasoning about some system [Mares]
18780 Standard disjunction and negation force us to accept the principle of bivalence [Mares]
18782 The connectives are studied either through model theory or through proof theory [Mares]
18783 Many-valued logics lack a natural deduction system [Mares]
18784 In classical logic the connectives can be related elegantly, as in De Morgan's laws [Mares]
18786 Excluded middle standardly implies bivalence; attacks use non-contradiction, De M 3, or double negation [Mares]
18785 Consistency is semantic, but non-contradiction is syntactic [Mares]
18787 Three-valued logic is useful for a theory of presupposition [Mares]
18788 For intuitionists there are not numbers and sets, but processes of counting and collecting [Mares]
18789 Intuitionist logic looks best as natural deduction [Mares]
18790 Intuitionism as natural deduction has no rule for negation [Mares]
18791 In 'situation semantics' our main concepts are abstracted from situations [Mares]
18792 Situation semantics for logics: not possible worlds, but information in situations [Mares]
18793 Material implication (and classical logic) considers nothing but truth values for implications [Mares]