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Single Idea 18832

[filed under theme 4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic ]

Full Idea

On an intuitionistic view, neither the truth-value of a statement nor any other mathematical entity can be given as the final result of an infinite process, since an infinite process is precisely one that does not have a final result.

Gist of Idea

Mathematical statements and entities that result from an infinite process must lack a truth-value

Source

Michael Dummett (Elements of Intuitionism (2nd ed) [2000], p.41), quoted by Ian Rumfitt - The Boundary Stones of Thought 7.3

Book Ref

Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.199

A Reaction

This is rather a persuasive reason to sympathise with intuitionism. Mathematical tricks about 'limits' have lured us into believing in completed infinities, but actually that idea is incoherent.

The 15 ideas with the same theme [logic which uses 'provable' in place of 'true']:

18832 Mathematical statements and entities that result from an infinite process must lack a truth-value [Dummett]
18073 Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Dummett, by Kitcher]
18122 Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
18074 Intuitionists rely on assertability instead of truth, but assertability relies on truth [Kitcher]
15430 Is classical logic a part of intuitionist logic, or vice versa? [Burgess]
15431 It is still unsettled whether standard intuitionist logic is complete [Burgess]
13715 You can employ intuitionist logic without intuitionism about mathematics [Sider]
18789 Intuitionist logic looks best as natural deduction [Mares]
18790 Intuitionism as natural deduction has no rule for negation [Mares]
13249 (∀x)(A v B) |- (∀x)A v (∃x)B) is valid in classical logic but invalid intuitionistically [Beall/Restall]
8708 Double negation elimination is not valid in intuitionist logic [Friend]
17925 Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan]
17926 Rejecting double negation elimination undermines reductio proofs [Colyvan]
18798 It is the second-order part of intuitionistic logic which actually negates some classical theorems [Rumfitt]
18799 Intuitionists can accept Double Negation Elimination for decidable propositions [Rumfitt]