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Single Idea 18788
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
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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
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Inconsistency doesn't prevent us reasoning about some system
[Mares]
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18780
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Standard disjunction and negation force us to accept the principle of bivalence
[Mares]
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18782
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The connectives are studied either through model theory or through proof theory
[Mares]
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18783
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Many-valued logics lack a natural deduction system
[Mares]
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18784
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In classical logic the connectives can be related elegantly, as in De Morgan's laws
[Mares]
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18786
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Excluded middle standardly implies bivalence; attacks use non-contradiction, De M 3, or double negation
[Mares]
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18785
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Consistency is semantic, but non-contradiction is syntactic
[Mares]
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18787
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Three-valued logic is useful for a theory of presupposition
[Mares]
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18788
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For intuitionists there are not numbers and sets, but processes of counting and collecting
[Mares]
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18789
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Intuitionist logic looks best as natural deduction
[Mares]
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18790
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Intuitionism as natural deduction has no rule for negation
[Mares]
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18791
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In 'situation semantics' our main concepts are abstracted from situations
[Mares]
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18792
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Situation semantics for logics: not possible worlds, but information in situations
[Mares]
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18793
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Material implication (and classical logic) considers nothing but truth values for implications
[Mares]
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