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Single Idea 17923
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
]
Full Idea
For intuitionists, all but the smallest, most well-behaved infinities are rejected.
Gist of Idea
Intuitionists only accept a few safe infinities
Source
Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3)
Book Ref
Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.7
A Reaction
The intuitionist idea is to only accept what can be clearly constructed or proved.
The
21 ideas
from 'Introduction to the Philosophy of Mathematics'
17922
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Reducing real numbers to rationals suggested arithmetic as the foundation of maths
[Colyvan]
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17923
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Intuitionists only accept a few safe infinities
[Colyvan]
|
17926
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Rejecting double negation elimination undermines reductio proofs
[Colyvan]
|
17925
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Showing a disproof is impossible is not a proof, so don't eliminate double negation
[Colyvan]
|
17924
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Excluded middle says P or not-P; bivalence says P is either true or false
[Colyvan]
|
17928
|
Ordinal numbers represent order relations
[Colyvan]
|
17930
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Axioms are 'categorical' if all of their models are isomorphic
[Colyvan]
|
17929
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Löwenheim proved his result for a first-order sentence, and Skolem generalised it
[Colyvan]
|
17931
|
Structuralism say only 'up to isomorphism' matters because that is all there is to it
[Colyvan]
|
17932
|
If 'in re' structures relies on the world, does the world contain rich enough structures?
[Colyvan]
|
17934
|
Proof by cases (by 'exhaustion') is said to be unexplanatory
[Colyvan]
|
17933
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Reductio proofs do not seem to be very explanatory
[Colyvan]
|
17935
|
If inductive proofs hold because of the structure of natural numbers, they may explain theorems
[Colyvan]
|
17936
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Transfinite induction moves from all cases, up to the limit ordinal
[Colyvan]
|
17937
|
Mathematical generalisation is by extending a system, or by abstracting away from it
[Colyvan]
|
17938
|
Mathematics can show why some surprising events have to occur
[Colyvan]
|
17939
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Mathematics can reveal structural similarities in diverse systems
[Colyvan]
|
17940
|
Most mathematical proofs are using set theory, but without saying so
[Colyvan]
|
17941
|
Infinitesimals were sometimes zero, and sometimes close to zero
[Colyvan]
|
17942
|
Can a proof that no one understands (of the four-colour theorem) really be a proof?
[Colyvan]
|
17943
|
Probability supports Bayesianism better as degrees of belief than as ratios of frequencies
[Colyvan]
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