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Single Idea 17922
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
]
Full Idea
Given Dedekind's reduction of real numbers to sequences of rational numbers, and other known reductions in mathematics, it was tempting to see basic arithmetic as the foundation of mathematics.
Gist of Idea
Reducing real numbers to rationals suggested arithmetic as the foundation of maths
Source
Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.1)
Book Ref
Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.5
A Reaction
The reduction is the famous Dedekind 'cut'. Nowadays theorists seem to be more abstract (Category Theory, for example) instead of reductionist.
Related Idea
Idea 10572
A cut between rational numbers creates and defines an irrational number [Dedekind]
The
21 ideas
from Mark Colyvan
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]
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17926
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Rejecting double negation elimination undermines reductio proofs
[Colyvan]
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17925
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Showing a disproof is impossible is not a proof, so don't eliminate double negation
[Colyvan]
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17924
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Excluded middle says P or not-P; bivalence says P is either true or false
[Colyvan]
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17928
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Ordinal numbers represent order relations
[Colyvan]
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17930
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Axioms are 'categorical' if all of their models are isomorphic
[Colyvan]
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17929
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Löwenheim proved his result for a first-order sentence, and Skolem generalised it
[Colyvan]
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17931
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Structuralism say only 'up to isomorphism' matters because that is all there is to it
[Colyvan]
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17932
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If 'in re' structures relies on the world, does the world contain rich enough structures?
[Colyvan]
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17934
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Proof by cases (by 'exhaustion') is said to be unexplanatory
[Colyvan]
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17933
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Reductio proofs do not seem to be very explanatory
[Colyvan]
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17935
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If inductive proofs hold because of the structure of natural numbers, they may explain theorems
[Colyvan]
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17936
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Transfinite induction moves from all cases, up to the limit ordinal
[Colyvan]
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17937
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Mathematical generalisation is by extending a system, or by abstracting away from it
[Colyvan]
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17938
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Mathematics can show why some surprising events have to occur
[Colyvan]
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17939
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Mathematics can reveal structural similarities in diverse systems
[Colyvan]
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17940
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Most mathematical proofs are using set theory, but without saying so
[Colyvan]
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17941
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Infinitesimals were sometimes zero, and sometimes close to zero
[Colyvan]
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17942
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Can a proof that no one understands (of the four-colour theorem) really be a proof?
[Colyvan]
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17943
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Probability supports Bayesianism better as degrees of belief than as ratios of frequencies
[Colyvan]
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