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Single Idea 17937

[filed under theme 15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind ]

Full Idea

One type of generalisation in mathematics extends a system to go beyond what is was originally set up for; another kind involves abstracting away from some details in order to capture similarities between different systems.

Gist of Idea

Mathematical generalisation is by extending a system, or by abstracting away from it

Source

Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.2)

Book Ref

Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.87

The 21 ideas from 'Introduction to the Philosophy of Mathematics'

17922 Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan]
17923 Intuitionists only accept a few safe infinities [Colyvan]
17926 Rejecting double negation elimination undermines reductio proofs [Colyvan]
17925 Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan]
17924 Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan]
17928 Ordinal numbers represent order relations [Colyvan]
17930 Axioms are 'categorical' if all of their models are isomorphic [Colyvan]
17929 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 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 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 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]