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Single Idea 17935

[filed under theme 14. Science / D. Explanation / 2. Types of Explanation / m. Explanation by proof ]

Full Idea

It might be argued that any proof by induction is revealing the explanation of the theorem, namely, that it holds by virtue of the structure of the natural numbers.

Gist of Idea

If inductive proofs hold because of the structure of natural numbers, they may explain theorems

Source

Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1)

Book Ref

Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.82

A Reaction

This is because induction characterises the natural numbers, in the Peano Axioms.

The 5 ideas with the same theme [proofs which also reveal why something is true]:

13231 Explanatory proofs rest on 'characterizing properties' of entities or structure [Steiner,M]
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]
17934 Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan]
17942 Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan]