In his book “Introduction to Mathematical Philosophy” Bertrand Russell seems to reach the conclusion that mathematical induction is a definition and not a principle. In essence he states that mathematical induction works because of its definition.
The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid. Some believed it to be really a case of induction, in the sense in which that word is used in logic. Poincaré considered it to be a principle of the utmost importance, by means of which an infinite number of syllogisms could be condensed into one argument. We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle… (*)
This is perplexing to me (perhaps because I am just beginning to study higher math). If we assume no knowledge of the principle of induction, it still works. In other words, changing the definition does not change the fact that it works.
To use his example: If quadrupeds are defined as animals having four legs, it will follow that animals that have four legs are quadrupeds; However, calling them bipeds does not change the fact that they have four legs.
It seems that induction works as a consequence works of the properties of the natural numbers not because of the definition of induction itself. Is this correct?
(*) Introduction to Mathematical Philosophy, 33