# Principle of mathematical induction

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

• If I recall correctly in Peanos' axioms of the natural numbers, induction is an axiom. If so, there may exist other numbers which share the other properties but not the inductive property. Commented Mar 24, 2015 at 20:41
• "It seems that induction works as a consequence works of the properties of the natural numbers not because of the definition of induction itself" but how do you define the natural numbers without induction?
– mrp
Commented Mar 25, 2015 at 20:25
• Roughly: Induction is an axiom if you use PA to define the natuals, and a theorem if you're using a set theoretical basis. The fact that there are different axiom systems that can make sense and lead to a certain conclusion you find obvious is a fact of life in mathematics. Commented Jun 11, 2015 at 12:44
• I don't know exactly when Russell wrote that, but he lived in the era (and participated to that era!) where the notions of axioms, definitions, theorems, were made as precise as we know them today. So keep in mind that "definition" and "principle" in the given quote may not mean exactly the same as they do today. Commented Nov 2, 2017 at 3:36

Let us look at the rest of the paragraph you quote.

... We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle. There are some numbers to which it can be applied, and there are others (as we shall see in Chapter VIII.) to which it cannot be applied. We define the “natural numbers” as those to which proofs by mathematical induction can be applied, i.e. as those that possess all inductive properties. It follows that such proofs can be applied to the natural numbers, not in virtue of any mysterious intuition or axiom or principle, but as a purely verbal proposition. If “quadrupeds” are defined as animals having four legs, it will follow that animals that have four legs are quadrupeds; and the case of numbers that obey mathematical induction is exactly similar.

Thus Russell claims that the natural numbers are defined precisely through mathematical induction. Instead of natural numbers, let us now say something where mathematical induction applies. Your claim that

it seems that induction works as a consequence of the properties of the natural numbers, not because of the definition of induction itself

would then in Russel's view translate into

it seems that induction works as a consequence of the properties of something where mathematical induction applies, not because of the definition of induction itself

but that doesn't make much sense, does it?

You have to compare Ch.3 : Finitude and Mathematical Induction, with the modern treatment is set theory.

See e.g :

an inductive set is a set containing $\emptyset$ and "closed under successor".

Thus, the induction principle holds "by definition" for any inductive set.