The principle of mathematical induction, which is actually a deductive principle, is used extensively in number theory, and it would seriously wreck mathematics if there were sound objections to it. Here, I argue that there are, as part of my Christian humbling programme for the sciences. Human hubris and intellectual pride needs to be put in its place so that the grace of God can once more rule the hearts of men. But, to business.
The principle of mathematical induction is a proof method for proving that a proposition P(n) holds for all natural numbers 0, 1, 2, 3, … . It is like a series of dominoes knocking the next over. And, it is said to work because the axiom of induction is one of the Peano axioms for arithmetic:
If φ is a unary predicate such that:
• φ(0) is true, and
• for every natural number n, φ(n) being true implies that φ(S(n)) is true,
then, φ(n) is true for every natural number n.
To conduct a proof, first prove the base step, P(0), that the property holds for the number 0, or 1. The second step is the inductive step where one makes the assumption that P(k), is true for some k. Then one considers P(k+1), and shows that P(k) implies P(k+1), thus concluding that the proposition is true by the principle of mathematical induction.