I think I have been guilty of taking for granted, along with many others, that Mathematics involves operations and calculations which don’t involve one in inductive reasoning, but rather represents a sheerly deductive enterprise. However, I was just reading about the Collatz conjecture, according to which if one takes any natural number and runs the following operation on it, then one will always end up eventually coming to the number 1; the function looks like this:
- for any natural number n (if n is not ‘1’ ):
- if n is even, then divide by two
- if n is odd, then multiply it by 3 and add 1 (3n+1)
The end result will be 1. Now, this is considered a ‘problem’ by Mathematicians. However, it led me to think that, since the ‘problem’ is precisely that we do not see clearly why this conjecture should be true, we presume that it is true not for any deductive reason, but rather because we ‘induce’ the conjecture from having done the operation on various numbers various times. Thus, in the study of Mathematics, Mathematicians appeal to inductive reasoning to advance the discipline.
I think that’s interesting.