Tarski wrote:Scientific reasoning is surely not just inductive reasoning anyway. We appeal to clarity, applicability, beauty, intuitiveness, and an indeterminate host of things.
I was just reading a couple of articles by Chaitin and he seems to be saying that mathematics may also depend on things like comprehensibility / compressibility. Otherwise, we could just add a bunch of complex truths as axioms. In fact he suggests that the realm of possible mathematical truths is so large (infinite) that we can't possibly express it all in a formal way--relating to Godel's incompleteness theorem.
In the articles he mentions Leibnitz as well as Occam with regards to scientific laws, and theories.
Also, as a side comment, I would be careful when using mathematical induction for proofs. It is all too easy to get sloppy and prove that 0 = 1:
Pa) 0 = (1 - 1)
Pb) 0 = 0 + 0 + 0 + 0 + . . .
Tc) 0 = (1 - 1) + (1 - 1) + . . .
Td) 0 = 1 - 1 + 1 - 1 + . . .
Te) 0 = 1 + (-1 + 1) + (-1 + 1) + . . .
Tf) 0 = 1 + 0 + 0 + 0 + . . .
Tg) 0 = 1
Of course, the proof fails because the sequence does not converge.
While I see no issues with the toy example of glowing red cubes, I do think that we often get into trouble easily when reasoning about the infinite. As we had fun trying to explain to Rob last year, properties of finite sets do not necessarily carry over to transfinite sets. Just because property X holds in a set of size 1, and set of size n+1 for 1 <= n < infinity, does not mean that property holds for for sets of infinite size. While there is no place to point to in a transfinite set where it moves from finite to transfinite (by adding or removing one element at a time), that does not prevent us from using induction to specify infinite sets such as the set of integers, rationals, or reals or indeed some of the relationships / properties within them. We just need to be careful when we do it.