My friend wrote this, I do not really have the expertise to evaluate it, but this is what he said:
Utilizing the raw power of math itself, we will say that a proof of size n for statement m is one in which, starting from base mathematical axioms, and counting the number of times we substitute an axiom, or apply a reduction rule as a step, and arriving at the statement m. Math proofs are basically like a game of telephone, but with very specific rules (which are always followed) about how you change the message from person to person. The rules ensure that if the message started out true, it is still true after the change and it is passed to the next person. Imagine you could change any 1 word but the sentence had to remain truthful (this isn't a valid math rule here, but it gives you the jist of what the actual rules do) If there is a sequence of passes that result in a desired message, and you started from a true message, then your target message is also true, because none of the passes from one person to the next allowed the message to go from true to false.
So now we have a very concrete way of coming up with a number, we'll just run every possible proof in a branching manner for n steps and take the function we proved to grow fastest!
Does this make sense, and if it does, what would be the growth rate of this function?