Leaderboard

There has been objection in the past to substituting 'A=A' for 'A is A', and validly so. In math, 3 is 3 and 3=3 reduce to the same, because in number, every instance of 3 is exactly the same. The meaning of 'A is A' is 'a thing is itself'. In number, the referent is an abstraction. The number stands in for the relationship of a group to one of its members taken as a unit. Using the membership/relationship/group/unit notion, should make the transitive property of (a+b)=(b+a) seems like an exercise in mental gymnastics.

Fine, but a rant is neither rational nor persuasive nor interesting. If your frame of understanding and reference is Left versus Right, then there isn't much I can say. It really only ever came from the French Revolution to distinguish between those who supported the monarchy and those who did not (more or less). It wasn't that bad of a distinction for a while because so much of European political reality was monarchy. But by these days, it's all kinds of confusing. Not to mention Objectivism never tried to be a left or right philosophy (which is how it can actually have elements of leftist politics). Even if the article is wrong, nothing sought to support authoritarianism, control over lives, skepticism as a theory of knowledge, collectivism, things like that. Being critical of a theory does not tacitly support every single adversary of the theory.

I didn't say or imply you might try to derail it.

Mathematical induction.

I have to emphasize that I am not a scholar on Hilbert, mathematics, or philosophy, so my explanations are not necessarily always perfectly on target, and at a certain depth, I would have to defer to people who have studied more extensively than I have. And I don't mean necessarily to defend Hilbert's philosophical notions in all its aspects. That said, however, here's a stab at answering your question: I think what Hilbert has in mind is the distinction between a) reasoning with symbols that are taken as representing particular numbers and b) making generalizations about an infinite class of numbers. For example, if 'a' is a token for a particular number, then the truth of 'a+1 = 1+a' cannot be reasonably contested as it can be concretely verified - it is finitistic. For example, for the particular numeral '2', the truth of '2+1 = 1+2' cannot be reasonably contested as it can be concretely verified. On the other hand, where 'A' stands for any undetermined member of entire infinite class of numbers, then 'A+1 = 1+A' (which is ordinarily understood as 'for all numbers A, we have A+1 = 1+A') cannot be verified concretely because it speaks of an entire infinite class that we can't exhaustively check. Therefore, some other regard must be given the formula. And that regard is to take it as not "contentual" but as "ideal" but formally provable from formal axioms (which are themselves "ideal"). And it is needed that there is an algorithm that can check for any purported formal proof that it actually is a formal proof (i.e., that its syntax is correct and that every formula does syntactically "lock" in sequence in applications of the formal rules); this is what Hilbert has in mind as the formal "game". Then Hilbert hoped that there would be found a formal proof, by using only finitistic means, that the "ideal" axioms sufficient for ordinary mathematics are consistent. Godel, though, proved that Hilbert's hope cannot be realized.