User blog comment:Vel!/Yudkowsky on googology/@comment-74.105.183.206-20140322081702/@comment-5529393-20140330122519

I see what you are saying, but I have to disagree that some mathematical types "have no sense of wonder or awe". I think if someone studies mathematics for its own purpose (rather than say, just to get your engineering degree), they must have some sense of the beauty and wonder of mathematics, and find some sort of joy in the patterns and consequences that arise, be it random graphs or spectral sequences. They just don't "get" the pursuit of large numbers, which, on the face of it, doesn't lead to a greater understanding of the underlying fabric of reality, which is the primary purpose of mathematics. So I think you go to far to say mathematicians are "dry and boring" because they don't see the wonder of large numbers; they see the wonder in other things.

I've tried to learn a little bit of proof theory, but it's rough going. Generally an ordinal analysis of a theory T is done by performing a cut elimination and showing that the lengths of derivations can be bounded by a particular ordinal. I'm not sure  whether it is important that the ordinal representation system be tailored to the theory T that we are analyzing, or whether any reasonably elegant ordinal representation system will do. Even if it's the latter, I'm somewhat dubious that we can make any significant breakthroughs without understanding the technical details of the formal theories whose proof theoretic ordinals we are trying to surpass. For example, there is apparently a major technical hurdle to reach an analysis of "stability". It's quite likely to me that all our ordinal notations stay below stability, and without some knowledge of the "cage" that we are locked in, we will blissfully make extension after extension within the cage, unaware that there is even something that we need to break out of.