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

I suspect that the third objection of being 'silly' is really just a cover for a host of issues. Consider this: mathematics has two fundamental components ... one must first create a mathematical system with suitable terminology and axioms ... and then one must analyze this system to form theorems. Generally speaking, creating the mathematical system is the fun and easy part ... analyzing it is the sometimes frustrating and hard part. Consequently mathematicians are weary of endless definitions because it seems like a frivolous distraction to proving theorems. Googology is essentially an extended exercise in definitions!

Consider the format of most mathematical papers. First the mathematician will introduce a few terms, definitions,  axioms, and notation, then they will state a conjecture, then they will spend the rest of the paper proving the conjecture.

Now imagine a conversation between a mathematician and a googologist. The googologist says he has an amazing mathematical system to demonstrate. The mathematician seems intrigued, so the googologist begins ... defining some basic notation ... ' and thus we can easily define the googolplex ' ... 'uh huh' says the mathematician, unsure where this is going. ' next we can define the grangol, greagol, gigangol, gorgegol, and gulgol ... to go further we generalize our notation. Here are some new terms ... hyperions, deutero-hyperions, trito-hyperions ', 'okay ... yes yes ... mmm hmm ' the mathematician nods hoping to get to the point ...

several hours later ' ... and thus we reach ''godsgodgulus, blasphemorgulus, agoraphobia, ... and transmorgrifihgh ... ''and now here we need a new formulation ... ' -- finally the mathematician loses his patience ' enough! What's the point of all this!? You've just been giving definition after definition ... what's your conjecture?! What are you trying to prove ?!?! ' . The googologist looks bewildered and confused, 'prove? I'm not trying to prove anything. I'm trying to see how far we can take this line of development. Now if you'll draw your attention to Rule set III, I believe we can expand-- ', 'I've heard enough! I have to get back to work...' blurts out the mathematician and hastly leaves.

Anecdotally, I once told my professor about up-arrows (this was early 2005-2006, so I was just getting back into large numbers). After explaining how the notation worked the professor said "So why are you interested in this stuff?". 'Why am I interest?!' I thought, 'isn't it self evident? It's just fascinating in it's own right'. But I knew what he was getting at. He was looking for a "point". So I told him "I want to find out how to reduce the amount of information needed to express these numbers". "Ah!" the professor said as if he now understood. Now it was a challenge, now it was something to "solve" rather than something merely explore and experiment with.

This I never understood about some mathematical-types. There is no sense of wonder and awe. There is just numbers to crunch, theorems to prove, structures to analyze ... but why? For what purpose, if they are not already fascinating for their own sake? The creation/exploration stage is an important stage in mathematical development. How else will one develop mathematical research unless one first scouts for interesting areas for further research. The thing that has always appealed to me about mathematics is it's promise of inexhaustible exploration. I suspect that even the stodgiest of mathematicians began their career as wide-eyed enthusiasts, eager to explore the endless realms of mathematics. After years of hard work and being besieged by difficult problems on all sides they have simply forgotten what it was that originally sparked their interest. Or maybe they're just boring like that! It is certainly a stereotype that mathematics is dry and boring, and so mathematicians should naturally be dry and boring too! (... and yet many colorful characters are mathematicians ... Ronald Graham, John Conway, and John Nash to name a few ... ).

I almost feel like those who see nothing but the "trivial" in googology have become almost too rational, having forgotten how to enjoy things as ends in themselves and not always as means to something else. All they see is a "trivial" demonstration that numbers go on indefinitely. "Yes", they say "You can ALWAYS MAKE A LARGER NUMBER! So can we please move on?!", as if we we're somehow in doubt or confusion on this point and were trying to prove there is a largest one (okay there maybe some crazy people who do think there is a largest number ... but these tend to be ultrafinitists, not googologist's). Somehow they miss that we are not trying to create a largest number, and that would actually be a ''bad thing! We don't want there to be a largest number, which is why infinity is verboten''. It's just a way for noobs to say ... 'okay, game over, I win, moving on...'. Rather we just want to keep creating larger and larger numbers and continue exploring the numberscape...

So in short, mathematicians probably secretly view googology as a guilty pleasure. That's why they'll dabble in it as long as it's "just for fun" as in The Book of numbers, Mathematical Snapshots, and Mathematics and the Imagination, or only if it's connected to some "serious" problem in mathematics, such as Graham's Number and TREE(3). But taken on it's own, in all seriousness, it must be dismissed as silly, and frowned upon lest mathematicians waste more time on it than is seemly. But that's where us cranks come in I suppose :)

So I think it's mostly a combination of (1) and the fact that googology is largely definitions for their own sake. Now mind you, it's not all fun and games. Googology has an analytical component too. There is finding the larger of two numbers, proving which of two functions "grows faster", and there are various theorems that can be developed for creating approximations and bounds. All of this is the equivalent of theorems in googology. Still you can see that googology is one part art and one part science. There really is no good mathematical reason to come up with names for these numbers. The notations we create suffice to "name" these numbers, and in fact are far more efficient at doing so. However part of the fun is creating names and naming-schemes.

Googology will probably always remain fundamentally esoteric. Yet there might be one way in which googology might make itself more widely recognized. If we were to extend the computable functions far enough we might devise highly extended countable ordinal notations which might be of use to professional mathematicians in proof theory. In this way there might be a useful upshot to all this number generating. The irony here is that we googologists have borrowed the tools of mathematics and large numbers that result as merely a by-product. We could return the favor if we generated something useful to mainstream mathematics as a by-product of our number-generating. That may be reason enough to fully formalize BEAF, because it might one day lead to a very powerful ordinal notation.

-- Sbiis.ExE