The Generalist – On Alexander Grothendieck | Hacker News Reader

The Generalist – On Alexander Grothendieck

thebigquestions.com

107 pointsbenbreen18 comments

I've been devoting myself for a while now to an approach to thinking that seems along the lines of Grothendieck's project (as described in the blog), where difficulty should only come in defining the framework to work within, while work within the framework is easy.

In defining a framework, fundamentally you are coming up with a set of types and describing how things of these types are able to interact with one another; if it's a framework that's easy to work with, the number of types, relations, and operations is small (relative to other formulations). If this framework is concise, but the universe of things described by it is large, then the types must be very general. Therefore, using an approach of finding an elegant framework, rather than mastering the complexity of an inelegant framework, relies heavily on one's ability to generalize.

My issue with spending a lot of time thinking this way is that it gives me a sort of uneasy feeling... What are these super abstract generalizations? From what I can tell, this mostly comes from wondering whether with all this 'zooming out' we are getting closer to truths about the universe, or our own minds—and the latter possibility feels a little suffocating.

Anyone know any books, essays, whatever taking up that issue?

https://en.wikipedia.org/wiki/Category_theory

Category theory is probably the epitome of what I'm talking about. Isn't it somewhat disturbing that what we can ground so much mathematics in something that so closely resembles a structure in the human mind?

Thinking about it directly again has given me a slightly different perspective, though. Maybe this resemblance of extremely general structures to human mental structures just relates to facility of expression, and isn't necessarily moving us further from objectivity—it's just easier for us to express our concepts as parameterizations of a structure that resembles the abstract structure of human concepts. The different parameterizations we can supply constitute all possible human concepts—but the important thing is that some of those concepts will be valid descriptions of objective phenomena, and those are the ones we attempt to supply.

Yes, isn't it. But we shouldn't be too surprised. The universe appears to me to profoundly self similar.

> Category theory is probably the epitome of what I'm talking about. Isn't it somewhat disturbing that what we can ground so much mathematics in something that so closely resembles a structure in the human mind?

No. First of all, it resembles a structure human minds can hold, not most structures human minds invent. It took work to get to category theory, thousands of years of it. Also, if the human mind didn't contain some small kernel of ability to accurately generalize from the real world, it wouldn't function as a living creature's mind, so we can trust, in the sense of a proof of existence, that evolution built us to reason correctly sometimes.

Those two conclusions don't seem mutually exclusive. The universe created the mind in the most broad sense. The mind is basically designed to learn patterns in the universe. Getting closer to one should get your closer to the other.

I spend a lot of my time thinking about how to construct the universe in as general a way as possible. Every now and then I feel like I 'level up' and the picture becomes clearer. Then usually about the same time I learn something new from mathematics that shows me I am following a path already travelled. Or in this case following a path closely related to a path well travelled. I think the truths about our own minds are intimately connected with truths about the universe. I suspect the mystery of consciousness is very similar to the mystery of the universe. Understand one and you understand the other. For me the ultimate mystery is explaining why there is pattern rather than no pattern. Why is there self similarity, why is the universe compressible? Why is it that within everything there is a place where this exists and why does 'this' have the structure it does. Can we invoke the anthropic principle,or is there something like population dynamics within totality. I am currently following an interesting line, starting from the thought experiment - If you recorded every moment of the universe on a physical representation, a hard drive for instance with perfect fidelity. Would there be any difference between the representation and the universe itself? I don't think there would. This is leading me to try to recast what I know of mathematics without the notion of process, to visualise it all it terms of structure only. And to to truly grok that our notion of computing is a method for finding a piece of the internal structure of a larger structure. Within a structure constructed from self similar elements. Just ramblings, sure it's non sense, or perhaps trivially obvious, but am trying to connect with someone :)

Your project makes a lot of sense to me—same kind of questions I've been wondering on: why patterns, why self-similarity, etc. I'm westoncb[at google's mail service] if you want to discuss sometime.

"If you recorded every moment of the universe on a physical representation, a hard drive for instance with perfect fidelity. Would there be any difference between the representation and the universe itself? I don't think there would."

There would be, because even if you could record with perfect fidelity the rest of the universe, at the very least, you can't record that hard drive on itself.

Lovely writing. This is better than most of what popular science writers have produced on the subject.

Can anyone who understands sheaves clarify the whole "point is a place where all functions are constant" analogy?

You actually don't need all the machinery behind sheaves to understand this! The general concept is that of a terminal object in a category. In the category of sets, the terminal objects are the singletons: given a set S, there is only one function f:S -> {a}... send everything in S to a! In the category of topological spaces, the terminal objects are the single-point spaces.

That was my first thought as well, but the author directly linked this notion to sheaves and also was talking about mapping from points.

I guess this is because the structure sheaf of the spectrum of a field is a constant sheaf.

Yes, this is exactly what I intended.

"What else falls when you drop it?" A duck!

Exactly! So logically if Grothendieck also fell when dropped he was a witch!

http://en.wikipedia.org/wiki/Duck_typing

Reply on news.ycombinator.com