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Data structures as topological spaces (2002) [pdf]

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Re: Data structures as topological spaces (2002) [pdf]

#21

Post by anon291 »

aleph_minus_one wrote: post #14
anon291 wrote: post #3

Earlier quoted context omitted.

Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (l…

> Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then: > https://en.wikipedia.org/wiki/Grothendieck_topolog…

> If you argue with more generality: why not consider sites (and, relatedly, topoi) instead of topological spaces then:

I thought linear programming would be something everyone knows, and I am not the original author so I can't speak for why they chose topological spaces instead of anything listed here. I think their e-mails are on the paper. Perhaps e-mailing them will help elucidate their choice.

Re: Data structures as topological spaces (2002) [pdf]

#23

Post by bmacho »

mathgradthrow wrote: post #19
anon291 wrote: post #3

Earlier quoted context omitted.

Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (l…

Topologies are necessarily continuous

This is not even false.

Re: Data structures as topological spaces (2002) [pdf]

#24

Post by skhunted »

mathgradthrow wrote: post #19
anon291 wrote: post #3

Earlier quoted context omitted.

Graphs are discrete, topologies are potentially continuous. Moreover, you can do different things with them such as create homeomorphisms to another topology much more easily than you can create bijections between graphs. In general, continuity lets you assume things that are impossible in discrete spaces. For example, many optimization problems are really easy in continuous spaces but really hard in discrete ones (l…

Topologies are necessarily continuous

You and OP are using the word continuous in two different contexts. Generally one would not say that the integers with the trivial topology is continuous. It’s a discrete space with a topology. But when someone says a space is continuous generally they mean not discrete.

Re: Data structures as topological spaces (2002) [pdf]

#25

Post by whosthatguy »

> 1. selects a sub-collection B of A whose elements match the path pattern β,

> 2. computes a new collection C as a function f of B and its neighbors,

> 3. and specifies the insertion of C in place of B into A.

This sounds a lot like the presentation of comonads as directed containers (e.g. https://arxiv.org/abs/1408.5809).

Re: Data structures as topological spaces (2002) [pdf]

#26

Post by xanderlewis »

penteract wrote: post #13
xanderlewis wrote: post #11

Earlier quoted context omitted.

It would have been slightly weird terminology even back then; it seems to have always been much more common to use the word 'children'.

I suspect that English isn't the first language of the authors and there has been a translation from a language where all nouns are gendered. The phrase "an hexagon" also suggests that they don't pronounce 'h'.

Ah — that makes sense.

Re: Data structures as topological spaces (2002) [pdf]

#27

Post by tudorw »

hackandthink wrote: post #16

Stephen Wolfram's ruliad seems to be some sort of concurrent topological computations. There are some beautiful pictures. https://content.wolfram.com/sites/43/2021/11/1110swimg46.png https://writings.stephenwolfram.com/2021/11/the-concept-of-t...

Fascinating, so would this mean an LLM is an approximator of a certain proportion of a 'ruliad'? Apologies for using GPT here, but it's a bit beyond my math to make a statement like that without reaching for my sidekick..."in a broad sense, one could conceptualize a Large Language Model (LLM) as an approximator of a specific, limited subsection of the ruliad" so I guess so, ish?

Re: Data structures as topological spaces (2002) [pdf]

#29

Post by m3ndax »

anon291 wrote: post #20
mathgradthrow wrote: post #19

Earlier quoted context omitted.

Topologies are necessarily continuous

Perhaps the confusion is that I should have said topological spaces can be continuous. There are discrete topological spaces. Topologies (which I believe is typically used to refer to the collection of open sets in a topological space) are not functions or relations themselves, so I'm not sure a useful notion of continuity applies there, but if I'm wrong, please inform.

There isn't really such a thing as a 'continuous topological space'. Technically speaking, continuity is a property of functions between topological spaces. I think you're being tempted to use the terms continuous and discrete in a more colloquial sense mapping more to uncountable vs countable/countable and finite perhaps. But yeah, you really wouldn't use the term continuous to describe a topological space or a topology.
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