Chu Spaces: A Visual Introduction | Hacker News Reader

Chu Spaces: A Visual Introduction

adelelopez.com

81 pointsprathyvsh27 comments

This blog post is very well written. It is also the platonic ideal of "articles about category theory that I love to hate."

The article concludes by saying:

> Hopefully I’ve convinced you that Chu spaces are indeed a mathematical abstraction worth knowing. I appreciate in particular how they provide such a concrete way of understanding otherwise slippery things.

But, I'm not convinced. What problems do Chu spaces allow me to solve that I couldn't solve before? Do they at least facilitate solving some problems more quickly or easily? If not – how do they help me understand the various "slippery things" better? How does this improved understanding manifest itself in terms of novel theorems (or novel proofs of known theorems)?

For most abstractions in mainstream mathematics, I can give plenty of concrete answers to those questions. But having read this article, I still don't know the answers for Chu spaces.

(I would further suggest that if someone is trying to sell you on a mathematical abstraction, but can't answer those questions convincingly, you should reject the claim that the abstraction has value.)

Does everything need to have an application in order to be useful?

What problems does music solve? Do poems facilitate solving some problems more quickly or easily? And if not, are they useless?

Listening to music can help us focus and work harder. Poetic experiences let us appreciate life and understand people around us. Some things give practical utility and some aesthetic value. Neither is more important than the other. We have to eat to survive, but that is only the bare minimum.

I definitely appreciate things more that provide valuable utility or some other derivative value. You're right that some works are appreciated beyond their utility, for how they are experienced.

I do think there may be utility in these things even when the pattern is more apparent than the application. A good poem makes us feel something. I just wouldn't get the same feeling from a math problem unless it solved something for me. Not to diminish your appreciation for this, just that I find a poem more accessible.

I was with you about valuing things that are utilitarian until you mentioned math. Solving math problems isn't really akin to the other things you mentioned. After all, you don't read poetry to solve a poetry problem and walk away feeling as if you benefited from it. Hopefully not.

But poetry is a fantastic analogy, and I think you may have selected that subconsciously. A great moment of seeing something mathematical with total clarity is exactly like the moment you suddenly understand a poem and have access to what the poet meant. Math doesn't actually have problems and doesn't need to be solved. It exists whether you figure it out or not.

What's sort of neat about this is that it physically demonstrates that fact.

Having an application definitely help spreading the concept. It is my first time hearing the term Chu Space, without an application, I have hard time motivating myself why should I read the article. We are more welcoming to new things when we see similarity to something we already knew. We use metaphor, analogy for this purpose - to establish conceptual linkage.

As for music, I do think it fits the human nature (not necessarily a "problem") of longing for company. Or put it other way, we can't have our mind "empty" for long. The practice of "mindful emptiness" is meditation imo.

Isn’t being useful the same thing as having at least one application?

Music is entertaining and enjoyable to listen to. Poems can communicate mental states and feelings succinctly.

> Does everything need to have an application in order to be useful?

By definition yes. Other things can be enjoyable, of cultural or historical interest, or just interesting pieces of knowledge, but to claim that something is useful is to claim it has applications.

The desiderata I listed are generally considered the point of introducing new abstractions in mathematics. I'm open to the idea that there are other ways in which abstraction could be useful, but still, I haven't seen this case made for Chu spaces.

Sometimes I see potential for growth in situations derived from a mathematical model. Think big O applied to a business opportunity or team dynamics O(x) means linear growth. O(x^2) means exponential growth. You can use the model to analyze your current situation and describe what the next level looks like.

I don't know what the potential of chu spaces is either. But I look at this concept like I do at articles about categorization and graph theory. It may not make sense now, but we may encounter a situation where these models that make sense mathematically help guide us in certain situations.

> O(x) means linear growth. O(x^2) means __exponential__ growth.

That's not a great start for a conceptual framework, is it?

O(x^2) is quadratic, not exponential (which would be O(2^x) or the like.)

Chu spaces can be used to formally model non-interweaving concurrency: https://citeseerx.ist.psu.edu/viewdoc/download%3Bjsessionid=... It’s useful in theoretical computer science

According to the blog post, Chu spaces can be used to model essentially anything! I want to know whether that modeling is actually useful. (Also, that link is broken for me.)

The blog post is pretty introductory but you might find these resources more interesting:

http://chu.stanford.edu/

http://boole.stanford.edu/pub/ph94.pdf

>What problems do Chu spaces allow me to solve

Based on this article, Chu spaces feel like a representation of a state space, and exploring limits to the relationships between state spaces while keeping certain invariants.

I'm not sure how high up the abstraction pile this thing sits, but it seems pretty high. That implies that a "practical" application is going to be hard to motivate because you're talking about invariants not of your tools, or even of the tools someone used to make your tools, but the tools someone else used to make those. (This is the general problem with category theory: everyone wants to out-meta each other so badly they don't notice when the discussion gets so rarefied that most everyone has already passed out.)

Well, I'm not a mathematician just a coder, so a lot of the proof went over my head... but my immediate thought was that this is a great formal architecture you could use to bake some complicated logic you derived from symbolic regression or a neural network into sets of bitmasks (or multibit / "color" masks). Laying these over each other could solve certain problems instantly by shining light through the physical structure. I guess any logic can be baked through a series of masks or gates and that isn't new, but the spatial format here is novel. I could even see some VM emulating these spaces as a way to speed up otherwise more processor intensive repetitive algorithms.

Chu spaces generalise the idea of concept-attribute table in formal concept analysis [1], by allowing the possession of an attribute by a concept to not simply be yes or no. Topological spaces are already a generalisation, where concepts are points and attributes are the open sets, but sometimes even topological systems are not general enough for the modelling you want to do.

Vaughan Pratt (the same Vaughan Pratt who was a cofounder of Sun), has done a lot to interest people in Chu spaces and his page might be of interest [2].

[1]: https://en.wikipedia.org/wiki/Formal_concept_analysis

[2]: http://chu.stanford.edu/

Hold on a second. Calling topological spaces a generalization of a concept-attribute table is a pretty ahistorical take. They were introduced to generalize metric spaces and other concrete examples.

Do you have a link to where Pratt (or someone else) has applied Chu spaces to facilitate solving a problem (either practical or theoretical)? Because that is the crux of the issue.

That page mentions Steve Vickers' book Topology via Logic, which motivates the study of topology in computer science via this application.

What application? What problem is solved?

By application, I meant the theory can find application in another problem domain.

Vickers doesn't discuss it, but this idea of modelling properties as open sets was used in Edalat's implementation of exact real arithmetic.

Very similar to the Channel Theory of Barwise and Seligman.

https://jacoblee.net/occamseraser/2011/06/06/introduction-to...

I was skeptical until I saw the applications, especially representing groups. Not entirely sure how distinct these ideas are from studying power sets or how to really think about what "color" means here.

It's a pretty cool abstraction. No idea how to use it but worth thinking about.

It was interesting that you can represent groups, but what can you do with that representation? And the exponential growth in the number of states seems to make them unwieldy to work with.

I was not familiar with this. The wikipedia article helps some, and this page has lots of links:

http://chu.stanford.edu/guide.html

I did some experiments with chu spaces some years ago in trying to understand their application to physics simulation, I'd forgotten most of what I learned, cool to see this on HN.

Do you remember any interesting papers / articles on the topic? Googling brings me back to this comment!

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