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What Is Applied Category Theory? (2018)

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Re: What Is Applied Category Theory? (2018)

#61

Post by Garlef »

a1369209993 wrote: post #58
Garlef wrote: post #16

Earlier quoted context omitted.

> The whole thing is just about arrow composition! There's much much more to it. For example, a version of the yoneda lemma also holds for metric spaces (instead of a set of arrows between to things, you simply have a number indicating a distance between two things). Here's how I like to think about the yoneda lemma: If you have some kind of objects you want to talk about, one way to do this is by relating these obje…

> a number indicating a distance between two things That *is* a arrow. Arrow composition is adding up the distance along a path.

The correct term would be "composition in an enriched category". The morphism objects in this context are usually not called arrows.

I think the other poster meant "elements in the set of morphisms" when they said "arrows".

The difference is the following: Metric spaces "are" categories enriched over the real numbers, ordinary categories are categories enriched over the category of sets. So in one case the morphism objects are sets while in the other case they are real numbers. "Arrows" then refers to something internal to the morphism object. The distance (a real number) between two points in a metric space does not have any internal structure.

Re: What Is Applied Category Theory? (2018)

#62

Post by user-the-name »

gnramires wrote: post #54
user-the-name wrote: post #33

Earlier quoted context omitted.

Maybe for a value of "has applications" that doesn't necessarily include anyone actually applying it. As a generalisation, I don't think anyone by people into category theory actually cares about or applies category theory.

I was about to agree, but then I realized it's not immediate that Set Theory has practical applications either. However, Set Theory is essential to build mathematics. Taking points from: https://mathoverflow.net/questions/10334/what-practical-appl... Analysis, Number theory, most of modern mathematics rely on set theory, even computer science in the definition of Turing machines and dealing with infinities there are…

It CAN be useful, you say, but is it actually? Is anyone using it? Beyond a handful of exceptions, I would say no.

Set theory actually IS used as the foundation of other work. But category theory isn't, as far as I know.

Re: What Is Applied Category Theory? (2018)

#63

Post by johndoe42377 »

Koshkin wrote: post #53
johndoe42377 wrote: post #30

Just a nice set of ideas to be used for social signalling. In with type systems, a typeclass is all you need. The mantra is "to be an X (being substituted for an X) is to be able to perform (implement) such and such actions (or have this or that biochemical properties). It is that general, that deep, it could be even seen in molecular biology. Category theory, on the other hand, is just a few nested abstract concepts…

There is nothing wrong with abstract thinking. It is indeed useful. It is, in fact, one of the things that separate the higher animal species like humans from the rest.

This is the social signalling I am talking about.

Re: What Is Applied Category Theory? (2018)

#64

Post by myWindoonn »

Garlef wrote: post #16
xelxebar wrote: post #14

Earlier quoted context omitted.

Not addressing your question, but the Yoneda Lemma is kind of a charlatan. On first reading, it seems magical and deep, but once you grok the proof, it feels like a relatively trivial observation. The whole thing is just about arrow composition! In a way, once you're on the other side, the Yoneda Lemma feels a bit like a checkpoint during the accimatization period where your brain gets used to thinking in categories…

> The whole thing is just about arrow composition! There's much much more to it. For example, a version of the yoneda lemma also holds for metric spaces (instead of a set of arrows between to things, you simply have a number indicating a distance between two things). Here's how I like to think about the yoneda lemma: If you have some kind of objects you want to talk about, one way to do this is by relating these obje…

I like your example and perspective. I would just add on that it is not facile to do this for metric spaces, but leads directly to constructive solid geometry [0], where we render images of complex solid objects by exchanging the object for a signed distance function [1], a function which indicates how far the object is from any point in the space.

[0] https://en.wikipedia.org/wiki/Constructive_solid_geometry

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

Re: What Is Applied Category Theory? (2018)

#65

Post by johndoe42377 »

myWindoonn wrote: post #41
johndoe42377 wrote: post #30

Just a nice set of ideas to be used for social signalling. In with type systems, a typeclass is all you need. The mantra is "to be an X (being substituted for an X) is to be able to perform (implement) such and such actions (or have this or that biochemical properties). It is that general, that deep, it could be even seen in molecular biology. Category theory, on the other hand, is just a few nested abstract concepts…

Every type theory gives a categorical logic [0]. Category theory is all about describing the structures which definitely exist around mathematical objects even if we don't acknowledge them very often. This isn't social signalling; I'm not posting under my real name and I'm not trying to get accolades. This is mathematics; we teach it to each other. [0] https://mikeshulman.github.io/catlog/catlog.pdf

I would claim that such structures are imaginary, not structures at all, and the whole field is a sect (a socially constructed movement based on sectarian consensus which regards "knowledge" of details of abstractions, which does not make any sense outside of sectarian contexts).

Let's talk the most basic algebraic laws. Yes, there is indeed no difference in adding 2 apples to 3 apples, or adding 3 of them to 2. The question is from which pile to start, and the result is the same. (Multiplication, being just repeated addition, is also obvious - there is really no difference which side of a rectangle comes first).

Notice, that there is still a fundamental connection to reality. Adding C to O has exactly the same properties. There is no difference which comes first.

The identity element is more tricky, because nothing of this sort existed in nature. Nature has distinct start and stop sequence, and does structural pattern matching in the literal sense.

So, we would superimpose an element upon reality which lacks it, the same way as it goes with zero. Let's say, that trying to add what's can't be added (by physical properties) produces no change to the original element, and it is equivalent to addition of a zero.

Generalising this noop we will get something similar to identify.

1 for multiplication, is natural , while an identify matrix is an artificial construction.

Okay, this is all common sense. The important thing that there is literally nothing deeper than this shaky generalisation which we call monoid. First, because only addition and multiplication are "true monoid".

Protein production by an enzyme is almost it, but no identity.

There is no identities outside your head. A list is a monoid because of added '() - the empty list, which, again, does not exist anywhere. There is no such thing as an empty DNA sequence, empty molecule, etc.

I could go on, but maybe you already see where it goes. Everything is imaginary and too abstract to have any applicable, meaningful context.

Re: What Is Applied Category Theory? (2018)

#66

Post by myWindoonn »

johndoe42377 wrote: post #65
myWindoonn wrote: post #41

Earlier quoted context omitted.

Every type theory gives a categorical logic [0]. Category theory is all about describing the structures which definitely exist around mathematical objects even if we don't acknowledge them very often. This isn't social signalling; I'm not posting under my real name and I'm not trying to get accolades. This is mathematics; we teach it to each other. [0] https://mikeshulman.github.io/catlog/catlog.pdf

I would claim that such structures are imaginary, not structures at all, and the whole field is a sect (a socially constructed movement based on sectarian consensus which regards "knowledge" of details of abstractions, which does not make any sense outside of sectarian contexts). Let's talk the most basic algebraic laws. Yes, there is indeed no difference in adding 2 apples to 3 apples, or adding 3 of them to 2. The…

This sounds facetious. After all, the humble empty set is not physical either, and yet most set theorists will claim that it is Platonically real.

More generally, mathematics is a social construction, yes. The word literally means "things we teach each other", and the defining quality of mathematical facts is that they are non-obvious but can be verified for oneself without any additional help or context.

Denying the usefulness of category theory will only make theoretical physics harder. It doesn't make physics any simpler.

Edit: I was going to just link you to the Encyclopedia of Philosophy, but I'll spell out the definitions explicitly instead.

A formal logic is a system with some propositions and some deductive rules. Each rule takes a (family of) propositions and sends them to new propositions, changing their syntax but not their truth. Rules may be composed associatively, and for each proposition, there is a trivial identity rule which changes nothing.

This is a category, right? So every formal logic has a corresponding category whose objects represent its propositions and whose arrows represent its rules.

Re: What Is Applied Category Theory? (2018)

#67

Post by bordercases »

myWindoonn wrote: post #66
johndoe42377 wrote: post #65

Earlier quoted context omitted.

I would claim that such structures are imaginary, not structures at all, and the whole field is a sect (a socially constructed movement based on sectarian consensus which regards "knowledge" of details of abstractions, which does not make any sense outside of sectarian contexts). Let's talk the most basic algebraic laws. Yes, there is indeed no difference in adding 2 apples to 3 apples, or adding 3 of them to 2. The…

This sounds facetious. After all, the humble empty set is not physical either, and yet most set theorists will claim that it is Platonically real. More generally, mathematics is a social construction, yes. The word literally means "things we teach each other", and the defining quality of mathematical facts is that they are non-obvious but can be verified for oneself without any additional help or context. Denying the…

If you have to appeal to Platonism to make the "realness" of a concept less controversial, your claim will not be strong, as Platonism itself is controversial, and by definition supposes the existence of things which cannot be perceived. This makes it more difficult to argue for the realness of a concept, not less, since any means of "observing" the existence of that concept must be indirect to a Platonist.

It is also inconsistent to claim that mathematics is a social construction, while agreeing with these Platonic mathematicians enough to use their view as evidence.

Re: What Is Applied Category Theory? (2018)

#68

Post by johndoe42377 »

bordercases wrote: post #38
johndoe42377 wrote: post #30

Just a nice set of ideas to be used for social signalling. In with type systems, a typeclass is all you need. The mantra is "to be an X (being substituted for an X) is to be able to perform (implement) such and such actions (or have this or that biochemical properties). It is that general, that deep, it could be even seen in molecular biology. Category theory, on the other hand, is just a few nested abstract concepts…

Hi Karma.

This is the first time someone recognised me. By writing style, I suppose.

Re: What Is Applied Category Theory? (2018)

#69

Post by bordercases »

johndoe42377 wrote: post #68
bordercases wrote: post #38

Earlier quoted context omitted.

Hi Karma.

This is the first time someone recognised me. By writing style, I suppose.

I've read a lot of your work and appreciate it. I don't agree with all of it. But all of it is good.

And indeed you have a distinct style.

Re: What Is Applied Category Theory? (2018)

#70

Post by myWindoonn »

bordercases wrote: post #67
myWindoonn wrote: post #66

Earlier quoted context omitted.

This sounds facetious. After all, the humble empty set is not physical either, and yet most set theorists will claim that it is Platonically real. More generally, mathematics is a social construction, yes. The word literally means "things we teach each other", and the defining quality of mathematical facts is that they are non-obvious but can be verified for oneself without any additional help or context. Denying the…

If you have to appeal to Platonism to make the "realness" of a concept less controversial, your claim will not be strong, as Platonism itself is controversial, and by definition supposes the existence of things which cannot be perceived. This makes it more difficult to argue for the realness of a concept, not less, since any means of "observing" the existence of that concept must be indirect to a Platonist. It is als…

Okay, but like, every logic has a corresponding category, right? Please focus on the maths and stop struggling; you're only making things harder for yourself.
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