Category Theory For Beginners: Everyday Language
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- Опубликовано: 8 фев 2025
- In this video I discuss how to represent everyday language using category theory. The idea behind this approach (ontology logs) is simple, but extremely powerful. The basic idea is to have objects representing indefinite noun phrases and arrows representing functional relationships. This approach allows us to encode ideas from everyday language (English language in this case) into concrete mathematical statements that can be represented in category theory. For example, we describe how the notion `a mother is older than her child' can be encoded in category theory. I also describe how limits like products, equalizers, and pullbacks (and also colimits like coproducts) occur in simple cases involving everyday language. I also show how ontology logs can be used to represent computations.
In the last scene I use recursion to represent a function h which takes an input (n,x) and outputs (0,f^n (x)) where f^n (x) is the element of S obtained by taking x and applying f to it n times. This can be seen as follows. We call the bottom left inclusion function in0. We call the right inclusion function in1. We call the top inclusion function in2. When n is greater than 0 we can think of (n,x) as belonging to the top left type. Since (h after in2)=k, we have
(h after in2)((n,x))= h((n,x))= (k((n,x)) = h((n-1,f(x))
in this case. Also, since (h after in0)=in1 we have
(h after in0 ((n,x))=h((0,x))=in1((0,x))=(0,x). So if n is greater than zero than h((n,x))= h((n-1,f(x)). Also h((0,x))=(0,x). So when we start with a positive n, we have that h keeps applying f to the right entry, while reducing the left entry by one. This continues until the left entry is zero, when it outputs the result.
For more on ontology logs (ologs) you can follow the links below to Spivak and Kent's paper on ontology logs, or Ryan Wisnesky's dissertation on Functional Query Languages with Categorical Types.
arxiv.org/pdf/...
dash.harvard.e...
Thank you for explaining category theory for non-mathematicians. I really appreciate the substitution steps you make.
Thank you, these ontology logs are very interesting, and your last example with commutative diagram for a function is exciting!
Also thanks for making your topology videos as well man!
So, if I understand well : human is the co-product of man and women, when "couple" is the product of man and woman. Right ?
Exactly
This is so fascinating, Iabsolutely love youre videos :)
Could you create an "ontology log" regarding the language of algebra? I feel like looking at algebra as a language would be a refreshing point of view compared to modern algebra based on set theory/operations, what do you think? :)
Also, are you part of the >implying we can discuss mathematics facebook group?
Hi Ollie, Yes I am also very excited about ontology logs. I have made a few. It is amazing how they allow you to see things in a new light. If you would like to discuss ontology logs, and future directions to take these ideas in, please feel free to drop me an email. You can find my email address by following this link.
sites.google.com/site/richardsouthwell254/home/about
I have just submitted a request to join that facebook group.
The Facebook group is generally good at having like proper discussions but we do have a lot of in jokes and occasionally memes, your videos are like precisely the kind of thing someone would post to the group to go 'hey guys check this guys videos out they are awesome' so you might fit in well, idk :)
People in the group aren't like super egotistical or like condescending either, it's a nice bunch of humans
Okay thanks man I might drop you an email in future!
Peace out!
can we say that composed word, like, for example "topology" or "homology" or "topography" are exponents ? I'm not sure if it's exponents or produtcs but I think exponent is better intuitively because we "apply" one word to the other and it recalls me about currying.
After having thinked to it, I think yes, topology is an exponent. A product would be any "logy" on any "topos", and exponent would be all the "logy" on all "topos".
Supeeeer! =) Thank you!
I thought "yields when paired with wife" meant henpecked.
Why can't also a married man be a categorical product?