I had previously written off Category Theory, but this nudges me a little bit towards thinking there might be a There there?
I like matrices, and I like graph theory, and they are very practical and useful practically everywhere for modeling work, that’s part of why I like them. Suddenly I wonder if category theory could work similarly?
Reading this and the comments, it seems like the most important thing at the center of the field might be whatever the “Yoneda Lemma” is… I checked, and Wikipedia’s summary agrees that this is important.
In mathematics, the Yoneda lemma is arguably the most important result in category theory.
I did not read thoroughly, but, squinting, it seems like maybe the central result here (at a “why even care?” level) is that when you throw away some assumptions (like assumptions that generate set theory?) about what would be a good enough foundation for math in general… maybe the Yoneda Lemma shows that set theory is a pretty good foundation for math?
Is this wrong?
(If the idea is right, does the reverse hold? Like.. does non-Yoneda-Set-compatible “mathesque stuff” totally broken or aberrant or Intractably Important or something? Maybe like how some non-computable numbers are impossible to know, even though they contain all of math. Or maybe, similarly to how non-Euclidean geometry contained spherical and hyperbolic geometry, maybe “non-Yonedan” reasoning contains a menagerie of various defeasible reasoningsystems?)
The CTWTB sequence has been fun so far! If a good explanation of Yoneda and its meanings was a third or fourth post like the ones so far, that would be neat <3
Just wondering—did you ever get around to writing this post? I’ve bounced off many Yoneda explainers before, but I have a high enough opinion of your expository ability that I’m hopeful yours might do it for me.
I had previously written off Category Theory, but this nudges me a little bit towards thinking there might be a There there?
I like matrices, and I like graph theory, and they are very practical and useful practically everywhere for modeling work, that’s part of why I like them. Suddenly I wonder if category theory could work similarly?
Reading this and the comments, it seems like the most important thing at the center of the field might be whatever the “Yoneda Lemma” is… I checked, and Wikipedia’s summary agrees that this is important.
I did not read thoroughly, but, squinting, it seems like maybe the central result here (at a “why even care?” level) is that when you throw away some assumptions (like assumptions that generate set theory?) about what would be a good enough foundation for math in general… maybe the Yoneda Lemma shows that set theory is a pretty good foundation for math?
Is this wrong?
(If the idea is right, does the reverse hold? Like.. does non-Yoneda-Set-compatible “mathesque stuff” totally broken or aberrant or Intractably Important or something? Maybe like how some non-computable numbers are impossible to know, even though they contain all of math. Or maybe, similarly to how non-Euclidean geometry contained spherical and hyperbolic geometry, maybe “non-Yonedan” reasoning contains a menagerie of various defeasible reasoning systems?)
The CTWTB sequence has been fun so far! If a good explanation of Yoneda and its meanings was a third or fourth post like the ones so far, that would be neat <3
A good explanation of Yoneda is indeed the third planned post… assuming I eventually manage to understand it well enough to write that post.
Just wondering—did you ever get around to writing this post? I’ve bounced off many Yoneda explainers before, but I have a high enough opinion of your expository ability that I’m hopeful yours might do it for me.
Still haven’t gotten around to it.