@julesh
Applied Compositional Thinking
Relays (1)
- wss://relay.ditto.pub – read & write
Recent Notes
In this paper we improve the bound on the value of Dylan's constant 𝐷. This value, first defined by Dylan [1963], is the number of roads that a man must walk down.
We briefly recap the history of Dylan's constant. Although it is not a priori obvious that a man must walk down a finite number of roads, a relatively elementary nonconstructive proof that 𝐷 is finite, using the pigeonhole principle, was quickly given by Marrow, Stookey and Travers [1963]. The first explicit bound, a stack of exponentials of height 5, was found by Wonder [1966]. This bound was significantly improved by Baez [1976] to 𝐷 ≤ 420 using methods of higher category theory. Although this bound remains the best known currently, Adams [1980] conjectured that 𝐷 = 42 based on extensive computer calculations.
We briefly recap the history of Dylan's constant. Although it is not a priori obvious that a man must walk down a finite number of roads, a relatively elementary nonconstructive proof that 𝐷 is finite, using the pigeonhole principle, was quickly given by Marrow, Stookey and Travers [1963]. The first explicit bound, a stack of exponentials of height 5, was found by Wonder [1966]. This bound was significantly improved by Baez [1976] to 𝐷 ≤ 420 using methods of higher category theory. Although this bound remains the best known currently, Adams [1980] conjectured that 𝐷 = 42 based on extensive computer calculations.
Why are there open source 3d printers but no open source 2d printers? Surely a dot matrix printer wouldn't be *that* hard to make and it would be able to handle 99% of things most people need
In theory a dot matrix printer could print 1 page in the time a laser printer could print like 20 pages, but in reality a dot matrix printer could print 1 page in the time that a laser printer can tell you that it's out of cyan
Why are there open source 3d printers but no open source 2d printers? Surely a dot matrix printer wouldn't be *that* hard to make and it would be able to handle 99% of things most people need
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More than 6 years after noticing it I have *finally* understood the Bayesian chain rule (*) intuitively
If you have conditional distributions P(y | x) and P(z | y), the Bayesian chain rule really just says
P(x | z) = ∫dy P(y | z) P(x | y)
Only if you expand P(z | y) using Bayes' law you get the term P(y), which is the pushforward of the prior. It's exactly like writing the differential chain rule in Leibniz notation,
dz/dx = dy/dx dz/dy
which similarly notationally suppresses the pushforward compared to the Lagrange/Jacobi notation
(g.f)'(x) = f'(x) g'(f(x))
* https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.24
If you have conditional distributions P(y | x) and P(z | y), the Bayesian chain rule really just says
P(x | z) = ∫dy P(y | z) P(x | y)
Only if you expand P(z | y) using Bayes' law you get the term P(y), which is the pushforward of the prior. It's exactly like writing the differential chain rule in Leibniz notation,
dz/dx = dy/dx dz/dy
which similarly notationally suppresses the pushforward compared to the Lagrange/Jacobi notation
(g.f)'(x) = f'(x) g'(f(x))
* https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.24
Downside of LLM culture that I have not seen mentioned anywhere before: The best way to understand something is to explain it to someone else, and people don't ask people to explain things anymore
The paradox of the stone: "Could God create a stone so heavy that even he could not lift it?" was solved by Grothendieck by introducing an infinite tower of gods. The God of the n'th universe can only...
@nprofile1q... There is a better resolution using type-in-type: an omnipotent God can create a stone so heavy that they can't lift it, and then they can lift it
Animist as in *everything* has a categorical soul if you listen careful enough. Not just three things.
@nprofile1q... What is this heresy? Category theory is not superior to the Son or the Holy Ghost, but equal to them
nostr:nprofile1qy2hwumn8ghj7un9d3shjtnyd968gmewwp6kyqpq6hx49ptzmru6djkmefdzdtx8dep9jup92hf8sngjcxk2h9h7msysa03n8d - if the rings are "too infinite", like the ring of continuous real-valued functions o...
@nprofile1q... I wonder if there's some model theoretic universality result that all finite (or not too big) commutative rings embed into one specific commutative ring that involves numbers, like the ring of rationals or something like that?
There are fascinating connections between the Riemann zeta function and music theory. I'll probably write a paper about this, but I can't resist talking about a little piece of the story. I will *...
@nprofile1q... I wonder what this looks like for rings that have very little to do with numbers, like boolean rings or rings of functions
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I begrudgingly agree with everything Macron said. I for one welcome our new French centrist overlords
https://www.bbc.com/news/articles/ce8n1zdnpd3o
https://www.bbc.com/news/articles/ce8n1zdnpd3o
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@nprofile1q... I never made the connection before but something similar probably happened over the last few decades in fields that use applied calculus because of Wolfram Alpha and similar tools, and nobody seems to particularly mind
@nprofile1q... I'm preparing to move my stuff off Godaddy and OVH are my leading contender, do you still recommend them?