As in take the subset of the given maximal atlas that is C^{r+s}

Uniquely extends the given structure so long as it is at least C^1

If M is a C^r manifold then the tangent bundle TM is most naturally a C^r-1 manifold (and more generally the k-jet bundle is most naturally a C^r-k manifold), but in fact any C^r manifold has a unique structure as a C^r+s manifold for any s (including s = ∞)

Have a look at the books by Laura Alcock (Mathematics Rebooted) and Secrets of Mental Math by Arthur Benjamin, maybe

Lots of it is simply adjacent to the Chinese room fallacy, a Turing machine is _just_ a mindless calculator that does state transitions

I wonder if there are versions of finitism that would assent to the theorems that are true in all boolean toposes, and then further wonder if there is a reasonable theory for the "free" boolean topos (essentially HOL + EM) that embodies such principles

It perhaps doesn't hold up conceptually, I meant something like the distinction between something like a physical problem in which some part of the structure is constrained to e.g. lie on a particular surface, and the universality of a law

I suppose I meant that you can have an equation which is thought of as a constraint as distinct from a law per se, but the idea of a law is universalizable and sometimes implies a necessary either cause and effect or relational attribute and so is a modal concept

When we say an equation is a “law” do you take that to have extra modal meaning whereby “in all worlds where such and such is so, then it must be thus”?

Honestly makes me think someone in the uk has their priorities sorted

im just a davidad fan, even given the milieu he exists in, I guess

A natural evolution in leadership for our Safeguarded AI programme Kathleen Fisher, CEO of ARIA, interviews Nora Amman and David “davidad” Dalrymple to mark a leadership transition in the Safeguarding AI programme.

best I can offer is: vimeo.com/1178834361

before bed I watched the ARIA video of davidad and then I had a dream where he was elected a member of parliament and I was in the chamber to hear his maiden speech

W

For some reason I type it wrong just about every time (not helped by my emacs spellchecker flagging the correct spelling as a typo)

I think public pricing is behind almost all the arguments for economic efficiency underlying the market economy’s ability to find the ‘best price’

I think ‘slop’ as in “something that takes less effort to produce than to consume” is a pretty good rubric for all kinds of wrong uses of all kinds of technology or process outcomes

PLEX: Normalization for Refinement Types | Proceedings of the ACM on Programming Languages Refinement types often use SMT solvers to automate program verification. However, since SMT solvers are first-order, verification of properties that requires higher-order reasoning is not possible. Pr...

It's out btw

dl.acm.org/doi/10.1145/...

same applies to ‘social media psychosis’ obv

Endless amounts of any form of social cognitive feedback is clearly deleterious to most human judgment and that definitely includes too much “That’s real. And it matters.”, without feedback from reality

Obviously anyone with their senses about them runs the numbers and thinks this means they need ~three meeting per day just to cycle through all direct reports in a year and sees that is going to be a bottleneck that cannot work and has never worked

I believe the ‘AI psychosis’ thing is a common kind of related collection of mental phenomena related to many emerging technologies which cause people to lose rational judgment

dudes rock

Saying it about Habermas is just utterly bizarre, and obviously based purely in cultural affect

guy involved in actively ruining the near-term 50 years: I’m worried about fragility of our species

everyone contemplating this question should read @gregegansf.bsky.social's magnificent Permutation City

I would love to one day read the Quantum Fields and Strings book, I return to it every few years and several parts make more or less sense

On top of that there are different structures we could forget and so those lead to different types of specialization

This gives us the idea that a forgetful functor is like a specialization, but there are certain circumstances where there is a natural symmetry making it that there are multiple candidates for being a forgetful functor e.g. the Vect_ℂ -> Vect_ ℝ

I had in mind something like in the original post we have a forgetful functor R-Mod -> AbGrp, which has a left adjoint constructing the free R-module from a group, and for R = ℤ, that is an adjoint equivalence, but already is says that a forgetful functor is a kind of specialization