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A message for my fellow texture-sensitive suncream haters
Try Neutrogena sunscreen!
I hate doing brand recommendations and it's like twice as expensive but for the first time in my 20+ years of using suncream, I can put it on and feel fine? My skin doesn't feel like it's burning, there's no desire to peel it off. Even the cream one, not the aerosol! It feels like a moisturizer, so not pleasant, but basically completely neutral. This has genuinely saved me this summer
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I'm in the slightly unusual situation (for me at least) where I'm stuck on two different problems, but which are entirely different. The end results I'm looking for are... maybe not similar but closely related, but one involves trying to geometrically understand certain 3-balls in 5-manifolds, and one involves trying to understand wtf is up with a certain class of fibre sequences of spectra.
In a way it's nice that each day I can decide whether I feel like doing category theory or not. But also why do I have to know what a Quillen equivalence is.
mmm i love technical lemmas
russian maths book haul i guess
Someone just put on the arxiv something which is like 60% something I had that I didn't think was publishable. Admittedly they do do something more with it, but other than that construction their work is also strictly less general than mine. Odd! Slightly unsure what to do about this situation.
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My review of spending the week just doing homotopy theory:
It's fine. I think I enjoyed it mostly because I was making progress and getting answers, bc that is a big change from the rest of the stuff I've worked on. It's nice to be doing actual calculations again.
On the other hand, the answers are kinda meaningless to me? Like, I had to compute some k-invariant, and now I know the answer, but it doesn't mean anything. I can't poke at it, yknow? I wanna be able to poke my shapes
Anyone have a good modern textbook for (basic) stable homotopy theory? I need to be able to check/cite some stuff that I've broadly absorbed but never learnt formally.
Based on this post, I'm gonna try to explain why I am not really interested in logic. I explicitly want people to argue back! I'd be very happy to be convinced that I'm wrong.
1. First and foremost, its barely Maths. We use logic in maths, sure, but a lot of logic isn't itself maths. That would be like saying that maths is a subfield of physics, just because it's its most well-known application.
Perhaps this is naive, but to me maths could be loosely and circularly defined as "the studied structures* of mathematics, and their applications to science". Here a structure is a Thingy With Axioms, and implicitly includes working in ZFC if there are sets. Maybe you're allowed to care about the continuum hypothesis, but if either assuming it or its contradiction gives better results (meaning that it allows us to prove more things with applications to the real world), then we should just do that. That's maths now. You can study whatever perverse axiom system you like, but that isn't maths. If you care about interpretation, it's Philosophy; if you don't, it's Logic. Maths cares about both intrinsic properties and interpretation of a particular set of "realistic" axioms, and the useful structures we define from them. I also don't think proofs are true if and only if they _are_ fully rigorous, we just the community to agree that they could be. That's good enough.
This is definitely circular, because what about set theory? But yeah, what about set theory? What is up with that?
2. The "interesting/surprising" results aren't that interesting anymore. Gödel is just the previous generation's Milnor/Freedman. I grew up with incompleteness, it doesn't surprise me. It's been a key part of maths since forever, as far as I'm concerned. I've been aware of exotic and non-smoothable manifolds for as long as I've been aware of manifolds, pretty much, so it isn't surprising that they exist. It's just a technical construction exercise. Same as Banach–Tarski isn't a paradox, it's just the statement that some sets aren't measurable, which isn't weird actually! Maybe there are some good results in there, but are they worth it? Not from the logic/model theory courses I watched my friends take.
On this note, I am excited to see what this is for future generations. Maybe they'll get more and more specialised, but I hope something else fun, well-known, and surprising is proven in our lifetimes, that our children will think is natural and boring.
3. Don't you people like fun?? I get that there's some fun in proving a surprising result, but the actual process?? Do you not like intuition? A physical vibe?? Maybe this is the topologist in me, but even the algebraists and analysists are always drawing shapes and diagrams of some kind, and talking about symmetries and actions. Maybe forcing is powerful, but has it any whimsy? I think not.
That's probably it. Maybe someone can explain type theory and HoTT in a way that doesn't make me go "that's maths not logic" and also "ok that's interesting". But Emily Riehl explained it to me and kinda agreed that isn't really logic at all, so I don't have high hopes.
