2026-08-26T16:18:57Z https://nostr.now jcreed https://nostr.now/npub1jxf7pzy53pjcqsvzyglscnjamewcru5zcg9chp8jec30p4977s9qmp0dxv https://files.mastodon.social/accounts/avatars/000/018/832/original/918310099fb11f81.png https://files.mastodon.social/accounts/avatars/000/018/832/original/918310099fb11f81.png https://nostr.now/nevent1qqsgqfzvrfqwt5xt20a5d7zcq64gx7xwwf9ys54q8zthlv66ekqutlgzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5zfqxdg love 2 wake up at 5:30 and doomscroll and find out I should urgently add some entries to /etc/modprobe.d<br/><br/>(cf. <a href="https://news.ycombinator.com/item?id=48053623">https://news.ycombinator.com/item?id=48053623</a>) 2026-05-08T10:04:39Z https://nostr.now/nevent1qqsx4m082mlvmx3rftc3s7zlkwd0a5laurn5ctdg30wxnj4dj9ca3rgzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5sh2lwe falsehoods programmers believe about time: if you have two weekly recurring meetings which<br/>- both happen at the same time every week<br/>- don&#39;t conflict this week<br/>then they will not conflict next week 2026-03-09T13:19:06Z https://nostr.now/nevent1qqsprsqwdcw5t85gd02le2ayvk4xgc3dju32cp4syrww5vqaspyejvczyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q55dpx3w In reply to <a href='/nevent1qqs2rl9us42whg5ljc8sxaxz9jnaw0a66l59n8vm7wgsjxuu40nm68qyh62h6'>nevent1q…62h6</a><br/>_________________________<br/><br/>I think I went to the wrong kindergarten and so didn&#39;t learn about all those things early enough... I skipped straight to lecture 2, where it relies only on things absolutely everyone can be relied on to understand intuitively, like categories and functors 2026-02-05T17:36:04Z https://nostr.now/nevent1qqspxs7jxxezlrud8qkp907ajyqsr53lummmawsj68aedzp95ehvz7qzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5yhfgfw In reply to <a href='/nevent1qqsydg9pgudpvlxktn2pn04jxmeyvx806rz6phfwcgju29kqypfjehsv7wcpw'>nevent1q…wcpw</a><br/>_________________________<br/><br/><a href="https://adebray.github.io/lecture_notes/m392c_Raskin_AG_notes.pdf">https://adebray.github.io/lecture_notes/m392c_Raskin_AG_notes.pdf</a> is an exact link to the notes I was reading; iiuc, it&#39;s student-written scribe notes and not, like, a textbook, but still very valuable! 2026-02-05T17:30:39Z https://nostr.now/nevent1qqsgpcjn4skyxr50j09adcxh4vxaz0tr8r2pjayy6fpac7z2z9y6hdczyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5cza8sy In reply to <a href='/nevent1qqsg8j3lfec6xwj8aa80y682c75zumw7ynpcjv6e882s83zxvkwadhsyyswyh'>nevent1q…swyh</a><br/>_________________________<br/><br/>yes, Raskin starts right away by saying that a &#34;space&#34; is a functor ComRing -&gt; Set, and affine schemes are yoneda of rings, and eventually defines a scheme to be a particularly nice space (i.e. a sheaf in the big zariski site that is locally affine) <br/><br/>I find all this much more digestible, but that&#39;s proably because I digested a lot of category theory earlier in my life... 2026-02-05T17:25:41Z https://nostr.now/nevent1qqsgx04s4v8jn04ysnqyn3df3cpywkmludszjq6gpceuy3h2rjmpkmszyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q55u2waw In reply to <a href='/nevent1qqsxxahzn4xjm49ynx7psw38awwntr4yzvmn5fs5zfhsdeehau7hl4cvh39wl'>nevent1q…39wl</a><br/>_________________________<br/><br/>then I still don&#39;t quite understand why you&#39;d think to study maximal ideals. It all gives a feeling of a field of study that is presented after all the dust has settled and people have figured out the right definitions to work with, which is fine, but I always like the exercise of trying to work out a story (even if ahistorical!) of how one &#34;could have invented it for oneself&#34;. 2026-02-05T17:15:37Z https://nostr.now/nevent1qqsxxahzn4xjm49ynx7psw38awwntr4yzvmn5fs5zfhsdeehau7hl4czyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q59ml0ds In reply to <a href='/nevent1qqsv02gvrxnvz00cn00vx4lm87jcuzll4xg7zfu252cyxq7jz63hezscsy7xg'>nevent1q…y7xg</a><br/>_________________________<br/><br/>yes, absolutely, strong agree! as I note in the thread, the functorial point of view actually makes a lot more sense to me as a natural thing to study, and I&#39;m trying to work backwards and understand why people thought about the prime spectrum. Perhaps I made a historical error in attaching *Grothendieck&#39;s* name to the cleverness involved in studying prime ideals in the first place? I&#39;ve read some things about why prime ideals are &#34;like maximal ideals but better behaved&#34;, but... 2026-02-05T17:13:37Z https://nostr.now/nevent1qqszsj2wfx02l3aqmllelzh3mvsnlrq8rp7yg2szrfsu70uyzqfzqnszyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q577qz4q In reply to <a href='/nevent1qqs8cxd0nuqv5chwa3eu9e55ugyhhqulql44fd4dyphwjejptwsc82cgsklx3'>nevent1q…klx3</a><br/>_________________________<br/><br/>Thanks! I&#39;m making do without even having had a commutative algebra class to sleep through! Just trying to piece things together from Hartshorne and Vakil and Raskin&#39;s UT Austin Algebraic Geometry course (<a href="https://adebray.github.io/lecture_notes/">https://adebray.github.io/lecture_notes/</a>) which I found much more approachable from the point of view of already being familiar with categories/functors/sheaves, and less so geometry 2026-02-05T16:51:25Z https://nostr.now/nevent1qqswpfx644xlta9wl6t6d6fd7cvnvj3lysrz5q3rkfjqp99l206v6uszyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5a95ma6 In reply to <a href='/nevent1qqs2m8y7fwvqh3f79ff0yw3q4h33mnq3lnh99p5m39ntav08g79230csrvwcr'>nevent1q…vwcr</a><br/>_________________________<br/><br/>right, you&#39;re absolutely correct that the naive reading of that can&#39;t possibly be true... I&#39;m just trying to sketch out anything at all that might give me some traction to believe that maps-into-fields and prime ideals could be naturally/mechanically/universally derived from the study of commutative rings, instead of requiring grothendieck-level cleverness! 2026-02-05T16:19:18Z https://nostr.now/nevent1qqswk74shrpgz2thg30l6rwd5qykt840fzmr4774z0vdllgnenhxzdqzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5knqjkf In reply to <a href='/nevent1qqsd9g3th0f85qtgkj4pqtmramlw7r89trrrql5k8awea2mw6hneuyqjh8ds7'>nevent1q…8ds7</a><br/>_________________________<br/><br/>very loosely speaking, is there some result that says: &#34;if you know everything important about all the maps R → k for fields k, then you know everything &#34;geometrically important&#34; about all the maps R → S for all rings S&#34; 2026-02-05T16:13:37Z https://nostr.now/nevent1qqsd9g3th0f85qtgkj4pqtmramlw7r89trrrql5k8awea2mw6hneuyqzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5ghjg7s In reply to <a href='/nevent1qqs2k3tfsy32l3k5pqeyskx8nu69eynhw56ax9tuuw2xud356x6hgaqjscw00'>nevent1q…cw00</a><br/>_________________________<br/><br/>hmm interesting. So wait, is there a nice clear theorem you can state which says something about the &#34;sufficiency&#34; of prime ideals in characterizing the geometry?<br/><br/>Like, the proposition you say about the colimit is something I believe, but it already presumes that you&#39;ve decided to take the colimit over all fields. So if you commit to fields, prime ideals pop out as a natural concept... but can you get to fields from just committing to functors ComRing -&gt; Set? 2026-02-05T16:11:22Z https://nostr.now/nevent1qqsqryancnt2kjv2ucvn689g9e9eywapsyt7n908gxeaq7923ywgrsszyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q575jw00 In reply to <a href='/nevent1qqszdyqcgq2gnhltkwgdhufeqcc35yj7vhuu43ahe9xgtmdj9z2mkfcml60n7'>nevent1q…60n7</a><br/>_________________________<br/><br/>But indeed R_𝔭/𝔭R_𝔭 and K(R/𝔭) are isomorphic! They&#39;re both the residue field.<br/><br/>The points of Spec(R), the prime ideals of R, are in some sense the final stops in the possible journeys of R towards becoming more field-like by taking elements of it and forcing them to be zero or invertible. 2026-02-05T13:49:02Z https://nostr.now/nevent1qqszdyqcgq2gnhltkwgdhufeqcc35yj7vhuu43ahe9xgtmdj9z2mkfczyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5pyp82x In reply to <a href='/nevent1qqsdmp5pqsnssvewy8htuwkf26mqn2umgk6g4uypaqrfvtjqcpl5jage5ydnl'>nevent1q…ydnl</a><br/>_________________________<br/><br/>&#34;Force the zero elements first&#34; means taking the quotient R/𝔭, and then inverting everything that&#39;s nonzero. This would be the field of fractions K(R/𝔭) of R/𝔭.<br/><br/>&#34;Force the invertible elements first&#34; means first computing the localization R_𝔭, (which by definition adjoins inverses to everything outside of 𝔭) and then computing a quotient by the image of 𝔭 under the universal map R → R_𝔭, which is 𝔭R_𝔭, the unique maximal ideal of the local ring R_𝔭. So we get R_𝔭/𝔭R_𝔭. 2026-02-05T13:48:00Z https://nostr.now/nevent1qqsdmp5pqsnssvewy8htuwkf26mqn2umgk6g4uypaqrfvtjqcpl5jagzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5c2a43x In reply to <a href='/nevent1qqsvud7lwshewczdar3nqx8f3xtnrzp4g6uq54ttzmneunna294kn0stuhf4x'>nevent1q…hf4x</a><br/>_________________________<br/><br/>The condition on a prime ideal that rs ∈ 𝔭 implies r ∈ 𝔭 or s ∈ 𝔭 now seems less arbitrary and unmotivated to me. By demorgan, it&#39;s the same as saying that if r is to be made invertible, and s is to be made invertible, then surely rs must be invertible too.<br/><br/>How do we construct the residue field? We could first force the zero elements, then the invertible elements, or vice-versa. 2026-02-05T13:47:44Z https://nostr.now/nevent1qqsvud7lwshewczdar3nqx8f3xtnrzp4g6uq54ttzmneunna294kn0szyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q59khcfs somehow finally the construction of the residue field of a prime ideal clicked for me.<br/><br/>The story that makes sense to me is: a prime ideal 𝔭 of a ring R can be understood as intending a partition of R into two halves, namely 𝔭, the elements you want to be zero, and R\𝔭, the elements you want to be invertible.<br/><br/>If we force everything in a ring to be zero or invertible, then we obtain a field. This is the residue field. 2026-02-05T13:47:25Z https://nostr.now/nevent1qqs8f7sqr539q5hpxtaqfqv5e3dj0drz0ad2vgxquc8s208ahdr47fgzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5whtm96 In reply to <a href='/nevent1qqswvl6v5c72q9s4e936p4c8gv8hhcunumz95xh46vqzed908dkjx8q5knwsx'>nevent1q…nwsx</a><br/>_________________________<br/><br/>makes sense! but you meant infimum, yeah? 2025-08-28T14:30:46Z https://nostr.now/nevent1qqsw73sastgxkh9u0r5hrm5mcf99u0eafy7hxg4n29gsrn7dqq4p7hszyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q54aa9re In reply to <a href='/nevent1qqsfks5amgcwxrukma2vdrwlwwdjvhf59zz0glvz0r48u8xls3mxjmg5ftgl4'>nevent1q…tgl4</a><br/>_________________________<br/><br/>what I&#39;m saying I think is just agreeing with the fact that cleverly picking a good target shape for optimal Rupert-refutability makes sense! 2025-08-28T13:56:30Z https://nostr.now/nevent1qqsfks5amgcwxrukma2vdrwlwwdjvhf59zz0glvz0r48u8xls3mxjmgzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5wrwjyc In reply to <a href='/nevent1qqsyg9qhln5tefk6cxtul7yvf8pnhejltz7y4gnx0dy7crx85gkxa8c9c3qcv'>nevent1q…3qcv</a><br/>_________________________<br/><br/>the snub cube is also a polyhedron for which numerical evidence suggests it&#39;s probably not rupert --- but it lacks point symmetry (if x is in the set then -x is) --- which enables a really helpful lemma that Steininger and Yurkevich mention in a previous paper that means you don&#39;t have to worry about translations in the &#34;fitting through&#34;, which cuts down the dimensions of the search space from 7 to 5, a huge win 2025-08-28T13:54:38Z https://nostr.now/nevent1qqsqwn92rravylygpaeetqkgqd6ttztqu4druz8gk9f5vxsnpuutztszyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5rs7ugh In reply to <a href='/nevent1qqst6pfx76dqeu5veuhcdumwgpg6n98kzdxg8y0k05mmattg36t7pwsz94mzl'>nevent1q…4mzl</a><br/>_________________________<br/><br/>I never would have guessed that the Jacobian determinant would come out so simple; did you have any other intuition that led you to try that, or was it just fortunate? 2025-08-24T09:29:40Z https://nostr.now/nevent1qqswlyemsn8fc28y6k5hnrdp0gatplks7jj7pv6u3w2jzmqnjd24njszyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q58ps62m In reply to <a href='/nevent1qqs24yxx53jnlvsuuy6yqk6vz2jxrtnfkne22nc4ha4qwpqpxgx47ncx0uecz'>nevent1q…uecz</a><br/>_________________________<br/><br/>So nothing stops us from doing some trivial linear algebra to find where these lines intersect the square&#39;s exterior and computing the triangles&#39; areas directly... but if you try integrating over the choices of P it isn&#39;t as lovely and simple as the |μ-λ| you get for the Jacobian determinant. 2025-08-23T23:48:07Z https://nostr.now/nevent1qqs24yxx53jnlvsuuy6yqk6vz2jxrtnfkne22nc4ha4qwpqpxgx47nczyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5pwpyfr In reply to <a href='/nevent1qqs0ts0fh8g94vgt9lvjycy0srsgekpyr08k3xhjeah4sf23xm5e3gq5pck7g'>nevent1q…ck7g</a><br/>_________________________<br/><br/>there is a nice purely geometric way of characterizing, given one point P within the square, which points Q would lead to opposite-side vs. same-side scenarios.<br/><br/>In the image, pink is &#34;if Q is here, then same-side&#34; blue is &#34;...then opposite side&#34;. The regions are bounded by the lines through P and some corner of the square.<br/> <img src="https://files.mastodon.social/media_attachments/files/115/080/738/125/369/827/original/649e61d261b61ac9.png"> <br/> 2025-08-23T23:45:58Z https://nostr.now/nevent1qqsvw0w73ddw4mwzh66a4usjsw2x7q0rcplzf48kv43rzh2mnwh6v8czyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5xtydw5 In reply to <a href='/nevent1qqsy4y2jhy8qa7d8zl8ul6w5srakm94lf5mhelanf5c65accrlg5fwgrd9n9c'>nevent1q…9n9c</a><br/>_________________________<br/><br/>...which is a feeling of kind of taking a pass at Relearning All of Math with different glasses on, which make certain issues much clearer.<br/><br/>constructivism: &#34;I&#39;m not just proving something is *true*, I&#39;m seeing if I can *construct* its truth.&#34;<br/><br/>proof assistants: &#34;I&#39;m not just convincing a *person*, I&#39;m convincing a hyperoptimized pedanticness machine&#34; 2025-07-02T14:33:02Z https://nostr.now/nevent1qqsy4y2jhy8qa7d8zl8ul6w5srakm94lf5mhelanf5c65accrlg5fwgzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5rlhw7w One of the reasons my excitement around lean is snowballing lately --- in its role of a proof assistant that actually formalizes a bunch of Real Math in a way that seems to be getting a decent amount of community traction --- is that it&#39;s reproducing the feeling I had when first learning about constructive logic, and when I first learned about proof assistants at all... 2025-07-02T14:30:53Z https://nostr.now/nevent1qqspftnurjmh789dh682zuhle7kvg4dh659ngvt3dcq7uy723emlqegzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5chw59h In reply to <a href='/nevent1qqsqx6qa2mv2cn4ux9yztu707v6dmwlg2g3zkp5veq9tqvrdx4prkdq0jn0ch'>nevent1q…n0ch</a><br/>_________________________<br/><br/>in any case I&#39;m very much on board with the idea that these kinds of metaphors are probably ubiquitous in basically all languages 2025-03-31T20:37:31Z https://nostr.now/nevent1qqsqx6qa2mv2cn4ux9yztu707v6dmwlg2g3zkp5veq9tqvrdx4prkdqzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5h8r3gh In reply to <a href='/nevent1qqsw46yhuv96gvesnd0qnl5n6w7v407lkxhzcy3me6f4434vg8zvhcq2wyg7x'>nevent1q…yg7x</a><br/>_________________________<br/><br/>yeah there might be a kind of Citizen Kane effect where the absolutely reasonable and correct parts of Lakoff&#39;s argument sailed right by me because I didn&#39;t realize that they were nontrivial points at the time it was written<br/><br/>and actually come to think of it (it&#39;s been a long time since I read either work) it may have been &#34;Where Mathematics Comes From&#34; (Lakoff &amp; Núñez) that is sticking in my mind as something I disagreed with a lot 2025-03-31T20:36:35Z https://nostr.now/nevent1qqs8yphpqr8ju2mxj042vdu3q085nuf7zjq0t82c8dtnu85fun28jcczyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5dykzrh In reply to <a href='/nevent1qqstgltc6w4lmgm9d6mm2fm6yeuz6q8e272zq8cqqvmpl6mkrapzzuqvpqc90'>nevent1q…qc90</a><br/>_________________________<br/><br/>this whole thread is so refreshingly reasonable. I feel like (but forgive me if I&#39;m putting words in your mouth OR perhaps I misunderstood Lakoff) you&#39;re also tacitly dunking in a way that I enjoy on Metaphors We Live By, which I always felt like it was somehow... off? in attaching so much MeAnInG to metaphorical associations between e.g. space and time. Like, yeah, bruh, we chose to map past onto backwards as an arbitrary convention. It&#39;s not much more profound than that. 2025-03-31T19:55:19Z https://nostr.now/nevent1qqsfa57yletffgkxvhr9vgj22283up3pg78jf3mdzy3s8d62egvqhcqzyzge8cygjjyxtqzpsg3r7rzwth09mq0jstpqhzuy7t8z9ux5hm6q5nhvrm9 Is there a canonical notion of &#34;dependent applicative functor&#34;?<br/><br/>If the universe of types is named U, then I imagine wanting imagine wanting a map F : U → U with, as usual for an applicative,<br/> pure : A → FA<br/> ⟨*⟩ : F(A → B) → FA → FB<br/>but I&#39;d also like a notion of &#34;global section&#34;<br/> Γ : FU → U<br/>so that I can do &#34;dependent function application under F&#34;<br/> ⟨*⟩d : F((x : A) → B x) → (a : F a) → Γ ((pure B) ⟨*⟩ a)<br/><br/>Is this a thing? Does it arise out of nicer axioms than these, maybe? 2025-01-22T18:15:04Z