Mathematics is music of formal structure Category theory is the linguistics of mathematics Anyone else cries from joy when learning math https://www.youtube.com/watch?v=VYQVlVoWoPY https://www.reddit.com/r/askmath/comments/1e5563j/where_is_this_math_wrong_settling_a_bet/ pokud člověk vezme že ten tvar tý první posloupnosti konverguje na tvar kruhu, tak vlastnosti prvků posloupnosti se liší od vlastností limity posloupnosti, a objem v limitě je objem kruhu a nemělo by to rozbít pi, což vytvoří kontradikci s tím proposed řešením, plus i když člověk vezme že to v limitě nekonverguje na tvar kruhu (což ale dle mě konverguje, protože v nekonečnu není ta jagedness, ta je jen v konečnu), tak by se tahle ekvivalence ještě měla rozbít když člověk vezme v potaz derivaci členů posloupnosti toho prvního tvaru co v limitě taky má debatable problémy konvergovat k derivaci kruhu aneb myslím že ten meme jenom konečně iteruje posloupnost bez toho aby vzal tu limitu v nekonečnu s jinými vlastnostmi než co mají členy tý posloupnosti v konečnu 😄 Bayes' theorem is theoretically and practically simultaneously so profound and empty that it's unbelievable I am sometimes thinking, if we want AI to discover some very out of distribution novel highly abstract mathematics, maybe one way could be doing an open ended search in the space of toposes? Topos theory is a branch of category theory that generalizes notions of inclusion and logic, which are traditionally based on set theory. It studies different mathematical universes, toposes (topoi), with their own laws of how mathematical objects within them behave, an example of such universe is sets, but there are many more. https://x.com/burny_tech/status/1967090536784834714 https://www.youtube.com/watch?v=gKYpvyQPhZo https://www.youtube.com/watch?v=o-yBDYgUqZQ Topos Theory for Generative AI and LLMs I found a nerd snipe on the intersection of AI and topos theory, paper from 10 days ago, but i dont know how legit/rigorous/practical/etc. it is Topos theory is a branch of category theory that generalizes notions of inclusion and logic, which are traditionally based on set theory. It studies different mathematical universes, toposes (topoi), with their own laws of how mathematical objects within them behave, an example of such universe is sets, but there are many more. https://www.arxiv.org/abs/2508.08293 Category theory is a very abstract and general mathematical language. It's a general theory of mathematical structures and their relations, it reveals how different kinds of structures from different subfields of mathematics are related to one another and share similar structure, allowing translation of some proofs from one subfield to another. It's mathematics of mathematics, or linguistics of mathematics. The fact that the universe and platonic realm is analyzable by the language of math got me interested in math In the spirit of using Turing complete NAND/XOR gates for everything ︀︀All elementary functions (sin, cos, tan, exp, log, powers, roots, hyperbolic functions, π, e, and even basic arithmetic) can be generated from just one binary operator\: ︀︀eml(x, y) = exp(x) − ln(y), plus the constant 1. https://arxiv.org/abs/2603.21852 https://x.com/i/status/2043756177083826377 General category theorists vs people studying something like zygohistomorphic prepromorphisms https://wiki.haskell.org/index.php?title=Zygohistomorphic_prepromorphisms : "Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time." - Freeman Dyson, Birds and Frogs I don't think there will ever be such a thing as "math is solved" Many new solved problems open even more problems category theory is building mountains between mountains if all mathematics is one mountain, but there are ridges and local peaks, then category theory reveals the global structure of the many peaks and ridges, instead of local structures on some one peak or ridge Ale nedávno jsem zrovna přemýšlel proč jsou new age lidi tak obsessed sacred geometrií Jsou to v podstatě krásný matematický symetrie, je to krásný umění a theraphy tool https://en.wikipedia.org/wiki/Sacred_geometry , a já miluju fraktály z podobných důvodů "there are deep links between logic and computing (https://en.wikipedia.org/wiki/Curry–Howard_correspondence) and the analogy here is that an argument going in a circle is analogous to a program cycling through states in a circle. Though people use "circular argument" to describe a few different types of poor reasoning, and I think more commonly it means when one uses your conclusion as a premise, which is going to be different to halting stuff." there's also this What A General Diagonal Argument Looks Like (Category Theory) https://www.youtube.com/watch?v=dwNxVpbEVcc I also need to look into these connections more https://x.com/bronzeagepapi/status/1931627351940764116?t=XO1ydeeEFNrBew0S9pat_Q&s=19 man I wanna understand all the context around infinity topos so much https://ncatlab.org/nlab/show/(infinity%2C1)-topos https://en.wikipedia.org/wiki/∞-topos There are so many more higher mathematical abstractions making big connections that I haven't deeply understood yet and I want to understand them all But not now probably, but later I will Now my path is more towards deeper understanding of mathematical and physics theories of deep learning (or AI and intelligence more generally) And towards deeper understanding the mathematics of the standard model of particle physics, general relativity, and maybe attempts at their unifications There's so much more math describing everything in the physical and platonic world that I want to understand I will need more than one lifetime and more than this limited biological lifeform for understanding it all. Category theory is the linguistics of mathematics (its great if it gives useful insights and results, plus its super euphoric to connect everything ) https://en.wikipedia.org/wiki/Curry–Howard_correspondence https://ncatlab.org/nlab/show/Science+of+Logic but i should stay away from category theory and not get captured by its lovely euphoric connecting of everything again, like in the past, because i want to do concrete empirical applied engineering math more now, or theoretical math that gives more concrete results, instead of swimming in euphoric abstractions lol Functions describe the world? https://www.youtube.com/watch?v=PAZTIAfaNr8 functions are just special case of a morphisms in the category of Sets... its actually morphisms in general that describe the world! probabilistic logical hypermetagraphs is interesting language Everything is a mathematical structure Theorems and conjectures are moves within a conceptual space, but definitions create the space. >51 being divisible by 17 is disgusting Zajímalo by mě proč to tak působí. Asi tipuju protože 51 na první pohled má moc charakteristik prvočísel (lichý číslo, nekončí 5, je to podobný 31, 41, 61, 71 prvočíslům,...), a navíc dělitelnost 17 (což je navíc prvočíslo) taky skoro vůbec neřešíme, takže to není tak "natural" (tím jak to není tolik v arsenálu naší intuice v tomto kontextu), a nějak nás nenapadne dělitelnost 3? Takže mozek to nejdřív hned uvidí vzor že je to prvočíslo, a pak dostane signál, že udělal chybu v předpovědi, což tvoří kognitivní dissonanci, když zjistí že to tak není, protože to rozbíjí hodně silný obvody tvořící intuici kolem prvočísel, co rádi generalizují některý vzory, a to je nepříjemný. To mi taky připomnělo Borwein integral, kde taky máme tendenci generalizovat vzor, ale v realitě ta generalizace do nekonečna nefunguje, a je to taky podobně nepříjemný https://en.wikipedia.org/wiki/Borwein_integral https://imgur.com/z9KLZPT Nebo jsem včera přemýšlel v kontextu teorii kategorií v čistý abstraktní matice: existuje kategorická kvantová mechanika a kategorický deep learning jako dva odděleny odlišný obory, tak by mě zajímalo, či někdy někdo dělal něco na intersekci, tím že existuje dost věcí na intersekci fyziky a deep learningu, a tím že teorie kategorií je taková matematická lingvistika matematiky. Ale nic jsem nenašel na intersekci. Kontext: https://en.wikipedia.org/wiki/Categorical_quantum_mechanics https://arxiv.org/abs/2402.15332 Momentálně zkoumat rovnice všeho druhu popisující různý věci z reality je největší část mýho smyslu života Často si přeju aby matiku světa lidi kolem viděli s podobným úžasem, ale většina lidí tak prostě bohužel nefunguje https://www.youtube.com/watch?v=dtjb2OhEQcU& >jak dát rovnice víc přistupné na školách co tě napadá co se týče větší přístupnosti? např víc rovnice derivovat, míň memorizace? sem tam taky přemýšlím o tom jak to vědění líp předat nebo se mi hodně líbí interaktivní vizualizace těch rovnic nebo někdy přemýšlím jak matiku vysvětlovat v jazyce co je míň matematický, víc přirozený, pro většinu lidí víc intuitivní, atd., přes např popisování co nejvíc konkrétních případů co nejvíc konkrétně, nebo hledání analogií, nebo zjednodušenin, apod., ale tam se pak ztrácí ta matematická/fyzikální rigoróznost, formálnost, obecnost, přesnost, úplnost, univerzálnost, jazyková jednotnost, apod. ale rozhodně jsem například pro ukázaní co nejvíc konkrétních případů té obecné rovnice či obecně matematický struktury miluju když je tam přidaná motivace proč ta rovnice nebo matematická struktura vůbec vznikla tak jak je, a ne jen matematická derivace, ale i její účel, nebo nějaký hlubší důvod i went through Category Theory for Programmers lectures I used to idealize the pure functional programming paradigm i think its mathematically so euphoric! but in practice pure functional programming is.... well.... its just much easier to do most stuff in python or other languages in as simple way as possible without all this fancy functional stuff but its still euphoric when you do one big functional composition for example but monads are still super cool to me and when it comes to category theory in computer science, i still wonder what will come out of categorical deep learning in practice functional programming theory was my gateway drug into category theory more generally i really recommend these lectures that I went through, it's category theory explained in big part using Haskell, but it also has pure general category theory in it, and its also very philosophically satisfying https://www.youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_ Bartosz Milewski is the best missionary for the functional programming cult >So we want to get to this highest possible level of abstraction to help us express ideas that later can be haha yes, this sums it well category theory, the ultimate formal abstraction i'm trying to study concrete physics at this very moment! so, i will not try to do a small category theory sidequest now! category theory berries can be eaten at other time! I also want to eventually look more deeply into topos theory, a branch of category theory [Toposes - "Nice Places to Do Math](<https://www.youtube.com/watch?v=gKYpvyQPhZo>) Topos theory studies different "mathematical universes", toposes (topoi), with their own laws of how mathematical objects within them behave. An example of such universe is sets, but there are many more. But I know so little about topos theory so far, so far I just watched these introductory lectures https://www.youtube.com/watch?v=o-yBDYgUqZQ In another words, topos theory is a branch of category theory that generalizes notions of inclusion and logic, which are traditionally based on set theory I am sometimes thinking, if we want AI to discover some very out of distribution novel highly abstract mathematics, maybe one way could be doing an open ended search in the space of toposes? But it would be extremely hard. And I'm not sure if that makes enough sense. I found a nerd snipe on this intersection of AI and topos theory, paper from 10 days ago, but i dont know how legit/rigorous/practical/etc. it is https://www.arxiv.org/abs/2508.08293 i agree here, i wanna see the architectures from categorical AI papers tested more too! https://imgur.com/2ILREYl Learning more math to be able to fully appreciate alien mathematics by human mathematicians and even more alien mathematics of future AIs What is the infinite limit of mathematics? https://youtu.be/cwiXMpnHkhc?is=U3CsQBzU-nJAKe1v my escape lately is reading and doing math, the (mostly) nonvague, technically precise, optimal, neutral, perfect language of patterns, where things (mostly) just make sense Mathematics is Lego where the blocks are abstract patterns