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Principia Mathematica is modern and insightful
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- glimshe 2mo agoIf you can read this book cover-to-cover, you're an absolute hero. Sometimes I wonder if they inserted a big logical error in the middle just to troll people under the assumption nobody would bother to read it.
- gumby 2mo agoYou mean you don’t have a framed, signed, bug-bounty cheque from Alfred North Whitehead on your wall?? More seriously, there is indeed a huge logical error at the heart of the whole enterprise but it was not discovered until much later by Kurt Gödel.
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- voxadam 2mo ago>there is indeed a huge logical error at the heart of the whole enterprise but it was not discovered until much later by Kurt Gödel. Which leads us to our next borderline impenetrable book, Gödel, Escher, Bach by Douglas Hofstadter.
- analog31 2mo agoGEB was one of the books that inspired me to study math in college. It made math come to life in way that my high school courses didn't.
- black_knight 2mo agoI showed up to first day at university and an older student talked to me for like two minutes before declaring that I needed to read GEB. I dutifully went ahead and bought it, and I still work in logic today.
- m-hodges 2mo agoI read GEB cover to cover and haven’t stopped thinking about it for years. Not a brag, a nudge that it’s not impenetrable and more people should read it.
- voxadam 2mo agoIt's been years since I cracked open a copy, maybe I should give it another shot.
- floxy 2mo agoIf you want to get to the meat of the issue, without the extraneous stuff, you might also tackle: Godel's Theorem Simplified. https://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensler/dp/081913869X https://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensl...
- lanstin 2mo agoThe extraneous stuff is honestly fabulous, IMO of course. Just being able to listen to a four part fugue sensibly is a rare but accessible pleasure. And amusing dialogs as an instantiated dialectic for showing the truth synthesized from apparent opposites, is a great pedagogical learning.
- m-hodges 2mo agoExtremely agree. The extraneous stuff is what makes the book magical.
- anthonygd 2mo agoI read it on my honeymoon 25 years ago. That book sticks with you.
- debo_ 2mo agoAre you still married?
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- scubbo 2mo agoI'm surprised to hear that that was your perspective! I felt that it dealt with otherwise-opaque topics in a very approachable way.
- annzabelle 2mo agoMy brother's favorite book in 6th grade was Godel, Escher, Bach. Why, yes, he works as a compiler engineer.
- suslik 2mo ago> Gödel, Escher, Bach by Douglas Hofstadter. I gave that book to my mathematician grandma, and she found it so boring she couldn’t finish it - “All this stuff was known for decades”. True anecdote.
- eru 2mo agoWell, it's a pop-sci tome, not a research article. However, it is pretty dated these days.
- taybin 2mo agoHow is it dated?
- jcranmer 2mo agoA large chunk of the book is philosophizing about AI and the nature of the mind and intelligence. And when it's betting on the AI that existed pre-AI winter... yeah, that part is quite dated.
- taybin 2mo agoYou should see his book "Fluid Concepts and Creative Analogies, which is entirely about his experiments with AI in The Fluid Analogies Research Group with his grad students. Although I wonder if those techniques could be paired with LLMs somehow.
- eru 2mo agoEh, that's just a pop-science tome. Nothing impenetrable about it.
- onraglanroad 2mo agoI'd have thought so too, but when I went looking for a copy in Waterstones and failed to find it under popsci section, or in maths, I tried the enquiry desk. After a few attempts where I eventually tried "goddle s-chair batch" we found it. It was under Western Philosophy.
- sorokod 2mo agoTo each their own, I read GEB several times and enjoyed the intellectual challenge and the humour. Hofstadter wrote a followup book: I am a strange loop. https://en.wikipedia.org/wiki/I_Am_a_Strange_Loop https://en.wikipedia.org/wiki/I_Am_a_Strange_Loop
- taybin 2mo agoHe wrote a number of follow-up books. I love GEB, but the follow ups were often disappointing in surprising ways. He has another book about the beauty and challenges of translating poetry, but it’s actually about the sudden death of his wife and it's been too sad for me to finish.
- steppi 2mo agoThis is commonly believed, but Gödel didn't identify a logical error at the heart of the whole enterprise, he proved astonishing theorems revealing limitations of any sufficiently powerful formal system. One can kind of think of the Principia as a science experiment to find the extent to which known mathematics could be proven from foundational axioms that could be thought of as "laws of logic". To make their system work, Russell and Whitehead themselves had to add extralogical axioms, such as their Axiom of Reducibility [0] and the Axiom of Infinity, giving empirical evidence (but not a proof) that "laws of logic" alone were not enough. They were also aware of limitations in their own system, such as the inability to define the cardinal $\aleph_\omega$ [1]. Like the article says, what they did was ahead-of-its-time, and a monumental influence on all subsequent work on formal systems, including Gödel's work, regardless of whether Russell and Whitehead achieved their initial aims. [0] https://en.wikipedia.org/wiki/Axiom_of_reducibility https://en.wikipedia.org/wiki/Axiom_of_reducibility [1] https://www.gutenberg.org/files/78255/78255-h/78255-h.htm#Page_190 https://www.gutenberg.org/files/78255/78255-h/78255-h.htm#Pa...
- gumby 2mo agoThe flaw was the belief that it was possible (though they were not sure that they two could pull it off). Gödel showed that it was not an impossibly difficult task but an actually impossible task.
- steppi 2mo agoA failed hypothesis is not the same thing as a logical error. Would they have been more free of flaws if they believed it was impossible and thus didn't try? We'd all be the poorer for it
- jibal 2mo agoIt cannot be overstated how flawed your own statements here are. You shouldn't be talking about logical error regarding a work of logic when you don't even know enough logic to know what a logical error is. And there was no "flaw" in attempting to resolve foundational paradoxes in set theory, primarily to address Frege's error and deal with Russell's paradox, or in attempting to prove that all mathematics can be derived from pure logic. Seeking to demonstrate a hypothesis that is later shown to be erroneous is not "flawed", else the entire knowledge-seeking enterprise is "flawed". And switching from "logical error" to the very vague notion of a "flaw" is goalpost moving that looks a lot like bad faith. I won't comment further.
- robobro 2mo agoThat's not how to spell Ludwig Wittgenstein!
- steppi 2mo agoWittgenstein didn't find logical flaws in the Principia and deeply admired it. He found flaws in Russell's follow up work on Epistemology, "The Theory of Knowledge."
- jibal 2mo agoUtter nonsense ... there is no known logical error in PM. Gödel proved that Russell and Whitehead's goal was unachievable but that's a totally different matter. OTOH, Russell found a logical error at the heart of Frege's work, and PM fixed it by introducing the theory of types.
- kjellsbells 2mo agoI used to wonder how likely it was that the printers made some typesetting errors. Who among us could, say, type a thousand pages of APL symbols without introducing a bug?
- inigyou 2mo agoapocryphally a typesetter saw "make x as small as possible" at the end of a math problem to be typeset, and did exactly that
- taneq 2mo ago“Find x” <— here it is!
- gjm11 2mo agoThe version of this story I heard is in Littlewood's "A Mathematician's Miscellany"[1] and it's a sigma rather than an x. But he tells it as something that happened specifically to him -- he wrote a memo that ended with "thus sigma should be made as small as possible", and that bit was absent but there was in its place a very very tiny sigma. Unless he's outright lying, I think this one actually happened! [1] The more recent edition is titled "Littlewood's Miscellany"; I am fairly sure this story is in both the older and the newer version.
- WillAdams 2mo agoThere's a reason mathematics was known as "penalty copy" and was notoriously difficult to typeset and even more difficult to turn a profit on. For a deep dive into both ends of that, see the history of publication of Knuth's TAoCP where the text was originally published traditionally by setting metal type on a composition machine (to the extent possible), then compositors would add the additional characters and spacing material necessary to compose the equations and so forth so as to lay out a galley (which would then be proofed/corrected) --- a successive edition was then typeset using an early imagesetter, which looked so ghastly that DEK considered giving up, but when informed that the imagesetter was controlled by a computer declared, "I am a computer scientist, I can fix that." and expected to knock out a typesetting system over his next sabbatical.... Roughly a decade later, TeX 1.0 was released.... the current version is 3.141592653 (with new versions adding another decimal place as the version tends towards \pi) --- while we're still waiting on the full publication of Vol. 4, it is widely considered that TeX was worth the delay.
- keltor 2mo agoIt was required reading for my Logics class in undergrad. Pretty sure it was also on the optionals (aka required) for my Set Theory class as well. It's also pretty typically a part of History Of Mathematics and Philosophy of Mathematics courses.
- derrida 2mo agoNo it's not. No it wasn't. And you did not read it. EDIT: source: took logic as undergrad + wrote on the tractatus which required a lot of pre-reqs to understand. 0 chance a course at undergrad level ever assigns principia mathematica. I don't care if you went to yale or oxford or ecole normale ... 0 chance. Most charitable interepretation: some pages of it + was on a bibliography. not required reading. if feel embarrassed, that is the consequence for lieing. There is such a thing as intellectual honesty.
- mathisfun123 2mo agoI'm with you - I hate when people exaggerate their bonafides beyond all belief
- derrida 2mo agoLLMs giving some people way too much confidence to conceptually shoot from the hip hehe - “effort to refute bullshit is order of magnitude more than to refute it”.
- my-next-account 2mo agoDerrida, go back to your grave, you messed up the quote!
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- derrida 2mo agoHa! Thanks. True! I'll leave it (p.s. this whole comment section is wild - the green & getting downvoted accounts? They said the most credible things of anyone )
- sergevar 2mo agoInterestingly, there was a Show HN last year formalizing PM in Lean (https://news.ycombinator.com/item?id=43797256 https://news.ycombinator.com/item?id=43797256), and the Principia Rewrite project (https://www.principiarewrite.com https://www.principiarewrite.com) verified all 189 propositional logic theorems (sections 1-5) in Coq against the original proof sketches
- WillAdams 2mo agoFor an accessible introduction before beginning this, consider his _Introduction to Mathematical Philosophy_: https://en.wikipedia.org/wiki/Introduction_to_Mathematical_Philosophy https://en.wikipedia.org/wiki/Introduction_to_Mathematical_P... and for ease of reading see the various PDF versions at: https://people.umass.edu/klement/imp/ https://people.umass.edu/klement/imp/
- zote 2mo agosimilarly the work itself is available here: https://people.umass.edu/klement/pom/ https://people.umass.edu/klement/pom/
- hasley 2mo agoOf you prefer an even more entertaining approach and a very gentle introduction into the topic, I recommend the comic "Logicomix" which tells Russel's journey (though not historically correct all the time for story telling reasons). https://en.wikipedia.org/wiki/Logicomix https://en.wikipedia.org/wiki/Logicomix
- bramadityaw 2mo agonew comics recommendation! thanks!
- eternauta3k 2mo agoJust got this from the library, it's a real page turner. Heard about it in this excellent interview: https://www.typetheoryforall.com/episodes/goedel-s-incompleteness-theorems https://www.typetheoryforall.com/episodes/goedel-s-incomplet...
- igravious 2mo agoLogicomix is novel, and done well, but flawed … it's deficiencies lie in what it leaves out which may come across as an unfair charge but in this case the charge is warranted. There is a more historically correct and less orthodox work waiting in the wings for whosoever should attempt it.
- voidhorse 2mo agoI have a copy and like it much. However, i was always partial to Frege's Begriffschrift. His notation was really creative. It's a shame Russel's deflation of that project has sentenced it to the rubbish heap of history.
- igravious 2mo agoThe Begriffschrift has in no way been consigned to the rubbish heap of history. What gave you that impression? It is seminal. That it had one unresolved paradox in its set-theoretic foundations does not scupper the philosophical insights, nor the creative notation, nor the more-or-less novel approach of conjoining mathematical functions and logic to give us predicate logic (apologies for this brutally simplified sketch) i like to think of Frege and the Begriffschrift like this Boole: logic + algebra = algebraic logic Frege: logic + functions = predicate logic ergo, if Boole is rightly deified then so should Frege regardless of minor infelicities (which prompted type theory anyhow) -- again, apologies if this is totally misleading
- voidhorse 2mo agoHey man, I completely agree. I love the begriffschrift and recognize the very important role it played in the history of logic. The reason I say it's in the rubbish heap is because basically only experts and people with strange hobbies know about it. The set of people familiar with the Principia is vastly greater than those who know what the Begriffschrift is. That's what I meant, not that it wasn't important, but that it's importance is now under recognized and under appreciated.
- cubefox 2mo agoYou are confusing his book Begriffsschrift ("concept notation"), where he invented what became modern predicate logic, and his later work "Grundgesetze der Arithmetik" (I/II) which (unsuccessfully) tried to derive arithmetic from purely logical notions, and which made heavy use of the Begriffsschrift.
- voidhorse 2mo ago
- TimorousBestie 2mo agoInstead of spending time beating one’s head against Russell and Whitehead, I would advise reading Homotopy Type Theory (aka the HoTT Book). Dependent types are cool and mind-expanding, but higher inductive types are downright mind-altering. The Little Schemer/Typer could be used as a preparatory text to gear one up for HoTT. It also has the advantage of being a bit more applicable to functional programming languages, maybe even more so than Mac Lane’s Categories for the Working Mathematician (which I sometimes see suggested to mathematically-inclined Haskell novices).
- js8 2mo agoI tried to read HoTT. First chapter on type theory is great and pretty easy to follow. The second chapter, I got completely lost. I don't remember why, maybe they fixed it since. But I find univalence axiom intriguing. I am interested in different approach to types, using triage calculus, which is more "materialist" than "structuralist" - type is given by the structure of the (quoted) term in normal form (unlike lambda calculus, triage calculus makes quoting easy). And I feel like univalence is related to quoting, something like if the two quoted terms are equal under "standard self-interpreter", then they are equal.
- leonidasrup 2mo agoI would highly recommend "PROGRAM = PROOF" by Samuel Mimram. It covers everything from pure lambda calculus through dependent type theory up to homotopy type theory. In comparison to the HoTT book, the book "PROGRAM = PROOF" is oriented less towards mathematicians more towards programmers. It contains also a short introduction to OCaml and Agda. The book can downloaded from the authors web page: https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/teaching/pp/course.pdf https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/teaching... https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/publications/ https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/publicat...
- zmgsabst 2mo agoHoTT distinguishes equality from equivalence. Univalence says that equality is equivalent to equivalence, ie, formalizing the notion of when we can use equivalence rather than equality as a step in a proof. In practice, we often only care about proofs “up to equivalence”. A way to think about this: - equality is an identity map - equivalence is an isomorphism For example, 2 in Z and 2 in R do not have an identity map between them — but do have an isomorphism. I think the key insight of univalence is not collapsing equivalence into equality — but allowing it to remain a second truth relation. We don’t want 2 in Z to be equal to 2 in R (because we collapse type distinction), but we do want them to be equivalent — so we can do equivalent reasoning about arithmetic in R to reach conclusions about Z.
- makerdiety 2mo agoSo... the ancient childish attempt to prove mathematics using mathematics (Gödel's Incompleteness slew the challenger) can be used to help me be a better TypeScript programmer? I learned something new today.
- bulbar 2mo agoWhy the belittling language? You actually can prove the completeness and consistency of portions of mathematics. While axioms were known in ancient times, only Hilbert started the whole "prove Mathematics" thing. How else would you prove mathematics and why would that be childish to use math? The limitations discovered were quite surprising back then.
- makerdiety 2mo agoYikes, guys/girls. I got downvoted to -4 points for a misunderstanding or something. Because the author of the website would probably agree with my simple point that although the Principia Mathematica tried to do the impossible, there is still utility for its value as a programming self-teaching resource for serious students of computer science. Wow. Yeah. You guys ironically didn't just throw out the baby with the bath water thing. You burned me at the stake like a witch for heresy. Due to your cognitive biases and distortions. You guys are Imperium of Mankind coded or something?
- gjm11 2mo ago> I got downvoted to -4 points > You burned me at the stake like a witch for heresy. I think you should try to get a better sense of proportion. Also: > my simple point that although the Principia Mathematica tried to do the impossible, there is still utility for its value as a programming self-teaching resource I don't know what your original intention actually was, but your comment read to me very much as (1) implying that the OP was claiming that PM is useful for making people into better Typescript programmers (which OP very much does not claim) and (2) making fun of the OP for making such a claim while (3) calling the enterprise of which PM was a part "childish". All of which seems to me like rather the sort of thing that does deserve downvoting to -4, though for what it's worth I didn't downvote you.
- data_maan 2mo agoIt always amazes me how a random dump of someone who read the first 40 pages of PM attracts dozens comments on HN. This really must be a very math-starved community of people who wanted to learn math but never quite could.
- laichzeit0 2mo agoTwo thoughts on someone who went out of their way to learn math: 1. If you can already program, the worst thing you can do is think of mathematics as learning a programming language. It is not, and you will waste your time being frustrated with things like syntax and notation. You get “used to” mathematics by doing it, and it’s something on its own. Just go with it. It’s ok to be confused. 2. Do the exercises, and stop asking for “solution manuals”, the point is to get you thinking and the struggle is most important part, not whether you got it “right”. Again, I think this is a programmer centric way of looking at things: “how do I know it’s right if I can’t compile it”. Maybe that’s why programmers like the foundations of mathematics. Like if somehow they could just go to the bottom of things, the assembler/machine code of sorts, the whole enterprise would make sense. Counterintuitively, the really great mathematicians of yore, did mathematics before it was anywhere close to formalized.
- futune 2mo agoI think your latter comment is kind of analogous to people writing python (or any high-level language) without understanding assembly. I think maybe that reduces the mystery a bit?
- nitsuaeekcm 2mo agoFor those who aren't familiar with the great but tragic story of Principia and Russell's quest for the foundation of math (spoiler: there is none), there's a really great graphic novel called Logicomix https://en.wikipedia.org/wiki/Logicomix https://en.wikipedia.org/wiki/Logicomix I haven't read it in probably ten years, but it's one of those books and stories I spend an inordinate amount of time thinking about, for whatever reason.
- emil-lp 2mo agoThe foundation of math is (mostly) ZFC.
- igravious 2mo agoIt is not. The foundation of math is contested -- but afaik it is widely held that HoTT is the, erm, hottest contender to the throne https://en.wikipedia.org/wiki/Homotopy_type_theory https://en.wikipedia.org/wiki/Homotopy_type_theory
- qbit42 2mo agoThere is not a single foundation - you can choose. The differences are rarely important for working mathematicians though. Most know enough of ZFC to get by and ignore foundations tbh
- ogogmad 2mo agoMost foundations are in a sense equivalent. In that sense, "ZFC" is as good a choice as any. I think there might be some confusion around the different meanings of the word "foundation": A foundation is a formal system that suffices, somehow, to encode virtually all of known mathematics. The reason why people (including me!) are interested in other "foundations" like HoTT is because they try to build the same mathematics as ZFC from a different set of building blocks, despite them eventually arriving in the same place. In the case of HoTT, it reduces mathematics to homotopies and fibrations, while also making those weighty-sounding concepts seem easy. If you're interested in homotopies, fibrations, cohomology theories etc. then HoTT is a really helpful way to better understand those concepts.
- black_knight 2mo agoI miss modernism!
- tristramb 2mo ago"Principia Mathematica is an odd book, worth looking into from a historical point of view as well as a mathematical one. It was written around 1910, and mathematical logic was still then in its infancy, fresh from the transformation worked on it by Peano and Frege. The notation is somewhat obscure, because mathematical notation has evolved substantially since then. And many of the simple techniques that we now take for granted are absent. Like a poorly-written computer program, a lot of Principia Mathematica's bulk is repeated code, separate sections that say essentially the same things, because the authors haven't yet learned the techniques that would allow the sections to be combined into one." - Mark Dominus (https://blog.plover.com/math/PM.html https://blog.plover.com/math/PM.html)
- danilafe 2mo agoThis was my first thought when I saw the article.
- bazoom42 2mo agoHave someone refactored it into a more concise and modern version?
- theonemind 2mo agoIt seems like a frontier model LLM could probably do it in day, probably less. Someone would have to read and correct it, though
- jonjacky 2mo agoPossibly pertinent: Principia Mathematica Maps and Table Site (PM-MATS): https://principia.lib.uiowa.edu/about.html https://principia.lib.uiowa.edu/about.html - more description in my top-level comment in this page.
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- lordleft 2mo agoIt blows my mind that Russell invented (formalized) types. Such an elemental concept, but so useful.
- layer8 2mo agoRussell’s types aren’t really the same notion as types in programming: https://planetmath.org/russellstheoryoftypes https://planetmath.org/russellstheoryoftypes
- d4rkp4ttern 2mo agoAn interesting fact I learned while reading The Dream Machine[1], is that Principia was the basis of Newell, Simon and Shaw’s Logic Theorist (1956), considered to be the “first AI program”. Amusing and amazing to see this in the context of today’s Erdos-slaying LLMs. Quoting from Wikipedia: https://en.wikipedia.org/wiki/Logic_Theorist https://en.wikipedia.org/wiki/Logic_Theorist Logic Theorist is a computer program completed in 1956 by Allen Newell, Herbert A. Simon, and Cliff Shaw.[1] It was the first program deliberately engineered to perform automated reasoning, and has been described as "the first artificial intelligence program".[1][a] Logic Theorist proved 38 of the first 52 theorems in chapter two of Whitehead and Bertrand Russell's Principia Mathematica, and found a new and shorter proof for Theorem 2.85.[3] [1] https://press.stripe.com/the-dream-machine https://press.stripe.com/the-dream-machine
- vixen99 2mo agoTangential but for those who don't know it, Whitehead's Science and the Modern World (1925) is a fascinating read.
- pngwen 2mo agoYou might be interested in Kurt Goedel’s extended book review wherein he proves that Principia cannot do what it sets out to do, nor can any such system. I do teach PM when I teach theory of computation, but largely to tell the story of how we discovered the limits to computation.
- lioeters 2mo ago> Goedel’s extended book review wherein he proves that Principia cannot do what it sets out to do Perhaps it's this one: On Formally Undecidable Propositions of Principia Mathematica and Related Systems https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propos... - PDF: https://monoskop.org/images/9/93/Kurt_G%C3%B6del_On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems_1992.pdf https://monoskop.org/images/9/93/Kurt_G%C3%B6del_On_Formally...
- radford-neal 2mo agoThe notation for avoiding parentheses is interesting, and I've thought that it might be useful in programming languages. To illustrate, suppose you have a non-associative operator $. Rather than write a$(b$c), you can write a$.b$c - the . makes the $ before it be lower precedence on the right side. More dots make things be even lower precedence. So, for example, a$b .$: x$y .$. p$q means (a$b) $ ((x$y) $ (p$q)) At least, that's my recollection. It's been over fifty years since I read (significant parts of) it...
- layer8 2mo agoIn what way do you think this is useful over parentheses?
- radford-neal 2mo agoIt's better visually. No dot, ., :, :., ::, ::., :::, etc. have increasing visual weight, which shows which are the more top-level operators without having to match up parentheses.
- layer8 2mo agoI see. It’s not clear how it would interact with operator precedence, however. For example, what would a * . b + c . + d * e do?
- radford-neal 2mo agoYeah. I'm not sure how one would handle that. (Can't remember if Whitehead & Russell had any inherent operator precedence.) So may be most useful if you don't have inherent precedence, as might be the case if you're not dealing with the standard math operators. Alternatively, one might say the dots override inherent precedence. So a *. b*c+d*e means a * (b*c+d*e) with the inherent precedence disambiguating the right operand of *.
- sergius 2mo agoThis book is an interesting approach to The Principia: Magnificent Principia (2013), by Colin Pask https://devontrevarrowflaherty.com/2014/08/26/book-review-principia/ https://devontrevarrowflaherty.com/2014/08/26/book-review-pr...
- zual 2mo agoit seems that someone wants to traduce the PM in lean here : https://github.com/l-pommeret/Principia-Mathematica https://github.com/l-pommeret/Principia-Mathematica (probably with the use of llms)
- titanomachy 2mo ago"traduce" doesn't have the same meaning in English as it does in Spanish. The translation of "traducir" is "to translate".
- jonjacky 2mo agoPrincipia Mathematica Maps and Table Site (PM-MATS): https://principia.lib.uiowa.edu/about.html https://principia.lib.uiowa.edu/about.html "The goal of this project is to make clear structural connections between different parts of Principia and to make analyzable data about the theorems, definitions, and primitive postulates in its text. We do this by providing three digital tools ..." For example here is their take on the celebrated proof in PM that 1 + 1 = 2 https://principia.lib.uiowa.edu/?n=110.643&n=110 https://principia.lib.uiowa.edu/?n=110.643&n=110
- 1vuio0pswjnm7 2mo agoDomainname optional https://216.92.24.179/ftp/ https://216.92.24.179/ftp/
- scoofy 2mo agoAs a former analytic philosophy student, it's always a bit strange and encouraging for me to see stuff like this show up in CS/Tech forums. We need more philosophy now that we are dealing with the implication of "intelligent" machines.
- mdrmdr 2mo ago[dead]