Principia Mathematica is modern and insightful • Hacker News Board

Re: Principia Mathematica is modern and insightful

#121

Post by asimpletune » Thu, Aug 13, 2026, 2:44 PM UTC

makerdiety wrote: post #82

bulbar wrote: post #49

Earlier quoted context omitted.

Why 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.

Yikes, 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 b…

Sometimes it's worth revisiting the site guidelines on comments https://news.ycombinator.com/newsguidelines.html#comments

I'm guessing people downvoted you for snark.

Re: Principia Mathematica is modern and insightful

#122

Post by js8 » Thu, Aug 13, 2026, 2:46 PM UTC

jerf wrote: post #108

js8 wrote: post #51

Earlier quoted context omitted.

I 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 lamb…

This is probably a reasonable example of a case where an AI can really help out as an endlessly patient assistant to answer your personal questions in a conversational format. It is possible it may get something wrong but as long as you keep beating on the wrongness you should eventually be able to work out what it is, and in its own way that would become possibly the best learning exercise there is. And of course, w…

Yes, maybe.. I read it like a decade ago though, the AI didn't exist then. Although my current interest in triage and lambda calculus is also fueled by AI.

Re: Principia Mathematica is modern and insightful

#123

Post by lanstin » Thu, Aug 13, 2026, 2:51 PM UTC

m-hodges wrote: post #8

voxadam wrote: post #5

Earlier quoted context omitted.

>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.

I 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.

I had my copy signed by Prof Hofstadter; a treasure to this day. Led to a maths degree, years of organ playing classes, a career in software, and a Zen practice going on 20 years. And I'm pescatarian. I even wrote a few haiku when my babies were little. Hmm, time for another listen to a Tocatta and Fugue.

Re: Principia Mathematica is modern and insightful

#124

Post by lanstin » Thu, Aug 13, 2026, 2:54 PM UTC

floxy wrote: post #22

voxadam wrote: post #13

Earlier quoted context omitted.

It's been years since I cracked open a copy, maybe I should give it another shot.

If 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-Gensl...

The 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.

Re: Principia Mathematica is modern and insightful

#125

Post by lanstin » Thu, Aug 13, 2026, 2:57 PM UTC

karmakurtisaani wrote: post #56

WillAdams wrote: post #17

Earlier quoted context omitted.

There'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 spaci…

> the current version is 3.141592653 Another instance of a "clever" joke that becomes annoying very fast.

Or delightful if you used to have pi digit contests in math camp as a middle schooler.

Re: Principia Mathematica is modern and insightful

#126

Post by ux266478 » Thu, Aug 13, 2026, 3:02 PM UTC

ogogmad wrote: post #104

igravious wrote: post #63

Earlier quoted context omitted.

It 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

Most 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 mathema…

> Most foundations are in a sense equivalent. Therefore, "ZFC" is as good of an answer as any.

This doesn't follow. The sense in which they are equivalent is that they are equifinal, which doesn't mean isomorphism or even homomorphism. It's a meaningful thing in theory, but not in reality. Otherwise, Turing tarpits wouldn't be a thing.

Every foundation occupies a unique region of proof space. Your foundation, and everything that goes into it, doesn't just affect the shape of what's accessible to you in native semantics, it also effects the way you move through this space. This means by changing foundation, not only can we prove things that we otherwise couldn't in theory (in native semantics), it also means we can prove things we otherwise couldn't in practice (what embedding other foundations as object languages doesn't get you). You can recognize a little bit of this in that it makes some things seem easy, but that's an extremely trivial case of what this relationship implies.

It's all just tools in a toolbelt. Treating them like immutable, universal truths is worth tolerating merely out of human limitation, because it's a lot of work to build intuition for a foundation. If we're talking about philosophy of mathematics though? No, it would be a mistake to pretend like choice isn't meaningful. It is extremely meaningful, and there's a lot to be gained out of realizing they're actually just highly specialized tools. Something to grab when it's useful, and throw away when it's not.

Re: Principia Mathematica is modern and insightful

#127

Post by rramadass » Thu, Aug 13, 2026, 3:07 PM UTC

eru wrote: post #102

rramadass wrote: post #87

Earlier quoted context omitted.

My point is that each of the material above (and i forgot to add the original Quantum Mechanics papers) were major watershed moments in science/mathematics and hence are not easy to read/understand. You need a good background in the subject matter(and mathematics) and/or somebody guiding you through them. So what i do is try and find books which are written for the "educated common reader" by an expert who guides you…

> My point is that each of the material above (and i forgot to add the original Quantum Mechanics papers) were major watershed moments in science/mathematics and hence are not easy to read/understand. I must dissent. Einstein's Annus Mirabilis papers are really quite approachable. You don't need a book to guide you through. Though if you are having fun with the book, more power to you! Enjoy!

We have to agree to disagree then. You definitely need a good background in Physics to understand them.

For a general reader, even though they might not get the whole thing they can still get an appreciation for the whole from the commentary/details added by the editor.

The book contains the English translations of the 5 papers with Einstein's original introduction and a short discussion on each, a informative introduction and a foreword by Roger Penrose.

Re: Principia Mathematica is modern and insightful

#128

Post by lanstin » Thu, Aug 13, 2026, 3:09 PM UTC

rramadass wrote: post #37

derrida wrote: post #12

Earlier quoted context omitted.

No 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 embar…

Well said. Thanks for calling out these sort of posers and charlatans on HN. We should not tolerate these people if we are to discuss/argue/motivate interesting/hard subjects productively. I automatically discount anybody on HN (until i have looked at their profile/comment history/any personal bio websites etc.) who claim they have read/studied a) Euclid's Elements b) Newton's Principia c) Maxwell's Treatise on Elect…

As an Oxford undergrad I read a lot of Bourbaki on topology, to supplement the lectures; they were recommended by the lecturer. I loved their style of writing and the cleanliness of the approach. I have not been assigned Gödel's paper but have read them in classes where the proof was demonstrated. Maths is hard tho, I agree on that, and maths reading very hard, and reading these books outside of class where you are forced to keep going till you understanding is only rarely worth it outside of professional life.

Re: Principia Mathematica is modern and insightful

#129

Post by theonemind » Thu, Aug 13, 2026, 3:17 PM UTC

bazoom42 wrote: post #117

tristramb wrote: post #77

"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 f…

Have someone refactored it into a more concise and modern version?

It seems like a frontier model LLM could probably do it in day, probably less. Someone would have to read and correct it, though

Re: Principia Mathematica is modern and insightful

#130

Post by js8 » Thu, Aug 13, 2026, 3:31 PM UTC

TimorousBestie wrote: post #119

js8 wrote: post #51

Earlier quoted context omitted.

I 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 lamb…

> 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). Interesting, dropping this link here for others: https://treecalcul.us/

Yes. In type theory, the term carrying a type is a metalogical notion. In triage calculus, you can define a typechecking program that operates on a quoted term and normalizes only if the term typechecks.

This means in triage calculus (unlike in lambda calculus, which lacks means to quote programs) you can include expected input data types in your programs.

You can also construct any type theory syntactically by putting together a set of terms in triage calculus which only normalize when composed with correct types.

In triage calculus, you can then study types and propositions as any other programs - using self-interpretation. But I believe, as I detail below, a univalence principle is needed, to postulate the equivalence of metalogical triage calculus and its representation within triage calculus.