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Principia Mathematica is modern and insightful

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Re: Principia Mathematica is modern and insightful

#101

Post by fn-mote »

TimorousBestie wrote: post #28

Instead 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 […])

FWIW, I am very against this recommendation. That book is needlessly opaque. I don’t know a good recommendation for category theory, but that isn’t it.

Re: Principia Mathematica is modern and insightful

#102

Post by eru »

rramadass wrote: post #87
eru wrote: post #41

Earlier quoted context omitted.

Einstein's 1905 Annus Mirabilis papers seem like they easiest of the bunch to just read through. I just pulled up 'Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen', the one about Brownian motion, and read the whole thing. It's only 12 pages and fairly accessible; more prose than equations. (Of course, if you don't read German, you should…

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!

Re: Principia Mathematica is modern and insightful

#103

Post by jcranmer »

taybin wrote: post #94
eru wrote: post #40

Earlier quoted context omitted.

Well, it's a pop-sci tome, not a research article. However, it is pretty dated these days.

How is it dated?

A 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.

Re: Principia Mathematica is modern and insightful

#104

Post by ogogmad »

igravious wrote: post #63
emil-lp wrote: post #59

Earlier quoted context omitted.

The foundation of math is (mostly) ZFC.

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 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.

Re: Principia Mathematica is modern and insightful

#105

Post by [deleted] »

TimorousBestie wrote: post #28

Instead 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,…

[deleted]

Re: Principia Mathematica is modern and insightful

#106

Post by danilafe »

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…

This was my first thought when I saw the article.

Re: Principia Mathematica is modern and insightful

#107

Post by fultonn »

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…

> who claim they have read/studied a) Euclid's Elements

Of the lot, Elements feels misplaced.

Lots of people actually do read Elements as part of their course of study. It's niche but there's a whole cottage industry within academia for that sort of thing. There are probably over a dozen institutions that have either a degree program or a core curriculum that is organized around original texts, with Euclid usually serving as the math distribution of that sequence. So running into people who have read (big chunks of) Elements is not that uncommon. That's true even IRL outside of online discussions forums on thread topics that likely select for such people.

My impression is that this is not really true of the other examples. Except maybe Godel's proofs; I do think a sufficiently motivated instructor could pull a decent chunk of college students through the original text in a semester. Probably better ways to spend everyone's time, though.

Re: Principia Mathematica is modern and insightful

#108

Post by jerf »

js8 wrote: post #51
TimorousBestie wrote: post #28

Instead 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,…

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, with the math proofs coming through from AIs lately, I wouldn't guarantee you'll see that much wrong stuff. I expect it would be at a low enough rate to keep you learning... after all, anyone who has had serious math education knows the human teachers aren't always completely correct either and there is the occasional impromptu exercise of everyone staring at the board and trying to figure out what went wrong with the demonstration.

Re: Principia Mathematica is modern and insightful

#109

Post by radford-neal »

The 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...

Re: Principia Mathematica is modern and insightful

#110

Post by wwweston »

anthonygd wrote: post #20
m-hodges wrote: post #8

Earlier quoted context omitted.

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 read it on my honeymoon 25 years ago. That book sticks with you.

I thought the recommended honeymoon reading was Knuth’s _Surreal Numbers_
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