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Principia Mathematica is modern and insightful

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Re: Principia Mathematica is modern and insightful

#131

Post by ogogmad »

ux266478 wrote: post #126
ogogmad wrote: post #104

Earlier quoted context omitted.

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 ev…

> This doesn't follow. The sense in which they are equivalent is that they are equifinal, which doesn't mean isomorphism or even homomorphism

That's what I meant. I also tried to provide one justification (out of many) for why looking at other foundations is still useful.

Re: Principia Mathematica is modern and insightful

#132

Post by rramadass »

my-next-account wrote: post #96
rramadass wrote: post #92

Earlier quoted context omitted.

I have tried to read some articles and watch some videos "explaining" Godel but never really understood it. Everyone seems to be explaining the mechanics of what Godel did but explaining the Why is lacking i.e. What was it in mathematics that got him even thinking on these lines in the first place? Can this problem be demonstrated with a simple toy axiomatic formal system? How did he hit upon his approach? Answers to…

I think that Gödel did all of this stuff because of David Hilbert basically posing the challenge to make maths' foundations consistent and complete. https://en.wikipedia.org/wiki/Hilbert's_program

And the reason for Hilbert's program? The problem of "Russel's Paradox" which is a contradiction in naive set theory - https://en.wikipedia.org/wiki/Russell%27s_paradox (Note that there were other paradoxes too).

Hilbert's idea was that by completely formalizing mathematics on a axiomatic/deductive basis, one can mechanically derive proofs so that you don't run into paradoxes/contradictions.

But then Godel showed such a formal system applied to basic mathematics can never be complete (if consistent) and never prove its own consistency.

Re: Principia Mathematica is modern and insightful

#135

Post by layer8 »

radford-neal wrote: post #109

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 le…

In what way do you think this is useful over parentheses?

Re: Principia Mathematica is modern and insightful

#137

Post by incorporeal »

fultonn wrote: post #107
rramadass wrote: post #37

Earlier quoted context omitted.

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 se…

Seconding that that one stood out to me. Actually studying at-least large portions of it is typical at a handful of liberal arts colleges that favor the "great books" approach, at least a couple of which have pretty good reputations and are likely to have turned out some folks who work in tech (maybe the programmer next to you... maybe your manager's manager), plus it's pushed in great books home learning programs that surely at least a fair number of people have credibly attempted, even if the overwhelming majority of those who start such programs don't complete them (and it's usually very early in those programs, so even those who gave it a real shot but abandoned it before getting far were likely exposed to quite a bit of Euclid, though maybe they dropped off before On Conic Sections or other texts common in those reading sequences).

... plus it's relatively approachable as such things go, and short enough that closely reading most or all of it isn't a crazy idea, and it was recently-enough widely used as an actual textbook that between that and ongoing modern interest in its use in that capacity, there are tons of study-oriented editions of it floating around and still being published. I mean hell "recreational mathematics" is a thing and lightly-annotated-and-updated Euclid's a pretty solid text for people with that kind of interest to noodle on, with bonus historical appeal since it's super-old and also is assumed background for all educated people into at least the early 20th century, so pops up all the time in historical writing and literature.

Now, Newton? That's more like it. Nobody reads a large amount of his mathematics unless they're some variety of mathematical historian.

Re: Principia Mathematica is modern and insightful

#139

Post by jonjacky »

Principia Mathematica Maps and Table Site (PM-MATS):

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

Re: Principia Mathematica is modern and insightful

#140

Post by incorporeal »

tirutiru wrote: post #74
nonameiguess wrote: post #58

Earlier quoted context omitted.

I don't know the exact curriculum and I'm sure it's changed over the years, but one of my girlfriends from back in the day really did go to a school that had one like this. St. John's College, which has two campuses in Annapolis, MA and Santa Fe, NM. They had no majors and everyone learns by reading the classics directly. They also have to learn classical Latin and Greek and read many in their original languages. I d…

I heard about St. John's from a twitter thread and find it deeply baffling. It's as if a group of monks wanted to keep the quadrivium and trivium but their clock stopped at the 16th century. One of their faculty proudly said they study analysis by reading Descartes! Which I thought was a highbrow joke but nope, dead serious. There's a reason that 'standing on the shoulders of giants' is a thing. Dive into the classic…

I think their deal is taking seriously the "college is about learning how to learn" thing, and direct engagement with the output of people regarded as greats in their fields on the assumption that, when possible, that's a good idea for obvious reasons (whether that's true or not in some rigorously-provable way, I can't say, but the reasons one might suspect that it is seem clear enough)

Some report it's pretty damn effective at that and leads to an impressive breadth of intellectual confidence in tackling material of almost any sort, but IDK. Anecdotes.

IIRC (it's been a while since I looked into their programs) they do a lot of supplemental reading of newer papers, articles, and book excerpts, and tend to used updated notation when it makes sense. Plus all their classes are heavily discussion-oriented so the reading is potentially enhanced and brought "forward" by whatever their instructors and peers bring to class in their heads.

> Take one of the easier problems from Rudin. Prove that a continuous function from the unit interval [0,1] to itself has a fixed point. I wonder how a student immersed in the 'classics' would even begin to tackle this.

I don't think they tend to train mathematicians, and I think for most students (graduating from any college or university) they never, ever, ever touch the specifics of their more-advanced e.g. math classes (I think this is true even for most programmers or engineers or what have you) any time in the entire rest of their lives, to the point that entirely forgetting most of that stuff by a decade or so later and suffering for that not at all is utterly typical. How much does it matter for students who aren't going into extremely narrow vocations that they come out of them unable to perform this specific task, without first needing to study further?

The usual defense of this fact is "well it's about learning how to learn, expecting the actual content to ever matter for any but a teensy tiny proportion of the students is unreasonable" in which case... see the rest of the post.

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