"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?
Possibly pertinent: Principia Mathematica Maps and Table Site (PM-MATS):
https://principia.lib.uiowa.edu/about.html - more description in my top-level
comment in this page.
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…
The Annotated Turing is great. Thanks for mentioning the others, I'll check them out. I've read a little of Einstein's original work (in longish excerpts) and it was a nice enough read I'd welcome more of it, especially with expert guidance.
This book is an interesting approach to The Principia: Magnificent Principia (2013), by Colin Pask https://devontrevarrowflaherty.com/2014/08/26/book-review-pr...
The article is about Russell and Whitehead’s Principia, not Newton’s.
Interestingly, there was a Show HN last year formalizing PM in Lean ( https://news.ycombinator.com/item?id=43797256 ), and the Principia Rewrite project ( https://www.principiarewrite.com ) verified all 189 propositional logic theorems (sections 1-5) in Coq against the original proof sketches
I believe the Principia Rewrite is at https://principia-rewrite.org/ .
> 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.
Fair enough. I just wanted to elaborate, because usually that specific phrasing justifies the opposite. I did mention you partially acknowledged the meaningfulness, but I felt like the point needed to be made stronger. The politics around foundations obfuscates a lot of their utility. I'm sure you're aware the tendency for randoms in a mathematics department to roll their eyes when you pay lip service to other foundations. Usually, it's not even about a sense of pragmatics, but irrational identity-protectionism and ZFC dogmatism. Things like HoTT, or any branch of TT, are percieved as "cute, but not something with any real usecase. Not like my perfect ZFC!"
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…
To me the mechanics and the why are closely intertwined. If you feel like self-referentiality is a way to demonstrate a problem (this is the “why”), it is not a long step to the mechanics of encoding. The work is in creating the theorem / contradiction from that point, but in the big picture, the approach doesn’t have to come from nowhere.
> the mechanics and the why are closely intertwined
But the "why and what" must always be explained first even if it is incomplete; since that is the problem we are trying to solve. With logic it is even more important since you can follow a proof from one step to the next but by the time you reach the conclusion you have lost the connecting thread to the starting point (for most general folks) i.e. "you have missed the forest for the trees". This is why many folks feel lost when doing mathematics as mere symbol-pushing.
>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.
Eh, that's just a pop-science tome. Nothing impenetrable about it.
I'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.
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…
The Annotated Turing is great. Thanks for mentioning the others, I'll check them out. I've read a little of Einstein's original work (in longish excerpts) and it was a nice enough read I'd welcome more of it, especially with expert guidance.
You might also want to look at The Annotated Godel by Hal Prince which i mention in another comment.
We should read the original great works but with intellectual honesty to accept that some/most of it might be over our heads (depending on one's background) and not feel ashamed about it. The point is to not be completely clueless/ignorant but at least have some correct ideas even if it is a simplification/approximation. It is just one stage in the learning process and as we re-read little-by-little over time we begin to understand more and more. This is real education.
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?
It'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.