So... 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.
glimshe 4 hours ago [-]
If 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 4 hours ago [-]
You 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.
steppi 2 hours ago [-]
This 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.
Wittgenstein 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."
4 hours ago [-]
voxadam 4 hours 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.
m-hodges 4 hours ago [-]
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.
eru 33 minutes ago [-]
I read it cover to cove back in the day and enjoyed it. But I'm not sure more people should read it today.
If they do and enjoy it, good for them! But many parts haven't aged all that well.
However, I can still very warmly recommend 'The Pleasures of Counting' to this very day.
buildsjets 2 hours ago [-]
I’m working on it every day during lunch break. Good old hardcopy.
anthonygd 3 hours ago [-]
I read it on my honeymoon 25 years ago. That book sticks with you.
debo_ 50 minutes ago [-]
Are you still married?
voxadam 3 hours ago [-]
It's been years since I cracked open a copy, maybe I should give it another shot.
floxy 2 hours ago [-]
If you want to get to the meat of the issue, without the extraneous stuff, you might also tackle: Godel's Theorem Simplified.
GEB 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.
scubbo 3 hours ago [-]
I'm surprised to hear that that was your perspective! I felt that it dealt with otherwise-opaque topics in a very approachable way.
annzabelle 2 hours ago [-]
My brother's favorite book in 6th grade was Godel, Escher, Bach.
Why, yes, he works as a compiler engineer.
suslik 1 hours 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 32 minutes ago [-]
Well, it's a pop-sci tome, not a research article.
However, it is pretty dated these days.
3 hours ago [-]
eru 34 minutes ago [-]
Eh, that's just a pop-science tome. Nothing impenetrable about it.
I 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?
WillAdams 3 hours ago [-]
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 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.
inigyou 3 hours ago [-]
apocryphally a typesetter saw "make x as small as possible" at the end of a math problem to be typeset, and did exactly that
taneq 51 minutes ago [-]
“Find x” <— here it is!
keltor 3 hours ago [-]
It 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.
mathisfun123 2 hours ago [-]
link the syllabi for the classes
debo_ 48 minutes ago [-]
What if they link the axioms for the classes and leave it as an exercise to the reader to derive the syllabi?
derrida 3 hours ago [-]
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 embarrassed, that is the consequence for lieing. There is such a thing as intellectual honesty.
nimih 2 hours ago [-]
Honestly, other than the length and tedious presentation, I don't really think the material in the Principia Mathematica is outside the reach of an advanced undergraduate. As a point of reference, MIT's capstone mathematical logic course[1] has a syllabus that requires at least as much mathematical maturity, and it wouldn't really surprise me that much to see it as an ancillary or excerpted text.
That said, even if the OP was assigned the text at some point as an undergraduate, I remain a bit doubtful it was actually read.
> As a point of reference, MIT's capstone mathematical logic course[1] has a syllabus that requires at least as much mathematical maturity,
The textbooks they use in that course are written in modern notation and are accessible to a knowledgeable reader; neither can be said of the Principia Mathematica. The archaic syntax is a serious issue.
rramadass 39 minutes ago [-]
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 Electricity and Magnetism d) Einstein's 1905 Annus Mirabilis papers. e) Principia Mathematica by Russell/WhiteHead f) Godel's Theorem g) Bourbaki's mathematics books etc. etc. They might have browsed it out of curiosity but that is not the same as reading/studying it.
Actual conceptual mathematics/science is intrinsically hard even ignoring the archaic language/notations.
As a good example; the Nobel-prize winning physicist S.Chandrasekhar wrote Newton's Principia for the Common Reader where he explains a subset of the principia (only dealing with gravitation) using modern notation and language. He himself found it quite hard and thus the "common reader" in the title is somebody who has had a good course in calculus and has the motivation to put forth the effort in understanding it.
eru 22 minutes ago [-]
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 get yourself a translation.)
I don't think I'm smart enough to casually read and understand original works on General relativity, but the Annus Mirabilis work seems much simpler. The famous E=MC2 paper is only three pages.
About Gödel: if you are interested in the theorems, and not necessarily their original presentation, you can get plenty of rigorous modern treatments. It's very common for mathematicians to work out simpler proofs and more appealing presentations of famous results over time.
See https://dn721807.ca.archive.org/0/items/uber-formal-unentsch... if you want to give one of Gödel's work a go. It's only 26 pages. Footnote 48a is especially interesting. Overall the prose is crisp, but the notation is rather archaic to modern eyes.
I agree with your general sentiment, and your heuristic in general.
mathisfun123 3 hours ago [-]
I'm with you - I hate when people exaggerate their bonafides beyond all belief
derrida 3 hours ago [-]
LLMs 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”.
TimorousBestie 2 hours ago [-]
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 to mathematically-inclined Haskell novices).
voidhorse 3 hours ago [-]
I 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.
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/
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.
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 [1] https://www.gutenberg.org/files/78255/78255-h/78255-h.htm#Pa...
Which leads us to our next borderline impenetrable book, Gödel, Escher, Bach by Douglas Hofstadter.
If they do and enjoy it, good for them! But many parts haven't aged all that well.
However, I can still very warmly recommend 'The Pleasures of Counting' to this very day.
https://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensl...
Why, yes, he works as a compiler engineer.
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.
However, it is pretty dated these days.
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.
It's also pretty typically a part of History Of Mathematics and Philosophy of Mathematics courses.
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.
That said, even if the OP was assigned the text at some point as an undergraduate, I remain a bit doubtful it was actually read.
[1] https://cfreer.org/18.515/
The textbooks they use in that course are written in modern notation and are accessible to a knowledgeable reader; neither can be said of the Principia Mathematica. The archaic syntax is a serious issue.
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 Electricity and Magnetism d) Einstein's 1905 Annus Mirabilis papers. e) Principia Mathematica by Russell/WhiteHead f) Godel's Theorem g) Bourbaki's mathematics books etc. etc. They might have browsed it out of curiosity but that is not the same as reading/studying it.
Actual conceptual mathematics/science is intrinsically hard even ignoring the archaic language/notations.
As a good example; the Nobel-prize winning physicist S.Chandrasekhar wrote Newton's Principia for the Common Reader where he explains a subset of the principia (only dealing with gravitation) using modern notation and language. He himself found it quite hard and thus the "common reader" in the title is somebody who has had a good course in calculus and has the motivation to put forth the effort in understanding it.
(Of course, if you don't read German, you should get yourself a translation.)
See https://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstei...
I don't think I'm smart enough to casually read and understand original works on General relativity, but the Annus Mirabilis work seems much simpler. The famous E=MC2 paper is only three pages.
See https://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstei...
About Gödel: if you are interested in the theorems, and not necessarily their original presentation, you can get plenty of rigorous modern treatments. It's very common for mathematicians to work out simpler proofs and more appealing presentations of famous results over time.
See https://dn721807.ca.archive.org/0/items/uber-formal-unentsch... if you want to give one of Gödel's work a go. It's only 26 pages. Footnote 48a is especially interesting. Overall the prose is crisp, but the notation is rather archaic to modern eyes.
I agree with your general sentiment, and your heuristic in general.
- “effort to refute bullshit is order of magnitude more than to refute it”.
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).