Comment by glimshe

18 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.

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glimshe

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gumby  18 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  16 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.

    [0] https://en.wikipedia.org/wiki/Axiom_of_reducibility [1] https://www.gutenberg.org/files/78255/78255-h/78255-h.htm#Pa...

  • jibal  13 hours ago

    Utter nonsense ... there is no known logical error in PM. Gödel proved that Russell and Whitehead's goal was unachievable but that's a totally different matter.

    OTOH, Russell found a logical error at the heart of Frege's work, and PM fixed it by introducing the theory of types.

  • robobro  14 hours ago

    That's not how to spell Ludwig Wittgenstein!

    • steppi  14 hours ago

      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."

  • voxadam  17 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.

kjellsbells  17 hours ago

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  17 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  17 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

    • gjm11  4 hours ago

      The version of this story I heard is in Littlewood's "A Mathematician's Miscellany"[1] and it's a sigma rather than an x. But he tells it as something that happened specifically to him -- he wrote a memo that ended with "thus sigma should be made as small as possible", and that bit was absent but there was in its place a very very tiny sigma. Unless he's outright lying, I think this one actually happened!

      [1] The more recent edition is titled "Littlewood's Miscellany"; I am fairly sure this story is in both the older and the newer version.

keltor  17 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.

  • derrida  17 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.

    • rramadass  14 hours 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.

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    • nimih  16 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.

      [1] https://cfreer.org/18.515/

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