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Comment by gumby | Hacker News Reader

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.

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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...

The flaw was the belief that it was possible (though they were not sure that they two could pull it off). Gödel showed that it was not an impossibly difficult task but an actually impossible task.

A failed hypothesis is not the same thing as a logical error. Would they have been more free of flaws if they believed it was impossible and thus didn't try? We'd all be the poorer for it

It cannot be overstated how flawed your own statements here are. You shouldn't be talking about logical error regarding a work of logic when you don't even know enough logic to know what a logical error is.

And there was no "flaw" in attempting to resolve foundational paradoxes in set theory, primarily to address Frege's error and deal with Russell's paradox, or in attempting to prove that all mathematics can be derived from pure logic. Seeking to demonstrate a hypothesis that is later shown to be erroneous is not "flawed", else the entire knowledge-seeking enterprise is "flawed".

And switching from "logical error" to the very vague notion of a "flaw" is goalpost moving that looks a lot like bad faith.

I won't comment further.

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.

That's not how to spell Ludwig Wittgenstein!

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."

>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.

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.

I showed up to first day at university and an older student talked to me for like two minutes before declaring that I needed to read GEB. I dutifully went ahead and bought it, and I still work in logic today.

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.

It's been years since I cracked open a copy, maybe I should give it another shot.

If you want to get to the meat of the issue, without the extraneous stuff, you might also tackle: Godel's Theorem Simplified.

https://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensl...

The extraneous stuff is honestly fabulous, IMO of course. Just being able to listen to a four part fugue sensibly is a rare but accessible pleasure. And amusing dialogs as an instantiated dialectic for showing the truth synthesized from apparent opposites, is a great pedagogical learning.

Extremely agree. The extraneous stuff is what makes the book magical.

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

Are you still married?

finally we shall be able to answer the age old question - does Gödel, Escher, Bach: an Eternal Golden Braid stick with you better than a spouse!

Godel, Escher, Bachelor?

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

I’m working on it every day during lunch break. Good old hardcopy.

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.

What hasn't aged well? I don't think any of it is dated to any particular time period.

GEB is not a difficult read. It is delightful!

It is a popular science book which catches the vibe of mathematical logic in an excellent way. It is not a textbook, nor a piece of research. It's all vibes, but high-quality vibes. If you are in the right headspace it can be really inspiring!

I had my copy signed by Prof Hofstadter; a treasure to this day. Led to a maths degree, years of organ playing classes, a career in software, and a Zen practice going on 20 years. And I'm pescatarian. I even wrote a few haiku when my babies were little. Hmm, time for another listen to a Tocatta and Fugue.

I'm surprised to hear that that was your perspective! I felt that it dealt with otherwise-opaque topics in a very approachable way.

My brother's favorite book in 6th grade was Godel, Escher, Bach.

Why, yes, he works as a compiler engineer.

> 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.

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.

You should see his book "Fluid Concepts and Creative Analogies, which is entirely about his experiments with AI in The Fluid Analogies Research Group with his grad students. Although I wonder if those techniques could be paired with LLMs somehow.

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.

It was under Western Philosophy.

To each their own, I read GEB several times and enjoyed the intellectual challenge and the humour.

Hofstadter wrote a followup book: I am a strange loop.

https://en.wikipedia.org/wiki/I_Am_a_Strange_Loop

He wrote a number of follow-up books. I love GEB, but the follow ups were often disappointing in surprising ways.

He has another book about the beauty and challenges of translating poetry, but it’s actually about the sudden death of his wife and it's been too sad for me to finish.

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