Name: From Frege to Gödel
ISBN: 9780674324497

What Hacker News says

pronFeb 2017

> Can you link to some of the references you cite? That seems like an interesting read.

Sure.

It's best to start with Jean van Heijenoort's seminal From Frege to Gödel A Source Book in Mathematical Logic, 1879-1931[1]. It includes the original texts by Hilbert, Brouwer and many others, but the text I personally found most interesting was Hermann Weyl's, which I quoted here[2] nearly in full (in the parent comment I quoted some relevant passages from Brouwer and Hilbert; in fact, it was as part of that Reddit discussion that I found the references and learned what little I know of the philosophy of mathematics).

I've also found Juliet's Floyd's discussion of Turing's (and Wittgenstein's) mathematical philosophy[3] fascinating. She pinpoints how and why Turing viewed the philosophy of mathematics the way he did, how that view led him to the discovery of computation, and why people like Gödel and others found that to be such a profound philosophical breakthrough.

The Stanford Encyclopedia of Philosophy's entry on the philosophy of mathematics gives a good overview[4].

> The intuition about computable functions is that they are precisely the functions whose output for any given input depends only on a finite part of the input.

Once you know what computation is, you can define it in many ways. But why would that definition coincide with what we call computation? You can define foo to be such functions, but why are you calling them computable? And, BTW, even the very definition of finite possibly (I'm not sure about this point) requires reliance on the physical due to the multiple models of FOL and the problems with SOL.

> The really interesting thing about the Church Turing thesis is that there are so many different models of computation that capture precisely this idea

It is absolutely trivial to come up with a super-Turing mathematical model (e.g. just use reals, as in "pick the supremum of this bounded set of reals", or use a Turing machine with an infinite number of heads). All those other models are somehow based on capturing an intuitive notion of the human thought process, which is completely physical, and therefore, it is not surprising that they coincide. Turing was just the first who was able to give the notion a precise mathematical meaning. If you go back to the inception of those ideas, from Brouwer's intuitionism, Hilbert's formalism and even to the far older notion of the algorithm, you see that they're all tied to the capabilities of a physical human mind.

It's true that not all of those ideas actually mention physics because not all thinkers necessarily believed the mind to be completely physical, but they're all based on the idea of a limited mind, which I think we can safely call physical.

[1]: http://www.hup.harvard.edu/catalog.php?isbn=9780674324497

[2]: https://www.reddit.com/r/programming/comments/5k1v04/is_math...

[3]: https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf

[4]: https://plato.stanford.edu/entries/philosophy-mathematics/

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pronJan 2017

I would recommend Andrew Hodges. Read this to get a taste: Did Church and Turing Have a Thesis about Machines?[1] (spoiler alert: probably)

For the history of logic, Jean van Heijenoort's[2] annotated anthology, From Frege to Gödel[3] is fascinating.

And, BTW, I think your hunch (about Whig history) is correct.

[1]: http://www.turing.org.uk/publications/ct70.pdf

[2]: https://en.wikipedia.org/wiki/Jean_van_Heijenoort

[3]: https://www.amazon.com/Frege-Godel-Mathematical-1879-1931-Sc...

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ehudlaAug 2016

A nice book to pair with GEB is /From Frege to Godel: A Source Book in Mathematical Logic/. Certainly also better for someone who might find GEB a bit too idiosyncratic.

And, of course, if you want to simply read the proof, there's Nagel's /Gödel's Proof/.

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daviddaviddavidJan 2016

I have been through a slew of logic texts. By far, my favorite for both theory and practice is Computability and Logic by Boolos, Burgess and Jeffrey.

http://www.amazon.com/Computability-Logic-George-S-Boolos/dp...

Another very good book on nuts and bolts proofs in both propositional and predicate logic is Beginning Logic by E. J. Lemmon.

http://www.amazon.com/Beginning-Logic-E-J-Lemmon/dp/09151445...

And a great collection of primary texts covering the history of modern logic is From Frege to Godel..., edited by Jean van Heijenoort.

http://www.amazon.com/Frege-Godel-Mathematical-1879-1931-Sci...

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daviddaviddavidFeb 2023

Here is a the letter from Russel to Jean van Heijenoort in response to the latter's request to print the correspondence between Russell/Frege in "From Frege to Godel" (https://plato.stanford.edu/entries/russell-paradox/).

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As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege’s dedication to truth. His entire life’s work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known. (Quoted in van Heijenoort (1967), 127)

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dtornabeneMar 2016

Could you post some links on the soundness comment? I share a fellow siblings comment of a little skepticism? But I'm neither a professionally nor scholarly logician, though I've read and owned (a couple of times) Van Heijenoorts From Frege to Godel, so feel free to throw some meat my way if you follow up with this comment.

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