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Can you link to some of the references you cite? That seems like an interesting read. As for computability being a physical notion, I can't speak to the motiva

by fmap 10y ago

Can you link to some of the references you cite? That seems like an interesting read. As for computability being a physical notion, I can't speak to the motivation of Church and his students, but I do know that there are characterizations of total computable functions in domain theory which make no mention of physics. 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. The really interesting thing about the Church Turing thesis is that there are so many different models of computation that capture precisely this idea (the surprising part is that these very simple models are expressive enough to cover all computable functions).

pron 10y ago
> 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 http://www.hup.harvard.edu/catalog.php?isbn=9780674324497 [2]: https://www.reddit.com/r/programming/comments/5k1v04/is_mathematics_the_oldest_legacy_system/dbv2gz6/ https://www.reddit.com/r/programming/comments/5k1v04/is_math... [3]: https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf [4]: https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/
pron 10y ago
P.S. I also strongly recommend the excellent 1988 paper by Robin Gandy, The Confluence of Ideas in 1936, published in The Universal Turing Machine A Half-Century Survey (ed. Rolf Herken), 1995[1] (pp. 51-102). Gandy gives a full historical background of who knew what in 1936, what the general state of knowledge was at the time, and the reception of both Church's and Turing's papers. [1]: https://www.amazon.com/Universal-Turing-Machine-Half-Century-Computerkultur/dp/3211826378 https://www.amazon.com/Universal-Turing-Machine-Half-Century...

by fmap 10y ago

pron 10y ago

> 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 http://www.hup.harvard.edu/catalog.php?isbn=9780674324497 [2]: https://www.reddit.com/r/programming/comments/5k1v04/is_mathematics_the_oldest_legacy_system/dbv2gz6/ https://www.reddit.com/r/programming/comments/5k1v04/is_math... [3]: https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf [4]: https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/

pron 10y ago

P.S. I also strongly recommend the excellent 1988 paper by Robin Gandy, The Confluence of Ideas in 1936, published in The Universal Turing Machine A Half-Century Survey (ed. Rolf Herken), 1995[1] (pp. 51-102). Gandy gives a full historical background of who knew what in 1936, what the general state of knowledge was at the time, and the reception of both Church's and Turing's papers. [1]: https://www.amazon.com/Universal-Turing-Machine-Half-Century-Computerkultur/dp/3211826378 https://www.amazon.com/Universal-Turing-Machine-Half-Century...