Comments on: mathematical formalism, computing, and the question of truth http://thepoliticsofsystems.net/2010/02/26/mathematical-formalism-computing-and-the-question-of-truth/ Thoughts on Power and Software Wed, 11 Aug 2010 16:43:18 +0000 http://wordpress.org/?v=2.9.2 hourly 1 By: bernhard http://thepoliticsofsystems.net/2010/02/26/mathematical-formalism-computing-and-the-question-of-truth/comment-page-1/#comment-2728 bernhard Sat, 22 May 2010 20:26:02 +0000 http://thepoliticsofsystems.net/?p=60#comment-2728 Hello mk, thanks for your comment! I do believe however, as I wrote in the post, that the idea that Turing's work on the Entscheidungsproblem is "based on the rejection of Hilbert's principles" is flawed, because it only takes a very narrow historical window into account. Certainly, Turing put another nail into the coffin of an idea of mathematics where every problem is in principle solvable, but his work was done inside of Hilbert's formalist paradigm. The idea that mathematics is nothing but symbol manipulation without any referent to an external world as judge of correctness was most forcefully defended by Hilbert - and the Grundlagen der Geometrie indeed became a model for 20th century axiomatic mathematics - and both Gödel's and Turing's work is, in that sense, a continuation rather than a break. Sure, Hilbert's program would never be completely fulfilled, but the logical and metamathematical space opened by the German mathematician is precisely the one that would later give birth to the computer... For the second part of you argument, I completely agree, but the real problem is probably not the logical question but the sociological attraction of what Gallison calls mechanical objectivity... best, B. Hello mk, thanks for your comment! I do believe however, as I wrote in the post, that the idea that Turing’s work on the Entscheidungsproblem is “based on the rejection of Hilbert’s principles” is flawed, because it only takes a very narrow historical window into account. Certainly, Turing put another nail into the coffin of an idea of mathematics where every problem is in principle solvable, but his work was done inside of Hilbert’s formalist paradigm. The idea that mathematics is nothing but symbol manipulation without any referent to an external world as judge of correctness was most forcefully defended by Hilbert – and the Grundlagen der Geometrie indeed became a model for 20th century axiomatic mathematics – and both Gödel’s and Turing’s work is, in that sense, a continuation rather than a break. Sure, Hilbert’s program would never be completely fulfilled, but the logical and metamathematical space opened by the German mathematician is precisely the one that would later give birth to the computer…

For the second part of you argument, I completely agree, but the real problem is probably not the logical question but the sociological attraction of what Gallison calls mechanical objectivity…

best,
B.

]]> By: mk http://thepoliticsofsystems.net/2010/02/26/mathematical-formalism-computing-and-the-question-of-truth/comment-page-1/#comment-2725 mk Thu, 20 May 2010 08:25:51 +0000 http://thepoliticsofsystems.net/?p=60#comment-2725 First, Turing's thesis that underlies Computer Science is based on the rejection of Hilbert's principles. The thesis has proved that mathematics is not powerful enough to uncover truth as it was earlier thought to be. Second, the problem solving using computers has two sides. While the formal side is based on logic pure mathematics, the empirical side is subject to interpretation because it has to do with practical applications that needs to engage with human reality. The latter is really outside the realm of mathematical reasoning; otherwise, we would not have read Penrose's Emperor's New Mind or about Searle's Chinese Room Experiment! First, Turing’s thesis that underlies Computer Science is based on the rejection of Hilbert’s principles. The thesis has proved that mathematics is not powerful enough to uncover truth as it was earlier thought to be. Second, the problem solving using computers has two sides. While the formal side is based on logic pure mathematics, the empirical side is subject to interpretation because it has to do with practical applications that needs to engage with human reality. The latter is really outside the realm of mathematical reasoning; otherwise, we would not have read Penrose’s Emperor’s New Mind or about Searle’s Chinese Room Experiment!

]]>