tag:blogger.com,1999:blog-3722233.post4833099759906403906..comments2024-04-19T18:30:53.405-05:00Comments on Computational Complexity: Can you feel the machine?Lance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-3722233.post-70984959165767129042024-03-30T14:11:38.664-05:002024-03-30T14:11:38.664-05:00Diagonalization is a constructive process. Going f...Diagonalization is a constructive process. Going from double negative to positive is the nonconstructive part.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-36690574770134869732024-03-21T08:58:55.677-05:002024-03-21T08:58:55.677-05:00I don't disagree at all. I eschew the word ...I don't disagree at all. I eschew the word 'cheating,' as I don't want to choose sides when studying the history of maths.<br /><br />I believe Brouwer would have agreed with the "cheating" connotation; he went for purely procedural maths only (to put it very crudely). Wittgenstein and Poincare are other examples.<br /><br />As a comp. scientist, however, I can appreciate the many diagonal arguments in the literature. <br /><br /><br />Edgarhttps://www.blogger.com/profile/14255710142894208595noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-15736806105651740202024-03-21T07:12:10.369-05:002024-03-21T07:12:10.369-05:00"Cantor's famous diagonal argument contai...<i>"Cantor's famous diagonal argument contains a procedural component"</i><br /><br />All mathematicians are somehow "cheating", they use procedural arguments while pretending that maths ontology is timeless.<br /><br />This is especially clear with Russel paradox: <br />Okaaay, here is the set of sets which don't contain themselves (which sounds a pretty natural concept), Oh! Dang! Where do I do I stash it now?<br /><br />CHEATING with time!!!<br /><br />Without such "cheating" you could say goodbye to all diagonalizations, including Godel incompleteness and the halting problem. :-D<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-25772814928493416772024-03-21T03:29:19.446-05:002024-03-21T03:29:19.446-05:00'Feeling the machine' is awesome. I believ...'Feeling the machine' is awesome. I believe this 'operational' way of doing mathematics on a completed infinitely large object/tape/matrix (that is, a procedural way of thinking, yet with a static ontology) is due to Turing et al., and especially Cantor.<br /><br />In my understanding, Cantor's famous diagonal argument contains a procedural component that explains how to construct the anti-diagonal. It pertains to a completed matrix, which represents a full-fledged bijection between finite strings and bit sequences. This, essentially, presents a Platonic backdrop.<br /><br />So, arguably, we see Cantor -- and, later, Turing even more so -- apply causal reasoning (= feeling the machine) in a non-causal (Platonic) landscape.<br /><br />Fusing the causality of a (mathematical) machine with the non-causality of (mathematical) Platonism is something Cantor was permitted to do, since he had idealistic leanings -- according to at least one Cantor scholar (i.e., Jane).<br /><br />We need the static backdrop to diagonalize out of a completed class of, say, linear bounded automata. We employ the procedural approach to construct a mathematical machine situated beyond the full-fledged class.<br /><br />This post wonderfully captures the essence of abstract mathematics in comp. science.Edgarhttps://www.blogger.com/profile/14255710142894208595noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-45380824624135557422024-03-20T11:57:43.407-05:002024-03-20T11:57:43.407-05:00I meant "to me". Fixed.I meant "to me". Fixed.Lance Fortnowhttps://www.blogger.com/profile/06752030912874378610noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-45046598084753747482024-03-20T11:52:28.581-05:002024-03-20T11:52:28.581-05:00"... Information doesn't sing do me but I..."... Information doesn't sing do me but I can see the compression." <br />sing to me or do me?<br />Is this a typo or I'm missing something? located at (n-1)th paragraph.<br /><br />great read!rezanoreply@blogger.com