http://lists.econ.utah.edu/pipermail/marxism-thaxis/2005-March/018429.html
[Marxism-Thaxis] Les Shaffer on Kurt Gödel Jim Farmelant farmelantj at juno.com Wed Mar 16 11:40:48 MST 2005 Previous message: [Marxism-Thaxis] Does Gödel Matter? Next message: Re: [Marxism-Thaxis] Les Shaffer on Kurt Gödel Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] -------------------------------------------------------------------------------- --------- Forwarded message ---------- From: Les Schaffer <schaffer at optonline.net> To: Marxmail <marxism at lists.econ.utah.edu> Date: Wed, 16 Mar 2005 11:37:41 -0500 Subject: [Marxism] Re: godel etc (was ...) Carlos A. Rivera wrote: > I argue that incompleteness in mathematics and uncertainity in quantum > mechanics actually point to materialist dialectics. As dynamic, > never-ending systems, they exhibit the same continous struggle that > dialectics call, while firmly footed on a materialist grasp on reality. > > Yeah, postmodernists eat your heart out!!! the thing that impresses me this time around is expressed nicely by R B Braithwaite in his Introduction in the Dover edition of Godel's paper: "... Godel, in this paper which established his two great theorems by methods which are constructive in a precise sense, on the one hand showed the essential limitations imposed upon constructivist formal systems (which include all systems basing a calculus for arithmetic upon "mathematical induction"), and on the other hand displayed the power of constructivist methods for establishing metamathematical truths." Godel's efforts in 1930 depended on a clever arithmetization of metamathematical statements (so-called Godel numbering), which Braithwaite likens to Descartes handling of problems in geometry by the introduction of coordinate systems and a reduction to algebra. by (crudely speaking) mirroring arithmetic in statements about arithmetic, he was able to establish limitations in the formal program of Hilbert and others. In some fundamental sense human (mathematical) activity cannot be reduced to formalism alone, such formal systems are incomplete. Somehow this incompleteness was interpreted by philosophers and popularizers as a fundamental attack on the structure of mathematics itself. but is any marxist here surprised to learn there is more to sensuous (mathematical) activity than formal reasoning? i doubt it. in some way what Godel really demonstrated was that the formal metamathematical systems created at the turn of the 20-th century were not so meta and outside of the system they were judging, so-to-speak, as had been supposed. it takes more than arithmetic to practice true arithmetic. > "The question whether objective truth can be attributed to human > thinking is not a question of theory but is a practical question. Man > must prove the truth - i.e. the reality and power, the this-sidedness of > his thinking in practice. The dispute over the reality or non-reality of > thinking that is isolated from practice is a purely scholastic question." one of the intriguiging aspects of Godel's paper was the construction of a __formally__ undecidable proposition that was demonstrably __true__. Braithwaite again: "The undecidability of some arithmetical propositions within the deductive system S may be classed among the syntactical metamathematical characteristics of the system S (represented by the calculus P [Les: the formal system P]) for the reason that this undecidability derives from the undecidability of some formulae within the calculus which represents S. Deductive systems, unlike calculi [Les: formal systems] have also semantical metamathematical characteristics; in particular their propositions have or lack the semantical property of being true -- what Godel in his introductory Section 1 calls being "correct as regards content" (inhaltlich richtig). Connecting the syntactical property of being provable with the semantical property of being true ... gives an additional kick to the undecidability in S of g {Les: g is the formally undecidable but true proposition] -- by adding that g is true. ... This metamathematical argument, which combines semantical with syntactical considerations, establishes the truth of an arithmetical proposition which cannot be proved within S. In his introductory Section 1 Godel intermingles semantical with syntactical considerations in sketching a proof of the undecidability of g ... The distinction between what is syntactical and what semantical was not made explictly until a year or two later (by Tarski, whose work included rigorously establishing unprovability theorems that were semantical) ... " while thinking about Godel's work this afternoon i was reminded of long lunches i had with a computer science professor at Cornell back in the mid-80's. he had a ton of $$$$$ from the US military for investigating programming systems that could be proven errorless via machine. this was at the time of Reagan's Star Wars and the (rather mushy) criticism that it could never work in practice because it was too complex a system to be trusted to work without testing, but almost by definition could never be properly tested short of a real nuclear attack, an event the system purportedly was designed to prevent. I checked recently and it seems my former lunchtime companion is still receiving millions of military dollars to develop automated procedures for producing error-free and secure software/firmware for military applications. if you think Godel should have put a stop to all this, think again. The military-industrial complex aside, far from undermining the foundations of mathematics, Godel succeeding in opening a whole new avenue of investigation. via the work of Tarski, Barkeley Rosser (father of marxmail alumni J Barkeley Rosser), Church/Turing, and Gregory Chaitin, we now have deep connections between mathematics, computational systems, and, more recently, physics. Les Schaffer This message has been scanned for malware by SurfControl plc. www.surfcontrol.com _______________________________________________ Marxism-Thaxis mailing list Marxism-Thaxis@lists.econ.utah.edu To change your options or unsubscribe go to: http://lists.econ.utah.edu/mailman/listinfo/marxism-thaxis