Re: [math-fun] Re: radio math
----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Saturday, October 27, 2007 12:44:41 PM Subject: Re: [math-fun] Re: radio math On 10/27/07, Henry Baker <hbaker1@pipeline.com> wrote:
Cantor-type diagonalization is the major tool Goedel's work and in
computational complexity, so Cantor's work does find a role in certain parts of computer science.
It's an interesting challenge to figure out what role diagonalization
might play in "real" life. I'm not entirely convinced that the similarity of the Cantor diagonal argument to the Russell paradox / Goedel proof / Cretan liar et al is sufficient grounds for drawing the latter into a discussion about infinities. Perhaps it might be argued that diagonalisation leads to a new object in an open-ended system (e.g. the continuum in the cardinals); but in a closed system it leads to a contradiction. At any rate, I am convinced that diagonalisation by itself does impinge very immediately on "real life", in a manner which does not seem well appreciated. There is an obscure corner of mathematical logic and computer science known as "situation theory" [one reason for its low profile may well be that none of its practitioners will actually admit to being a situation theorist!]. Situation theory employs axiomatic set theory to model the concept of truth in a closed universe where the active agents are capable of self-reference. The central revelation is that (loosely) in such situation, either there is are questions whose answers are unknown to some agents, or else there are answers about which agents must disagree. There is a very readable account of all this material in Jon Barwise & John Etchemendy's book "The Liar". Failure to appreciate the ramfiications of this theorem have resulted elsewhere in the generation of an enormous amount of otherwise (possibly) well-informed and entertaining twaddle --- a situation which will doubtless persist, whatever I say! Fred, This sounds interesting. Can you give us an example of how situation theory impacts ordinary life? Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com
On 10/28/07, Eugene Salamin <gene_salamin@yahoo.com> wrote:
Fred,
This sounds interesting. Can you give us an example of how situation theory impacts ordinary life?
Gene
It's a long time since I looked at this --- I'd probably still be ignorant of it if my daughter hadn't done her Ph.D. in it </ END OF BOAST> --- so I hope I'm not going to commit too many gross errors. My previous "loose" assertion was a trifle too loose for comfort: I'll try again. In a universe containing agents who are capable of self-reference and belief, there must exist statements which are disbelieved by some agents, but are nonetheless true. It's easy enough to construct artificial examples along the lines of the Cretan liar of course, but to a layman theyare obviously flawed by failure to correspond to any conceivable real-life situation --- in a more refined formal system (type theory, well-founded set theory) they may indeed be meaningless (not well-formed). Trying to answer Gene's question at the time, I concocted this "parable of the good driver". The well-wheeled citizens of Great Wittering conduct a survey to establish those qualities required for safe car (automobile) driving, and use it to rank themselves in order of merit. It transpires that the only relevant feature is that a driver should not overestimate his own technical abilities; on this basis they publish a list, in which the terminally unassuming J. Alfred Prufrock is rated the safest driver around. Now everybody else can look at this list and say to themselves "Yup, that's right --- JAP is the best in the business". But at the moment he reads it and agrees with it, it becomes untrue! There are plenty of objections one might make to this scenario: the simplistic rating criterion, the time-dependency, the ambiguity of defining "best" with respect to many variables, etc. Nonetheless, we routinely make similar real-life assessments of this type --- and apparently rational people routinely disagree violently about their outcome. Also one marvels how successful, intelligent and capable businessmen, politicians, and others in time almost inevitably succumb to episodes of irrational hubris, leading to self-destruction. I find it intriguing to speculate that perhaps simple logical necessity might be driving this behaviour! Fred Lunnon
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Eugene Salamin -
Fred lunnon