From: ben.usenet@bsb.me.uk   
      
   Richard Heathfield  writes:   
      
   > Still, let me lay my somewhat arid sense of humour aside for a moment and   
   > take one serious crack at explaining the rationale that lies beneath my   
   > previous 'contributions' to this thread.   
   >   
   > To argue for a mathematical model (such as a Turing machine) that never   
   > halts necessarily and /obviously/ entails the claim that a mathematical   
   > model can exist in perpetuity, for if the model ceases to exist, so does the   
   > Turung machine.   
      
   That is not at all obvious.  In fact, it seems like a deliberately   
   perverse interpretation of what a mathematician means by a theorem.  The   
   TM model exists at the moment, and theorems about TMs are about what we   
   know about that model now.  The TM that just "writes blank and goes   
   right" can be proved to "not halt" right now.   
      
   Aside: I use "scare quotes" because I don't like metaphors that can be   
   taken too literally.  I'd write out the model, and give the state   
   transition function, but that would add too much bulk.  Just note that   
   when I say "not halt" I mean that the sequence of configurations arising   
   from the state transition function is not finite (i.e. can proved to map   
   onto to a proper subset of itself).   
      
   > But such models themselves exist only in the minds, writings and   
   > inventions of mathematicians and scientists, and all the artifices and   
   > devices of thinking creatures will end with the heat death of the   
   > universe and the consequent inability for devices to do   
   > work.   
      
   But the statement, made now, that "this TM right here:  does not halt" is not a statement about what will be   
   known and understood for all time.  It is simply a statement about what   
   we know now.  2+2=4 is a reasonable thing to assert now, given our   
   current shared agreement about the symbols and basic arithmetic.  To   
   challenge it on the basis that there will one day be no minds to know   
   any arithmetic is to misinterpret what the claim is.   
      
   > Therefore, to argue that an unending program and its   
   > encompassing mathematical model can exist necessarily requires one to   
   > argue, in essence, for a non-corporeal and /intelligent/ life force to   
   > persist after death not only of the individual but of the entire   
   > universe.   
      
   I've never heard anyone else claim that a model needs to "exist" for any   
   more time than it takes to make statements using it and have them   
   understood by whoever we are talking to.  You appeared to dispute a   
   claim made about some model we could reasonably be assumed to share.   
      
   We might all suffer profound mathematical amnesia tomorrow, but I was   
   still correct to claim, yesterday, that not all TMs halt, just as you   
   were wrong, yesterday, to dispute it.  We won't know, tomorrow, what on   
   earth we were talking about, much less the right and the wrong of it,   
   but at the time we made our contradictory claims, we did.   
      
   > I don't know of many computer scientists who would be   
   > prepared to argue for such persistence (because most computer   
   > scientists I know are atheists). I conclude that a computer scientist   
   > who argues that an unending Turing machine is possible is very likely   
   > to be suffering from cognitive dissonance.   
      
   In my experience, religious people are not very good an understanding   
   what's going on in the minds of atheists.  And while the reverse is also   
   true, most atheists have stopped trying.   
      
   Very specifically: I don't argue for an unending Turing machine!  The   
   notion need only exist for as long as it takes for a few theorems to be   
   proved about that notion -- about the same times it takes for a number   
   of people to pop up on Usenet to say that they have a refuted those   
   theorems.   
      
   I see mathematics as an exploration of rules: set up some rules and see   
   what can and can't be the case about the things named in the rules.  If   
   we play chess, and you win, that fact remains a fact that we can talk   
   about (and even pass into history) as long as the words and rules remain   
   understood.  That not all TMs halt was a game that was concluded last   
   century, but we both still know the rules.   
      
   > If you choose to reply, I will of course read your reply with interest, but   
   > I may well not reply in turn because I intend to at least attempt to refrain   
   > from contributing further to this thread, as I, too, am making too much of a   
   > trifle.   
      
   Well I thought this part of the was interesting.  I find it curious,   
   though, that you see the whole meaning of mathematical truth as a   
   trifle.   
      
   --   
   Ben.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)