XPost: rec.arts.books.tolkien   
   From: jcb@inf.ed.ac.uk   
      
   On 2014-06-05, Stan Brown  wrote:   
   > On Thu, 5 Jun 2014 11:47:24 +0000 (UTC), Julian Bradfield wrote:   
   >> Most (rational) people take as an axiom "nothing is true unless it's   
   >> evidenced"   
   >   
   > No, rational people take as an axiom "Nothing is _proved_ without   
   > evidence."  There are things that are true but not proved: Godel   
   > showed that this is just how the universe works. The (non)existence   
   > of a deity is one of those: there is insufficient evidence to decide   
   > the quetsion either way, but clearly any given deity must either   
   > exist or not.   
      
   Sigh.   
   Proof does not require evidence, in its *mathematical* sense: a proof   
   *is* (incontrovertible) evidence of the truth of the thing it proves,   
   assuming that the proof system is sound.   
   Gödel did not show that there are things that are true but not proved:   
   he showed that *any given (suitably powerful and well-behaved) formal   
   proof system* cannot prove all true sentences. The Gödel sentences are   
   true, because we prove them to be true outside the formal system we're   
   studying.   
   The existence of God is not analogous to this: it is more analogous to   
   the equally profound result  that there are sentences whose   
   *truth* is independent of any given formal system. For example, the   
   Axiom of Choice (AC) is a hugely useful, but very dangerous,   
   mathematical axiom. Mathematicians are divided about whether it's   
   true, and nobody could ever prove it's true. Then Cohen showed that   
   set theory without AC is not capable of establishing either AC or its   
   negation - you can choose to do mathematics either accepting or   
   denying AC, and your choice affects what you can prove and how you can   
   prove it.   
   So mathematicians no longer argue about whether AC can be proved - we   
   know it can't - but they still argue about whether it's "true", by   
   bringing up evidence and counter-evidence.   
      
   "proof" in everyday English does not mean mathematical proof - it   
   simply means "supported by enough evidence that I no longer think it   
   worthwhile to doubt the truth".   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)