[FOM] What is a proof?
John T. Baldwin
jbaldwin at uic.edu
Mon Oct 27 21:49:46 EST 2003
Stephen G Simpson wrote:
>John T. Baldwin 27 Oct 2003 writes:
>
> > (Of course it would only be terminological if you wanted to reserve
> > proof for the weak notiion ;-) used by fom and `good' proof for the
> > stronger notions that I am suggesting.)
>
>Dear John,
>
>I imagine your ;-) is intended as an indication that we are not to
>take your statement too seriously. And, you have ackowledged the
>usefulness of reverse mathematics.
>
My wink is just twitting you on terminology. I am just giving a partial
formalization of what stands a good mathematical practice.
I am in the midst of trying to understand some work which is
burdened by proofs which are usually correct in the sense of fom; but
are not proofs in the sense I am
speaking of -- they contain at least 30 or 40 % correct statements which
are not used for the Lemma or Theorem at hand.
So the difference between a proof and a good proof is very real to me at
the moment. I gave a formal
definition, not at all vague, of one requirement of relevance.
This is a perfectly good formal notion which would improve the
explanatory value of a proof while not affecting its verifability
(not psychologically but formally ) one whit.
Simpson writes
>
> Definition. Let T be a theorem of second order arithmetic, Z_2.
> Let P be a formal proof of T within the formal system Z_2. We say
> that P is *explanatory* if the conjunction of the fintely many Z_2
> axioms used in P is logically equivalent to T, over RCA_0.
>
>This rigorous notion of *explanatory proof* may not do everything you
>want it to do, but it is good enough to make some useful distinctions.
>
>
As I said before.
It makes some useful distinctions; it does not come near to doing
everything I want it to do .
>
>
More information about the FOM
mailing list