[FOM] Query for Martin Davis. was:truth and consistency

Aatu Koskensilta aatu.koskensilta at xortec.fi
Mon Jun 16 02:35:35 EDT 2003

Bill Taylor wrote:

[snip points directed to Sazanov)
> [---]  Let us imagine that someone, around 1900,
> first proved that 1893....21 was prime.    He used only methods known for
> centuries, just took somewhat more time about it than anyone else ever did.
> I can ask:  "Before he proved it, was it TRUE?"
> VS can only interpret this as "was it provable?"  But no, not even that,
> because I doubt he'd want to admit the Platonic meaning of "provable but
> not yet proved" any more than he would of "truth".     So he can only
> interpret the question as "before that had it been proved?", which we all
> agree is trivially false.   So he is virtually reducing math to history;
> and actually only to attested history, not even events themselves.

I believe that a very important maxim of healthy discussion is to 
picture the most plausible view for your opponent. I don't think you've 
done that here. Of course, I might be guilty of the same with your post 
here :)

We could think, for example, that "truth" and "false" are concepts used 
thusly: if \phi is a statement and |- \phi, then \phi has always been 
true and is true. If phi is a statement and |- \phi then \phi has always 
been false and is false. All statements are (by stipulation) either 
false or true, but we are under no obligation to know the turth value of 
any specific statement. Also, if at any point we discover new methods of 
proof (so that |- becomes |-') we are to retroactively classify 
statements as true or false so as to agree with |-'.

A formalist (in the Hilbertian sense) could say that truth and falsity 
as defined above are 'ideal' notions, i.e. they do not have a 
verificationistic (or finitistic) truth conditions, but they provide us 
with very *strong* method of producing finitistic theorems, similarly as 
one might hold (as Hartry Field does) that mathematics serves merely as 
a conservative (as regards to nominalistic statements) extension of 
nominalistic theories that shortens the derivations in an extreme fashion.

> Could we not ask him, "I have rolled this dice under a hat,
> now I look and see it is a 4; so was it a 4 just before I looked?"

Similarly, one could argue that the concepts of truth and falsity are 
ideal in the formalistic sense (as presented above) also as regards to 
empirical statements. Notice that there is no *contradiction* in 
assuming that all objects vanish when no-one's looking, it simply is 
silly; there's no reason to assume they vanish, so we opt for simpler 
"ideal" notions.

Similarly, no contradiction can arise even if we let the truth values of 
mathematical statements wiggle around as much as they please, so long as 
the theoremhood and truth agree on the provable points. Why one would 
want to do this is something I can't fathom. The formalist is not, 
however, forced to let the truth values oscillate randomly or be 
undefined; he can introduce truth as a "formal" notion as presented above.

> I think, with his views, he will have to say "no", or "I dont know",
> or "it is meaningless", or some other such response; where we would all
> unhesitatingly say "yes of course it was!"

Not necessarily. He might have at his disposal the ordinary notion of 
truth, merely recast as a formalist ideal notion.

P.S. I'm not saying that these are my views by any means.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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