FOM: The meaning of truth
V.Sazonov at doc.mmu.ac.uk
Wed Nov 8 10:30:50 EST 2000
Some late reaction.
Joe Shipman wrote:
> The following seems like it might be acceptable to Professor Sazonov:
> The only way that we could come to KNOW that there are arbitrarily large
> twin primes (which is the same as saying "TPC is true" as far as I'm
> concerned; the general truth predicate, as opposed to
> truth-in-a-given-model, has the property that saying " 'A' is true" is
> the same as saying "A") is by a mathematical proof. The only way we
> could come to know TPC is not true (that there is a largest twin prime)
> is by a mathematical proof.
Yes, if we consider TPC as a mathematical statement.
> By the Principle of Parsimony, we should therefore not introduce a
> notion of "truth" that is distinct from provability because it is not
> needed and accomplishes nothing for us.
> But I disagree with this (does Kanovei?). In addition to "sacred
> scriptures", I would also allow the possibility of empirical discovery
> of a mathematical-sentence-generating oracle which had never been known
> to emit a sentence that was known to be either false or inconsistent
> with its previous utterances.
I ignore mentioning the oracle. As to empirical confirmation
of some mathematical theorems or hypothesis, it is quite different
story. Empirical confirmation of Pythagorean Theorem considered as
a statement of Geometry-as-Physics is quite possible and reasonable.
But this cannot be considered as a scientific proof of Pythagorean
Theorem as a statement of Geometry-as-Mathematics. As I remember
such distinctions were clearly made by Hilbert.
Analogous considerations may be applicable to 4CC and to any other
mathematical statement having also some interpretation in the reality.
It is also a different story, whether mathematically proved theorem
having some interpretation in the reality will be confirmed by
experiments. It is desirable, and usually it takes place, probably
with some (even many!) exceptions. Let me recall my old question:
is log log n (of unary presented natural number n) a bounded
function (mathematically and practically)?
I do not know why so often here in FOM list these distinctions
are ignored or not clearly pronounced. Sometimes it is clear
from the context that the author is completely aware of these
distinctions, but sometimes I even do not know what to think.
Mathematical truth (as provability? or as information received
from the God?) is mixed in a bad cocktail with experimental
Will you, Professor Shipman, give me and us know what is your real
opinion on these distinctions. Do you agree with what I wrote above?
Otherwise I do not understand the above
> But I disagree with this
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