FOM: The meaning of truth

[Joe Shipman]

First of all, we can agree that the notion of true-in-a-model is
unproblematic, and for mathematicians willing to commit to an ontology
containing infinite sets like {0,1,2,3,....} the problem with statements
like Goldbach's Conjecture (henceforth GC) is purely epistemological and
the statement "GC is true but not provable" makes sense (and can be
fully formalized if we choose a proof system like ZFC).

Professor Sazonov does not like to hear about "the" integers and would
not accept the structure {N,0,1,+,*} as a well-defined object.  He
therefore has trouble understanding what it could mean for an
arithmetical statement to be "just true" without reference to a theory,
and does not see how we could make any sense out of "A is true" absent a
proof of A.  Thus, he finds statements like "It is possible that the
Goldbach conjecture is true but not provable" incoherent.  I am
sympathetic to this formalist position but do not agree with it.

Professor Kanovei, while still unwilling to accept the infinite
structure {N,0,1,+,*} as a complete, well-defined object, lists three
ways in which the statement "GC is true" can be given a meaning:

(a) "it has been correctly proved mathematically"
[Kanovei did not specify an axiom system, but I will assume that he has
one, and that he will accept statements correctly proven from it as
"true" but will not necessarily accept a statement proven from a
different axiom system as "true".  I would hope his axiom system
includes PA but do not expect that it includes ZF since ZF allows a
truth definition for arithmetical statements like GC.]

(b) "it is true as a fact of nature"
[Here Kanovei refers to counting pebbles, but more generally to
statements about the physical universe.  In the case of facts that can
be established by computation, this is only different from case (a) as a
matter of scale.  For example, Appel and Haken proved (in a traditional,
humanly verifiable manner) that the 4-color conjecture 4CC was implied
by a certain logically simpler statement S that a particular computer
program had a particular output, and then verified S by a computation on
a real physical machine.  Only the size of the computation prevents us
from regarding 4CC as "proven" in the traditional sense, because we can
only duplicate the experiment (running the computer) and not verify the
proof directly, but we are still justified in believing 4CC and calling
it "true" and even saying that we have established it to be true and we
know it to be true.  The justification lies in our scientific
understanding of the way computers work and their record of
performance.]

(c) "That it is true is given in a sacred script" [I suspect Kanovei is
joking here, thinking that nobody believes in the truth of a
mathematical statement because of a religious text.  But this is not so
clear.  A committed traditional theist will probably believe not only in
the existence of infinite sets (since infinity is a traditional
attribute of God) but also in arithmetical corollaries like Con(PA) and
maybe even Con(ZF).]


In the following, I will use the Twin Prime Conjecture (TPC) rather than
GC as an example, since it is Pi^0_2 and neither it NOR its negation
can, if true, be finitely verified so far as we know.  This will avoid
some confusion.

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.

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.  The degree of faith we would have in the
truth of the sentences it provides would depend on how good a physical
model we had for it; but for sufficiently plausible physical models and
sufficiently impressive oracular performance the epistemological status
of an utterance, while not attaining "mathematically proven", might
still qualify as "scientifically known".  We "know" the 4-color map
theorem to be true although no human has verified the proof because we
have a sufficiently plausible model of how our computing machines work
and they have a sufficiently good performance.  The only difference here
is that the physical experiment we run will not necessarily be
algorithmically representable so that a human cannot "in principle"
verify it as he could for the proof of 4CC.

-- Joe Shipman