FOM: The meaning of truth

[Kanovei]

> Date: Wed, 01 Nov 2000 14:38:53 -0500
> From: Joe Shipman <shipman at savera.com>
 
> ... 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 

For anybody "willing to commit to an ontology containing" unicorns, 
and also willing to be a scientist, the problem is to present a 
unicorn to colleagues. 
So, are you going to present "infinite sets like {0,1,2,3,....}" 
in "ontology", that is, as I understand, empirically existing, of sort ?
Are you also going to claim that those "sets" are determined enough to 
claim that any statement "like GC" is true of false in the same manner 
as 1=1 is true and 0=1 is false ? 
This means that you will finally let everybody to have a look at the 
notoriously invisible platonical "bats" flying between pages of 
philosophical books since centuries ago.  

> 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,

Why don't you explain to Professor Sazonov what does 
or "could" it mean, in order to relieve the poor guy from his 
troubles of understanding ?

> Professor Kanovei, while still unwilling to accept the infinite
> structure {N,0,1,+,*} as a complete, well-defined object

If in the ontological sense, yes, very much unwilling.
It is since Renaissance if not earlier that anybody in the world 
of science claiming of having seen a unicorn was kindly asked to 
present an evidence. I only follow this pattern.

> (a) "it has been correctly proved mathematically"
> [Kanovei did not specify an axiom system 

I wrote "proved mathematically". 
Choice of axioms is not what was the subject of discussion. 

> 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.]

Has this anything to do with the ontological truth ? 

> (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, 

Computation is a method of mathematical proof

> 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.  

What this has to do with the point of discussion ? 
The 4CC proof can be and is regarded as mathematical. 
Do you mean that it is not ? 
Of course we have to believe that the computers are not faking us 
and also, to that extent, that dosen or so of specialists who really and 
fully understand the recent proof of FLT are not just plotting in order to 
get fat grants, but I am pretty sure we are in these beliefs much less 
credulous than you who believe in   
"an ontology containing infinite sets like {0,1,2,3,....}".

> (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. 

Read carefully, I wrote about "sacred scripts", not about any religy 
nor any deity nor a someone's confession, of course. 
By those I mean, to be concrete, the known metaphysical thesis of 
Goedel's misinterpretators, which is nothing but a sacred script of 
a branch of modern pseudophilosophy  of mathematics. 

> 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).]

A "committed traditional theist", if also a scientist, first of 
all separates scientific matters from matters of faith. 

> 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.

Both of the two can be verified only by mathematical proof, which, 
with respect to not-GC, can be just a computation provided 
an unexpected counterexample is found. 

> 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

The notion of "truth" in the system mathematician vs. mathematics 
(ie, ontological truth, so to speak) is well known: true is what 
has been mathematically proved. 
If somebody wants to give any other notion this must be done in 
scientifically correct manner, at first. Is it needed and how it 
will be useful is another matter.

> But I disagree with this    

You disagree because you do not argue in scientifically correct manner as 
the following habbracadabbra shows.

> 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.  

So you add a like of medieval folktales to "sacred scripts", here we go...

> 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".  

A mathematical statement qualifies as "scientifically known" if 
it is mathematically proved. The proof can include computation, 
in particular, made by a computer, provided the circumstances of 
the computation are mathematically verifiable and reproducible. 
In addition, another program can convert the computation in a 
formal proof in ZFC (in the 4-color case, probably, in PA), and yet 
another formally check the correctness, so I see no problems to 
qualify proofs that include human-uncomputable computation in this way.  

> We "know" the 4-color map
> theorem to be true although no human has verified the proof 

This is wrong, the proof is verified because experts verified 
the program, observed the repetitious runs of computation on 
independent computers, and have sufficient knowledge as how those 
computers work.

> 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.
 
Any kind of experiment, which cannot, in principle, be 
transformed into a mathematical proof (the 4-color proof can) 
cannot qualify as a proof (of a mathematical statement) 
to a sound mathematician.   

V.Kanovei