FOM: ultrafinitism again

[Kanovei]

> Date: Wed, 8 Nov 2000 02:18:53 +0100
> From: Robert Black <Robert.Black at nottingham.ac.uk>

This is my last letter to this discussion. 

I began by the claim that (ontologically) there is NO 
mathematical statement true but not provable. 

Despite this thesis was almost unanimously criticized, 
NO counterexample was presented, 

other than in the assumption 
of "ontological" existence of *standard model of N* 
(in which case there is simply nothing to discuss in virtue 
of straightforward application of Goedel's theorem) 
or equivalent assumptions where the standard morel is hidden 
behind mathematical facts based on it. 
I quit with a hope that the failure to deny my thesis 
(other than in the abovementioned metaphysical assumption) 
will give the opponents some matherial to think about the 
nature of mathematics and its relationship with reality. 

The rest is an answer to Mr. Robert Black, it is by necessity 
rather long because so was Mr. Black's very interesting letter.

> interesting Russian
> tradition in  the philosophy of mathematics with which I associate the name
> of Yesenin-Volpin and which I know too little about. 

If you "know too little about" why don't you just write 
"Y-V tradition in the philosophy of mathematics" ? 

Prior to the rest, let me say once again that I am not at all  
a philosopher, or, if you want, my philosophy as mathematician 
can be formulated as follows. 

i)   any statement must be supported by some evidence, 
     otherwise it is disqualified as scientific statement
ii)  criteria of taking evidence into account are different 
     in different branches of science
iii) mathematics, which pretends to give an "absolute" knowledge, 
     requires "absolute" evidence, the only known form of this is 
     called mathematical proof.  
iv)  mathematicians admit different kinds of partial evidence, 
     in particular gathered by methods of natural science, 
     but statements supported by a partial evidence are 
     called "conjecture", "hypothesis". 
v)   the above is well known to any mathematician

> ... what sense I take it to be 'ultrafinitist'.
> 
> 1.  The only things which really exist are concrete material objects, and
> there are (or for all we can know may be) only finitely many of them.

There are 3 major (kinds of) things that "exist", philosophically, 
(A) material things, 
(B) ideas which belong to the social life of society (e.g. mathematical 
    theorems, Goedel's theorem, its misinterpretations, etc.)
(C) individual "thoughts" of any human 

There are different "professional" philosophies which tend to 
treat interrelations between (A),(B),(C) wrong way. 
To the matter of the discussion the most 
important example of those is NATURPHILOSOPHY, which means that 
things of kind (B) are illegitimately mixed with those of (A) while 
properties of (B) as ideal objects are assumed to be also their 
properties in the domain of (A). In scientific word this usually 
takes the form of following a prespeculated dogm in research. 

Examples of naturphilosophy follow. 

Example 1. A dogm that the Earth is the immovable center of 
universe forced Ptolemy to come up with a strange theory of 
"epicycles" to explain the orbitation of planets. 

Example 2. Goedel's theorem, assumed as "true" in the world (A) 
by the "true-non-provable" fom-subscribers. 

Example 3. Phrase 
"there are (or for all we can know may be) only finitely many of them" 
(see above)
which relates a mathematical concept of finiteness to the world (A) 
where it AT LEAST has to be explained as what is its (finiteness') 
intended (A)-meaning

> 2.  Therefore, any truth (in the full-blooded, correspondence sense of
> truth) is true in virtue of facts about these objects. 

Everything (A)-true is true by virtue of facts. 

> Further: any set of
> axioms which has no finite models thus has, strictly speaking, no models at
> all. 

As mathematical (B)-fact wrong, as (A)-fact meaningless until the  
involved notions are explained.

> Sentences of mathematics purporting to quantify over an infinite realm
> of abstract objects must be regarded as fictional, 

they are "living" in (B), not in (A), they become fictional 
when a naturphilosopher puts them in (A) without an accomodation.

> for there are no
> abstract objects, and there is no infinite realm of anything.  

A mess of (A)- and (B)- meaning. 
To ask me do I accept "infinite realm of anything" in the (A)-sense 
please first explain what does it mean in the (A)-sense, until then 
the question is qualified as senseless.

> 3.  When we say that a sentence of mathematics is 'true', all we mean (or
> all we should mean) is that it has a formal proof (in ZFC, say). Such a
> formal proof is a finite material object, 

1st of all there is an idea of proof, living in (B), 
that is, a proof as it is traditionally understood. 
For me Brauer's hedgehog theorem is an established mathematical 
fact despite nobody has converted its proof to computer-checkable 
(A)-form. 
Proofs can be, in principle, formalized, i.e. converted to 
something which begins to obey (A)-rules. 
(There are details here, a program still has its idea, hence 
some (B)-side.) 
I can understand that the formalized proof is *finite* only 
in the sense that it has finite number of symbols, not in 
any ethernal preloaded Platonic sense of finiteness that you may 
have in mind. 

> and thus its existence (or
> perhaps its possible existence?) can be a matter of objective truth in the
> full-blooded correspondence sense. 

Mathematical ideas can be subject of full formalization 
and full formal verification (there are of course known 
problems with this, cf. Mr. Bauer's post), but the 
formalizability normally adds little if anything to their (B)-values 
(4-color case needs separate comments which I am not ready to)

> Professor Kanovei will no doubt let me know if the above three points are
> not a reasonable summary of his position. (I don't myself accept (1), but
> (2) seems to follow from (1) all right, and once one has swallowed (1) and
> (2), (3) is pretty natural.)

I tried to give a reasonable summary, which therefore is NOT 
the above three points of yours. 
The problem is that we may speak in different languages. 
This leads me to (C)-matters. I fancy, western 
scientific life has so strong roots in scholastics, that the idea 
of Platonic numbers, naturally, (B)-dwellers, and the like, 
has almost genetically 
penetrated in (C) of a western scientist as something about (A). 
Scholastics was practically unknown in Russia, unless in the 
form of hesychasm (if this has anything to do with scholastics), 
so I am absolutely not precharged to pay tribute to this concept 
while you may find this so wild that your first idea is to 
immediately qualify me as a kind of dropoff using academical 
synonimic definitions like "super"-something. 

> Now: the reason I call the position ultrafinitist is that it is a position
> which only allows for the existence of a bounded, finite number of objects.
> It is thus much more restrictive than the potential infinity of
> intuitionism or of Hilbertian 'finitism'.

I call the position "materialism of a scientist". 
It seems that you are more happy to call it "ultra"..., so be it, 
you know better.
Anyway I would readily accept "infinity" in world (A) if you show 
me one and explain me what does it mean being infinite and 
demonstrate that it holds, in scientific way.  
Sentences like "I believe in it", "I follow <name>'s teaching in that" 
do not substitute a demonstration.

> Note that on this position incompleteness results come out as trivial. Even
> theories which are classically complete will come out as incomplete, since
> there will be sentences just short enough to fit into the universe, but
> such that neither their proof nor their disproof is short enough to fit
> into the universe.

Again and again, (B)-things are moved to (A), their meaning lost. 
You need to specify the (A)-meaning of "theory", first of all, 
if you want to discuss this at (A)-level.

> Now I take Kanovei further to be saying that Goedel's results must be seen
> as *mathematical* results, e.g. the second incompleteness theorem for ZFC
> does not really say that if ZFC is consistent (i.e. if there are no
> physical objects fulfilling certain conditions) then Cons(ZFC) is not
> provable in ZFC (i.e. there are no physical objects fulfilling certain
> other conditions), all it really says is that a certain sentence of
> arithmetic, viz. Cons(ZFC) -> not-Prble ([Cons(ZFC]), *is* provable in ZFC.

We must understand all the time are we talking in terms of (A) 
or of (B). 
Goedel's theorem is a mathematical result, i.e. (B). 
A principle known to any educated engineer says that 
(A)-applicability of ANY mathematical result needs validation 
of correspondence between mathematical and real circumstances 
(is friction taken into consideration, are all significant 
influences accounted for, etc.). 
This validation can be short in case of 2+3=5 and can be long in 
other cases. Goedel's theorem belongs to the "other" category. 
In fact I would take it as making sense that 
the second incompleteness theorem for ZFC
does say that if ZFC is consistent (i.e. if there are no
objects fulfilling certain conditions) then Cons(ZFC) is not
provable in ZFC (i.e. there are no objects fulfilling certain
other conditions), with the reservation that at some level 
of understanding *objects* may be (B), at another level of 
understanding may be (A), the key point is to always know 
what are we talking about, (A) or (B), here is the point where 
the naturphilosophical mistreatment easily occurs. 

> What I do not really understand about this position is the relation between
> the formal result and our expectation (which surely Professor Kanovei
> shares) that if we could find a physical object which was a proof of
> Cons(ZFC) in ZFC, then we could also find a physical object which was a
> proof of 0=1 in ZFC. 

I do share this, as a hypothesis or expectation, moreover, 
I am close to consider this as an (A)-fact, at least, after 
some clarifications which take into account effects connected 
with quantum-mechanical properties that may be related to 
cosmologically big quantities, in general, I would  
like to be able to evaluate *sizes* of proofs, but this seems 
to be "doable" because everything is localized on 
Goedel's proof, so a specialist has to check whether there is 
an expansion of length somewhere. I just don't know.
 
> More generally, I don't understand how Professor
> Kanovei thinks formal mathematics gets applied.  

Any educated physicist knows how. 
Observe facts, create a theory, check whether it fits to 
a bigger massive of facts, if this fails try again. 

If indeed you ask why in general (B) and (C) in any way 
are correlated to (A) we have to appeal to natural history. 
The origins of (B) may go to, say, 1 my ago. Those of pre-humans 
who had the idea of a tiger as a slow harmless friendly plant-eater 
eventually perished, survivors were those who had proper idea 
of tiger -- this is how (B) was being formed. 
(I hope you will not ramify the discussion by asking how I know 
that tigers and the natural history itself is not just a   
feverish nightmare of the "Ich".)

> The other problem which I raised is the possibility that accepted
> mathematical proofs might fail to be formalizable in ZFC in the sense that
> the full formalization would be too long to fit into the universe. 

This is an interesting issue but not what I carry of primarily. 
I accept the (B)-existence of mathematical proofs (as perhaps 
all mathematicians practically do), so if a proof is formalizable 
for computer verification - fine, otherwise also is not red-alarming. 
But the issue has to be studied by specialists (I am not). 

V.Kanovei