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

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Mon Jun 9 16:06:07 EDT 2003

Bill Taylor wrote:
> ->>Do you admit the reality of 10^10^10^10 in the same sense (whatever that is)
> ->>that you admit the reality of 3 and 7 ?   Would you put these three numbers
> ->>on the same footing, ontologically.
> ->
> ->In a sense yeas, in a sense not.
> Aha.  So you can give me a definite maybe on that one, then?   ;-)
> I was expecting an answer of that type, though I was hoping for a better.
> I'm not interested in this system or that system, as in your reply.
> My query was quite specific.  I specified "ontologically".  If you
> insist that this has no meaning in the context, fine, but you could
> have at least said so.
> If you will not answer the question as asked, then there is no point
> going any further.   I should have known an ultrafinitist would
> be canny enough to avoid a definite answer.   ;-)

I replied to this question in that way how it makes sense to me. 
You omitted (although mentioned) the essential details of my answer 
which you do not like: 

> I'm not interested in this system or that system, as in your reply.

This way it is easier to characterize me as canny. If it is canny for 
you, what can I do?  For me everything DEPENDS on "this system or 
that system". You want something ABSOLUTE instead of looking for 
a solid ground in the reality of formal systems. Of course, for 
peoples with such polar starting views it is difficult to get 
mutual understanding. One (who?) does not feel other. At least, 
long time ago I have had views like yours. I can recall what it is. 
But I already cannot return back. 

For more clarity (and replying also to the posting of Robin Adams 
<Robin.Adams at stud.man.ac.uk> which I received just this moment), 
it is necessary to stress that a formal system, like PA ( + FOL 
on which it is based) is a FINITE and FEASIBLE object. I mean 
that its description comprises several lines of a text together 
with simple explanation how to use the rules of PA ( + FOL). 
This is like a finite (actually, non-deterministic) computer 
program (without any input) with a rather simple semantics,  
so simple as that of Turing machine. (Of course, additionally 
we have some informal description of the intuition about natural 
numbers related with this formalism. This involves some imagination.) 

To be more practical, FOL should be extended, of course, by some 
rules of abbreviations. Then we can generate step-by-step various 
derivations. Each of them will be, of course, of a feasible length. 
It is unnecessary to think too much about "all" such derivations 
which we can potentially generate. We have some, and can get more. 
We should rather think whether we generated something intuitively 
interesting and meaningful. 

I realize, that this is rather idealized picture because we usually 
present our derivation in a semi-formal way. But we usually believe 
that a semi-formal derivations can be transformed to quite formal one. 
If we do not feel for a particular semi-formal derivation that this 
is possible then we actually have a doubt that our reasoning was 
rigorous enough. We continue to work on this semi-formal derivation 
until this feeling arises. 

I should also note that I am not an ultrafinitist, I am rather a 
formalist in that sense as I described in FOM some time ago 
(cf. also below). By some reasons, including those inherent in 
my formalist views, I am interested also in any possibility to 
formalize the concept of feasible numbers. As I wrote, formalist 
(= mathematical) view assumes, in particular, formalizing ANY 
idea which is possible to formalize. 

> ->What (as you say) "disinformed" me? Some deeper that in the school
> ->things like Goedel's theorems, especially on incompleteness and
> ->Goedel/Cohen proof on independence of CH, what demonstrated (to me)
> ->that both N and continuum are vague concepts,
> WOW!  That's a mighty leap from a standing start!   I have sympathy with
> the view that CH-independence and the like could make you newly unsure
> of exactly what sets were; indeed this is my own case to some extent.
> I am less sympathetic to it making one uncertain about the reals, though
> even there I can see some vague concern.  But N !?  I quite fail to discern
> the psychological processes that could take you from CH-independence
> to unsureness about N.

Goedel incompleteness theorems lead me to my doubts on N. 
Proofs of independence CH lead me to the doubts on the 
meaningfulness of the powerset operation and on the general 
concept of the universe of "all" sets. All of this leads me 
to formalist views which are based on using concrete (feasible) 
formal deductions (and makes my doubts on N even stronger). 
Otherwise we have a vicious circle which was also pointed 
out recently by Dana Scott. 

> But everyone is different; and if you say it happened to you, then I'm
> sure we all believe you.
> ->Yes, yes! What are we ever talking about when discuss about
> ->this N? I know - this is some illusion. This is, I believe,
> ->a more honest position than to assert that there exists
> ->some idealized unique N. By which way unique? Illusion or
> ->idealization are something about which uniqueness makes
> ->no real sense. This is rather from religion-like views
> ->on which science should not rely.
> First of all math is not science, though I know it is usually included
> so in continental thought.  (I have heard this is mostly a linguistic
> problem, that the German word for "science" means more like what we Anglos
> would call "studies".)

I consider mathematics as a science on and an engineering of 
formal systems considered as devices strengthening our thought 
and intuition. We have a general idea on what a formal system is 
which leads to a lot of PARTICULAR formalisms (each being a finite 
system of rules) already created and potentially possible. Thus, 
we have some, quite concrete material to investigate. 

> Secondly, though it is mere mudslinging to call it religion, there is
> undoubtedly an aspect of metaphysical thought in FOM.  However one might
> like to remove it as much as possible, it seems to me (and most of us)
> that metaphysical objections to N are so absurd as to amount to throwing
> out the baby with the bathwater.

Which baby? Formalist view throws nothing, except things like 
the doubtful "standard" N, etc. We gain a lot - the freedom of 
non dogmatic thought, a wider view on mathematics. 

> ->I feel the difference, but in a sense they are equally ullusive,
> ->i.e. ALL of them are illusive.
> Then you might as well say, so are small deductions from simple premises
> illusive, so are small numbers, 

I already wrote you that numbers like 3 and 7 are not so 
illusive for me, unlike the "standard" N. The same I wrote 
on small deductions from simple premises (as on syntactical 
objects). That is why concrete formalisms may serve as a solid 
fundament for mathematics. 

so are any ideas, so even are material
> objects like tables and chairs and stones.  This cheap undergraduate
> radical skepticism is ultimately a dead end.  Declaring something
> to be illusive is no argument.
> ->> As I see it, the real problem with formalism, is that you deny the reality
> ->> of all of N, but somehow re-admit the reality of all derivations from
> ->> a system of logic.

Of course, NOT! See above and my other recent postings. 
Particular (feasible) formal deductions are quite concrete 
objects like this text. Admitting the reality of all 
derivations from a system of logic is essentially the 
same as admitting the reality of N. Here is the vicious 
circle (if to consider "all" these derivations as something 
on which N or ZF universe are based). We can easily break 
down this circle by admitting only concrete derivations. 

"All" derivations in formal system is a subject of 
metamathematics and, therefore, this is a quite different story. 

> ->
> ->Not all (I do not know what means `all' here), but only those
> ->we can really do.
> Which changes with time, understanding and technology.   So you want to
> make math reality dependent on biology and technology?   

Who knows, what will happen with mathematics when computers 
will be able to make non-trivial deductions (not just routine 

This is possibly
> a tenable philosophical view, but is far beyond what most of us would
> consider to be true mathematical thought.   You will soon be saying,
> may have already said,   "Before computers were invented  1893....21
> wasn't prime, but now it is".   What a strange and unpalatable conclusion!

It is your conclusion, not my. My formulation would be: 
"Before computers were invented  1893....21
wasn't proved to be prime, but now it is proved".

> ->> The latter [ Th(PA) ] is more complicated than the former  [N] ,
> ->> and just as big, so why regard it as more fundamental?
> ->
> ->Not so complicated. Say, PA consists of several axioms
> ->and one axiom schema. It is based on FOL based on several
> ->(schematic) rules. We easily understand how to use these rules.
> ->All of this is quite concrete, unlike N,
> Wrong.   You are comparing apples and oranges.  

Of course PA and (illusive, vague) N are things of different nature. 
But they are comparable in some sense. The former is simple (concrete), 
the latter is illusive and, of course, not simple. But the former 
may serve as a basis to make the first vague ideas on N to be more 
clear and allows us to make reasoning on N (even though N is still 
vague). As I already wrote, without formalisms like PA (this could 
be school semi-formal considerations on N) we would have N as 
something like {one, two, many}. 

> You should compare 
> the operations above with the simple operation of adding one to a
> (written down) number and writing down the answer.   I'm sure even you
> will agree that the latter is a LOT simpler and more fundamental than
> the former!

Of course, a (suitably formalized) metatheory of, say, PA and theory 
of N (i.e. PA itself) as formal systems have different complexity. 
And so what? Why should I compare them in this discussion? 

Formalist view (as I understand and described it) is BASED on 
theories like PA, not on metatheories. 

> ->I am not sure that I will convince many participants of FOM
> ->as the previous experience shows.
> I'm sure everyone will agree with that, at least.

I am sure people will eventually reject meaningless and useless 
ideas such as the "standard" N.  (I hope, it is clear that I do 
not mean here the rigorous definition in ZFC of the model of PA 
with the same name.) 

> ------------------------------------------------------------------------------
>                Bill Taylor       W.Taylor at math.canterbury.ac.nz
> ------------------------------------------------------------------------------
>                Philosophical clarity would have the same effect
>                on mathematics as sunlight does on potato shoots
> --------------------------------------------------------------------------

Here I agree. 

Vladimir Sazonov

More information about the FOM mailing list