[FOM] Question for Vladimir Sazonov

Mon Aug 14 19:04:31 EDT 2006

I wanted to stop posting (and still intend to) because of the lack of 
time, but...

Quoting Bill Taylor <W.Taylor at math.canterbury.ac.nz> Fri, 11 Aug 2006:

> I have a question for Vladimir Sazonov, who wrote:
>
>> (This is why we do not even notice the vagueness of N.
>> Just do not ask questions on vague features of N.)
>
> Can you elaborate on this please; specifically, what are some of the 
> questions
> we might (but are "not allowed to") ask about the vague features of N.
>
> Could you, perhaps, supply a definite list of (say) five such questions?

Sorry, it will be a bit long because a specific view on the natural 
numbers should be presented first.

We traditionally consider that N is "generated" by counting 
0,1,2,3,"and so on". Here the most important part "and so on" should be 
described, and the description is, of course, circular. Here a 
reference to the abstraction of potential infinity should stay. Usually 
what is the essence of this abstraction is either not explained well or 
the best what I know is presented as a description like the following 
one:

to be able to add 1 to each already generated natural number,

to be able to iterate this, which leads to addition operation,

to be able to iterate addition (multiplication),

to be able to iterate any operation arising in this (which exactly 
"this??) way,

to be able to iterate any our ability to iterate(?),

etc.,

"iterated etc".,

etc, etc, etc. ...

This is my version of the description from the Encyclopaedia of 
Mathematics published in Russia seemingly at the end of 1970th. It is 
just a vague idea on non-restricted iteration of iterations ... of 
taking successor.

Note, that it is considered here that the mere ability to add 1 is 
insufficient! Evidently, this is related with the idea of feasibility. 
Why do we need mentioning the potential infinity at all? This is not 
only to assert the closure of N under successor which seemingly does 
not guarantee its closure under + and *, etc. If we would postulate 
here only the closure under successor we would rather generate a proper 
initial part F of "feasible" numbers of N. Really, the problem is that 
we cannot count too long, even if we assume the closure under 
successor. Postulating this closure of N rather originates from our 
inability to determine what is the last possible step in taking the 
successor and is only a way to assert this inability, so this almost 
does not extend our abilities to count: just a very minimal step of 
idealisation. Of course further stronger iterating of our mental 
abilities (and allowing to use +, *, exp, etc.) should lead to longer 
initial parts of N than F. (For example, I have mentioned in one of my 
posting the class P (extending F) of polynomial numbers closed under + 
and *. Assume also closure under <. So, we can have longer and longer 
extensions of F approximating N.)

All these "etcetera" in description of Potential Infinity actually 
assume in some way (or are strongly based on) "feasible number of 
steps" because this is related with our mental activity which is, of 
course, feasible (some real actions of our physical brains).

Anyway, our ancestors have found a way how to "collapse" all these 
"etcetera" into (or to make some "jump" to) some formal theories of N 
like PRA or PA which are quite comfortable to work with. (For PRA this 
seems more straightforward. For the case of PA or HA ? Heyting 
Arithmetic ? we seemingly should understand the iteration as higher 
type primitive recursion and use something like Goedel Dialectica 
Interpretation or go along the idea formulae-as-types. What should be 
new here is suitable incorporation of the concept F of feasible 
numbers.) Therefore, once a formalisation (PRA, PA) has been found 
which ignores any new ideas related with feasibility, the illusion 
arises that we have a unique row of natural numbers described by these 
theories (with any vagueness or possible non-uniqueness of N happily 
ignored due to inability to express this in the traditional first-order 
formal languages), that we have a solid concept N of the row of natural 
numbers. But what is really solid and is the base of our "crystal 
clear" illusions are these formalisms. They are quite formal and rigid, 
and we have *developed* a very good intuition allowing to work even 
without explicit reference to these formalisms. But the real concept N 
(the set of natural numbers) behind these formal systems is 
nevertheless extremely vague being generated by some not very clear 
mental process like the above description of potential infinity with 
all vagueness inherited from this generating process.

Now, I think it should be clear what kind of "vague" questions can be 
asked on so generated N. Something like the following:

* Does the initial part F (of feasible numbers) of N has the biggest number?

* What is the (precise?) upper bound of F in N?

* Which are further properties on F (assuming it has no biggest number) 
and also on the Medium and Small numbers, etc. (formally) defined in 
terms of F (see my paper "On feasible numbers")?

* Further questions on the structure of the whole N generated according 
to the above description of abstraction of potential infinity. For 
example, which are longer (than F) initial parts of N corresponding to 
stages of the above process and whether and how they exhaust N. How 
"long" is N generated this way? Is N uniquely defined by this 
abstraction/process?

* In which precise sense the abstraction of potential infinity 
generates N? Is this abstraction uniquely understood or there are 
various its versions leading to different versions of N? (Something 
like mentioned above difference between primitive recursive functions 
vs. functionals over N.) How the abstraction of potential infinity 
should be properly formulated so that the generated N be a model of a 
given arithmetical theory (PRA, PA, arithmetical part of ZFC, etc.)? 
How to understand the term "model" above? Note that we are in a 
framework rather different form ZFC where the traditional model theory 
"resides". Say, F is not a set in the ordinary sense of this word.

* Are different versions of N, N?, ? (arising according to different 
clarifications of potential feasibility) linearly ordered by inclusion? 
Or may be they are branching? (Who knows?) If they all are linearly 
ordered, is there any sense to ask whether the longest N does exists 
(what could be probably called the standard model of arithmetic). By 
the way, I strongly doubt that such a longest N exists (or can be 
constructed).

* Which of these questions are stupid and which are meaningful 
(formalisable)? At least F is indeed formalisable in a definite sense 
and some questions on it have unexpected/unusual formally derivable 
answers.

I hope we should agree that these questions are really about N (taking 
into account its genesis, how its idea appears and is developed in our 
minds).

The traditional formalisations of N consists, in particular, in

(i) FORMAL restricting the natural human language so that the above 
questions become illegal and in

(ii) FORMAL description of axioms and proof rules allowed to (legally) 
derive (legal) theorems.

But what prevents us from appropriate including into formalisation of N 
the other informal ideas (of potential infinity) related with the 
genesis of N thereby adding to N an additional structure (say, of 
initial parts of N, like F, with some closure properties)?

There is the opinion that mathematical concepts like N, Euclidean 
Geometry, continuity, etc. can exist BEFORE and INDEPENDENTLY of any 
formalisation. I strongly disagree with this. Before formalisation we 
have only extremely vague idea on a concept which does not deserve to 
be called mathematical. Only *in the course of and due to 
formalisation* we obtain something really mathematical. Of course, this 
is a process, and it can have some stages. The limit stage is some 
contemporary formal system like PA and ZFC. But even in the case of 
Euclid when the contemporary standard of formal system was not 
developed yet, we already had quite reliable and sufficiently 
formal/rigorous approach to the Geometry. The process of formalisation 
of the Euclidean Geometry was practically finished by Hilbert.

As to the natural numbers, the very counting process, especially when 
the decimal notation was invented (with the algorithms of addition and 
multiplication) served as a kind of formalisation. Some other templates 
of reasoning on N were suggested and repeatedly used. All of us in our 
childhood go through the way of our ancestors in developing the idea of 
counting to its formalisation. And this process is not so trivial. Ask 
your children (say, at 4, 5 as I asked once my daughter) whether the 
biggest number exist or not. This is not so self-evident. Then we do 
some of the above steps of potential infinity which are also not so 
self-evident. We approach to the Induction Axiom somewhere at the end 
of the school. Moreover, this is usually enforced by the teacher. The 
teacher tells us: DO IT THIS WAY. This is just a kind of a FORMAL or 
MECHANICAL TRAINING, of course provided also with supporting informal 
comments. Some our "stupid" questions or opinions (like "is there the 
biggest number?" are usually interrupted as illegal instead of telling 
us that the alternative is also possible, but it is non-developed yet 
and the standard way is already known and proved to be quite nice and 
useful from the point of view of theory and its applications. Instead 
of such explanation we usually listen something about (absolute) truth 
or even that "natural numbers were created by God".

On the other hand, formalisation of F and some other similar concepts 
related with the informal generation of N according to the (mental 
process of) potential infinity would allow us to investigate in a quite 
mathematical (rigorous, formal) manner what has been formerly 
considered as illegal. One of the main points here is that N has a 
richer structure than that described, for example by PA. So, let us try 
to formalise and investigate this structure.

Feasibility concept is interesting because it is actually about our 
real (feasible) computational practice. This does not exclude any 
traditional theories of Theoretical Computer Science based on 
traditional formalisms. This is hopefully just a new interesting angle 
of view which is related in some way with Complexity Theory, but has a 
different, foundational flavour. Instead of asking what are feasibly 
computable (typically understood as polynomial time computable) 
*functions* over N we ask ourselves what are feasible (and other such 
kinds of) numbers in N.

Vladimir Sazonov

P.S. As to related discussions on the size of the Universe and 
appropriate physical considerations, I should note that this is not the 
most important question *for me*. It is just enough to note that 
physical abilities of people and computers are rather restricted. 
Otherwise no idea on abstraction of potential infinity would arise at 
all. It arose even without any knowledge of and reference to these 
physical facts. Although the fact that numbers like 2^100 and 2^1000 
are practically non-feasible follows from some physical experiments and 
theories, this can be also demonstrated by more elementary 
considerations. (I remember some such considerations from an old 
popular mathematical book for children demonstrating how huge is 
exponential - at that time I was not so excited about that.) After 
saying that, NO MORE COSMOLOGY AND TECHNICAL PHYSICAL DETAILS ARE 
NEEDED FOR FURTHER CONSIDERATIONS (I mean considerations of that style 
which I am trying to suggest). Feasible number theory is not a 
cosmology or physic - it only has some remote relation to that.

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