FOM: Was sind und was sollen die Zahlen

Vladimir Sazonov sazonov at
Tue Nov 4 18:48:52 EST 1997

I am a newcomer of this FOM list. Sorry for any possible 
misunderstanding of its goals. Also, sorry for my English.

First, I would like to mention the following attractive to me 
fragments from different postings of Vaughan Pratt:

> Mathematics is like art and fashion, it is defined by its producers and
> consumers and it has its trends and staples.  Everything changes in
> mathematics, some faster than others.

Harvrey Friedman replies:

> Basic classical mathematics is relatively unchanged. The classical number
> systems are unchanged. So what do you mean by this?

Again Pratt:

> Even on earth, let alone Quasar X9, mathematicians disagree wildly among
> themselves as to "was sind und was sollen die Zahlen".

> ... proof and truth
> as equal partners, not proof as the servant of truth.

Also Neil Tennant <neilt at> wrote:

> [Please do not reply that the lesson of Goedel's incompleteness theorem
> for arithmetic is that there could be 'alternative arithmetics'.  For
> in response to such a suggestion one can make two fatal points.
> First, ALL these arithmetics would have to contain the same
> quantifier-free statements such as '7+5=12'.  Secondly, we all agree
> that the independent Goedel sentence for a given (intuitively sound)
> arithmetical theory is TRUE in the intended model.]

I do not agree, because the Goedel's consistency statement does 
not have precisely (if we could ever have any hope on a precision 
here) the intended meaning. Actually, what is *the* intended 
model/meaning of PA, etc.?  Of course, so called `standard 
natural numbers' are characterized uniquely, up to isomorphism, 
in the framework of ZFC. But what is proper relation of these 
`standard natural numbers' to natural numbers from our `real 
world'?  (Cf. the discussion below.) Alternatively, why should we 
consider only ZFC (or PA) perspective as the unique way of 
formalizing the `naive' notion of natural numbers which is given 
to each of us BEFORE any learning this concept in school or in 
our universities. For example, my daughter once asked me (when I 
counted "1,2,3,...,100,...,1000, and so on") why not use one 
sufficiently big number which will be `enough' for us. In a sense 
she was right. (This is a way to describe PTIME computability 
in terms of recursive arithmetic relativised to such a finite row 
of natural numbers.  However, this story slightly deviates from the 
following below considerations, but is related.)

Joe Shipman <JSHIPMAN at> wrote:

> Not all mathematics is logically prior to physics; as Godel saw
> we could get new axioms from physics.  In my thesis (TAMS 10/90)
> I showed some theories of Quantum Mechanics were independent of
> ZFC, and in the '92 PhysComp Proc. discussed how a noncomputable
> number might be measurable (to more digits than ZFC decides).-JS

In connection with these notes and instead of debating whether 
7+5=12 (I think that this question is simultaneously too simple 
and too difficult to be resolved/understood) I would like to 
suggest to FOM more concrete discussion on non-traditional 
`alternative' Arithmetic Systems, specifically, on Feasible Numbers 
[cf. an old paper by R.Parikh in JSL, 36 (3) 1971 and more recent 
my paper in LNCS, Vol.960 also available by; of course, 
Yesenin-Volpin and some other authors should be mentioned, too; 
there are also very interesting related notes in the book 
"Constructivism in Mathematics", 1988 by Troelstra and van 

Let us consider so called *feasible numbers*, i.e. natural 
numbers allowing to count sets of objects which *really exist*, 
say, the number of symbols in a real computer file or the like.  
The `semi-set' of these numbers is denoted by F. Intuitively, F 
is closed under successor (F+1\subseteq F) and even under 
addition (F+F\subseteq F). Evidently, 0,1\in F. Now, let us 
denote by log n the integer part of binary logarithm of n. Then 
we may check (say, by using a computer) that for each concrete n 
\in F

	log log n < 10					(*)

where 10 is, of course, `ten' (8 is also good). Anybody who 
doubts are welcome to give counterexample!

This is some *experimental low of arithmetic* with n represented 
in unary notation as any *real* sequence 111...1 which any of us 
could print on a keyboard of a computer. Moreover, if you have 
any other computer which sends to the given computer (evaluating 
log log n) arbitrarily big (but physically existing, i.e. 
feasible) input n then the result will be the same, i.e.  (*) 
still holds. Of course, this *physically true low* (which is very 
much like Newton Mechanic lows or commutativity and distributivity 
lows of Arithmetic) contradicts to what we know from our teachers: 
that logarithm is unbounded function or that mathematical induction 
axiom is inevitably true for the numbers, etc.

Is this only a non-serious childish question on big and small 
numbers or a real possibility of some different `Feasible 
Arithmetic'?  Is it possible any formalization of such an 
arithmetic (let only some oversimplified version for a while) as 
rigorous and consistent as Peano Arithmetic? Yes, it is!  Now I 
can say: "Nothing starnge", because the above axioms are `true' 
in a reasonable sense. But this was not so evident at the 
beginning, however the solution proves to be not very difficult. 
(The details, which are actually essential and interesting, are 

As I know, the first rigorous approach 
to feasible numbers with analogous but essentially weaker axiom 
than (*) was suggested by Rohit Parikh (cf. op.cit.). The case of 
(*) follows the main idea of Parikh and gives more precise upper 
bound 2^1024 (i.e.  approximately 2^1000) for F than his original 
`non-elementary' exponential tower of exponential height.  Also 
our consideration differs from that of Parikh in imposing some 
feasibility restriction on abbreviating mechanisms implicitly 
used by mathematicians and logicians in First-Order Logic (cf.  

It follows that the number 2^1000 of all possible strings of a 
given length 1000, in the fixed two latter alphabet {0,1} can not 
be considered as feasible one according to the axiom (*). (It is 
only an `imaginary number'.) This seems to contradict to the 
ordinary opinion that this set is very concrete, or that all its 
elements `physically exist' in our world `altogether'. Of course, 
we can write (even by hands) some examples of such strings of the 
length 1000, say `010101010101...0101' or randomly, by using a coin. 
But how could we understand (if not stay `inside' any formal or 
semiformal system like PA or ZFC) what is a precise meaning of 
`the set of *all* such strings of the length 1000'. I believe 
that this is a genuine vague/fuzzy set, as well as F.  Is not 
this vagueness problem of this `set' analogous to the Continuum 
Problem (in my edition):

What does it mean precisely the set of *all* infinite binary 
strings (isomorphic to the powerset of the set of natural numbers)?
What is precise cardinality of this set? (Kantor)
Are all such strings `simple' or constructive(ible) in a 
reasonable sense or there are `random', very `complex' strings? 
(Goedel, Kolmogorov)

We know from Goedel and Cohen that these questions are 
non-resolvable in ZFC what witnesses (at least by my personal 
opinion) that continuum is just a vague set. We can prove very 
formally many interersting theorems on the continuum.  But this 
does not mean that it is not vague and uniquelly determined. 
Formally, its cardinality is 2^\aleph0, but this information is 
too poor. More or less the same holds for 2^1000.

There are additional interesting and debatable foundational 
issues arising in formalizing Feasible Numbers (cf. op.cit.).  
E.g., What is a formal axiomatic system?, What is a rigorous 
mathematical proof?, Which abbreviating mechanisms are actually 
used by mathematicians and can we assert without knowing this 
that formal logic was indeed properly and `completely' 
formalized? Which abbreviations are admissible in `feasible' 
mathematics/logic?, What is a contradiction?, What about 
mathematical truths such as (*)?  Is mathematical induction axiom 
true/reasonable?, How `expensive' is using Modus Ponens or Cut 

I interrupt to mention here (without comments) the following 
related note by Harvey Friedman:

> In the usual foundational setups, as I said earlier, no really interesting
> mathematical proof is cut-free.

I continue:
What about a model theory for `feasible' logic?  Is it possible 
corresponding Feasible (Nonstandard) Analysis or Algorithms 
Theory or Constructivism (compare with op.cit. of Troelstra and 
van Dalen)? etc.

Let me mention one effect which Feasible Arithmetic seems to 
explain:  It is our ability to see any picture on computer 
display (`Feasible Continuum') BOTH as discrete and continuous 
just by switching something in our brains.  (No optic, just a 
logic; Cf. op.cit.) It is interesting whether this will have any 
non-trivial/useful/illuminating consequences in description of 
physical lows concerning micro-world (consisting of elementary 
particles/waves) in terms of Feasible Arithmetic/Mathematics. 

Vladimir Sazonov
Program Systems Institute,  	| Tel. +7-08535-98945 (Inst.), 
Russian Acad. of Sci.		| Fax. +7-08535-20566
Pereslavl-Zalessky,		| e-mail: sazonov at 
152140, RUSSIA			|

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