[FOM] Predicativism and natural numbers

[Arnon Avron]

On Mon, Jan 09, 2006 at 03:04:12AM +0100, Giovanni Lagnese wrote:
 
> The concept "of lenght omega" is not initially available, because it is not 
> clear before you define it.

There is a point in providing definitions to concepts only if they are
done in terms of simpler, clearer, and more intuitive concepts. The 
impredicative "definitions" of the natural numbers you have in mind 
are not better for the purpose of clarification than Euclide's 
"definition" of a point as "that which has no lenth, breadth, or height".
The set of natural number was certainly "available" to Gauss before
it was "defined" in terms of arbitrary sets, and it is available (and clear)
even to small children even though (hard to believe!!) they never
heard its "definition" as the minimal set that contains 0 and is closed
under the successor operation.
 
  By the way, before you ask me to provide a definition of the
natural numbers, I need you to clarify for me the concept of
"definition", because this concept is not clear before you define it.
Without a definition of the concept of definition
this concept is not available to us, and so we are not able to
give any definition at all of anything.

> The concept "unending" is clear, but not the concept "of lenght omega".
> "Of lenght omega" does not mean simply "unendig". It means "minimal 
> unending". This is the point.

This may be *your* point, and I can do nothing about it. If you declare
that you understand the misterious expression "minimal unending" better
then the concept of a natural number,  I have to believe you, and accept
that we shall never be able to agree on the foundations of Mathematics,
on the meaning of its statements and about their certainty. Our minds
simply work differently.

> 
> > I can imagine no process which would enable me to
> > identify a structure which could play the role of the
> > power set of omega and which could not be further
> > enriched by a longer process.
> 
> I can say that you can not identify a structure which could play the role of 
> omega and which could not be further enriched by a longer process...
> 
Of course you can *say* so. It is not forbidden by the law. But even your
own formulation ("I can say", not "I say") hints that you do not believe 
it yourself. On the other hand I believe that you do understand
what Weaver was saying above, and you understand the difference.
If not, then again we can only agree that our minds work differently.

One more comment before I end. *Personally* (I am not trying to speak
here in the name of all predicativists) I think that the notion of a
natural number *is* definable in terms of more basic concepts. The basic
intuition behind the construction of the concept of natural number
is "and so on" (or the idea of "..."). The natural numbers consist of 
0, 0', 0'', and so on (or 0, 0', 0'', ... ).
Now in my opinion, the mathematical concept that is hidden 
behind this "and so on" (or " ... ") is the operation of formimg the
transitive closure of a given 2n-ary relation. This notion 
implicitly (and *naturally*) enters into all natural languages in many ways.
Thus the notion of ancestor is derived through it from the notion
of a parent, descendant from sun/daughter, "above" (in one
of its senses) from "on", etc. Now the intuitive meaning of
TC(P)(x,y) is:
 P(x,y) or \exists z P(x,z) & P(z,y) 
  or \exists z',z'' P(x,z') & P(z',z'') & P(z'',y) or ...
Hence its understanding is based on the intuition of " ... ". On the
other hand all natural uses of " ... " can be formalized using TC.
In my paper "Transitive Closure and the mechanization of Mathematics"
(in  "Thirty Five Years of Automating Mathematics", 
(F. Kamareddine, ed.), 149-171, Kluwer Academic Publishers, 2003)
I provided arguments and evidence why a first order language enriched
with TC is the right language for formalyzing mathematics. In this
language TC is taken as primitive  - like the connectives and the
quantifiers. And exactly like the standard connectives and 
quantifiers its meaning cannot be defined (in a noncircular way) -
it can only  be *explained*. 
 Needless to say, in a language with TC and = the notion of a 
natural number can easily be defined from 0 and successor - and this
definition (at least in my opinion) is a direct formal counterpart
of our real, intuitive notion of a natural number.


Arnon Avron