[FOM] predicative foundations

[Eray Ozkural]

On 2/17/06, Aatu Koskensilta <aatu.koskensilta at xortec.fi> wrote:
> There is no reason to think that talk about possible chairs renders the
> word "chair" meaningless, and anyhow usually by possible chair one
> means a physically possible chair. Physically possible marks can't
> serve as a justification of N, only possible marks that are, in effect,
> just assumed, posited or intuited to have essentially the properties of
> the natural numbers. If someone harbors doubts about N there is
> absolutely no reason to think that "marks on paper" type talk will take
> them away.

In the last portion of my post, I had explained the following:
if you think of "N" as something existing independently from human
mathematicians, of course the possibility of anything, (or any modal
argument in general), will not help you much. However, if you think of
N as a human abstraction, something that exists by virtue of neural
"marks" which we have come to develop in thousands of years, surely
a theory of marks will help.

At this point, let me assert that a theory of X has to incorporate the
notion of "possible X". For instance, if I have a theory of rockets, the
sentences in my theoretical language will mention possible rockets. In
the same fashion, Turing talked of what marks may be made by a finite
mechanism (i.e., theory of computation).

However, Turing's formalization and conception of computation, by talking
about possible marks, is perfectly admissible, and explains everything
about computation. On the other hand,  you said that a similar account
cannot explain numbers, because it seems according to you a number has to
be a unique, universal, independent entity. Then, this entity would
have to be a "bag of possible marks", which seems wrong (and I agree
that it is wrong!)

In fact, it is that very definition and philosophy that is in trouble.

I am trying to explain that if one divorces his mind from the
ancient Pythagorean view of mathematics, then one might
start seeing how marks, and in general computation, fully
explains natural numbers, as useful pieces of our imagination.

> Of course, there's nothing in the least doubtful about N in the first
> place and the point of my remark was simply that Weaver's description
> of N in a "metaphysically uncontroversial way" is just an illusion:
> there are perfectly coherent ways of raising controversy about Weaver's
> way of describing N and it is very difficult to imagine anyone doubtful
> about Weaver's form of predicativism being convinced by reference to
> "marks on paper".

First, let me mention that I am not in command of Weaver's
philosophy, thus I say the following not in defense of his views,
but my own views.

I think that perhaps the problem here is taking the metaphor
of mathematical existence too seriously, mistaking it for real existence.
When I say that a number exists, I mean that I am in possession of
such an abstract concept in my brain. When a Platonist says that a number
exists, he means something else altogether.

According to the former view (of mine), since N does not exist like a chair
or a table, it is useless to try to describe it as if it were a physical object.

Thus, allow me to briefly describe what may constitute a philosophical
basis for natural numbers with reference to marks.

By the most "metaphysically uncontroversial" view of the human mind,
human mind consists of electrochemical reactions in a nervous system.
When these reactions stop, the mind seems to go out of existence.

The marks on a paper (squiggles of 1,2,3) have no meaning unless
they are interpreted by the human mind. The human mind is also the
origin of these marks. Thus, there will be some representation or
structure in the mind which are the "numbers". Even if the human
mind is just a finite mechanism (and by all appearances it is), then
in general a natural number is representable in the human brain (trivially).

That is, a theory of marks is relevant assuming a computationalist
theory of mind.

> It is humanly impossible to create arbitrarily large marks or survey
> arbitrarily large finite sets of natural numbers.  By abstracting away
> limitations of time et cetera we obtain an interesting picture on basis
> of which we can justify  various sorts of mathematical principles of
> e.g. predicativism or intuitionism. It is also humanly impossible to
> survey infinite sets of natural numbers, but by abstracting away this
> limitation we obtain yet another picture which might motivate or
> justify other mathematical principles.

Indeed abstraction and generalization of natural numbers seems to
be a major motivation to do any mathematics beyond simple arithmetic
acts. (For instance, I will agree that the idea of a universal quantifier
was a major novelty for humans.)

The question then comes down to: what exactly is this picture?
How do we accomplish any degree of mathematical abstraction?

In my opinion, for instance with the definition of "a natural number",
a computational definition is sufficient, and thus this definition itself
is natural, because it may be conceived by a finite mind. Likewise,
a set of natural numbers is a language, and it is another robust
computational concept (fully conceivable). All of these are easily
explained by a theory of marks, i.e., computation.

Unfortunately real numbers cannot be explained likewise, and this is
not an arbitrary stopping point assuming the above criteria. However, in
another interpretation of conceivability, real numbers, too can be
explained by letting go of computability. In that sense, it may suffice
for the entire theory to be formal, without any essential meaning
or empirical basis for the axioms, whatsoever. (Though, I think
of formalism to be too loose for a philosophical basis, because
assuming a mechanical mind, any idea seems to become
mathematical)

--
Eray Ozkural (exa), PhD candidate.  Comp. Sci. Dept., Bilkent University, Ankara
http://www.cs.bilkent.edu.tr/~erayo  Malfunct: http://www.malfunct.com
ai-philosophy: http://groups.yahoo.com/group/ai-philosophy
Pardus: www.uludag.org.tr   KDE Project: http://www.kde.org