[FOM] Indispensability of the natural numbers

[Steven Ericsson-Zenith]

> Of course, *any* kind of argumentation concerning the nature of
> natural numbers and illusions on them could be considered as
> (meta)theorising in a very broad and *informal* sense, although
> I would not use the term "(meta)theorising" here.
>
> Do you realize that in the second paragraph above from my posting
> I mean rather a *formal* metatheory 

I do mean a *formal* metatheory. :)

> ... and confirm that "some
> metatheoretical considerations might illuminate some questions
> discussed here".

Yes, but I assumed your claim that mathematicians do not need 
metatheory means just that - while metatheory is perhaps the sole
domain of the students of metamathematics.  I acknowledge that
you did not entirely dismiss metatheory.

My point is exactly that the operating mathematician does need
a formal metatheory to improve mathematical development, I do not
disagree that this metatheory is insufficiently formalized 
today.

> Is this really a contradiction?

My observation of contradiction comes from the claim that the
only metatheory available to the operating mathematician is an
acceptance of the illusion in N - and the fair observation that 
this is inadequate. But I take this to imply that a strong and 
formal metatheory is necessary and should not be ignored as you 
seem to imply.

To be clear, methematicians need to understand how mathematical 
intuition works - They are not merely calculators because there
is something else going on that is not captured by the machine
as we understand it currently.

> If you know what to say on the following part below, please do that.
> My way of thinking is probably slightly different. But I presented
> something more concrete in my answer to Timothy Y. Chow. In particular,
> I tried to explicate why solidness of N (or, actually, of the meaning
> of "and so on") is illusive.
>
>
> > It seems to me, on further review, that you are missing an important
> > aspect of this question.
> >
> > An understanding of the nature of apprehension and the "process" of
> > prediction in inference - abduction, induction and deduction - in which
> > illusion is simply a part - demands that we provide comprehensive
> > metatheory. And in this manner we can, in fact, provide an exact
> > definition of what illusion is.

I do indeed have somethings to say on this, but I hesitate because these
are nascent thoughts.  The case of N is an interesting one and
I wish to consider it in more detail before directly refering to it.

My concern in general, however, is with all forms of prediction, 
such as the implication of N, and the formal nature of such inference.

This concern derives from the observation that the ontology in which 
mathematics lies is incomplete.

Godel, from my point of view, demostrates either a state of nature or a 
state of our mathematical ontology - I prefer to believe that it is we
that are fallible. This is essentially the argument put forward by 
Penrose but I prefer not to appeal to the failure of reductionism but
rather to insist upon reductionism.

A formal model that describes inference in a more satisfactory manner 
than the current informal intuition we demand would, I suggest, be
desirable.

With respect,
Steven