[FOM] Indispensability of the natural numbers

[Vladimir Sazonov]

Steven Ericsson-Zenith wrote:
I acknowledge that
> you did not entirely dismiss metatheory.

OK

> 
> My point is exactly that the operating mathematician does need
> a formal metatheory to improve mathematical development, 

In principle, yes. But the contemporary mathematical practice
does not seem to require this. Also, although any formal metatheory
is actually a kind of mathematical theory, we should be sufficiently
careful to distinguish mathematics from metamathematics to avoid
the evident vicious circle. In this sense, the last resort for us
will be not a formal metatheory but quite *naive* understanding
of what is a formal system and how it interplays with our intuition.
This is usually done just by training started at school lessons of
mathematics.


I do not
> disagree that this metatheory is insufficiently formalized 
> today.

I do not know what do you mean. I also could say something like
that... However, arithmetization of syntax and model theory seems
are quite sufficient for the "normal" mathematics.

> 
> 
>>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 

Again, I would not call this a metatheory. This is too informal.
I also do not think that such a discussion would be interesting
to the majority of operating mathematicians (even if it is
interesting to me). But this might be interesting to them in
the case of some foundational crisis to understand how to behave
in a new situation.

- 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.

I do not want really to object, but in the normal situation (of no
crisis) the ordinary intuition (making them not merely calculators)
works very well.

> 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.

I am not sure that I understand you properly. Proof theory and model
theory seems give a good answer to your question if you do not
mean something going beyond the ordinary paradigm. It seems you
really mean that, but I have no clear idea what.

> With respect,
> Steven

Kind regards,

Vladimir