[FOM] formalism

[V.Sazonov@csc.liv.ac.uk]

Quoting Rupert McCallum <rupertmccallum at yahoo.com> Sun, 05 Nov 2006:

>> The above definition does not require any metatheory. Formal systems
>> are assumed to be considered in a naive manner (to avoid the evident
>> vicious circle) as I described in another recent posting answering to

I think, I should clarify, that no metatheory is necessary to play with 
formal rules. Just like children play with Lego, domino, like we use 
key to unlock the door etc. ? quite naively.

> But what statements are accepted as known from this naive point of
> view?

If you succeeded to derive formally a theorem you definitely know about 
this using only a quite naive understanding what your formal system is. 
Additionally, you might have some idea what this formal system is 
about: you are also able to see (quite informally) that the axiom and 
proof rules (roughly) correspond to your imagination on the "world" 
this theory is "describing". You will probably conclude that derived 
theorems fit well in your picture of this imaginary "world". Otherwise, 
if something will go not according to your intuition and expectations 
you will think that it is an exception, like well-known counterexamples 
in analysis. Most probably, you will not consider these exceptions as 
sufficient reason to change the formal system and will learn how to 
co-exist with these exceptions. May be you even will eventually find 
these exception also natural in a sense. May be the formal system will 
somewhat change your intuition and vision of this imaginary "world". 
That is normal when your intuition is governed by formal rules.

That is basically all. In this sense formalism is even not about 
consistency of the formal systems you are working with. You just see 
that it agrees with your imagination (at least to some degree). You can 
discuss this with other people by appealing to their intuition. 
Probably you will find that you understand one another. Probably the 
illusion will appear that you are discussing about "the same" imaginary 
world (because you ground your discussion and intuition on the same 
formal system or a similar range of formal systems) and because your 
fantasies are somewhat related with (or are natural extrapolations 
from) our joint REAL WORLD.

If you want to prove consistency of one formal system in another one - 
do this as usually. The full existing mathematical practice (and 
probably much more) is included in this picture.

Edward Nelson considers the consistency of Robinson Arithmetic to
> be an open problem. Do you?

In principle, any sufficiently nontrivial formal system can be 
suspected to be contradictory, even if we have some imaginary world 
which this system seemingly describes. But our imagination is, in 
general, so vague. In some cases it seems solid enough and we are 
inclined to believe that the formal system is therefore consistent. But 
nobody can give a full, absolute guarantee. Any proofs of consistency 
are based on some other, stronger formalisms. This gives only a 
relative guarantee and depends on some (may be subjective) preferences 
to consider some theories as having a better intuitive background and 
believed as more reliable.

For me personally, it is sufficient to prove consistency of a new 
formal system in ZFC. If it seems impossible, the intuition plays the 
role of a guarantee.

Another thing, if by some reason I am interested in fantasies of a 
special kind such as feasibility.

Supposing a dispute arose between two
> formalists about that issue, wouldn't it have to be settled by a choice
> of metatheory?

If both like the same metatheory. . . But formalist point of view, in 
my understanding, is not and should not be related with some specific 
preferences to such or other formalisms and intuitions. It is quite a 
general view - a full freedom, except always being based on formal 
systems (once it is called mathematics, once it is not just a free art).

It seems to me that Nelson's preferences to some formal systems and and 
his intuitions on arithmetic are not a direct consequence of his 
formalist view on mathematics. Rather formalist view is a necessary (or 
desirable) condition for taking his preferences (related with some 
doubts in natural numbers). Formalist is not necessarily an 
ultrafinitist, or the like. It is just a general view on mathematics in 
its widest sense.


Vladimir Sazonov


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