FOM: geometrical reasoning, logic and proof

[Vladimir Sazonov]

Colin McLarty wrote:
> 
> kanovei at wmfiz1.math.uni-wuppertal.de (Kanovei)
> 
> wrote:
> 
> >There is apparently no any example of a
> >mathematical statement commonly accepted as a
> >true theorem but not deducible logically from
> >some (also commonly accepted) list of axioms,
> >say ZFC.
> 
>         Obviously every commonly accepted mathematical statement can be
> deduced from commonly accepted mathematical statements. It can be deduced
> from itself, for example.
> 
>         On the other hand, as soon as any recursive set of mathematical
> statements becomes commonly accepted, we find further statements which are
> not logically (i.e. first order) deducible from them and yet are commonly
> accepted.
> 
>         Peronally, I believe ZFC is consistent, and I believe most people
> who understand the issue agree. But I do not derive this belief (in the
> consistency of ZFC) from deduction in any formal theory.
> 

Adding new axiom to any fixed theory T (say, ZFC) is quite 
***different*** story. It is not deriving/obtaining a new 
truth on the subject of T. It is just creating a new theory, 
the process analogous to creating the original theory. And 
there is *no need* to mix these radically different things. 

Start from some intuitive (probably rarher vague - it does not 
matter - but desirably coherent) concepts, beliefs and what you 
want, formalize them in some (probably not unique) reasonable way, 
and get a theory T. After that, derive theorems according to 
axioms and rules of T and enjoy if these derivations will 
happen to be coherent with your initial crude intuition and 
especially if this theory will somewhat change your initial 
intuition and make it much stronger and "organized" than before 
you got T. Use your at present stronger intuition and the 
machinery which gives you theory T somewhere outside T (say, 
in physics and then in engineering: building bridges or 
nuclear reactors or space ships, electronic computers, etc.). 

If you want, repeat this process to get new theory T' (probably 
extending T). Consider many such theories. Interpret one theory 
in another if possible (and if it makes any sense). Enjoy again. 

Finally, we may be interested in the laws of human or mathematical 
thought according to which we have intention to extend a theory T, 
say, by Consis(T) or ZF by Choice Axiom. (I.e. why do we *like* 
Consis(T) and Choice Axiom?) We could analize how and why it 
happens. But I see no reason to consider that Consis(T) is derived 
from T in *any* sense or that this belongs to some Platonistic world 
of truths. 

That, by my opinion, is roughly what is doing "working mathematician", 
pure and applied one (essentially with no need of Platonism and 
absolute truth). 


Vladimir Sazonov
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