[FOM] Towards a Real Finitism?

[David F. Isles]

I'd like to second Podnieks' suggestion in his Dec. 28 FOM posting:

> Shouldn't a real finitism be more radical and starting positively? Couldn't
> we obtain a richer structure by replacing the traditional "linear" notion of
> natural numbers by the notion of FUNCTIONAL EXPRESSIONS?

In fact, in my short note "Questioning Articles of Faith: A re-creation of the 
history and theology of arithmetic" (Bulletin of Advanced Reasoning and 
Knowledge  vol.), I investigate some consequences of taking seriously  
 
> ...... that EXPRESSIONS ARE MORE
> FUNDAMENTAL THAN NATURAL NUMBERS? And that natural numbers are simply
> expressions based on 1 and +, i.e. 1, 1+1, 1+1+1 etc.?

As I see it, however, one is not led to this position by either an acceptance 
or rejection of actual (or potential) infinities but simply by the observation 
(belief?) that numerical notations are what one can see being used and that, 
like natural languages, that use seems to be rule based. Then, as happens in 
linguistic investigations, one can form a model ("functional expressions")of 
these notations and seek to investigate its properties.