[FOM] mathematics and ordinary language

[Vladimir Sazonov]

Martin Davis replying to Dean Buckner wrote:

> Yes indeed, the numbers ordinary folk use in
> counting are the same entities as those with which mathematicians deal when
> they speak of "natural numbers". 


Here I do not agree (cf. below). One of the questions is what 
means "the same". 


But does he [Dean Buckner] really imagine that it is
> simple minded statements about how many apples Johnny has if he picks 5 and
> eats 2 with which mathematicians are concerned. 


If this phrase is intended to note that mainly in such kind 
of considerations on natural numbers ordinary folk and 
mathematicians agree, then, this is OK for me, too.  

But in general, as I wrote many times, during mathematical 
formalization of any ideas these naive ideas are somewhat 
changing. By this reason only, mathematicians and the 
ordinary folk have different ideas on natural numbers, 
geometry, etc. 


Moreover, people may be quite different. Say, children 
before school education or other such people definitely 
have much more vague and even different ideas on natural 
numbers than mathematicians. I even would not say that 
mathematicians have this idea as completely clear one. 

Once I listened from a child that there should be a biggest 
natural number which is "enough" [for all our goals]. 
Even some quite educated non-mathematicians consider 
induction principle (and, say, reductio ad absurdum logical 
rule) as something extremely strange and unreasonable. 
The equivalent minimum number principle seems is acceptable 
by everybody, but it (as well as induction principle) assumes 
considering arbitrary properties of natural numbers. 
Only logicians (and not all mathematicians) have clear idea 
that these properties should be written in a framework of a 
formal theory. But we have a lot of such theories (PA, ZFC, 
ZFC + some additional axioms, etc.) The stronger is the theory, 
the longer raw of natural numbers it guarantees, and vice versa. 
Say, a simple arithmetical theory without induction (or with 
some bounded induction) and with the cut rule forbidden guarantees 
the existence of numbers only much less than 2^1000 which 
nevertheless are closed under successor and have no 
biggest number.  

Also, what exactly mathematicians/logicians/FOMers mean when 
they assume the idea of potential infinity for natural numbers? 
This idea is usually considered (in the ordinary language) as 
something self-sufficient. But, does it mean only closure 
under successor, or something more? In general, I consider the 
idea of potential infinity as highly unclear. But, if we agree 
that each time we should work in a formal theory which explicates 
this idea in some way, then no problem arises. All mathematicians 
do this either explicitly or implicitly, just de facto. 


I did not read the whole thread "mathematics and ordinary 
language". But I hope, the above notes might be useful. 


Vladimir Sazonov