[FOM] A question concerning continuous functions

[Vladimir Sazonov]

Ayan Mahalanobis wrote:
> 
> This is some comments on the recent posting on Arnon Avron.
> 
..........

> The root of the problem behind epsilon-delta definition is the lack of
> continuity in the definition. 

I did not follow the discussion in the detail. But, at least the 
last sentence seems to me strange. As I already wrote some time ago, 
I strongly believe that mathematical definitions, axioms, theorems, 
proofs, and any other formal things are nothing else as technical 
instruments helping and strengthening our thought and intuition, 
like any other devices helping us to move quickly, or to be stronger 
(levers, etc.). Do we ask why, for example, the cars are lacking foots, 
if they ran so quickly. Our formal (epsilon-delta, etc.) devices 
help very much to our intuitions, such as about continuity. 
We can realize which way it is happening, as Dana Scott explained 
in his recent posting. What else do we need in general? 


> I think the definition is so successful because classically we
> conceive real line as a set of points with the law of trichotomy, which
> makes it ** discrete ** but uncountable. Probably we need to look at our
> classical understanding of continuum more closely.

Of course, any other attempts to formalize any our vague ideas 
(here - on continuity) are welcome. But the result of ANY 
formalization will be always something different from what 
we had in the mind initially. We will NEVER have complete 
coherence between our intuitions and mathematical formalisms. 
The very words `vague' and `rigorous' say this for themselves. 
We also know this from our mathematical experience. 

> 
> Some thoughts, any comments welcome.
> 
> Cheers
> Ayan


Arnon Avron wrote:

.........

> I believe, accordingly, that a new synthesis of analog and discrete 
> reasoning should be taken as the most important foundational challenge. 
> My question above is, of course, connected with this challenge.

Why not? Looks quite interesting and promising. 
For example, how it is possible that a picture 
on a computer screen ia BOTH discrete AND continuous? 
A vague idea, but quite a real fact. Is this a contradiction? 
Is this possible to formalize? Or should we avoid formalizing 
such phenomena (related also with the vague ideas of big 
and small natural numbers and with reconsidering finitary 
arithmetic and even classical logic)? I guess, Arnon would 
say that he has something different in his mind. But for me 
this looks very close. Anyway, first, the initial vague ideas 
should be clarified leading to some first steps of formalization. 
I even unsure that formalization is a second step. Everything 
should happen simultaneously. Formalization is a way of 
clarifying the intuition. During this process even the 
initial ideas might be changed, or some pathologies in 
foramalization will arise. That is normal. These pathologies 
should not bother us too much, if the formalization is, 
nevertheless, sufficiently good. 


Best wishes, 

Vladimir

-- 
Vladimir Sazonov                        V.Sazonov at csc.liv.ac.uk 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.       http://www.csc.liv.ac.uk/~sazonov