[FOM] A question concerning continuous functions

[Giuseppina Ronzitti]

Arnon Avron wrote:

>  My immediate question is: can anybody provide me  with references to
> works which discuss this issue and present the reasons for accepting
> the identification of the two notions? (What I have in mind is
> something similar to the way the case for Church Thesis
> is presented in Kleene's "Introduction to Metamathematics" or in
> Shoenfield's "mathematical Logic"). I would like, of course, also  to
> hear any personal answer to (or thoughts about) this question
> that you might have.

Perhaps it might be of help Elliott Mendelson's article "Second thoughts about
Church's thesis and mathematical proofs", the Journal of Philosophy, vol
LXXXVII, no. 5 1990. In that article EM raises the general question of the
adequacy of some formal definitions to the 'corresponding'  intuitive notions
and lists some logical and mathematical "theses" which  are not usually called
theses but, in his view, deserves to be called so if CT is called a thesis. A
related  example is that of the "Weierstrass thesis" on the definition of a
limit of a function.

> 1) The fact that I mentioned Church Thesis  above is no accident. In
>    both cases it may be argued that an intuitive, but too vague a notion
>    has been given a precise, mathematically manageable definition. In
>    both cases (one may continue arguing) there is no hope of *proving*
>    the equivalence of the two notions (because of the inherent vagueness
>    of the intuitive one), but it is possible to present a convincing
>    evidence for the thesis that the precise definition captures the
>    intuitive concept.

Mendelson thinks differently : "The assumption that a proof connecting
intuitive and precise mathematical notions is impossible is patently false"
(p. 232).

> 2) In my opinion, the weakest point of Intuitionism, and the reason why
>    it will never be adopted by most mathematicians (even though many of
>    them adhere to the importance of constructive proofs) is that it
>    seems to totally ignore geometric intuitions and concepts. As far as
>    I understand (and please correct me if I am wrong), intuitionists
>    have completely abandoned the original, intuitive concept of
>    continuity in favor of (some constructive version of) the "official"
>    definition.

As far as I understand nobody can do math using *the original, intuitive
concept of
continuity*, if any. One needs (at least) a  linguistic translation.
As far as I know, 'the' intuitive concept of  continuity in intuitionistic
math is embodied in the so called Continuity Principle (what Kleene calls
"Brouwer's principle for numbers") and not in any definition of continuous
functions. As a consequence of CP, one proves that  a real function whose
domain of definition is the closed segment [0,1] *is* continuous. (cfr. W.
Veldman "On the continuity of functions in intuitionistic real analysis",
Report 8210, Mathematisch Instituut, katholieke Universiteit Nijmegen).
To my thinking, the weakest point of Intuitionism is its folklore which is
sometimes perpetrated by some people working in Intuitionism.

G. Ronzitti
Genova, Italy
(Europe)