[FOM] A question concerning continuous functions

[Arnon Avron]

There is a question concerning the definition(s) of
the concept of a "continuous function" which has bothered
me since I was a first-year student in mathematics. I
believe that this list might be the most appropriate place
to look for an answer.

  The question is the following: I think that the first
definition  of this notion that each of us
has encountered (and later has given to our own undergraduate
students, at least in the form of an intuitive explanation) is
that a real-valued function defined on some  interval is continuous
iff its graph can be drawn without lifting the pen from the
paper (or the chalk from the blackboard). Later, however, we
were given a completely different, analytic definition (using  epsilon,
delta, and alternating quantifiers - or something which is equivalent
to it if we assume the axiom of dependent choice). We then started to
work with this official definition, never returning to the original,
intuitive one again. Now what I find amazing is that I have never been 
given, (and neither have I found in any textbook I have seen) any 
argument why these two notions of a "continuous function" are in fact 
equivalent.  This does not seem obvious at all, at least to me . More 
precisely, it does seem very intuitive to me that a function which is
continuous in the intuitive sense should also be continuous according 
to the official definition, but I see no compelling reasons
to believe that the converse is true.

  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.

Let me explain why I think that this question is very relevant
and important to FOM:

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. In the case of continuous functions one may point
   to basic theorems like: the intermediate value theorem ("second
   Bolzano-Cauchy theorem"), the boundness of a continuous function
   on a closed finite interval, and the fact that it has a maximum
   and a minimum there. Note, however, that unlike the case of
   Church Thesis, here we also have strong evidence that the two
   notions are *not* identical (e.g. in the form of functions which are
   "continuous" but have no derivative in any point of their domains).

   I would like to point out again that while almost any textbook
   on computability defends Church Thesis in one way or another, authors
   of textbooks in analysis seem to feel free from such obligations. Why?

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.  They have ended up rejecting the classical theorems 
   mentioned above which provide (in my opinion) the main evidence for 
   accepting this identity in the first place... Let me add that this 
   phenomenon provides (for me) another evidence that the two notions 
   of continuity are not identical.

3) Returning to the comparison with Church Thesis, there is one
   big difference in that (at least in my eyes) the intuitive definition 
   of a continuous function  is *not* vague or imprecise  (certainly not 
   as vague and imprecise as the notion of a "mechanically computable 
   function" had been before Church Thesis was formulated). It is 
   completely sufficient, in fact, for conveying and understanding the 
   notion we have in mind. From this point of view, it is even superior 
   to the official definitions. True: it does seem useless for proving 
   properties of continuous functions or even for showing that the 
   elementary functions are continuous. This might be due, however, to 
   the inherent limitations of using discrete languages (whether formal 
   or informal) for describing and reasoning about continuous geometrical 
   objects and phenomena. I suspect (and this is no doubt a *BIG* 
   speculation) that the incompleteness phenomenon discovered by Goedel 
   is really due to the unbridgeable gap between our discrete languages 
   (and proofs) and our geometrical intuitions.

4) According to my best understanding, the main problem in the 
   foundations of mathematics is (and was) how to justify the use of 
   infinite objects in mathematics without destroying the absoluteness 
   and the necessity we like to ascribe to mathematical propositions. 
   Finite, concrete objects cause no difficulty here (even if there are 
   certainly philosophers who would be happy to argue to the contrary...). 
   Infinite objects *are* problematic. Their use involves strong 
   metaphysical assumptions that one cannot honestly take as necessary 
   and obvious. Unfortunately, the real numbers, the heart of modern 
   mathematics, are infinite objects according to any official definition. 
   Had these definitions told the whole story concerning the reals I would 
   not have hesitated to dismiss their use as "theology" (or at least 
   judge their use in science by pragmatical criteria, like we judge 
   physical theories). However, I do have a strong *geometrical* intuition 
   concerning the reals and their connection with finite line segments - 
   and the latter are for me finite, concrete objects (note that what I 
   call here "a finite line segment" was simply called "line" by Euclid).
   According to my views, therefore, it is only geometry that may be able 
   to provide a real justification and foundation for the real numbers. 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.