Brouwer's Continuity Theorem

A basic reference to the topic in this section is: Continuity and Infinitesimals
As far as the topics in this webpage are concerned, most emphasis is laid on he following two chapters: When considering Constructivism and Intuitionism, the discussion has been severely hampered by the fact that intuitionistic logic - especially its problems with the Law of excluded middle - has been over-emphasized, ever since the infamous debate between Hilbert and Brouwer took place. As a physicist, I find another achievement of Intuitionism far more intriguing than its contributions to Mathematical Logic.

After some search, we arrive at Brouwer's Continuity Theorem, boldly stating that Every total real function is continuous. Somewhat more elaborate: Any function which is defined everywhere at an interval of real numbers is also continuous at the same interval. With other words: For real valued functions, being defined is very much the same as being continuous.
Why is this so interesting? Well, because of the fact that, in i.e. physics, "neat", everywhere continuous functions are employed most of the time. Brouwer's Continuity Theorem might provide bottom-up evidence why this simply must be so, Maybe he intuitionist's continuum is indeed more resemblant to the real-world continuum (in physics) than the classical continuum has ever been. This is the main motivation for my "pink intuitionism" (in honour of Torkel Franzen + ).

The history of the Continuity Theorem is somewhat older than Brouwer, though. Weyl has already made statements which came quite close to it, such as: Above all, however, there can be no other functions at all on a continuum than continuous functions. After pointing out the absence of discontinuous functions on R, Weyl went on to say that: When the old analysis allowed the formation of discontinuous functions, it thereby showed most clearly how far it is from grasping the essence of the continuum. What one calls nowadays a discontinuous function, consists in fact (and this also is basically a return to older intuitions) of a number of functions on separated continua. Weyl illustrated his statement by the piecewise defined function f(x) which is x for x > 0 and -x for x < 0. This partial function, he remarked, can be extended to the total function |x|, but the function g with g(x) = 1 for x > 0 and -1 for x < 0 cannot be extended to a total function. Brouwer adds in the margin the far more sensible remark that a discontinuous function better : is a not everywhere defined function. The definition of the function does not have to be split in cases, as the pointwise discontinuous function shows. (Note: I got the information in this paragraph from the Internet)

Instead of considering Brouwer's Continuity Theorem as being "unreal" from a physical point of view, I would like to present the opposite argument here. On the contrary, having become familiar with the overly complicated details, associated with the classical understanding of continuity, doesn't mean that there cannot exist other points of view. Reconsidering Brouwer's Theorem, from the point of view of a physicist, may be helpful in broadening our mind. Last but not least: Finally, it has been shown that a natural notion of infinitesimal can be developed within intuitionistic mathematics (see Vesley [1981]), the idea being that an infinitesimal should be a "very small" real number in the sense of not being known to be distinguishable - that is, strictly greater than or less than - zero. This is far more natural (: according to physics) than Robinson's Nonstandard Analysis, where an extension of the (real) number system is required, in order to be capable to take infinitesimals into account.

Quoting from 'Smooth Infinitesimal Analysis'.
It provides a rigorous framework for mathematical analysis in which every function between spaces is smooth (i.e., differentiable arbitrarily many times, and so in particular continuous) and in which the use of limits in defining the basic notions of the calculus is replaced by nilpotent infinitesimals, that is, of quantities so small (but not actually zero) that some powe - most usefully, the square - vanishes. Since in SIA all functions are continuous, it embodies in a striking way Leibniz's principle of continuity: Natura non facit saltus.
It is obvious that SIA is another bottom-up approach which supports the view that in real-world continuum all functions are "differentiable arbitrarily many times". Moreover, infinitesimals are, indeed, a part of the real number system here.

Consider the following nowhere continuous function, the so-called Dirichlet function:       f(x) = 1 for x = rational ; f(x) = 0 for x = irrational .

Think about it. No physical experiment can be devised that could ever show a difference between rational an irrational numbers in a real(istic) continuum. Therefore, from a physical point of view, it is clear that the Dirichlet artefact is a complete nonsense function. Indeed, according to constructivist mathematics, there cannot even exist such a function, because it is non-unique. The latter stems from the fact that irrational and rational numbers cannot be distinguished in a strict manner, meaning that our Dirichlet function f is actually undefined everywhere on the real numbers. Intuitionism is entirely in agreement with physical reality in these matters.

The problem with classical mathematics is that it makes a sharp and rather unrealistic distinction between: equality of two real numbers and: continuity of real valued functions. In physical (and mathematical) practice, though, these two things are essentially the same. Suppose, for example, that you don't "have" yet the irrational numbers at your disposal and therefore you want to define √2 (square root of a number called two), all by yourself. Such a thing can be accomplished, with an algorithm, as follows.

  program wortel;
    Square Root of Two
  { eps : double = 1.e-15; }
    eps : double = 1.e-6;
    r,h,s : double;
    r := 1; h := 1;
  { Initialization }
    while true do
      h := h/2; s := r*r;
    { Binary search here }
      if s >= 2 then r := r - h;
      if s <= 2 then r := r + h;
      Writeln('sqrt(2) = ',r,' ; 2 = ',s);
    { Continuity requirement }
      if abs(2 - s) < eps then Break;
Thus, given any small number called ε (by employing binary search) we find a number r such that | 2 - r2 | < ε . And we call this number r the square root of 2 . One might ask what the inaccuracy in r is. With other words: what is the small number δ in | r - √2 | < δ ⇒ | 2 - r2 | < ε ? Reasoning goes as follows: | 2 - r2 | = | √2 + r ||√2 - r | = 2 √2 δ < ε . Gives δ ≈ ε / (2 √2 ) . Because of √2 < 1.5 , a sufficient accurate guess is: δ = ε / 3 . It is concluded that finding √2 is very intimately related to a procedure of the following form: find a δ such that for any ε : | x - a | < δ ⇒ | f(x) - f(a) | < ε . Here x is the unknown number (as usual), a = √2 , f(x) = x2 and f(a) = 2 .
Even superficial inspection reveals that we have re-discovered here the classical definition of continuity. With other words: √2 can be found only because there exists a function f(x) = x2 on the reals wich is continuous. And thus forces the number x to assume a certain value for f(x) = 2 . In My (not so) Humble Opinion, the gist of Brouwer's Continuity Theorem is that the very definition of a real number, essentially, cannot be distinguished from the continuity of associated real valued functions. Real numbers and real functions have to be defined in one big sweep, instead of the former by e.g. Dedekind cuts, and the latter by some other means, at some time later on.