As far as the topics in this webpage are concerned, most emphasis is laid on he following two chapters:

- 8. 'The Constructive Real Line and the Intuitionistic Continuum'
- 9. 'Smooth Infinitesimal Analysis'.

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.

Thus, given any small number called ε (by employing binary search) we find a number r such that | 2 - rprogram wortel; { Square Root of Two } const { eps : double = 1.e-15; } eps : double = 1.e-6; var r,h,s : double; begin r := 1; h := 1; { Initialization } while true do begin 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; end; end.

Even superficial inspection reveals that we have re-discovered here