[FOM] constructive analysis & isomorphism of R^2 with the Euclidean plane

Frank Waaldijk frank.waaldijk at hetnet.nl
Mon May 18 08:48:49 EDT 2009

Somewhat in response to Arnon Avron, Andrej Bauer and Harvey Friedman, I 
would like to comment on what Arnon wrote on May 12:

>  In his Book "Mathematics: Form and function" Mac-Lane
> says somewhere that one of the main shortcomings of intuitionistic
> mathematics is that it ignores geometric intuitions (I am
> relying on my memory here, I do not have a copy of the book).
> I fully agree. Thus geometric intuitions dictate that the mean-value
> theorem (if f(0)<0 and f(1)>0 then f(a)=0 for some a in [0,1])
> should be valid for any adequate notion of a "continuous function"
> (i.e. a notion which has pretensions to correspond to
> the idea of a function whose graph can be drawn with a pencil
> without leaving the paper). As far as I know, this theorem fails
> for the constructive theory of the reals (and continuous
> functions on them). If so, then the constructive R^2 (whatever
> it is) is even less entitled to be identified with the
> true Euclidean plane than the R^2 of classical set theory.

For his seminal (classical) topological work, Brouwer constructed a number 
of counterintuitive planar `curves', for instance one that is at every point 
the boundary of three disjoint open-and-connected spaces (the construction 
of such curves can also be very instructive for people wishing to understand 
what constructive math is about).

At that time, people working in planar topology had also been relying on 
their `geometric intuition' to arrive at theorems which were false, or 
theorems whose proof was faulty, all of which Brouwer demonstrated by his 
counterintuitive examples. All this in classical mathematics, by the way, 
since Brouwer had not yet fully developed his constructive theory of 

The intermediate value theorem doesn't completely fail in constructive 
analysis. You could even state that it holds -in a classically equivalent 
formulation. But Brouwer gave examples -much simpler than his planar curve 
examples- of continuous functions for which we cannot pinpoint the exact 
location where the intermediate value is assumed. In other words: we do not 
have an algorithm that for every continuous function f from [0,1] to [-1, 1] 
for which f(0)=-1 and f(1)=1 will approximate an x in [0,1] such that 
f(x)=0. (In computable analysis (RUSS) one can prove there cannot be an 
intermediate value algorithm that works for all continuous functions, in 
intuitionism a similar proof yields that the intermediate value theorem 
fails in its sharpest classical formulation. But there are very good and 
revealing alternatives.

These function examples are extremely instructive with regard to why certain 
computer algorithms cannot be expected to work, in the field of applied 
math. But they were unknown, and ignored in Brouwer's time. However, they 
should widen our geometric intuition. If anything, mathematics has shown 
that intuition is not a static phenomenon. By doing mathematics, we arrive 
at results which sharpen and expand our intuition. There is in my perhaps 
not so humble opinion no such thing as a common human intuition of  `the 
Euclidean plane', let alone `the true Euclidean plane'. Look at the history 
of the fifth postulate of Euclidean geometry, for another illustration of 

And do not forget, that according to modern physics, our physical real world 
seems to be not Euclidean at all. So in that sense our Euclidean intuition 
of space can be said to be distorted right from the beginning.

That aside, also constructively the intermediate value theorem is in the 
context of functions which correspond precisely to the idea of a pencil 
drawing a graph while never leaving the paper. In constructive analysis 
(BISH, INT, RUSS) the Euclidean plane can be defined in many ways, but the 
ways that I have seen so far all are readily constructively proven 
isomorphic to R^2.

It seems, after all these years and hard labour by very sharp minds, that 
still the old Hilbertian prejudice against constructive math prevails. Such 
prejudice blocks the understanding of the true issues which are being 
adressed in the constructive schools of math. Which is a shame, especially 
considering that

a) most of these issues make perfect sense in applied math, and give applied 
math a solid theoretical frame for what can and what cannot be approximated
b) most of the results in constructive math are classically valid, yielding 
usable algorithms
c) many examples are uncovered -like Brouwer's planar curves- which are of 
interest also classically

Finally, I would like to make the following personal observation. I believe 
that our physical real world is as of yet still largely beyond our 
understanding. Modern physics is in a sense -as it has always been- still 
only scratching the surface. Even the mathematical foundations of our 
scientific endeavors are still in debate, but this reflects on the fact that 
we cannot even answer the most simple questions about the `foundations' of 
our physical universe. Is the universe (whatever this means) finite? Is time 
continuous? And what do we mean by that, what is time exactly? Is space 
continuous? Is our universe computable, by which I mean: is every 
(infinite...if infinity `exists') sequence of numbers produced in nature in 
fact the same as one given by a corresponding Turing machine?

As long as these questions are opaque, to me it seems unwise to rely on our 
`intuition' alone. Anything that can further our grasp, our experience, our 
questions, even our mistakes about reality, mathematics, physics etc. should 
be welcome, I feel. In this sense, even although I'm personally convinced 
that constructive mathematics is in far better accordance with physical 
reality than classical mathematics, I also welcome classical math, because 
it provides other structures which could be informative.

It is a matter of focus, and of what issues deserve attention. To say that 
the constructive issues do not deserve attention is anyone's right of 
course, but the statement would impress me more if it is made from a good 
insight in these issues. Unfortunately, the Hilbertian prejudice like I said 
keeps prevailing, resulting in large numbers of students who are unfamiliar 
with these issues. That's why these paradigm things take so much time!

Sorry to be so long...thanks for the patience,
kind regards,

Frank Waaldijk

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