From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: Intuitionism questions and references.
Date: 29 Sep 2000 01:23:07 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8qp2vu$7g9$1@nnrp1.deja.com>,
lemma_one@my-deja.com writes:
|I occasionally hear people now and then mumble things about
|intuitionists and their ways. I might need to do a term project on some
|aspect of the history of mathematics, and if so, I might want to do it
|on this or something.

Intuitionism is basically Brouwer's name for his own philosophy, which
has been taken up to some degree by followers. To research it, you'd
basically be researching Brouwer and the influence he's had on others.

|Anyways,
|
|    (i) My discrete mathematics teacher from a year ago once said that
|intuitionists have a notation like "x#0" to mean that there is no way
|that x could not be 0, but that they still couldn't construct the 0. Is
|this accurate?

Not quite. This is sort of inverted. x#0 means "x is apart from 0".
Real numbers can be represented by Cauchy sequences of rationals r1,
r2, r3,... where r_n is within 1/n of the limit of the sequence, in
intuitionism just as in any other kind of mathematics. Saying that the
limit of the sequence is #0 is the same as saying that for some n,
|r_n| > 1/n. That's a tidy, positive statement. For x to equal 0, it's
sufficient that this doesn't happen for any n. That x=0 is the
negation of x#0. Then there's the negation of x=0, which gets denoted
with the usual inequality sign, not available in this character set,
like x=/=0. If you show that assuming that x=0 leads to a
contradiction, then that means x=/=0. It's not accepted by
intuitionists as being the same as x#0, because it doesn't give you
an n for which |r_n|>1/n; it just tells you that it's impossible that
every n satisfies |r_n|<=1/n.

There is a fairly well-known principle known as Markov's rule, which
says that for every x, x=/=0 -> x#0. Markov's school of constructive
mathematics would use this rule more freely than the intuitionists,
although they'd note when they'd used it.

As it happens, it seems like the only times when I've felt like using
Markov's rule while doing a constructive proof were times when I was
somewhat confused and didn't really need it after all. It's not a big
problem, since there's a metatheorem saying if you've proven x=/=0 for
a computable real x, then you can also prove x#0. But proofs arriving
at x=/=0 seem ordinarily to be just less elegant. If you see yourself
proving x=/=0 by contradiction, try to reorder the proof into a proof
of x#0 directly. In places where you show an equality implies another
equality, see whether it's just a contrapositive of a proof that an
inequality (apartness) implies another inequality.

I also like the looks of x=/=0 better (when I don't have to put it
in ASCII), and since apartness is the more important relation than
not-equality, I like to write x=/=0 for "x is apart from 0" in my own
notes, and ~x=0 or the like for "it is false that x=0", so as not to
need "x#0".

In a sense the ASCII notation I use, x<>0, is also apt because x being
apart from 0 really is constructively synonymous with "x is greater
than or less than 0", in contrast with x>=0 which in constructive
mathematics normally means "not x<0" rather than "x>0 or x=0".

|    (ii) Is it true that irrationals don't exist to intuitionists,

Irrationals exist for intuitionists; Brouwer certainly talked about
pi as an irrational. The usual proof shows that a sequence we use to
represent it is "apart from" any given irrational in a thoroughly
constructive way.

There are some widely quoted remarks by Kronecker where he says that
irrationals don't exist, or don't really exist, or something like that.
Since some of his best known mathematics is about algebraic irrationals
I don't think he meant this in as stupid a sense as it sounds. He's
also quoted as saying "God created the integers, all the rest is man-
made" (loose translation from the German). This is perhaps where the
notion that intuitionists don't believe in irrationals could get
started. Kronecker was not an intuitionist-- Brouwer was later-- but
is regarded as an early constructivist and hence related somewhat.

I've read recently that Kronecker actually said and wrote very little
about philosophy, and that a number of the remarks attributed to him
are from someone's notes on a particular lecture of his.

|nor
|do noncontinuous functions. (But then if irrationals don't exist, do
|they have continuous functions as well?)

Intuitionists have come to believe certain principles which other
constructivists, like Bishop and his students, have not used and not
necessarily believed in. There's a famous proof by Brouwer that every
real function on the real line is continuous. This followed his
intuitions, after all. The proof has been analyzed as following from
two principles of his, which contradict traditional mathematics.

Bishop describes the situation like this. F(R,R) is the set of
functions from the reals to the reals.

   Brouwer's contention that all elements of F(R,R) are continuous
   seems to contradict claims of certain recursive function theorists,
   who give examples of elements of F(R,R) that are not continuous.
   In both instances, the claims are based on extramathematical
   considerations. Brouwer analyzes all possible techniques for
   constructing elements f of F(R,R), and comes to the conclusion
   that all such f are continuous. The recursive function theorists
   analyze the possibilities for constructing real numbers, and come
   to the conclusion that they all possess a certain property (i.e.,
   they are all recursive). In addition, they show how to construct
   a discontinuous function on the set of recursive real numbers.
   These two positions are, in fact, compatible. They do not
   contradict each other, because it is possible to believe both (a)
   that all constructive real numbers are recursive and (b) that
   without making use of some unprovable hypothesis (such as the
   hypothesis that all constructive real numbers are recursive) the
   only elements of F(R,R) that can be constructed are continuous.
   Extra-mathematical considerations of both types (especially the
   first) are useful in indicating that we should not try to do
   certain things constructively, but they have no place in the
   actual development of constructive mathematics.

Apparently Brouwer would have disagreed with the last part.

|    (iii) Are intuitionists still around? If so, only recreationally or
|in the university system as well?

Michael Dummett is an academic philosopher who has long argued in
favor of their position. I somehow get the impression that there were
some Europeans still doing intuitionist-flavored analysis, but I don't
read much about it.

|    (iv) Do they accept the infinitude of the integers? How about the
|infinitude of the primes?

Yes, yes.

Keith Ramsay