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From: lambert@boring.UUCP
Newsgroups: net.math
Subject: Re: Nova's Mathematical Mystery Tour
Message-ID: <6353@boring.UUCP>
Date: Sun, 10-Mar-85 15:47:36 EST
Article-I.D.: boring.6353
Posted: Sun Mar 10 15:47:36 1985
Date-Received: Tue, 12-Mar-85 20:39:46 EST
References: <143@ihlpa.UUCP> <460@petsd.UUCP>
Reply-To: lambert@boring.UUCP (Lambert Meertens)
Organization: CWI, Amsterdam
Lines: 137
Keywords: intuitionism, constructivism, higher cardinals
Summary: Long article
Apparently-To: rnews@mcvax.LOCAL

Lev Mammel expressed surprise at Smullyan's statement that most mathemati-
cians were Platonists and so would believe that the Continuum Hypothesis
(CH) is either true or false, rather than that you could take your pick.
Chris Henrich paraphrases `Platonism' as: `believing that mathematics is
about some objective reality, not just participation in a game of manipu-
lating symbols'.

There is something funny here.  The schism in mathematics is one between
`classical mathematics' on the one hand, and intuitionism or constructivism
on the other hand.  The latter two represent different schools, but are re-
lated in their criticism of classical mathematics.  The protagonist of the
intuitionist school was Luitzen Brouwer, and his main target was David Hil-
bert, whom he attacked for his `Formalist' position, that is ... seeing
mathematics as a game of manipulating symbols.  This would make Brouwer,
rather than Hilbert, the more Platonic of the two.  But if mathematician A
chides mathematician B for being a Platonist, you can be sure A is an in-
tuitionist or constructivist.  (For a proper understanding it is necessary
to take into account that the great Brouwer-Hilbert debate was on the prop-
er *foundations* for mathematics.)

Anyway, my experience is (i) that most mathematicians are indeed Platonists
in the sense that they believe that any proposition is either `objectively'
true or `objectively' false, independent of the existence of effective
methods of verifying or falsifying it, but also (ii) that they are not
really aware of the contents of existing criticism and are not prepared to
parry it.  The discussion is complicated by the fact that no-one can define
the notion of `objective' here.  However, intuitionists or constructivist
certainly do not believe you can just take your pick (except where it comes
to definitions, but that is universally agreed).  This would be ill-advised
too, since the fact that we cannot currently prove or disprove some propo-
sition does not mean that we will not some day succeed in doing so.

The same must apply to classical mathematicians.  Even though CH is in-
dependent of the axioms of ZF Set Theory, it is conceivable (although high-
ly implausible) that someone will some day come up with new methods of
mathematical reasoning that are *obviously* valid, using which CH can be
decided.  The situation is different from the one concerning Euclid's Fifth
Postulate.  None of Euclid's axioms is `true', obvious or not.  There are
`geometries' in which there can be several lines through two given points
(e.g., great circles on a sphere).  However, a `mathematics' in which both
a proposition and its negation can be true is unacceptable to classicists,
intuitionists and constructivists alike.

A good question to discuss Platonism in mathematics is the question whether
the cardinality of the continuum (c) is greater than that of the natural
numbers (Aleph0).  This has some bearing on CH.  The usual definition of
two sets having the same cardinality is the existence of a bijection.  I
give a slightly different definition, that is equivalent to the original
one for classical mathematicians (who accept Zorn's Lemma or the Axiom of
Choice) but makes a difference to constructivists:

Define A <= B, for two sets A and B, to mean: there is an injective mapping
from A to B.  This relation is reflexive and transitive.  Define A ~ B to
mean A <= B & B <= A.  This relation is an equivalence relation.

As an example, take A to be the set of Goedel numbers of never-halting Tur-
ing Machines (TM's) and B to be the set of natural numbers.  Obviously, A
<= B.  But we can, for each natural number n, construct a TM that outputs
(in some coding) n.000000... (i.e., a decimal representation of n followed
by an infinite sequence of zeros).  So also B <= A, and therefore A ~ B.
However, we cannot effectively construct a bijection between A and B.  For
B would then be recursively enumerable.  Since the complement of B is also
recursively enumerable, this would then imply that the Halting Problem is
solvable.

Now consider the following `axiom': All real numbers are computable (mean-
ing: there exists a never-halting TM that outputs a decimal expansion).

Is this axiom false?  There is one thing I am sure of: you cannot come up
with a counterexample that will satisfy a constructivist.  For a construc-
tive definition of a purported counterexample real number can be turned
into the construction of a TM computing that number (or we would have
disproven Church's Thesis).

If one assumes the axiom to be true, then we have R~N, where R denotes the
set of real numbers, and N the set of integers.  So c = Aleph0.  The usual
diagonalization argument to show that c > Aleph0 does not work, because it
produces an uncomputable number.  It shows, however, that the Halting Prob-
lem is unsolvable.  (Assume it to be solvable.  Enumerate all never-halting
TM's, let d[i] be the i-th digit output by the i-th machine, and construct
a TM whose i-th digit is d[i] mod 5.  Let k be the number of this machine
in the enumeration.  Then d[k] = d[k] mod 5.  Contradiction.)  So the con-
nection between the two arguments is deeper than the surface.

A classical mathematician will now say: Although the diagonal number is not
computable, to me it is an acceptable definition.  But how can we interpret
this unless we assume the belief that the diagonal real number `exists' in
some sense.  But what is this sense?

What the constructivist and classicist have in common is that they see that
to assume that one can enumerate all `acceptable' real numbers in an `ac-
ceptable' way leads to a contradiction.

Now it is not uncommon that some assumption leads to a contradiction.
Famous examples are the assumption of the existence of the set of all sets
(what is its cardinality?), or of the set of all sets that do not shave
themselves.  (Interestingly enough, these contradictions are shown by diag-
onal arguments of again the same form.)  The usual way taken out is to ex-
plain these counterexamples away, by saying, e.g., "the set of all sets
does not exist".  It is not a priori clear why one cannot admit the set of
all sets.  However, it leads to a contradiction, so let us put some fences
around the roads that lead to this pit.  So there are fewer `acceptable'
sets than we would, naively, have thought.  It would not have been strange
if, in history, mathematicians would also have put fences around the defin-
ition of enumeration, leading to fewer `acceptable' real numbers.  (This
does, by the way, not automatically lead to the constructivist position.)
In particular, they could have taken the position: since the argument
showing that 2^n > n for each cardinal n leads to a contradiction if we
take n = Sjin, the cardinal of the set of all sets, there is something
wrong with the proof, and we must not admit unrestricted powersets.
This is not the way it happened.  Instead, it was decided to be liberal
here and assume higher cardinals than Aleph0.

Before I can talk about the limit of a given sequence, or the smallest
number with a certain property, I must show that that limit or that number
exists.  Not to require this leads to contradictions.  For example:  Let n
be the smallest number that is both equal to 0 and to 1.  So n = 0 and n =
1.  Therefore, 0 = 1.  In general, mathematicians must show that the ob-
jects they define exist.  So before we define Aleph1 as the smallest cardi-
nal exceeding Aleph0, we must at the very least show the existence of car-
dinals exceeding Aleph0.  This requires showing the existence of the unres-
tricted powerset of at least one infinite set.  But how is this done?  Why
is just assuming its existence any more acceptable than just assuming the
existence of the set of all sets?  Because one does not lead to a contrad-
iction, whereas the other one does?  That is a rather weak justification,
even if one could rigorously prove that the assumption would not lead to a
contradiction.  However, this has never been shown anyway, nor can it be
shown.  One can certainly play an interesting formal game with the higher
cardinals.  But is it more than a game?  For example, can we take the
statement (by H.W. Lenstra) seriously that pure mathematics = not-yet-applied
mathematics?  Is it conceivable that one day we might find an application
of Aleph1?  That day I too will believe it exists:-)
-- 

     Lambert Meertens
     ...!{seismo,philabs,decvax}!lambert@mcvax.UUCP
     CWI (Centre for Mathematics and Computer Science), Amsterdam