FOM: New book on foundations

[John Mayberry]

The Cambridge University press has recently published a book by me 
dealing with the foundations of mathematics:

	The Foundations of Mathematics in the Theory of Sets
	Encyclopedia of Mathematics and its Applications, Vol. 82
	J.P. Mayberry
	Cambridge University Press, Cambridge, 2000
	ISBN 0 521 77034 3 hardback

I think it might be of interest to members of the FOM list as it 
takes up many of the topics that have been discussed on the list, as 
well as others which haven't been but perhaps ought to be.
 	I have put up the preface and the first two chapters, 
together with an Analytical Table of Contents, on my website

	www.maths.bris.ac.uk/~majpm 

under "Recent Publications".
	Steve Simpson has suggested that perhaps I ought to say a few 
words here by way of an introduction. 
	The main argument of the book arises from my conviction that 
there is a central fallacy - let's call it the *operationalist 
fallacy* - that infects much of the thinking about the foundations of 
mathematics. In its simplest form it consists in the conviction that 
the natural numbers 0, 1, 2, . . . constitute the "raw data" of 
mathematics, that they are simply "given" to us as a unique infinite 
structure which can be characterised fully and rigorously as "the 
successive images of zero under repeated applications of the 
*successor operation*". On this conception, the principles of proof 
by mathematical induction and definition by recursion are simply 
"given" along with the natural numbers themselves, so that, in 
particular, these two principles can be accepted as legitimate 
without further justification: they are, in short, *self-evident*. 
	Another form of the operationalist fallacy is to be found in 
the view of formal syntax in which the "constructions" employed 
(which are really recursively defined functions) are somehow "given" 
immediately as self-evidently efficacious, and so do not require 
mathematical analysis or justification. The operationalist fallacy 
thus underlies what I believe to be the illusion that Formalism is a 
coherent foundational theory.
 	The first thinkers to acknowledge the operationalist fallacy 
were Frege and Dedekind, Frege in his definition of "following in a 
series" in the *Begriffsschrift* and the *Grundlagen*, and Dedekind 
in the theory of "chains" expounded in his monograph, *Was sind und 
was sollen die Zahlen?*. The clearest description of the fallacy *as* 
a fallacy, however, is to be found in Dedekind's "Letter to 
Kefferstein", and, moreover, Dedekind saw much more clearly than 
Frege did, the central significance of the principle of definition by 
recursion.
 	Zermelo also belonged to the anti-operationalist party, and 
in his article "Sur les ensembles finis et le principe de l'induction 
compl`ete" gave a purely set-theoretical derivation of mathematical 
induction, a derivation, what is more, that did not rest upon any 
assumption about the existence of transfinite sets. I think this work 
of Zermelo's is highly significant, representing as it does a kind of 
*finitary*, but nevertheless  *completely set-theoretical*, analysis 
of  natural number.
 	In fact, the lesson that emerges from the work of all three 
of these great pioneers of our subject is that the notion of "set" is 
prior to the notion "natural number" and we must employ it if we are 
to carry out a proper analysis of the principles of induction and 
recursion in natural number arithmetic, or in formal syntax, come to 
that.  But, surely, this runs counter to the widespread perception 
among the general public, and, indeed, among mathematicians as well, 
that set theory is the very quintessence of the "New Math", whereas 
natural numbers, the familiar things we count with, constitute the 
oldest part of our mathematical inheritance.
	 That widespread perception, however, is just a mistake. In 
Chapter 2 of my book, which I have posted on my web page, I have 
called attention to the fact that our notion of *natural number* is a 
relatively recent invention, and that the ancient notion of "number" 
(the Greek notion of *arithmos*, namely, a finite plurality composed 
of units) that it has replaced is essentially our notion of "(finite) 
set". What is more, the ancient notion is very much better as a 
vehicle for explaining the facts that underlie the science of 
arithmetic than our modern notion of "natural number".
 	But that is not all, for as I point out in my Preface (which 
is also posted on my web page), modern set theory itself is really 
best regarded as a refined and generalised version of classical 
arithmetic, though, to be sure, a *non-Euclidean* arithmetic, since 
it abandons Euclid's axiom (Common Notion 5) that "the whole is 
greater than the part" by assuming the existence of "numbers" (i.e. 
arithmoi, that is to say, *sets*) which have the same size as certain 
of their proper subsets. Indeed, I am convinced that the best way to 
look at the development of modern set theory is as a transition 
*within classical arithmetic* from the *Euclidean* conception of 
"finiteness" to the modern *Cantorian* conception. From that 
standpoint, all the central axioms of Zermelo-Fraenkel set theory - 
including the so - called Axiom of Infinity - are to be seen as 
*finiteness assumptions* and, as such, take on that air of 
self-evidence that is expected of "axioms" understood in the 
old-fashioned sense of  "basic truths".
	 What I am calling the "operationalist fallacy" here also 
arises, in a different guise, in the foundations of Cantorian set 
theory as well, where it takes the form of positing "transfinite 
processes" for defining functions (e.g. as in the "generation" of the 
universe of sets as a cumulative hierarchy by the "transfinite 
iteration" of the power set operation). In Chapter 5 I attempt to lay 
down axioms for conventional Cantorian set theory that avoid taking 
these supposed "processes" as fundamental.
 	Perhaps the most novel aspect of the book is the treatment, 
in Part 4, of what I call "Euclidean set theory" which is classical 
arithmetic, with its Euclidean conception of finiteness, but 
reformulated as a version of Zermelo Fraenkel set theory in which 
Cantor's (misnamed) "Axiom of Infinity" is replaced by an axiom to 
the effect that all sets are Dedekind finite. In the theory of 
"natural number arithmetic" that naturally arises in this theory 
there are natural number systems (i.e. simply infinite systems) of 
different lengths, for Dedekind's theorem that all simply infinite 
systems are isomorphic is no longer provable. If you want an idea of 
how to make sense of that, look at Part 4 of the Analytical Table of 
Contents that I have posted on my web site. To see it worked out in 
detail, of course, you have to turn to the book itself.

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John Mayberry
School of Mathematics
University of Bristol
j.p.mayberry at bristol.ac.uk
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