FOM: Recommended books

[Dean Buckner]

(1) The Foundations of Mathematics in the Theory of Sets
J.P. Mayberry,  Cambridge University Press, Cambridge, 2000
ISBN 0 521 77034 3

I strongly recommend this book.  This is not just because I was a student of
Jonh's in the 1970's, and had the opportunity to discuss a wide variety of
ideas with him in the great "Room K" seminars on Aristotle hosted by CJF
Williams at Bristol University in the early 1980's.  Many of the ideas
discussed here are entirely new.

Mayberry's central idea is that "the modern notion of a set is a refined and
generalised version of the classical Greek notion of number (arithmos), the
notion of number found in Aristotle and expounded in Book VII of Euclid's
Elements (p ix)".  That is, a number is just a number or set of of things.

The natural numbers, he says, are a recent invention.  Part of the interest
of the book is his working out of "Euclidean set theory" where he replaces
Cantor's assumption that the species of natural numbers forms a set by the
"natural" Euclidean assumption that every set is strictly larger than any of
its proper subsets.

The book is truly foundational.  We cannot use the modern axiomatic method,
he says, to establish the theory of sets.  "We cannot, in particular, simply
employ the machinery of modern logic, modern mathematical logic"

He cautiously challenges one the most important of the set-theoretical
axioms.  "Cantor's axiom, the so-called axiom of infinity - has scarcely any
claim to self-evidence at all, and it is one of my principal aims to
investigate the possibility, and the consequences, of rejecting it."


Mayberry (an early member of FOM) discusses some ideas in a posting (May 1 -
May 31, 2001), and part of the book is on his website.

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


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(2) Grattan-Guinness The Search for Mathematical Roots, 1870-1940.

This is best described as an interesting collection of papers, letters,
pictures and anecdotes that you might have found bundled together in a
garage sale, impregnated with caustic & often poisonous remarks from the
author, and occasionally more robust humour of the P.G.Wodehouse variety.  I
read it with great pleasure and fascination and warmly recommend it.  His
pastiche of Russell is particularly entertaining.

GG.  makes some particularly interesting and important proposals about
Frege.

1.  There were two Freges.  One was a mathematician who made an
insignificant contribution to the subject.  He has a minor place in the
complex family tree of modern set theory (p. 570).  The other (who GG calls
Frege' ) was "the founder of the analytic tradition", was an invention of
the anaytic tradition itself (he rightly singles out Dummett as one
culprit).  Thus neither of the Freges really existed.

2.  Frege (the real one) was not well-received in his lifetime for a number
of reasons including his awkward personality, his crude mis-statement of
opponents' views, his clumsy notation and the oddity of his ideas (GG pokes
much fun at the idea of the True and the False).  He thus dispels the
"artist in garret" myth put about by what he calls "the Frege' industry".

3.  Frege was an unoriginal thinker whose key ideas (it is implied) are
taken from sources he failed to acknowledge.  Some information entirely new
to me was that Frege's father Karl published a grammar book in 1862 in which
(amongst other things) it is argued that object-words can be formed from
concept-words by appending the definite article "the".  (I already knew that
the definite article idea is found in Lotze and Ueberweg, who were prominent
and well-known logicians of the period).  Thus the idea of the Eigenname may
have been the first philosophical ideas he took up (age 14) - as well as the
last one he
dropped (age 76).

I have a lot of time for these ideas, although (1) having been educated in
the Frege' tradition, I can say thagt Frege'ans read more Frege than GG
suggests (2) the entertainment of Frege as a writer is precisely the
childish polemics he takes up against all and sundry ("I can see my ideas
were not like coins dropped in the street, that anyone could pick up and
make their own") (3) while an awful lot of Frege's ideas are simply received
opinion, or ideas of the period, whose provenance Frege does not mention,
there a number of key concepts that are surely his alone, and Frege at his
best is I think both good and original, though GG would doubtless object
that the good ideas are not original, and the original ideas not good
(Johnson).




Dean Buckner
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