in the Irish Meteorological Service from 1964
to 1968. For most of that time he was a student of
Mathematical Science at University College, Dublin, where he was
particularly influenced by
P.G. Gormley, who turned him decisively to analysis, particularly
complex analysis. Resigning from the Meteorological Service to devote
himself to Mathematics, he was further influenced by T. J. Laffey
and E. C. Schlesinger, and decided to
continue his studies in the United States.
At Brown University, his horizons were expanded by H. Federer, B. Harris,
A. Browder, J. Wermer, B. Cole, W. Fulton and A. Landman, among others,
and his subsequent work focussed mainly on algebraic and geometric aspects of
real and complex analysis. In 1975, after two years at the
University of California
at Los Angeles, he was appointed Professor of Mathematics at Maynooth College,
He was elected to the Royal Irish Academy in 1980. While engaged in
teaching, research and administration at Maynooth, he contributed to
the mathematical and scientific community in Ireland and abroad, and
visited research institutes and universities in Canada,
France, Germany, India, Israel, Japan, Russia, Sweden, Spain,
the UK and USA, as well as speaking at many international conferences
and collaborating with many researchers.
Now Professor Emeritus of Maynooth University, he devotes himself to
research, writing and pro-bono activities, as well as
a modicum of outdoor activity and cultural pursuits.
He is married, with three
surviving children and seven grandchildren.
His home page.
up through the natural numbers, integers, rational
numbers, real and complex numbers, and we establish their
properties on the basis of some more basic
We try to avoid various extremes:
The theory is not a 'formal theory', in the strict,
logical sense. There are no axioms for the logic used.
It is thus a rigorous theory for actual people,
not for automata.
It does not try to do the impossible. You have
to understand that, because of the work of
twentieth-century logicians, we know that we cannot
place mathematics on the kind of foundation of which David Hilbert
dreamed. That is, it is impossible to give an account of analysis
(or of any other sufficiently-rich mathematical theory)
which is provably consistent. The only comfort
I can offer is that if the theory is inconsistent,
then it is already inconsistent by the end of the
chapter about the natural numbers,
We do not get involved in the technical discussion of
We avoid doing 'clever' stuff that violates
common-sense notions. For instance, we do not insist
that every object under discussion be a set
(or even a class).
Some of the ideas and concepts introduced are purely auxiliary, and you can forget about them once you have worked through this book.
Indeed, the entire book is designed to be read in a week or two,
and then forgotten, unless you are one of the people who
want to dig deeper still, and get really serious about logic.