FOM: Definition of f.o.m.
J.P.Mayberry at bristol.ac.uk
Sun Jan 18 13:57:11 EST 1998
Moshe' Machover's distinction between f.o.m. and ph.o.m. [Jan14] is of
fundamental importance. But I believe that our discussion will benefit
from a sharper formulation of what a f.o.m. must consist in. This will
be of relevance to the controversy over whether topos theory can
provide a foundation for mathematics.
I start from the observation that mathematics is about *proof*
and *definition*. This is what distinguishes mathematics proper from
other disciplines, such as physics, economics, and engineering, which
*use* mathematics. And it is because mathematics deals with proof and
definition that it requires foundations in a way that other sciences do
not. For mathematical proofs and definitions are intended to be
complete, absolute, and final - that, in any case, is the ideal that
mathematicians must strive for, even if it should never be perfectly
Mathematics, like a building, must stand upon its foundations.
Therefore, to expound the f.o.m. is to present the basic presuppositions
upon which mathematical proof and mathematical definition rest. These
fall under three headings
1.) ELEMENTS. These are the basic notions of mathematics: the basic
*concepts*, the *objects* that fall under them, and the basic
*relations* and *operations* that apply to those objects. These basic
notions neither require, nor admit of, proper mathematical definition,
but all other mathematical notions are defined, ultimately, in terms of
these basic ones.
2.) PRINCIPLES. These are the basic mathematical propositions that,
although true, neither require, nor admit of proof, and that constitute
the ultimate assumptions to which all mathematical proofs finally
appeal. As such, they ought to be, at least in some sense, obviously
and uncontroversially true.
3.) METHODS. These are the canons of definition and of argument that
govern the introduction of new concepts and the construction of proofs.
This is, of course, a restricted, perhaps a minimal, notion of what
counts as a f.o.m. But that makes it useful when we consider questions
such as whether Theory X can provide a f.o.m.
Martin Davis [Jan15] has called our attention to the true sense
in which the foundations of present day mathematics are to be found in
"Look at almost any current graduate graduate textbook in whatever
branch. It will begin with an introductory set-theoretical section on
"notation". This represents a consensus that set theory is the proper
Exactly! And the scare quotes on "notation" remind us that in such
introductory chapters the principles of set theory are typically put
forward in the innocent seeming guise of mere definitions or notational
conventions. ("The set of all subsets of a set S is called the *power
set* of S and is denoted by P(S).", "Let N denote the set of natural
Notice that what Martin Davis calls "the proper foundation"
here is not the formalized, first order axiomatic theory ZFC. Indeed,
it could not, as a matter of logic, be a formalized first order theory
of any sort, because the apparatus of set theory must already be in
place before you can explain how formal axiomatics works. Surely it is
obvious that no axiomatic theory, formal or informal, of first or
higher order, could, on its own, constitute a foundation for
mathematics. For it would be a central purpose of *any* foundational
theory to give an explanation of, and justification for, the axiomatic
method itself. Here when I speak of "axiomatic theories" and the
"axiomatic method" I have in mind, not Euclid's, but the modern notion
of axiomatics in accordance with which a system of axioms (e.g. the
axioms for groups) is seen as *defining* the species of mathematical
structures in which they hold true.
The consensus set theory that provides the foundations of our
mathematics is not, then, a formalized theory. To understand it you
have to grasp what a set is, what membership means, etc. But it is a
model of rigor nonetheless. And if you look carefully at the
presuppositions underlying it, you will discover that the Principles of
this consensus set theory - the basic assumptions that underlie
mathematical proof - correspond to the formal axioms of first order
ZFC. So if you do mathematics in the modern style you are working "in
Zermelo-Fraenkel" whether you realize it or not. It is the
non-formalized theory that is logically prior: formalized first order
ZFC is parasitic on it.
Now let me ask Colin McLarty: what are the Elements,
Principles, and Methods that constitute the category-theoretic or
topos-theoretic approach to f.o.m.? Give us a sketch of how *your*
introductory chapter for a graduate textbook in functional analysis or
topology might run.
Lecturer in Mathematics
School of Mathematics
University of Bristol
J.P.Mayberry at bristol.ac.uk
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