FOM: CAT inattention to f.o.m. issues
Robert Tragesser
RTragesser at compuserve.com
Sun Mar 15 09:02:13 EST 1998
Steve Simpson has repeatedly pointed
out the malappropriateness of category theory
and categorial logic for addressing f.o.m.
issues.
I have not seen that the
defenders of CAT address this issue with anything
like the care it deserves. An instructive recent
work by
Lawvere and Schanuel [Conceptual Mathematics:
A first introduction to categories, CUP 1997]
makes it clear that CAT can't address the deepest FOM
problem CAT's very virtues/powers raise (as I shall
explain).
Answering my own questions about f.o.m. goals,
this is surely a major one:
Reducing foundational/philosophical problems
to (delicate) questions about what (logically) depends
on what (and of course what is independent of what).
Bertrand Russell remarked in the Principles
of Mathematics that one of the great discoveries of
all time was just then emerging and clarifying: that
all of the concepts of mathematics could be represented
in the language of logic
.
This is important quite independently of any
logicist thesis. And we, now, after the development
of a formally rigous presentation of ZF are in a better
position to make the points:
[1] "all" mathematical concepts can be
logically modelled on the basis of a handful of primitive
expressions. When we have different presenations of the
same mathematical concept, the differences can be
preserved through different translations of the presentations
into the language of those primitives. [Friedman's point
that ZF is highly sensitive to/responsive to such
differences].
[2] The peculiarly "logical" character of the
primitives assures that whatever the failings of
logical rigor and logical detail of the defined,
the defining
expression composed of logical symbols permits an
exact and metamathematical investigation of logical
consequences (generally: logical dependencies).
[3] That "all" mathematical concepts go over (with
surprising ease for the experienced) immediately into
a language which is prime/optimal for carrying out in-
vestigations of what logically depends on what makes
logical languages optimal for investigating foundational
problems by the f.o.m. strategy (of reducing them to
problems of logical dependency) -- and for investigating
such issues of dependency among very different
appearing regions of mathematics.
ZF is as it were a powerful COORDINATE SYSTEM
for delicately representing mathematical concepts
and investigating logical dependencies. (There
are different logical coordinate systems, as for
example PRA. . . By investigating the logical
dependencies among the DIfferent logical
coordinate systems, and carrying out the investigations
within different logical coordinate systems gives one
very refined characterizations of logical dependencies.)
As Feferman and others have observed, there are two
very different uses for formal axiomatic systems:
One, the Noetherian,-- one chooses the axioms with
the intent of working within the system and means
for the axioms to embody the best and deepest insights
into the mathematics at issue.
TWO, one uses a formal-aximatic
system to metamathematically
investigate a branch of mathematics, but has no
intention of working within the system.
Hilbert's axioms for geometry are
borderlibne in the sense that
they permit the natural development of geometry
and also permit mmetamathematical investigations.
But Tarski's axioms for elementary geometry are
excellent for the latter (metamathematical
investigations) and hostile to the former
(the development of geometry within the system).
This distinction is part of a very general
distinction: a framing that is fruitful and insightful
for doing mathematics VERSUS a system which is
fruitful for f.o.m. investigations [investigating
what logically depends on what; the pursuit
of foundational/philosophical problems by
reducing them to questions of such dependency].
One can say roughly that ZF is not good for
doing mathematics; while category theory as
presented by Lawvere and Schanuel is rather nicer
from the point of view of doing mathematics --
THERE IS A MORE IMMEDIATE CONNECTION BETWEEN
THE PRIMITIVES OF THE LANGUAGE OF
CATEGORY THEORY AND THE STRUCTURES AND
PROCESSES ONE MEANS TO BE REASONIONG ABOUT..
The fundamental problem that both fomers
and CATs face is that repeatedly raised by Kreisel
(seconded by Feferman): What is there about
abstract methods (more conceptual methods) that
make them such a powerful source of understanding
and insight? E.G., there is a clear sense
in which looking at free groups in terms of
universal mapping problems gives one
a better understanding and appreciation than
the more concrete, set-closer characterization
in terms of generators, relations...
I see the foundational problem for CAT
people is to take on this problem.
At the same time I think they must admit
(as it is very clear that Lawvere and Schanuel
would readily and cheerfully admit) that category
theory is useless for investigating f.o.m.
questions (optimally delicate questions of
what logically depends on what).
Robert Tragesser
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