FOM: More on Maddy on method

Neil Tennant neilt at
Sat Feb 21 14:42:04 EST 1998

Maddy writes:
We can think that a particular theorem of algebra is provable from the
axioms of set theory (via definitions of the appropriate algebraic notions)
without thinking that we would ever have discovered that algebraic theorem
-- that is, without thinking that we would ever have thought of looking at
these particular sets in this particular way -- without thinking
'algebraically'.  That's what I mean when I deny that the methods of algebra
are the methods of set theory.  

OK then; this is compatible with the theory-reducibility claim of

	(algebra, analysis, geometry, topology, etc.) 
	can be reduced to 
	set theory

and very different from the situation concerning physics and other

	(biology, psychology, economics, etc.)
	can NOT be reduced to

Pen is of course right about the psychology of mathematical
inspiration, insight, discovery, inventive conceptualization, and so
on. One would hardly expect an algebraist to think only in terms of
the epsilon relation when conceptualizing actions of groups, kernels
of homomorphisms, quotient algebras and the like, and seeing his or
her way to interesting new algebraic results.

But is it not still mind-boggling and astonishing that every single
deductive twist and turn in any branch of mathematics, with however
highly compiled concepts might be thus employed, can in principle be
reduced, by appropriate modelling definitions, to trains of reasoning
that concern *only* sets and the membership relation? The mystery is
even more wonderful then we remark further that it can all be done in
*pure* set theory, describing this dyadic tissue of things teetering
on the empty set.

No such foundational claim can be made for physics. Moreover, in the
case of hypothesis-formation in *any* science, the basic conceptual
modules being juggled are likely to be highly compiled when compared
with the conceptual primitives of the science itself. That is, the
*method* of the science will appear to involve ways of thinking that
are not directly or immediately catered for in the primitives notions
that suffice for their eventual expression.

Adaptive evolution by natural selection strikes most people as "going
conceptually beyond" heritable phenotypic variation and differential
reproduction. Yet the former reduces to the latter. Equipped only with
the concepts of heritable phenotypic variation and differential
reproduction, Darwin's stunning insight into how phenomena such as
speciation might be possible could well elude one. To attain that
insight, one needs to be thinking in terms of adaptation and
reproductive success---concepts which, however, can be obtained from
those of heritable phenotypic variation and differential reproduction.
So Pen's point about mathematics has an analogue even within
evolutionary biology itself. 

Still less would one expect anyone to be able to attain Darwin's
insights if they were restricted to the concepts of quantum mechanics
and what is definable therefrom. Indeed, there could be no
quantum-mechanical recapitulation of the story as to how species
evolved. Biology is not reducible to physics.

There is, however, a set-theoretical recapitulation of every theorem
of algebra, geometry, topology etc.; even if they were first proved
without thinking solely in terms of sets and membership.

Insofar as mathematics is a motley, it is so *only* with regard to its
methodology of discovery; with regard to its ontology and resources of
proof, however, it can be revealed *via reduction* as the non-motley
of sets and membership (according to the set-foundationalist).

Strangely, no set-foundationalist has ever (as far as I know) made so
bold as to claim that algebra, geometry, topology, etc. are "folk
mathematics", ultimately false and in need of complete replacement by
set theory. Yet we hear claims from the so-called eliminative
materialists that "folk psychology" is ultimately false and in need of
complete replacement by an as-yet-unfinished neuroscience! Perhaps the
explanation is precisely that, via the reduction that everyone
acknowledges is possible (of mathematics to set theory) the mathematics
would be vindicated as *true*. Correspondingly, the eliminative
materialist's worry seems to be that any theory that is *irreducible*
to our best basic hard science of matter and energy in space and time
must be false in important respects.

The eliminative materialist, then, prefers the "lower level" theory to
the "higher level, folk" theory. But consider the analogue of this in
the mathematical case: the eliminativist would have to prefer *set
theory* to any mathematical theory that might turn out to be
irreducible to set theory! I doubt whether anyone would ever espouse
this kind of eliminativism in the event that a new branch of
mathematics turned out to be irreducible to set theory. Rather, we
would re-think what is to be counted as an adequate foundation for all
of mathematics.

What is it about non-set-theoretic mathematics, then, that ensures
that it would never be dismissed as "folksily false" upon defying
reduction to set theory?

Anti-analogously, what is it about ordinary psychology (talk of
beliefs, desires, intentions, etc.) that invites its (possibly
misguided) dismissal by some as "folksily false" upon defying
reduction to fundamental particle physics (or even neuroscience)?

Finally, what is it about evolutionary biology that has stopped anyone
from dismissing it as "folksily false" for defying reduction to
fundamental particle physics (or even chemistry)?

It seems to me that these are questions that would have to be
addressed in any "foundations of all subjects" in Harvey's sense.

Neil Tennant

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