FOM: sets and numbers
neilt at hums62.cohums.ohio-state.edu
Mon Nov 3 14:21:59 EST 1997
Vaughan Pratt writes:
Mathematics is informed by many structures, and I see no a priori
reason why any particular one of them needs to be an absolutely
mandatory component of the mathematics of every civilization.
But I wasn't saying that the structure of natural numbers "needs to be an
absolutely mandatory component of the mathematics of every civilization."
Those are Vaughan's words, not mine. My own words were
arithmetic comes in as a NECESSARILY POSSIBLE extension of
[any] conceptual scheme
by means of whose concepts one can distinguish and re-identify objects.
So I was careful to AVOID claiming that arithmetic was a mandatory
component. I was claiming only that the aliens would be obliged to
acknowledge the legitimacy of *extending* their conceptual scheme by
tacking arithmetic on to it.
Vaughan also retorts:
I thought foundations made sets more fundamental than numbers.
But it's not at all clear that foundations does this. It would depend entirely
on the foundation in question. Jon Barwise has already cautioned against being
moved by the `modelling' of mathematical objects and structures provided by sets, into thinking that sets are the most fundamental of all mathematical objects. Sets are certainly convenient and comprehensive (as an ontology for
mathematics), but that doesn't necessarily make them THE ONLY foundational
entities of mathematics.
I'd like to see more discussion of this point. Let's consider the following analogy:
natural numbers middle-sized objects
sets fundamental particles
set theory quantum theory
arithmetic observational evidence about middle-sized objects
Quantum theory is tested (and accepted) on the basis of its ability to
predict the observational evidence. Likewise, set theory is accepted
at least in part because of its ability (via suitable definitions of
number-theoretic concepts) to deliver arithmetic. We rely on our intuitions
and common-sense about the behavior of middle-sized physical objects; we
feel better acquainted with them than with fundamental particles. Likewise,
we rely on our intuitions about the natural numbers; we have a better grasp
of them, as mathematical objects, than we do of sets.
Epistemologically, the parallel seems quite thorough. But ontologically,
matters are different. The question "What are physical things, REALLY?"
tends to elicit from the theoretical physicist the type of answer that
Eddington gave about his table: they are congeries of tiny particles whirling
around in largely empty space. The things that REALLY exist `for the physicist'
are the fundamental particles. (Why else call them `fundamental'?) Moreover,
it's by going to the level of fundamental particles that the really beautiful
and simple regularities begin to emerge---unlike the messy behavior of
middle-sized objects. Middle-sized physical objects, on this view, are
"really only" composites of the fundamental particles.
And that's where the key point of disanalogy is to be located. WHY SHOULD
the foundational set theorist be allowed to claim that the mathematical
objects that REALLY exist are the sets, and that the natural numbers (say)
are "really only" sets of a particular kind?
Note that Paul Benacerraf long ago pointed out one reason for the
disanalogy: namely, that it is arbitrary which particular sets one identifies
as "the" natural numbers. [By contrast, any given physical object is composed
of exactly those fundamental particles that are (mereologically) IN IT.]
Another reason for the disanalogy is that the objects of arithmetic have
their own beautiful and simple regularities. Indeed, one of the challenges
for the set theorist is to come up with a scheme of outright and conditional
set-existence axioms of comparable beauty and simplicity. As I understand him,
Harvey Friedman is rightly concerned to do just this. But even if he succeeds,
the special status of the natural numbers will remain. I don't see any reason
to be prepared to "demote" them, or deprive them of foundational status, even
in the wake of a really beautiful and simple axiomatization of set theory.
This brings in another foundational issue: the way facts about the universe of
sets are needed in order to establish facts about "just the numbers themselves". Again, Harvey's work is relevant here. But rather than rush once again to
elevating sets over numbers, regarding the former as `more foundational than'
the latter, we might say the boot is on the other foot: to wit, all the
complexity of the set-theoretic universe is already implicit in our conceptions of *just the natural numbers themselves*. It seems to me that this is something
already hinted at by the downward Lowenheim-Skolem theorem and the Bernays-Ackermann theorem.
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