[FOM] 605: Integer and Real Functions
Mark Steiner
mark.steiner at mail.huji.ac.il
Sun Aug 30 15:50:40 EDT 2015
I would like to make some remarks on Harvey's posting on elementary
functions, based on my own reflections in:
Steiner, Mark. "Mathematics--Application and Applicability." In OUP
Handbook for the Philosophy of Mathematics, edited by Stewart Shapiro:
Oxford University Press, 2005.
I don't think that what I am going to say contradicts in any way the
interesting program that Harvey wants to pursue.
1. "Addition on N serves a very fundamental purpose. COUNTING finite
sets." I would rather say MEASURING ALL sets. Addition on N requires
another mathematical "miracle," as Harvey puts it: finite cardinals and
finite ordinals are the same. We can thus apply recursive definitions,
which derive from the ordinality of the natural numbers.
2. A similar remark goes for multiplication: xy is the (cardinal) number
of the union of x disjoint sets, which of which has the cardinal number y.
In the finite realm, since all the natural numbers are both cardinals and
ordinals, we can use ordinal arithmetic to make calculations which count
how many members the union has. In the infinite realm, we have no such
identification which makes even simple questions like: what is 2 to the
power aleph_0 notoriously hard, having no arithmetic or algorithm to
answer such questions. (My colleague, Saharon Shelah, tried to persuade
me that we should stop asking the question; it's not the right question to
ask.)
3. To say that multiplication is grounded in areas seems to be to get the
matter backwards: areas of what? We do not want to ground arithmetic in
one of its empirical applications, nor do we want to assume that space is
Euclidean. From Greek mathematics till Descartes, the numbers were not
closed under multiplication, the product of two linear magnitudes was,
indeed, a two dimensional magnitude.
4. I would now like to point out something about the concept of sets.
There is a common believe that set theory begins with infinite cardinals
or ordinals, and has no real application in elementary finite arithmetic.
I believe that this is incorrect: the difference between addition and
multiplication on the natural numbers is that addition has a simple
empirical application, namely measuring the size of "collections", i.e.
mereological sums of stable objections. It is clear from this why
addition is commutative. Multiplication does not have a natural model in
mereology, which is way school children don't understand why you can "add"
candies to candies but you can't multiply candies by candies. The numbers
x and y in xy play a different role; x counts not candies, but sets of
candies. The commutativity of multiplication can be proved using multiple
inductions, and I used to teach the proofs on the college level, but the
proof is somewhat "pathological," to borrow a term and no student could
understand why multiplication is commutative. Saul Kripke once told me
that it took him a year to figure out why 7 x 15 has to be equal to 15 x
7, etc., and then he was already five years old. Using the set
theoretical approach it is clear that multiplication can be thought of as
a function on the Cartesian product of two sets, and thus is clearly
commutative. If I am right, the concept of a set is embedded in
mathematical thought, at least by implication, at a very elementary level.
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