[FOM] Foundations/Philosophy

[Dean Buckner]

Friedman:

> The set theoretic foundations of mathematics is still the only workable
> universal foundations for mathematics that we have. Other candidates are
> either philosophically incoherent, or are naturally mutually interpretable
> with the set theoretic foundations.

That at least seems a reasonable criterion.  I agree: any alternative to set
theory has to be

(a) coherent (= philosophically coherent)
(b) not just mutually interpretable with standard set theoretic foundations
(c) (also) capable of explaining "ordinary mathematics"

And I don't see why any of these rule out the kind of story that Slater & I
have been proposing, at least not for certain, and I don't see why it
shouldn't, pace Martin Davis, be acceptable for discussion on FOM.

Let's start with "coherent".  The alternative story I have in mind (and
which aligns pretty well with Prof. Slater's story), is that sets, in our
natural conceptual scheme, are identical with their members - whereas in set
theory a set is something which contains its members.  So in plural theory
we expect to see statements like

    the people at the party = Alice Bob and Carol
    S = a & b & c
    S = the ( x: x was at the party)

and so on.  What's incoherent about any of these?  The first statement is
true iff "those people" refers to what "Alice Bob and Carol" refers to.  If
the people at the party were brokers then (by Leibniz) Alice Bob and Carol
are brokers.  Incoherent?  How?  And the beautiful limiting case is when
only one individual satisfies the description, thus

    the ( x: x was at the party) = Alice

whereas in set =-theory world we get the mysterious {Alice} - see Slater's
paper for more on this.  How "coherent" is{Alice}?  Who or what is this?
Has science anything to say about it?

OK, what about ("b) not mutually interpretable with standard set theoretic
foundations".  I've never quite understood this one, despite requests from
FOM readership for coherent explanation.  I understand it as, every theorem
in one system is a theorem in the other, and conversely, given some method
of translation between the two.  I dont see how we get this.  The simple
limiting case is {Alice} again.  We can have a theorem in each system
proving the existence of Alice, only in set theory will we also have a
theorem establishing {Alice}.  And to take Hartley's example, we have

    The pack of cards = the Hearts & Clubs & Spades & Diamonds
    the Hearts = the A, K, Q, J &c of Hearts
    &c

Since these are identity statements, by Leibniz again we have

    The pack of cards = the A, K, Q, J &c of Hearts, the A &c of Clubs &c

i.e. the pack of cards  = those 52 cards.  But as Hartley argues (p.8)

"But if [the pack] was a set in the current mathematical sense it could not
be both sets, since one set has 52 members, while the other has 4 members. "

I don't see any possibility of "mutually interpretable" here.

Finally (c) "(also) capable of explaining "ordinary mathematics" ".  There
has been a gauntlet thrown down for some time by Davis (explain a simple
proof in analysis using plurals) and by Friedman (explain a proof about
poker) which I have, I admit, not been able to show.  This is because,
however, there are a number of competing theories of plural reference and
quantification (Tom McKay's is another, Jonathan Lowe's is yet another),
which are very  foundational & important, and it's hopeless addressing the
more simple questions like proofs, until we have all discussed the more
difficult ones, like the basic assumptions.

For example, one basic disagreement among the pluralists is whether to have
infinite sets or not (I believe not, I believe the whole idea of infinite
set belongs to set theory alone).

But why is that an argument against any sort of discussion of the subject?
There were all sorts of sceptical arguments against Biblical creation
accounts right upo until the nineteenth century.  Were the sceptics right to
propose their arguments, without a complete and coherent & fully worked out
account of everything?  I'm not sure we have that even now.

Dean