Reasoning about Theoretical Entities
:

I wrote this book because I was concerned that the Philosophical literature contained many discussions of reductionism that expressed views about the feasibility - or infeasibility - of the enterprise without ever making clear what the reductionist claims were. For one party (the antis) there is much rhetorical benefit to be derived from painting the other party as unfeeling brutes who don't understand

I treat in detail two special cases of theoretical entities arising
from equivalence relations, namely cardinal and ordinal arithmetic.
There is an intelligible and plausible ontological stance to take in
relation to cardinals to the effect that there are facts about
cardinals, but no cardinals: facts about cardinals are just facts
about sets and mappings between sets. The fact that cardinal
multiplication is commutative is nothing more than the fact that *A
× B* and *B × A* are the same size. On this
analysis the only assertions that we can meaningfully make about
cardinals arise from predicates of sets for which equipollence is a
congruence relation. Although this is an attractive analysis (it
explains why ``is 3 a member of 5?" is a silly question, for example) it is
a bit too restrictive, in that it outlaws (for example) the relation
"|*x*| > min(*y*)" where *x* is a finite set of
naturals (and |*x*| is the cardinality of *x*). This is
*relative largeness* which we definitely want to keep). It turns
out that a more fruitful analysis identifies as cardinal arithmetic
those assertions about (naive) cardinals whose truth value does not
depend on a choice of implementation of cardinals as sets. This lets
in relative largeness but excludes 3 ε 5 as desired. To
complete the picture there is a completeness/preservation theorem for
a typing system. The typing system says there are two types:
cardinals and sets, a binary relation: ε and a unary function:
| |.
`* x = |y|*' is well-typed iff `*x*' is of type CARDINAL and
`*y*' is
of type SET. `*x* ε *y*' is well typed iff
`*y*' is of type SET. So we
can prove that something is well-typed iff its truth-value is not
affected by choice of implementation.

This completeness theorem is clearly a sensible result, and one we
should be happy with. However the act of turning over this stone has
revealed an interesting fact which--although perhaps not terribly
surprising--is certainly significant and worth noting. This
completeness theorem turns out to be equivalent to the axiom scheme of
replacement. It is surely remarkable that a philosophically motivated
way of merely *thinking* about cardinals (never mind proving
theorems about them!) commits us to a set-existence axiom: the
conclusion is that the decision to think of cardinal numbers in this
rather natural way commits us to replacement. I think this is the
strongest argument yet for replacement, and it appears to be
new--although Adrian Mathias has shown me a proof that if *x × y *
exists for all *x* and *y* and all implementations of ordered pair
then replacement follows.

This analysis has some new light to shed on the Burali-Forti paradox, but it is too involved to discuss in a short summary, and the real purpose of the book is of course to be a dry run for the much harder cases to be found in philosophy of mind (``mental states are logical constructions out of brain states''). If you want to start to get a handle on what claims like this might mean, then you should read this book. The blurb quotes one of the publishers' readers as saying ``Not since the days of Carnap's Aufbau has reductionism received such close attention...''

You can order it by visiting World Scientific . It has already been nominated for the British Society for the Philosophy of Science President's prize for the best recently published textbook in Philosophy of Science (by me admittedly - but seriously!!) A list of typos will be maintained here .

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