[FOM] Slater's recent comments
Randall Holmes
holmes at diamond.boisestate.edu
Mon Sep 29 13:40:52 EDT 2003
Dear FOM colleagues,
Slater asserts that he has "dealt with my point", but I note for the
record that this is not the case. He has asserted his views about the
proper interpretation of the grammar of sentences about numbers and
sets -- but not in such a way as to compel my agreement, which is not
forthcoming.
He says:
"But I also think there is no choice about the matter: the place for
'n' in '(nx)(x is in Y)' has a quite different syntax from the place
for 'Y', and any formalisation which does not respect this difference
is not going to mix up numbers and sets, since, for instance,
iota-n(nx)(x isin {{},{{}}}) is identical to the number 2, but is not
identical to {{}, {{}}})"
It is quite true that if you set up your syntax correctly, you will
not mix up numbers and sets (you will say about numbers only those
things which are appropriate to numbers and about sets only those
things which are appropriate to sets); that's what I meant by saying
that numbers are an abstract data type. This does not mean that
numbers are not sets; it means that you have avoided the question as
to whether they are or not (of course, it may be a very good idea to
do so!).
In computer science (whence I borrow the notion of abstract data
type), an abstract data type interface protects data being used for a
certain purpose from being handled in ways inappropriate to that
purpose. However, the objects of that abstract data type really do
have the hidden features which the abstract data type interface
protects. For example, one might not want to allow the user access to
the information as to whether one element of an object of type "set"
is stored in memory before or after another element of the "set",
because the only thing which is relevant to the abstraction "set" is
what "elements" it has (in particular, there is no canonical order on
its "elements"). Nonetheless, the elements of an implemented "set" are
stored in some order (please note that here I am not talking about
mathematical sets, but about a computer data type "set").
Applying the analogy, restricting ourselves to the language of
arithmetic (with its usual interface with the theory of finite sets)
restricts the contexts in which we can relate numbers to sets:
sentences like "the set A has 2 elements" are understood as natural,
while sentences like "2 is an element of 3" violate convention. Hazen
describes sentences like these as "don't cares". The point I am
making is that "don't cares" do not have to be meaningless: Quine, who
originated this usage (I believe) was equally open to the possibility
that "don't cares" may have truth values -- which we don't care about
(for the purpose at hand). In normal discourse about arithmetic and
finite sets (related in the usual ways) to ask whether 2 is an element
of 3 is to ask for private information about the way that the natural
numbers are implemented, which could lead to useless philosophical
disputes rather than to useful mathematics :-) But the fact that we
don't ask this question does not necessarily imply that it doesn't
have an answer (or, more confusingly, that it might not have different
answers for those who choose different implementations -- including
"true", "false", and "category mistake"). It may be true that there
is a best answer, but the question as to what the best answer is is
not a question of the usual theory of arithmetic and finite sets, and
also is not settled by the facts of grammar, which can be interpreted
in such a way as to support many positions.
It is an entirely defensible position that if numbers (or other
mathematical objects) are in fact some sort of objects, that whatever
objects the numbers _are_ are just that -- objects. Any object either
is a set or is not a set (if there really are sets). One could also
suppose (as Hazen has suggested) that the proper logic is a type
theory in which the question as to whether or not a "number" is a
"set" may not be asked; Slater's view appears to be something like
this. But the facts of grammar do NOT enforce such a view; the fact
that numbers are only used in certain ways in language can be
interpreted as indicating (following Hazen or Slater (?)) that we
suppose that there is a special type of numbers about which we cannot
ask the question as to whether they are sets, but it can equally well
be interpreted to mean that we postulate the existence of numbers for
a specific purpose, and that our talk of numbers is normally limited
to predicates and relations appropriate to that purpose (numbers are
an abstract data type and we do not concern ourselves with the details
of our implementations or the implementations used by others, when we
are talking about arithmetic). From this standpoint, the conventional
ZFC view that the von Neumann natural numbers implement all the
predicates and relations appropriate to natural numbers, and so can be
conveniently identified with the natural numbers is coherent -- though
clearly conventional rather than natural. The position which I have
described (to which I do not necessarily subscribe) that the Frege
natural numbers can be taken to be the natural numbers (with more
claim to naturalness), is also coherent, and not in any way refuted by
anything Slater has said (_contradicted_, but not _refuted_).
The type-theoretic view described by Hazen is definitely also
coherent. From this viewpoint, remarks like Slater's could be made.
I do not ascribe a view to Slater, because he has not clearly
explained what his view is.
I can't imagine what Slater means by the following:
"But premise 2 is false, since, following on from the above, the
number two is not the set of things with two members, and so it is
not the extension of the property of having two members - it is the
extension of the property of being 2!"
This hinges on the grammar of 2 in "the extension of the property of
being 2". If this means "the extension of the property of being
(identical to) 2 (noun)", then he is saying that 2 is a property which
belongs only to itself (???). If 2 is to be understood as an
adjective, then this doesn't contradict a "Fregean" view at all:
to be 2 (adj.) is to have 2 elements, and the extension of the
property of being 2 (adj.) is the set of all sets with two elements.
In either case, I can only construe this statement as confused (in one
case because it doesn't seem like a view anyone would want to defend,
and in the other because it agrees with the view it purports to refute).
The opinions expressed | --Sincerely, M. Randall Holmes
above are not the official | Math. Dept., Boise State Univ.
opinions of any person | holmes at math.boisestate.edu
or institution. | http://math.boisestate.edu/~holmes
More information about the FOM
mailing list