FOM: Alice, Bob and Carol

Dean Buckner Dean.Buckner at
Thu Apr 11 15:40:58 EDT 2002

>----- Original Message -----
> From: "Manolo Martinez" <schiphol at>

> Hi, Dean.
> I don't know if the following is of any relevance to the discussion, but
just in case:
> In order to analyze these two different sentences:
> (1) Ive met three tap dancers
> (2) Three and five are odd numbers
> You need, I would say, different conceptual apparatus because you make
different ontological commitments. In analyzing (1) it could very well be
that you can do without the concept of number, something as you proposed, in
the line of:
> (3) Exist x, y and z such that  (x, y and z are tap dancers) & (x <> y, y
<> z, x<>z)
> This very tentative analysis is probably insufficient; as you surely need
something more to be able to obtain
> (4) Ive met an odd number of tap dancers
> Which is clearly entailed by (1).

First, a card-carrying nominalist will not want to "quantify over" things
like number.  In "There were a large number of people at the dinner",
"large number" qualifies "people" rather than "large" qualify "number".

Similarly, in "there were an odd number of plates on the table", the term
"odd number" qualifies "plates".  Interestingly, there's an ancient
grammatical dispute which pedants love, as to whether we write "there were"
or "there was" a certain number of people, plates &c.  I suppose nominalists
prefer "there were", realists "there was".

To reply to your (indeed relevant) point.  Clearly if there were three
plates on the
table, and if any collection of three objects is an odd number of objects,
then it follows (logically) that there an odd number of plates on the table.
Your method of putting it rather begs the question.  You assume there are
these things you call "number".  I'm not sure there are any such things -
though I do agree that, as it happens, there are numbers of things in the
world, and that the number are just those things (as in "a number of people
replied to my posting").

You're bound to ask how we know "any collection of three objects is
an odd number of objects".  "Odd" is surely a general notion of some kind,
whereas "three", if I'm right, has some kind of singular content.

I haven't really thought hard about this, here's an idea.

(i) There is an odd number of things O iff either (a) there still remain a
number of things O' when two things are taken away from O, and O' is also an
odd number of things or (b)  O is just one thing.

(ii) "There are three things" means " there is one thing and there is
another thing and there is another thing".  This is an arbitrary definition,
but then "three", unlike 123, is an arbitrary name like "Percy".

(iii) We can substitute these arbitrary numerals for the longer expression
and conversely.

(iv) We can split any numerical proposition into component parts so that
e.g. "there is one thing and there is another thing" expresses the
conjunction of that p = that there is one thing and that q = that there is
another thing

(v) Define remainders as the number of things stated to exist by the
proposition that p, when we have split the proposition as above, and where
that q expresses the number of things taken away, and where that p&q
expresses the number of things from which the subtraction is made.

(vi)  We start with the proposition that there are three things,
substitute to get "there is one thing and another thing and another thing",
substitute again to get (p&q) that there is one thing and there are another
two things, then we are left with the proposition p that there is one thing,
the remainder.  This asserts that there are an odd number of things (because
one thing is an odd number of things)

(vii) Would take a bit more work to prove more general statements.  For
example, if there are an odd number of things X, and another thing besides
that, making X' things, then can you show X' things are not odd?

Dean Buckner

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