Possible worlds are useful constructs in
commercial databases, but numbers aren’t a problem; arithmetic is all
that’s necessary. So mathematizing the concept of 4 isn’t
very useful there. Number theory isn’t very useful to most companies.
So Numbers are the first thing I would drop from an FO, and use them only as
descriptive (adjectival) values in relations.
Rich AT EnglishLogicKernel DOT com
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Pat Hayes
Sent: Wednesday, February 11, 2009
Subject: Re: [ontolog-forum]
On Feb 11, 2009, at 10:41 AM, Chris Partridge wrote:
Last time you raised the question of numbers, and I responded, Chris Menzel
told me off for raising something NOT relevant. Note to ChrisM - PatH raised
1. Are there certain things like e.g. numbers and possible worlds?
2. Are they abstract or concrete?
Even the first may be controversial, but even if you agree that
exist, whether they are considered abstract or not is a different
(and arguably more controversial).
Whether they exist or not, I think you have to say that numbers
not concrete. You can't weigh seven.
As I think you must know there is a Fregean concept of number (where 2 is
the set of all sets with two members).
Well, you can't weigh a set, either.
This Fregean proposal has
supporters - e.g. Crispin Wright http://en.wikipedia.org/wiki/Crispin_Wright
There is debate about whether sets are abstract - David Lewis asks why we
cannot say the set of cars in the car park is located in the car park - and
if it is located, it cannot be abstract. Hence, there is an account of
numbers that says they are not abstract.
As you know, the fact that there is an account of X does not make that
account likely, intuitive, or even consistent with many other widely held
positions. Philosophy is full of minority reports. All the cases I've read of
treating sets as concrete seem to me to be cases of getting sets and mereological
sums muddled together. Lewis' example has a clear reply: the cars are located
there; all of them are located there; but it does not follow that the set is;
just as if we say that there are four cars in the car park, it doesn't follow
that four is in the car park. We can loosely say this, being careless (as we
all often are) about ontic commitment, but the intended meaning is clear. As
Barwise once asked me: look around the office: how many sets can you see? Question back to Lewis: are all the
subsets of the set of cars also in the parking lot? What about the power set?
Etc.. (Yes answers seem more and more silly; no answers have no rational
justification, if you accept that the first set is there.)
There seems to me to be gap between the mathematician's notion of
the common sense (or engineer's) notion.
Seems to me that the former is simply a precise and formalized version
of the latter. But we ought to distinguish the mathematician's notion (they
talk of "the natural
numbers" without blinking an eyelid) from the philosopher-of-mathematics'
notion, which is typically altogether more tenuous and abstract.
For historical reasons, mathematicians wanted mathematics to be devoid of
Actually I think for practical reasons. Trying to actually DO
mathematics more or less forces one to think in Platonic terms, or go slightly
So, if I say there are four books on the table - how
this four relates to the (set of?) books is unclear.
Its the cardinality of the set. This seems to me to be about as clear
as anything can be. Its much clearer than explaining the relationship of the
books to, say, their color or their authors.
(Ditto, the 4lb of
apples on the table.) Assume there is an abstract thing that is the number
four - how does it get related to the books/apples? Hope you see the
No, I don't. There is no problem here that I have ever been able to
discern. The answers to these questions are obvious, even to a child. The
relation between the books on the table and four is, that the number of them is four. There are four of them. If you count them, you will
get to four, then stop. The cardinality of the set of books on the table is
four. If you pair them up with the sequence of names "one",
"two", etc.,, you will find that that final item in the sequence is
"four". And so on. All these are different ways of saying the same,
obvious, thing. What mystery or problem is there here?
(For the record, this kind of finding problems where no problems exist
is what made me tire of academic philosophy. The real world is hard enough to
understand, without having to deal with ethereal non-issues arising from an
overdeveloped sense of awe at the most trivial observations. But carping aside:
whatever qualms one might have about the underpinnings of the last century or
so of mathematics, surely just on pragmatic grounds, it can hardly be gainsaid
that numbers are very useful: so useful, that asking anyone from a child counting
bricks to any engineer to do without them is asking too much to justify even
the most serious philosophical scruples.
Note: this is simple for the Fregean. It seems to me that engineers
would be happier with a Fregean approach for counting and weighing.
Maybe one cannot practically weigh seven (or locate it - as it is far too
Its not in the spatiotemporal universe at all. "Scattered"
isn't meaningful applied to a number.
But similar problems occur for things like the mereological sum
of all milk (or gold, or water). These are also too scattered, but seem to
me irredeemably concrete.
If you believe in mereological sums like this, then they are indeed
scattered. But it still makes sense to ask what is the total mass of gold in the
universe, for example. Such estimates do in fact exist, I believe, though they
are of course impossible to verify directly. But the total mass of seven?
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