Number (multitude) and magnitude are the
generic kinds of quantity. As continous and discontinous quantity, both are
necessary in SO. Quantities with their relationships make the subject
of mathematics. For more, see the Quantity article, http://en.wikipedia.org/wiki/Quantity
----- Original Message -----
Sent: Thursday, February 12, 2009 12:42
AM
Subject: Re: [ontolog-forum] standard
ontology
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.
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT
com
From:
ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Pat Hayes
Sent: Wednesday, February 11, 2009 1:53
PM
To: [ontolog-forum]
Subject: Re: [ontolog-forum]
standard ontology
On Feb 11, 2009, at 10:41 AM, Chris Partridge
wrote:
PatH,
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
this.
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
these
things
exist, whether they are considered abstract or not
is a different
question
(and arguably more
controversial).
Whether they exist or not, I think you have to
say that numbers are
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
modern
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 number and
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
ontological
commitment.
Actually I think for practical reasons. Trying
to actually DO mathematics more or less forces one to think in Platonic
terms, or go slightly mad.
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
problem.
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
scattered).
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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