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?
Pat
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