Re: Why should -1 multiplied by -1 be plus 1 and not -1

From: Jeffrey Ketland (ketland_at_ketland.fsnet.co.uk)
Date: 08/04/04 

Date: Wed, 4 Aug 2004 17:24:52 +0100

Andrew Boucher wrote in message ...
>"Jeffrey Ketland" <ketland@ketland.fsnet.co.uk> wrote in message
news:<ceq9r9$6e1$1@news6.svr.pol.co.uk>...
>> My point is that it isn't merely a model. It's an explanation of what the
>> integers are, based on abstraction, just as we explain what natural
numbers
>> are from an abstraction (Hume's principle).
>>
>> Frege provided an explanation of what natural numbers are. I.e., they are
>> cardinalities of finite sets, obtained by an abstraction principle,
>> sometimes called Hume's Principle, namely,
>> #A = #B iff there is a bijection from A to B
>> This is an explanation, by an abstraction principle.
>> More intuitively,
>>
>> There's a bunch of trees! There's a bunch of pebbles!
>> Look, there's a one-to-one correspondence between
>> the bunch of trees and the bunch of pebbles.
>> So, they have "something abstract in common".
>> Let's call these things "natural numbers".
>
>This is getting off-topic, but I object (to Frege!) in two places:
>(1) There is the ontological assumption in the Abstraction Principle,
>that there is a "something abstract in common", and

Yes, certainly. As you know, the idea goes back to Plato. This painting is
beautiful; and this one is too. They have "something in common", a universal
of some sort. That thing they have in common is Beauty. That's an
ontological assumption. After I first read about this idea when I was about
17 (in Russell's _Problems of Philosophy_), I went around telling people for
weeks!

>(2) Even if you can abstract, somewhere you have to suppose that the
>things you've abstracted are the things you've wanted to begin with.
>I.e. why are the abstracted entities the *natural numbers*?
>
>Usually the justification I've heard for (2) is that these things have
>all the same properties of the natural numbers one wants.

Well, that's a very abstract sort of structuralism. Yes, that would be
circular.

But Russell and Frege were very clear that the essence of natural number
concerns their application: how they are related to certain abstract
properties of finite collections, which are deemed "equivalent" when there
is a bijection between them. So, when, long ago, I first read Bertrand
Russell and he wrote that you can have a triple {t1, t2, t3} of trees and
another triple {p1, p2, p3} of pebbles and consider what they "have in
common". Well, what do they have in common? Three-ness! That's what the
number 3 really is. It's the abstract thing that all triples have in common,
just as Beauty is what all beautiful things have in common. Seems like a
perfectly good explanation of what the (natural) number 3 really is. In
fact, I've never heard any better explanation.
Furthermore, this explanation---essentially based on Hume's
Principle---accords precisely with the application of natural numbers, by
the scheme,
    there are n Fs iff #{x: Fx} = n

>But this
>seems to beg the explanatory question - the explanation of what the
>natural numbers are, only works if one has a pre-existing notion of
>the natural numbers and their properties, so it seems you haven't
>really explained anything.

I'd reply that we introduced the notion of natural numbers so that we could
talk about what equinumerous finite sets all have in common. That is their
raison d'etre (excuse the French!).

>- "What are the natural numbers?"
>- "They are these abstracted things."
     ... what equinumerous finite sets all have in common

>- "Why do you think those abstracted things are the same as the
>natural numbers?"

- "The raison d'etre of the natural numbers is that we introduced "the
number 3" as a name for what all triples have in common: Three-ness is the
common essence of all specific triples."

> [Structuralist] "Well, they have all the same properties of the natural
numbers, so
>they're just as good as the natural numbers."
>- <blank stare>

Right.
But that's a response to structuralism, and doesn't in any way undermine the
Frege-Russell view.
The real problem with the Frege-Russell view is that the collection of all
triples is a proper class ... This is why a purely abstractionist approach,
as pioneered by Wright, is attractive. Numbers are the abstract entities
"described" by Hume's Principle, precisely because of their application in
assigning cardinality to equinumerous finite sets.

--- Jeff

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