On 2/12/2013 1:19 PM, Dan wrote: > On Feb 12, 5:19 pm, Craig Feinstein <cafei...@msn.com> wrote: >> Let's say I have a drawer of an infinite number of identical socks at time zero. I take out one of the socks at time one. Then the contents of the drawer at time zero is identical to the contents of the drawer at time one, since all of the socks are identical and there are still an infinite number of them in the drawer at both times. But the contents of the drawer at time zero is also identical to the contents of the drawer at time one plus the sock that was taken out, since they are exactly the same material. So we have the equations: >> >> Contents of drawer at time 0 = Contents of drawer at time 1 >> Contents of drawer at time 0 = (Contents of drawer at time 1) plus (sock taken out of drawer). >> >> Subtracting the equations, we get >> >> Nothing = sock taken out of drawer. >> >> This is false, so infinity cannot exist. >> >> How does modern mathematics resolve this paradox? > > Your 'reified' equation doesn't reflect the reality of the situation . > You can assume each sock has a different atomic structure . Then the > situation would be different .The socks can only be identical as far > as you can observe . From the moment you took a sock , the remaining > pile is a different pile from the one that was before , regardless of > what you would like to think. > Nice.
In Aristotle's discussion of priority one finds the remark that to have the existence of two implies the existence of one.
> Even if we take socks to be fully identical and you're equations to be > true , what they say is that taking 'nothing' out of the pile of socks > has the same effect as taking a 'one sock' of the pile of socks . If > you can get an infinite number of socks , one sock might as well be > worth nothing :) . >
Such is the nature of value vs. fiat currency
> > Let's attempt to look at this another way : since no sock is supposed > to be different from another sock , equations must be , > ultimately ,referring to numbers , that is , quantity ,abstracting > individual existence . When I say 'two pears' , I abstract the fact > that they may be of different color . > > Your equations reduce to , basically : > > infinity = infinity > infinity = infinity + 1 => > > infinity - infinity = 1 - 0 = 1 => > 0 = 1 > > The problem is 'infinity' is not a proper quantity , not a number . > The reason it can't 'stand' as a quantity is it's defined as being > equal to a proper part of itself . (infinity = infinity + 1) . >
This can be augmented by the fact that "infinity" in set theoretic mathematics is a (transfinite) number only in the sense of being one object of a system whose relations are governed by an arithmetical calculus.
Mathematics has many numbers that are not quantities.
> Also , you can't make a choice from any number of 'absolutely- > identicals' . According to Leibniz's principle of 'identity of > indiscernibles' a number of 'absolutely-identicals' is a false > concept .
This is the paper with the contemporary counter-example to that statement.
It has a number of interesting references listed at the end.
> Metaphysically , there's , ultimately , only 'one' of > anything . From the moment you choose a sock from the drawer of > socks , and even, extending in time , forever before and after that > moment, that sock is and was no longer identical to the other socks. > That sock is 'chosen' , all the others are 'not chosen' .
Choice introduces a relational property against which the principle of identity of indiscernibles may be applied.
I would guess its general application involves three axioms to form such relational properties:
3)I am not it
> > If all the socks were white , your choice of sock acts as a 'red > dye' , forever marking the chosen sock as 'red/chosen' . So what your > equations really say is : > > socks = 'chosen sock' + 'unchosen socks' . > "socks = unchosen socks" is clearly false . > > Interesting thought experiment , nonetheless . >