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.
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 :) .
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) .
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 . 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' .
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 :