On Feb 12, 9:59 pm, fom <fomJ...@nyms.net> wrote: > 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. > > http://home.sandiego.edu/~baber/analytic/blacksballs.pdf > > 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: > > 1)I am > > 2)It is > > 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 . > > It is. And you did it quite well.
I've had a bit of time to clarify and summaries my argument :
Either the op's equations are referring to pure quantity, number abstracted from form (they are 'arithmetical') : "These two pears are equal to those two pears not because they are the same pears , but because they are 'two' " in which case the problem is that 'infinity' is not a well behaved quantity , ie. 'infinity - infinity' is not uniquely defined .
Or ,they're referring to individuated being : 'These tho pears you see now are exactly those pears you saw then' (they are 'set-theoretical') in which case my Leibnitzian argument that 'each sock is its own being' and 'the unchosen socks' cannot be equal to 'the whole socks' applies .
I'll have a look at that paper :). For now , at least , I still stand firmly by the identity of indiscernibles . You could take a 'theological' way of looking at this , as Cantor did with his sets ('God surely knows them' , 'God can surely discern the non-identicals') , or , at the other extreme, you can take a positivist view : ('why should we consider as non-identical two 'indiscernible' things? how can we even talk of them as 'two' is there is no 'discerning attribute' , either given by us or known to exist? If 'two things' cannot ever be discerned by us as different , then they are identical , ie. two 'references' to the same thing.)
I think the problem , the 'uncomfortable feeling' that most people have with 'the identity of indiscernibles' stems partly from the inherent vagueness of the predicate 'is' in language . Consider these two examples .
'My son's father' is 'me' . // This refers to 'Leibnitzian' identity, two references that stand for the same thing 'is; is a sort of equality .
I'd add that to understand clearly, we need to understand also the 'baudrillardian' distinction between reference and thing, ie 'The word "this apple" , and this apple , the apple.' , and that language is used and misused , as connecting the two .
This pear is a thing. This apple is a thing. //We cannot deduce here that 'this pear' is 'this apple' , as is the case were 'is' an equivalence relation . Rather, here , 'is' stands for a sort of 'set-theoretical' membership relation . I forget which philosopher first made this distinction . If you know please remind me . This translates as :
This pear is a member of 'Thing' . This apple is a member of 'Thing'. Where 'Thing' is 'the set of things' .
In conclusion , it's a shame not more people read Leibniz :) .