Paradox: objective meaninglessness of concept of similarity?

because real-world objects have an infinity of properties, you could cherry-pick as many properties as you wanted to counterintuitively "prove" that, for example, an orange is more similar to a roll of aluminum foil than to an orange segment. You can generate real-world proof of this concept by playing http://toast.topped-with-meat.com/connector/frenchtoast.html.

You might be interested inhttp://homepages.wmich.edu/~baldner/black.htm (original paper http://home.sandiego.edu/~baber/analytic/blacksballs.pdf which tries to disqualify Leibniz's identity of indiscernibles: two things are identical if and only if they share all the same properties. Max Black tries to argue against this by counter-example: imagine two spheres identical in all their properties. Aren't they still two rather than one and the same thing? This turns your question upside down to pick apart the notions of 'similarity' and 'identity' - two things may be different even if they are exactly similar, I don't know if this would be useful to you but it's an interesting angle I think. I'm not aware of any 'proof' that similarity is objectively useless. To my understanding it is actually useful in various logical languages and to philosophers and mathematicians alike. And real world objects have a variety of shared properties that are not considered relevant from certain perspectives.

(Of course, the basic ideas underpinning this overlap considerably with problems of induction dating all the way back to Hume. But I think Goodman was the first person, or at least the first major person, to relate it all to similarity).

In quantum mechanics, any http://en.wikipedia.org/wiki/Indistinguishable_particles, that is, they aren't really two separate things.

It's not exactly the same thing, but Radiolab did an episode called http://www.radiolab.org/2009/jun/15/ which was about randomness and coincidence. The first segment addresses basically what you are talking about, although not with math. This is more an-interesting-thing-that-may-be-on-topic than it is an answer.

Seconding forza -- I bet you're thinking of Goodman.

forza, thank you! I'm sure the recent discussion I saw was philosophical fallout from Goodman's paper. Bingo. mkdirusername, thanks. I am also interested in the (similar?) concept of the identity of indiscernibles. gauche, I heard most of that episode but don't recall that segment. I'll have to give it a second listen. alms, great game! I will try to get this played at my next game night.