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One of the more sophisticated ideas that we all seem to understand intuitively is identity. We agree that Sepulveda Street is the same street at the upper end of the San Fernando Valley as it is in Long Beach, 43 miles away. It's not the same asphalt, the same neighborhood, and even the traffic signs have changed, but you can drive from one end to the other without changing streets.
In Long Beach the name changes to Willow Street. Is it now a different street?
Except for some of my friends, we all agree that I am the same person I was ten years ago, even when I acknowledge that I've changed a lot. (Those same friends would probably argue that I'm never the same person.) But during that time almost every molecule of my body has been replaced and, even if all the electrons are just the one and same electron going back and forth through time, they're in different configurations and places each time.
When we say two things are equal, the same, or even identical, we are agreeing to ignore the known differences, the ones that aren't relevant to our purposes. The result of the comparison can be powerful (science and mathematics look for ways things are the same and the implications of similarity) or we can find ourselves far up the river without a paddle if the differences turn out to be relevant after all.
Saying things are the same or equal is also the source of confusion, both in mathematics and the world. I once agreed to meet a friend at Starbucks at the intersection of two streets. Trouble was there were two Starbucks at that intersection; he was at one and I was at the other. A quick cell phone call cleared up the confusion, but the relevant differences could have meant we never met at all.
The paradox of time travel (what if I go back and kill my grandfather, I won't be born and therefore I can't go back to kill my grandfather) suffers from this identity issue. You kill your grandfather on a different street even though it has the same name; it's on the other side of the freeway. The many worlds interpretation of quantum mechanics has some of this same flavor.
And in mathematics the issue of identity provides some of the stickiest problems. Formal math today tends to ban self referential statements in proofs because, after discovering that the barber I described above caused significant problems, mathematicians tried to shave him out of their basic axioms. Then Godel came along and showed that self referentiality can arise even when you start with the simplest of systems.
Not that I have the depth to be able to suggest a different program for mathematics, but perhaps a different way out of the problems of self referentiality might be more attention to relevant differences. The two parts of the sentence about the barber are usually taken to be two different ways of describing the 'same' object. Well, maybe there is just no way you can step into that same barber shop ever again.
- Sepulveda Street is the longest street in Los Angeles City and County.
- The headwaters of the Missippi River are in Minnesota. Why there and not at the headwaters of the Missouri River?
- Speaking of streets, there's a street near me that because of freeway construction has the same name as a street on the other side of the freeway but it no longer connects.
- Speaking of rivers, how about the idea that you can never step in the same river twice?
- In physics there's an idea that there is only one electron in the universe, traveling forward in time as an electron and going backward in time as an anti-electron.
- And, in the different world of mathematics, there's a village where the barber shaves everyone who doesn't shave himself.
1 Jul 2009
Copyright © 2009 Truck Smith