Apr 14, 2007

reality will make sense if you just find the right topos

Goodbye, logic and all that. Hello, topoi. [sub. req.]
But there is a price to pay, and it is precisely what the Kochen-Specker theorem warned of: the demise of simple truth and falsity. For all its drawbacks, Boolean algebra does at least allow every statement about our universe to be either true or false. Yet this turns out to be the exception among all the different types of algebras - including the one underpinning quantum theory. The logic associated with quantum topoi encompasses true, false and many shades of grey in between.

Does that mean we must accept a universe that is real, but about which any question will receive myriad answers, all of them true? According to Isham and his colleagues, the answer - appropriately enough - is both yes and no. If we are content to view reality through the window of classical physics, then we can enjoy straightforward true/false answers to our questions - as long as we avoid the realm of atoms. But if we insist on making statements about atoms, we must use the logic of quantum topoi and accept the existence of a whole host of realities, all as valid as each other.

And that might just be the start; after all, there are more topoi than just the standard and quantum ones. In a series of papers unveiled last month, Isham proposes an even more mind-bending idea: there may be myriad ways of viewing reality, each based on its own topos. Together with Andreas Doering of Imperial, he has shown that every physical system - from an electron to the whole universe - has a unique mathematical identity that dictates how it will appear when viewed through the prism of a particular topos.

Seen via the topos of set theory, an atom takes on its classical appearance with nice, well-defined properties. Viewed through the topos associated with quantum theory, it becomes altogether fuzzier and strange.

We needn't stop there. Why not opt for another topos? It could lead to a view of reality even more astonishing and successful than quantum theory. "What we're hoping is that topos theory becomes the basis for a whole new way of constructing theories", says Isham.
Postmodern sciency mumbo-jumbo, or a radical revision of the way we understand--or is it construct--reality?


(Yes, that's an interrobang.)


TheTachyix said...

Hey, what's the HTML tag for an interrobang? I want one!

Jim Anderson said...

I just cut-and-pasted from Wikipedia. Firefox did the rest.