Monday, 27 March 2006

The Local Meme, Part I

A couple of years ago I was chatting to an acquaintance and, like many people, he was fascinated by quantum mechanics. I resisted the temptation to suggest he learn about Hilbert spaces before trying understand the theory of observation in QM (so, right, you take this operator wotsit, and you apply it to the state vector, right, and ...) He did make me realise one thing, though: several of the great intellectual achievements of the last century have a common thread - the importance of the local. So at the expense of 99.9...% of the important stuff, here goes:

Special relativity overturns the primacy of the Newtonian observer and focusses attention on the relationship between different observers. What you can see depends on where you are and how fast you are going;

Quantum mechanics says that there is no information without observation. You interact with the system to measure it, and what you do changes the system. Again, there is no gros picture, no privileged observer (at least leaving aside the vexed question of the wavefunction of the whole universe).

The various incompleteness theorems of Gödel and others again assert limits to global knowledge: you can have this property of a proof system, but not that too. If you want all of arithmetic, you have inconsistency. (And interestingly these proof theoretic results now seem to be important for understanding the P = NP problem.)

And then there is constructive mathematics where to prove that the exists an x such that some property holds, we have to actually hold up that x and show it has the property - another local idea.

In Part II, a (probably jejune) way of thinking about how this fits together.


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