10.01.2006

About Time

Lee Smolin writes, near the conclusion of his new book The Trouble with Physics, that the key to cracking the problem of linking quantum mechanics with relavatistic gravity may be our notion of time. In classical physics, there's no way to identify what 'now' means. All times have equal status. This doesn't match our perception, so either the physics is wrong or our perception is wrong. The notion that time is just another dimension like the three spacial dimensions is also unsatisfying. Then there are the paradoxs. Can time have a beginning? If it does, what caused it to start--Aristotle's unmoved mover? Or is that nonsense?

It seems clear that to begin with we need a type of logic that can deal with time. Normal predicate logic has no such parameter. Smolin writes that there is a promising attempt in that direction coming from category theory, called topos theory. He elaborates on it in a New Scientist Article this week (subscription required), asking this question:

[I]s there a way to represent the laws of physics mathematically that retains the notions of the present moment and the continual unfolding of time? And would this allow us - or even require us - to formulate laws that also evolve in time?
And here's one approach to answering it:

There are logicians who have proposed alternative systems of logic that incorporate a notion of time unfolding. In these logics, what is true and false is assigned for a particular moment, not for all time. For a given moment some propositions are true, others false, but there remains an infinite list of propositions that are yet to become either true or false. Once a proposition is true or false, it remains so, but at each moment new propositions become decided. These are called intuitionalist logics and they underlie a branch of mathematics called topos theory.
I find this very interesting, partly because it resonates with something from computer science. Anyone who's taken a programming course and a logic course will see the following contradiction.

In logic, statements like "This statement is false." are problematic because they are paradoxical, allowing neither a true or false assignment. But over in the programming class, we write things like
toggle := !toggle
where the exclamation mark means logical negation. The two examples are completely analogous except for one thing. Logic is static in time, whereas computer programs rely on execution of statements sequetially--that is, in time. So in the first case, we have an unresolvable paradox that caused logicians a lot of grief in the 20th century. In the second, we get a perfectly normal computer assignment, which merely switches the truth value of the variable between true and false. In fact, if you write
while(1){
toggle := !toggle
}
you've created a clock that ticks back and forth. So could Smolin be right, and the understanding of time in physics depend on a mathematical logic that can handle time? The subject of temporal logic has been around for a while (see here and here, for example).

The connection between logic and computation is one that Smolin makes as well:

It is interesting that some physicists now propose that the universe is some kind of computer, because similar questions are being asked in computer science. In the standard architecture all computers now use, invented by the mathematician John von Neumann, the operating system never changes. It governs the flow of information through a computer just as an eternal law of nature is thought to guide physics. But some visionary computer scientists such as Jaron Lanier wonder whether there could be other kinds of architectures and operating systems that themselves evolve in time.
This approach still leaves some unanswered questions, even if it's possible to derive a physical theory from logical axioms. Which axioms are the right ones? How were they "chosen" by the universe? In other words, it still leaves us with an unmoved mover-type problem.

I have an idea that solves this problem, but it's probably not palatable to most scientists. The idea is that the universe didn't start logically, but evolved into one that is mostly logical. Wittgenstein said that we couldn't speak of what an illogical universe would look like, but I think we could. There are certain ways in which logic can break down. Like, the paradoxes in ordinary (non-temporal) logic prompting the construction of temporal logic. In this view, time is just the universes way of straightening out paradoxes! Although I'm not an expert on logic, I imagine that the flavors of temporal logic have their own paradoxes to be circumvented.

A related question then is, is the universe as we currently know it actually logical? Anybody who's read about double-slit experiments would surely wonder how we bend terminology to call that behavior logical. Could the universe be illogical at some level? If so, could we detect it? This is perhaps a scientist's nightmare.

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