Montague in the epiologue of his book, which I blogged about recently, argued that a marriage of psychology and physics is in order. His thesis is that our intuitions about the world are based on flawed systems that were only designed to survive and reproduce. There are several responses to his argument. The first is that while we do rely on intuition to do math and physics, the intuition is based on learned concepts more than our “primitive” intuitions. For example, logical inference itself is completely nonintuitive. Computer scientist Scott Aaronson gives a nice example. Consider cards with a letter on one side and a number on the other. You are given 2, K, 5, J and the statement every J has a 5 on the back. Which cards do you need to turn over to see if the statement is true? (Hint: If A implies B then the only conclusion you can draw is not B implies not A). Most college freshmen get this wrong. Quantum mechanics and thermodynamics are notoriously counter intuitive and difficult to understand. We were led to these theories only after doing careful experiments that could marginalize away our prior beliefs.
However, that is not to say that perhaps we’re at a stumbling block over questions like what is dark matter or why do we remember the past and not the future because of some psychological impediment. A resolution to this issue could reside again on whether or not physics is computable. Montague doesn’t think so but his examples do not constitute a proof. Now if physics is computable and the brain is governed by the natural laws of physics, then the brain is also computable. In fact, this is the simplest argument to refute all those that doubt that machines can ever think. If they believe that the brain is in the natural world and physics can be simulated then we can always simulate the brain and hence the brain is computable. Now if the brain is computable, then any phenomenon in physics can be understood by the brain or at least computed by the brain. In other words, if physics is computable then given any universal Turing machine, we can compute any physical phenomenon (given enough time and resources).
There is one catch to my argument and that is the fact that if we believe the brain is computable then we must also accept that it is finite and thus less powerful than a Turing machine. In that case, there could be computations in physics that we can’t understand with our finite brains. However, we could augment our brains with extra memory (singularity anyone) to complete a computation if we ever hit our limit. The real question is again tractability. It could be possible that some questions about physics are intractable from a purely computational point of view. The only way to “understand” these things is to use some sort of meta-reasoning or some probabilistic algorithm. It may then be true that the design of our brains through evolution may impede our ability to understand concepts that are outside of the range it was designed for.
Personally, I feel that the brain is basically a universal Turing machine with a finite tape so that it can do all computations up to some scale. We can thus only understand things with a finite amount of complexity. The way we approach difficult problems is to invent a new language to represent complex objects and then manipulate the new primitives. Thus our day to day thinking uses about the same amount of processing but accumulated over time we can understand arbitrarily difficult concepts.