Philosophers of science are a verbose bunch of people. Most often they discuss the foundational paradoxes of physics using common language. Linguistics of any kind, even the most precise sort, is a poor substitute for the mathematical rigor when applied to the description of the fundamental fabric of reality for two reasons. One is that it carries a semantic load, therefore subjecting the definitions to interpretation based on the meaning of the words used, and meanings are often different to different users. The second reason is that the languages we use are conditioned on human experience that is inevitably classical in nature and therefore is nigh on useless when applied to non-classical settings, such as those of Quantum Mechanics or General Relativity.
Most of the foundational issues of natural sciences are only part conceptual. At their core, the problems are, for the most part, mathematical in nature and can solely be dealt with in the language of mathematics. Tom Banks, of the University of California in Santa Cruz and a Professor of Physics at Rutgers discusses the issue using Quantum Mechanical Measurement Problem as an example:
Tom Banks (UC in Santa Cruz, Prof at Rutgers) says:
For every quantum state and every Hermitian operator, the math of QM allows you to calculate D, which is the density matrix corresponding to the state, A is the operator and n is an arbitrary integer. From these quantities you can extract a bunch of positive numbers, summing to one, which QM claims are the probabilities for this operator to take on each of its possible values.
The interpretation of this math is that if you prepare the system repeatedly, in the state D, and then measure A by coupling to a macroscopic system whose pointer points to a different value for each of the different eigenstates of A, then the frequency of observation of a particular value will be equal to the positive number extracted from the calculation. These predictions have, of course, been tested extensively, and work.
Many of the things one normally talks about in these discussions involve probabilities of HISTORIES rather than of observations at a fixed time. I can’t go into the details but GellMann and Hartle, in a series of beautiful papers, have shown that a similar sort of interpretation of mathematical quantities in QM in terms of probabilities of histories as valid when “the histories decohere”. The reason histories are more complicated is that they involve measurements of quantities at different times, which don’t commute with each other.
However, it’s important to note that GellMann and Hartle use only the standard math of QM (I’m not talking about their attempt to generalize the formalism) and that their interpretation follows from the probability interpretation at fixed time by rigorous mathematical reasoning.
Given that the math suggests a probabilistic interpretation of fixed time expectation values, which actually reproduces experimental frequencies, I can’t understand a statement that I shouldn’t interpret these things as probabilities just because they don’t satisfy some a priori rule that a philosopher derives from “pure thought” or “elementary reasoning”. The whole point of my post is that “elementary reasoning” is flawed because our brains are not subtle enough. The rigorous mathematical formulation of “elementary reasoning” is mathematical logic, and I think it’s quite interesting, (and IMHO the most interesting thing in this rather stale discussion), that that formalism contains the seeds of its own destruction.
The other interesting thing about David’s post is that it points up the drastic difference between the modes of thought of theoretical physicists and philosophers.
Theoretical physics has three parts. It begins with the assumption that there’s a REAL WORLD out there, not just activity going on in our consciousness, and that the only way to access that real world is by doing experiments. I mean experiment in the most general sense of the term. A macroscopic trace left on a distant asteroid in the Andromeda galaxy is an experiment for an observer on Earth, if it is in principle possible for some advanced civilization in the distant future to send out a spacecraft or bounce a beam of light off the asteroid to bring back information of that trace.
The second part of theoretical physics is mathematics.
We build a mathematical model of the world and compare it to experiment according to some well defined rules. In QM these rules are: calculate the probabilities of different events using the density matrix formula, and compare those probabilities to frequencies in repeated experiments on the same quantum state.
In quantum cosmology, a subject which is still under construction, we’ll never be able to repeat the experiment so we have to use the more subjective meaning of probability.
Finally, there’s the story we tell about what these results mean and how they relate to our intuition. It’s a very important part of the whole game, but we’ve learned something very interesting over the years. The story can change drastically into another story that seems inconsistent with the first one, even when the math and the experiments change in a controlled way, whose essence is contained in the word “approximation”. In math an exponential function can be approximated by a linear one when its argument is small enough. In experiment an exponential behavior can look linear when we don’t measure large values of the control parameter. That is, these two features of our framework can change and they change together in a controlled way, with a quantitative measure of the error of approximation.
Stories however, change in a drastic manner. Newton’s absolute space, and absolute time, Galileo’s velocity addition rule, etc. , if they’re taken as a priori intuitively obvious laws, simply do not admit the possibility of relativity. When I try to explain relativity to laymen, many of them have a hard time understanding how it could be possible that you could run to catch up with something and still have it recede at the same speed.
It’s easy for someone with a little math background to understand that the correct law of addition of rapidities for parallel velocities, becomes the velocity addition law when an exponential of rapidity is replaced by a linear approximation.
Philosophers are committed to understanding everything in terms of the story, in terms of words. This approach can work even for relativity, whenever classical logic works. Whatever philosophers call it: “elementary reasoning” , “common sense”, they’re committed to a world view based on a denial of the essence of QM, because QM is precisely the abandonment of classical logic in terms of the more general, inevitably probabilistic, formalism, which becomes evident when one formulates logic in a mathematical way. Just as we use the math of the exponential function to explain the reason that the velocity addition law looks so obvious, we use the math of decoherence theory to explain why it is that using “elementary reasoning” is a flawed strategy.
Let me end by recommending to you some words of Francis Bacon, which were quoted in an article by Stanley Fish in the NYT some years ago. Unfortunately, I’ve misplaced the computer file with my copy. Bacon, writing at the dawn of experimental science, complained about the ability of men to twist the meaning of words, which made getting the truth by argument impossible. He argued that the only way to get to actual truth was to do reproducible experiments, which could be done independently by different researchers, in order to assess their validity. Modern theoretical physics has another arrow in its quiver, which is mathematical rigor.
Words, the Story of theoretical physics, are still important, but they should not be allowed to trump the solid foundations of the subject and create unjustified confusion and suspicion of error where none exists.
It’s sad, but the architecture of our consciousness probably will not allow us to come up with an intuitive understanding of microscopic quantum processes. But mathematics gives us a very efficient way of describing it with incredible precision. To me, there’s every indication that QM is a precise and exact theory of everything, and will survive the incorporation of gravitational interactions, which has so far eluded us apart from certain mathematical models (the theory formerly known as String) with only a passing resemblance to the Real World. Attempts to force it back into the straitjacket of “simple reasoning” are misguided and have not led (and IMHO will not lead) to advances in real physics.