Monday, January 08, 2007

The Bell Inequalities and Proving a Negative

Wow. As usual great questions and really good discussion. I promise I'll weigh in on the atheism and faith question tomorrow; but today, I'll start with the easy ones -- logic and quantum mechanics.

Justin asks,

I frequently hear the phrase that one "cannot disprove a negative." Is this actually true or does it depend on the context?
In order to make sense of this, we need to understand what we mean by prove and disprove. There are two kinds of sentences -- those like "It's raining" whose truth or falsity depends upon the way the world is and sentences like "It's raining or it's not raining" whose truth or falsity is a matter of the form of the sentence or the meaning of the words and does not depend upon the way the world may or may not be. A proof of a sentence means showing that there is no way for the sentence to be false, that it must be true. A disproof is a proof that the sentence must be false. Can you show that the negation of a sentence must be false?

The second sort of sentence, those that are not contingent on the state of the world, are the sort of sentences that logicians and mathematicians deal in. Here, there are some examples in which you can show that a negative claim cannot possibly be true, but in the general cases -- the interesting ones -- you cannot do it with examples. Consider the negative claim, "There are no odd perfect numbers." A perfect number is one where you take all of the positive counting numbers that it divides into evenly, exclude the number itself, add up the rest and you get the number back. 6 is perfect because the only numbers that divide 6 evenly are 6 and 1, and 2 and 3. Ignore the 6, add up 1+2+3 you get 6. 28 is perfect: 1 and 28, 2 and 14, 4 and 7 are its divisors, so ignore the 28 and add up 1+2+14+4+7=28. There are therefore even perfect numbers, but since the times of ancient Greece, the question has been whether there are any odd ones. To this day, no one knows for sure. You could prove that an odd perfect number exists with a single example of one, but you could not disprove the existence of odd perfect numbers with even a huge list of odd numbers that are not perfect because the one that is lurking out there (should there be one) may just not show up on your list yet. Enumerating examples will never produce a disproof of the claim. You can't disprove a this way. There are other routes that may work -- showing that a result of there being odd perfects contradicts a sentence we have already proven must be true, what we call an indirect proof or reductio ad absurdum.

Then there are the sentences that depend upon how the world is. These cases are different because these are not the sentences of mathematics or logic which are based upon definitions, these require observation. These are the propositions of science and generally are not the sort of sentence for which there can be proof. For these sentences, there may be evidence which shows that a sentence is more or less likely to be true, but you don't get the sort of absolute certainty that comes with proof. There are, of course, cases like "You will see red" or "Your hand will hurt" which are open to direct experience and those may be shown to be absolutely true or false at the time, but general claims, the sort of positive claims that could be laws of nature, generally can be shown false by a counter-example but never true by instances. Take Newton's law of Universal Gravitation which specifies that any two objects with mass will attract each other with a certain pull based on how much mass and how far apart they are. If I find one example where Newton's equation does not hold, the law is false, but I can show thousands of cases where it holds and that will be good evidence that it is probably true, but I will never be able to show in this way that it must be true. The converse, then, is that negative claims could be shown true by examples, but never false by instances.

But sentences do not exist in vacuums (unless you shredded your dissertation, threw it on the floor, and then decided to clean up your apartment). We often determine whether a sentence is true or false relative to what else we have good reason to believe. These things might be false, but we have good evidence for them and so can use them, until shown otherwise, as a basis against which to judge further claims. In this way, we can ask for proof relative to other sentences. We can disprove a sentence on the assumption that some theory is true. For example, if for the sake of argument we accept quantum mechanics to be true, then Einstein's negative claim that "Quantum Mechanics is not complete because there are hidden variables" can be disproved. This is what the Bell inequalities did...

Not Einstein asks,
Can you explain the significance of Bell's inequalities to me in 2 paragraphs or less?
No. But it's my blog, I'll use more paragraphs if I want to.

Quantum mechanics is the physical theory that governs the behavior of matter and energy and is particularly necessary when dealing with very small things. Its central equation is what we call Schrodinger's equation into which you can put the description of a physical system at any given time and it will show you how the state of that system changes over time. Nothing strange there -- that's what all physical theories do for you.

The first strange thing, though, is that the description of the system that gets plugged in and pulled out of Schrodinger's equation is generally a state that we never, ever observe; it's what we call a superposed state. Consider a quantum egg in a quantum egg carton, there are twelve possible places for the egg to be. If we had a "free quantum egg" and did not look inside the carton, Schrodinger's equation would tell us that the egg is in a state of superposition made up of all twelve slots -- that is, the egg in some sense is in all of the slots, just not completely. If we applied a force that affected quantum eggs, we could make it so that the superposition was more heavily in certain slots and even not at all possible in others, but the egg would still be spread out among multiple slots.

The second strange thing is that whenever we open the egg carton, we always see the egg in one and only one slot, but we can never tell which one it is going to be. If the superposed state that we get from Schrodinger's equation rules out a slot, the egg will never be there, but for all the possible slots, there is a chance it may be there. We can get from the theory the probability that we will find the egg in a given slot, but there is absolutely no way in quantum mechanics to move beyond this probability to certainty.

This has pissed off many people, among them Albert Einstein, who argued that quantum mechanics is a good theory, a useful theory, a helpful theory, but it is an incomplete theory. This probabilistic element, he contended, is the result of an incomplete description of nature. There are other facts not included in the theory as it stands that determine which slot the egg will be in, we just haven't figured out yet what these hidden variables are. If and when we find them, we can use them to augment quantum theory and create a complete theory that will allow us to accurately determine where the egg will be.

In 1964, physicist John Bell worked on this question and figured out that there would actually be physically detectable consequences if the most natural sense of Einstein's notion of hidden variables were real. He worked out a set of equations called the Bell inequalities which would hold if there were no hidden variables and fail if there were. Turns out that they hold and so there is disproof for a certain sort of claim about hidden variables. Einstein's claim, in the way Einstein most likely thought about it, was shown by Bell to be wrong. A disproof of a negative claim...sort of...

Smart folks have shown that there may be ways to reformulate the notion of hidden variables so that a detour can be found around Bell's roadblock. This is what Phil is referring to in the comments (you are quick out the blocks, Phil, my friend). And we can use this to come back to Justin's original question about disproving a negative. Has Bell shown that there can't be hidden variables? No. Disproving a negative claim in the realm of contingent sentences is tricky business. He has shown that it can't be done in the way we would expect, but he has not -- as Phil correctly points out -- shown that it must be false. Do we have reason to believe that there are hidden variables? No. But we have not disproved it.