June 28, 2009

Proof of Uncertainty

Kurt Godel proved that in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system.
You’ve heard me say that logic invariably leads to faith, but don’t take my word for it. There’s considerable evidence to support the idea and people who love the hard sciences will especially appreciate it, because the confirmation comes from logic itself.

As an introduction, consider the 19th Century worldview and how it changed going into our modern age. Throughout the 1800s, there was great enthusiasm and optimism in the power of logic. Earlier scientific and mathematical discoveries—Newton’s classical physics, for example—had uncovered what seemed to be an elegant symmetry in nature, which gave intellectuals a view that all things could be understood in terms of scientific laws. In fact, people (the great Emmanuel Kant, included) believed the logic inherent in science would eventually lead to an understanding of the mind of God.

But cracks began to appear in this deterministic construct. One of the first was the development of non-Euclidean geometries that rejected a postulate used in the plane geometry we study in high school. The possibility that other postulates might be incorrect or of limited value ushered in an age of uncertainty that additional scientific and mathematical discovery only amplified.

The study of physics, for example, added to the disquiet. As physicists delved more deeply into the interactions of subatomic particles, they discovered that such relationships were based upon probabilities rather than strict rules as had been previously supposed. To make matters worse, one implication of the discovery was that the result of any subatomic interaction didn’t finalize until a sentient being observed it. To understand what this means, consider that old philosophical conundrum: If a tree falls in the forest and no one is there to hear it, does it make a sound? Well, according to one interpretation of quantum mechanics—the Copenhagen version first posited by the Nobel Laureate, Niels Bohr—the tree doesn’t even fall! Its subatomic particles remain in what is referred to as a probability wave until a sentient observer comes along. Thereupon the particles “choose” one of the infinite range of possibilities available to them. (My son, who is working on a physics PhD, tells me no one really believes that’s what happens, although it is consistent with the inexplicable results of various experiments. For more on this, see the book Schrodinger’s Kittens and the Search for Reality, by John Gribbin, which in my opinion is the best book about quantum mechanics written for lay people).

Perhaps one of the most significant discoveries that increased the uncertain worldview of our time was made by Bertrand Russell, a man who had once been a proponent of mathematical determinism. Russell discovered a logical inconsistency in set theory that is evident in the following question:
A man of Seville is shaved by the barber of Seville, if and only if, the man doesn’t shave himself. Does the barber shave himself?

The paradox can be described in a rigorous mathematical way, but consider the following simplification: Obviously, the barber of Seville either shaves himself or he doesn’t. Regardless, however, the result is illogical. If he shaves himself, he cannot be shaved by the barber of Seville, but since he is the barber of Seville and he shaves himself...well, you get the point. The problem, as Russell was able to distinguish, can occur anytime a set is an element of itself. For example, the set of men shaved by the barber of Seville (let’s call this A) is not an element of itself, because A is not a man shaved by the barber of Seville. However, let’s create another set (which we’ll call R) and include in it everything that is not in set A. Since set R is also not a man who is shaved by the barber of Seville, it is an element of itself, a characteristic often called self-referencing. This may seem like a trivial matter, but self-referencing can lead to serious logical problems and points to the limitations of logic.

Finally, let me describe a theorem that is simplistically sneaky, but has implications that have essentially put an end to strict mathematical determinism. I’m talking about Kurt Godel’s Incompleteness Theorem, which like Russell’s Paradox, has a rigorous mathematical rendering, but can be described in simplified terms.

Let’s say you create a computer that you call the Universal Truth Machine (or UTM for short) which you’ve programmed to tell the truth. Before approaching it, you write the following words on a sheet of paper:

The UTM will never say this sentence is true.

Now, you turn the UTM on, show it the paper and ask if the statement is true. What happens? First, the UTM can’t say the statement is true. Can you see why? If it does, the statement will be rendered false, which is contrary to the UTM’s programming. In a paradoxical way, the fact that the computer will not say the statement is true is your greatest evidence that the statement is, in fact, true. On the other hand, the UTM can’t say the statement is false, either, since as we’ve already demonstrated, that would be untrue. (Another way to look at it is if the UTM said the statement was false, it would render the statement true).
Again, the result seems contrived and trivial, but the math is sophisticated and full of implications. In short, Godel’s Incompleteness Theorem suggests that rational thought can never penetrate to the ultimate truth, or said another way: There are truths that cannot be discerned, or proved, strictly through the use of logic.


Matt's brain said...

Well, not to be overly nit-picky, but the Copenhagen interpretation doesn't make any references to human consciousness or sentient beings or anything. What it actually says is that the collapse of a particle's wave-function occurs when it is MEASURED.

The Copenhagen interpretation, however, doesn't define what actually constitutes a measurement, which is probably where the confusion comes from. But if you actually look at what happens in a real measurement, there is a lot more going on than the actual observation by a sentient being.

To find the location of a particle, for example, you have to scatter light across it, or make it interact with matter in some other way. A lot of things have to happen before some number appears on the experimental display. It is probably just these real quantifiable effects in which the microscopic system interacts with the macroscopic system that cause the collapse of the wave function- not the fact that someone looked at the display.

Alan Bahr said...

I thought you might say that. Whatever you physics types say in the safety of your own laboratories may not be what is published in books for lay people, but I remember seeing something written by Niels Bohr that kept referring to sentient beings, not measurements. In fact, I'm pretty sure the argument went: Measurement alone doesn't appear to collapse the wave function. Only a sentient being can do so, because the measuring instruments would be subject to the same quantum laws as the experiment is designed to test.

Here is a definition (from the Standford Encyclopedia of Philosophy) of the Many Worlds Interpretation, which I believe is a variant of the Copenhagen. As you can see, it requires a sentient observer to collapse the superimposition of states.

"Another concept (considered in some approaches as the basic one, e.g., in Saunders 1995) is a relative, or perspectival, world defined for every physical system and every one of its states (provided it is a state of non-zero probability): I will call it a centered world. This concept is useful when a world is centered on a perceptual state of a sentient being. In this world, all objects which the sentient being perceives have definite states, but objects that are not under her observation might be in a superposition of different (classical) states. The advantage of a centered world is that it does not split due to a quantum phenomenon in a distant galaxy, while the advantage of our definition is that we can consider a world without specifying a center, and in particular our usual language is just as useful for describing worlds at times when there were no sentient beings."

Matt's brain said...

So, here's something that Heisenberg said:

"Of course the introduction of the observer must not be misunderstood to imply that some kind of subjective features are to be brought into the description of nature. The observer has, rather, only the function of registering decisions, i.e., processes in space and time, and it does not matter whether the observer is an apparatus or a human being."

And this is from Steven Weinberg:

" The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe. But these rules are expressed in terms of a wave function (or, more precisely, a state vector) that evolves in a perfectly deterministic way... The Copenhagen rules clearly work, so they have to be accepted. But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wave function, the Schrödinger equation, to observers and their apparatus."

In other words, the quantum mechanical state of something being measured must be closely tied to the measuring device. If you could solve the Schrodinger equation for the system in conjunction with the measuring device (a VERY tall order, even by supercomputer standards), then you should be able to see exactly how the measurement forces the collapse.

On another note, physics exclusively addresses claims that are provable in nature. So if you read about something that doesn't appear remotely testable, then it isn't physics- it's philosophy. I think the prevalence of the Copenhagen interpretation is really due to the fact that it avoids these types of wonky explanations and only explains what has been observed in lab situations.

If you like, you could ascribe to one of the many interpretations that avoid the indeterminacy problem altogether, like the Bohm interpretation. This interpretation states that the state of a system must also contain a hidden non-local variable which determines the results of a measurement but cannot be calculated. This view seems much "nicer" to those who insist that God not roll dice, while still being consistent with observation (though it is completely useless from a physics perspective).

By the way, provability is exactly the problem with string theory. It may explain the universe in a beautiful and exciting way, but until it presents some testable claims, there is no way to prove its validity.

I think this discussion is actually proving your other point, though, since all this scientific pursuit ultimately ends up in one big question of faith.

Alan Bahr said...

Very cool. Thanks for the clarifications.