Some Things *ARE* Unknowable

By ScottN

Or... Why Is That Heisenberg Guy So Uncertain About Things?

WARNING -- THIS ARTICLE IS KIND OF TECHNICAL

Over on NitCentral, there have been some debates about the proposition that "G-d is unknowable". This is not part of those debates.

Instead, I am going to continue my writings about science in general. In that vein, let us discuss the concepts of uncertainty and undecidability. There are three major items that I will talk about in this context. The Halting Problem, Godel's Undecidability Theorem, and The Heisenberg Uncertainty Prinicple.

The Halting Problem essentially states that it is impossible to write a computer program (A) that takes a second computer program (B) as its input, and determines if B completes processing. This is a fundamental theorem of Computer Science, and has been known since the early days of automata theory. Thus, there are definite limits on the knowledge that we can derive from a mechanistic approach.

OK, mechanistic reasoning is out, what about a more intuitive approach? Surely pure math, wherein sometime brilliant leaps of logic are made doesn't have these limitations! Alas, it does. In 1931, Kurt Godel dropped a bomb on the mathematical world, as follows:

To every w-consistent recursive class k of formulae there correspond recursive class-signs r such that neither v Gen r nor Neg (v Gen r) belongs to Flg (k) (where v is the free variable of r).

Did you understand that? Good. I didn't expect you to either. In English, it essentially says that "All consistent axiomatic formulations of number theory include undecidable propositions" (Hofstadter D., Godel, Escher Bach: An Eternal Golden Braid, p.17). So how do we interpret that? A consistent system is one where it is impossible for a statement and its negation to be true simultaneously. In other words, "2 + 2 equals 4" and "2 + 2 is not equal to 4" cannot both be true. If they are both true, the system is inconsistent. So if we have a consistent system, it is possible to write a statement that is true, but cannot be proven.

Godel's genuis was to discover that by encoding the symbols of number theory into numbers themselves, and creating rules to manipulate those numbers, it is possible to write statements of number theory that have two meanings. The first is the literal meaning of the symbols themselves. However, if the numbers have been properly manipulated, there is a second "meta-meaning" to the statement, which can be obtained by reversing the encoding. Using this scheme, Godel was able to construct a statement (G) which had the meta-meaning "G has no proof". Since statment G was constructed using the proper formalisms and encoding on the literal level, the meta-meaning must be true. So it is possible to construct a statement which is true, but has no proof. Therefore in the mathematical world, there are unknowables as well.

Now that we've determined that the abstract worlds of computers and mathematics have unknowables, what about the real world? Sadly, the foundation of quantum mechanics is the fact that the physical world is unknowable as well. Werner Heisenberg discovered the principle that bears his name: The Heisenberg Uncertainty Principle. In essense, it says that for any entity, certain complementary properties cannot be known more accurately than a certain value. Specifically, the product of the uncertainties in these properties must be greater than Planck's constant h.

Generally, the properties discussed are position (x) and momentum (p). Writing it out mathematically, we get

Dx * Dp > h

Putting that into plain English, "the uncertainty in position multiplied by the uncertainty in momentum is greater than Planck's Constant". Now, h is incredibly small, somewere on the order of 6 * 10^-34 erg-seconds, so the Uncertainty Principle is unnoticable for macroscopic objects such as a grain of sand, you, me, your computer, the Empire State Building, etc... When you are standing still, you think you are not moving, and you have an absolute position. In actuality, you are bouncing around a sub-microscopic circle. But since you are so big, compared to these fluctuations, you don't notice it. On the other hand, for microscopic objects such as atoms, subatomic particles, molecules, or a politicians brain, the effect is quite large and noticable. This means, for example, if you could confine a single atom within a small (molecule sized) box, the atom would blur out, and you wouldn't be able to tell how fast or which direction it was moving. By confining the atom, you have reduced/limited the uncertainty in its position, so it has a corresponding increase in the uncertainty of its momentum.

The mathematics of this are simple and elegant, but I'm not going to post them here.

So, we have the fact that there are unknowable things in the world... Are there others? I don't know...

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