I recently finished reading A Brief History of Time (by Stephen Hawking). It was good. In it he mentions the Heisenberg uncertainty principle, and it lead me to down a rabbit hole of various thoughts, thoughts about things that we know we cannot know. So I decided I’d attempt to put them together into some sort of post.
1) Heisenberg’s Uncertainty Principle.
After looking through a couple of textbooks and some popular science books, I’ve decided Hawking’s is the best description, and will thus quote it below (the important bit is italicized at the end, if you don’t care to read all the stuff before):
In order to predict the future position and velocity of a particle, one has to be able to measure its present position and velocity accurately. The obvious way to do this is to shine light on the particle. Some of the waves of lights will be scatted by the particle and this will indicate its position. However, one will not be able to determine the position of the particle more accurately than the distance between the wave crests of light, so one needs to use light of a short wavelength in order to measure the position of the particle precisely. Now, by Panck’s quantum hypothesis, one cannot use an arbitrarily small amount of light; one has to use at least one quantum. This quantum will disturb the particle and change its velocity in a way that cannot be predicted. Moreover, the more accurately one measures the position, the shorter the wavelength of the light that one needs and hence the higher the energy of a single quantum. So the velocity of the particle will be disturbed by a larger amount. In other words, the more accurately you try to measure the position of the particle, the less accurately you can measure its speed, and vice versa. Heisenberg showed that the uncertainty in the position and the particle times the uncertainty in its velocity times the mass of the particle can never be smaller than a certain quantity, which is known as Planck’s constant. Moreoever, this limit does not depend on the way in which one tries to measure the position or velocity of the particle, or on the type of particle: Heisenberg’s uncertainty principle is a fundamental, inescapable property of the world.
(as an aside, it’s interesting to note that Heisenberg may be the reason that the Nazi’s never developed the atomic bomb before the Americans: after the war he would say that he could have built the bomb, but decided against it (for moral reasons), and played dumb. However, whether or not this is true, I believe, remains controversial)
2. Godel’s Incompleteness Theorem.
I found this description from a Philosophy of Science article from Nature:
“In 1931, Kurt Gödel proved that any mathematical system that includes enough of the theory of natural numbers contains statements that cannot be proved to be either true or false, and is thus incomplete. The general argument for the proof is based on Epimenides’ liar’s paradox — is ‘This sentence is false’ a true or a false statement? — but it replaces ‘false’ with ‘unprovable’.”
– P.-M. Binder, Nature 455, 884-885 (2008)
oh, and this one from A History of Mathematics – Carl B. Boyer:
“Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions … It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science – to devise a set of axioms from which all phenomena of the external world can be deduced.”
3. Turing’s Halting Problem.
“In computability theory, the halting problem is a decision problem which can be stated as follows: given a description of a program, decide whether the program finishes running or will run forever. This is equivalent to the problem of deciding, given a program and an input, whether the program will eventually halt when run with that input, or will run forever.
Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. We say that the halting problem is undecidable over Turing machines.”
(as an aside, I’d just like to mention that Alan Turing was chemically castrated as a cure for being homosexual. He later killed himself)
There might be more…but I’m not currently aware of them.
However, one other problem I wanted to mention is the Three-Body Problem. This isn’t necessarily unsolvable, but at this point, no one has an answer, which I think is sort of ludicrous…cause it’s only three-bodies.
The problem is in simply “taking an initial set of data that specifies directly or indirectly the positions, masses, and velocities of three bodies, for some particular point in time, and then using that set of data to determine the motions of the three bodies, and to find their positions at other times, in accordance with the laws of classical mechanics”
For example, if you were to isolate the moon, earth and sun from all the other planets and other gravitational forces (which would pretty much be the universe…I don’t think gravity has an upper bound), put them in motion, and tried to predict their future behaviour based on your complete knowledge of their current behaviour…we couldn’t do it. That just strikes me as shocking. There is so much complicated math out there…and yet there’s no complete solution to this problem.
I guess I could also point out chaos theory. Which every one knows to be a deterministic system, but if you alter one piece of information even slightly, you may get a completely different outcome then expected (relative to non-chaotic phenomena).
…hmmm, I guess the take home message from this is simply a humbling one: there are some things in this world that we can fundamentally never know (the first three I mentioned, not the last two). But, those things are very well defined…so don’t go taking this message as some sort of reason to believe in the supernatural or miracles or whatever. That’s just silly.
Edit: while previewing this post, wordpress directed my attention to a another one, which I found to be perfectly related and well written. I suggest everyone read the short post here.