Or phrased another way, what cannot be computed if we have infinite computational resources at our disposal? Is the answer "nothing"?
February 17, 2014
Maxwell's equations are taught in physics courses as the description of light and magnetism. But those equations require mathematical concepts (integrals and derivatives) that themselves are uncomputable. For example, according to Maxwell a light ray passing through space involves energy oscillating back and forth between the electric field and the magnetic field. Maxwell's equations describe how at every point in space if there is a changing magnetic field, this will give rise to an electric field. And if there is a changing electric field, this will give rise to a magnetic field. Solving these equations yields results that behaving like light beams travelling through space. The solutions have sin() and cosine() elements, two functions of time that are out of phase with each other.
Those functions require infinite precision in terms of time as well as infinite precision in terms of value. This is typical of math, the concepts in math can involve such imaginary, infinite precision. So there is the question, if the universe we live in is described by equations that themselves require infinite precision (and are therefore uncomputable by any known computer), can the equations really be a description of what is going on in the universe?
One thing we've observed about light coming to the earth, through telescopes. It seems that the further away an object is, the more its light is red-shifted. The common assumption is that means the object is moving away from us, and the further away it is the faster it is moving away. This holds true for pretty much every direction we look in. So people say, "Aha! This means there was a Big Bang where everything blew up, and now we're seeing the aftermath where everything is flying apart." But the Big Bang theory has so many glaring holes in it that it simply cannot be a valid explanation of what actually happened. There are objects in the universe older than the age of the universe as demanded by the Big Bang theory. Moreover there are very distant objects (quasars) that are unacceptably bright when considering their red-shift. Halton Arp demonstrated that they are actually much, much closer than commonly believed.
Ok, so we have a prevailing theory (Big Bang) that is part of the commonly accepted explanation of reality, and part of that is Maxwell's equations for describing how light moves around the universe. But Maxwell's equations depend on infinite precision which itself is uncomputable. And light rays that have been moving through space for a very long time seem to end up slowing down (red shifting). Might there be some other explanation for the red-shifting that will resolve all the issues?
Specifically, suppose the universe is computable, and there is not infinite precision anywhere (in time, or in space). Now that would mean that Maxwell's equations are only an approximation of the machinery that is active in the universe. Suppose the universe itself, operating with limited precision, exhibits waves moving through space that appear to remain unchanged (light) but suppose that there are effects that manifest themselves over very long periods of time. Specifically suppose a ray of light moving along appears to gradually slow down. This slowing down would appear identical to the red-shift associated with a light source moving away. Also suppose the slowing down is a necessary side effect of the fact that the universe is not operating with infinite precision. That is, suppose the slowing down in light frequency is a natural outcome of things like rounding errors in the computations.
Then we have a situation where the redshift we observe is not caused by an exploding or expanding universe, and so there is no need to postulate the Big Bang theory in the first place. So the universe can be very much older than we've been assuming, and so the objects in the universe that are so inexplicably old now will pose no problem at all. And finally, if the universe does not demand infinite precision in order for its operation we can feel a sense of relief that it is in fact computable.