View Full Version : The Limits of Math

2014-Jun-08, 08:32 PM
I've been reading Brian Greene's books and he says the math for describing even the simplest physical phenomenon in the real world is wildly complex, thus physicist use vastly simplified models. Since current math is insufficient to describe everything in the universe and a final theory, will humans ever be able to figure out the most complex physical features of nature? Would quantum computers be able to find the final theory, determine if worm holes can exist, etc.? How far off are we from this lofty goal?

2014-Jun-09, 02:55 AM
I agree and disagree.

Let's look at Newton's relativity and quantum mechanics as examples. On the fundamental level, they are both simple ideas and simply expressed mathematically. Newton says the force between 2 particles is the product of their masses divided by the square of the distance between them. Easy, simple, clean. The problem is when there are 3 particles(or more) instead of two. There is no single formula that describes all the particle's motions. This is called the three body problem and makes solving gravity for large numbers of particles extremely difficult. We can't solve this problem infinitely far into the future. We can, however, use computers to solve the simple equation we have for a very short time, and then do the calculation again for a short time, and again, and again. It's not perfect, but the accuracy is a question of how long you're willing to wait for the computer to crunch enough numbers.

As for quantum mechanics. It's a beautiful theory founded on a simple idea(the wave equation). The problem, again, is that it can only be solved exactly for two particles. In this case, an electron going around a proton(hydrogen). QM tells you everything you want to know about hydrogen to incredible precision. But to get answers for stuff like sodium atoms results in the same type of computer based approximation that we use to solve Newton's gravity.

That's why I disagree. The basic idea is not beyond math. It actually involves relatively simple math.

I agree because to be useful for the kinds of problems we are really interested in requires a lot of math tricks and/or sheer computer power.

My modern physics prof used to say physics has only solved two problems exactly. I don't even remember which he quoted as the first, but he referred to the solution for the hydrogen atom in QM as being the second. When he said 'problem', he meant a solution which holds to this day as an exact solution. In that case, Newton doesn't qualify because we now know about general relativity.

2014-Jun-09, 04:58 AM
It is not the math which is insufficient, it is our ability to do the computations. Adaptive optics is an excellent example of this. The principles of adaptive optics have been around for centuries but the computing power to compute the change virtually instantly has only been around for the last couple decades. That is why ground based telescopes rapidly overtook the HST in sheer resolving power.

Beyond that, it isnt the math which will be the limit of theory, it will be the limits of our ability to see what is going on. Math, and only math, in the form of quantum theory has turned quite a few stable elements radioactive. Tungsten is the one that comes to mind first. It wasnt until the theory said it should have a really long half life that anyone really checked.

It will always be our ability to do the experiments, to see what is actually happening, and to do the computations the math requires that will be the limit to our knowledge, not the actual mathmatics itself.