First, I would like to draw your attention to this article, about a processor that uses probabilistic calculations, and as such, runs 7 times as fast and at 1/30th the power of conventional hardware. The basic idea is, what information do you -really- need?
Which brings me to the point at hand. I am a math person, first and foremost. But for the last three years or so, I’ve also been pursuing physics a great deal. For the most part, if you know math, then physics is relatively easy. But, physics also provides a different perspective on math that I think is very important. In math, there is a tendency to get lost in symbolic manipulations, algorithms, problems of a specific type, and forget what the math is ‘actually’ saying. You can become entrenched. Blinded, almost. In physics, math is often treated as almost an obstacle to be dealt with. Something to be worked around to get at the ‘real’ answer. At least, the relevant answer. It’s fascinating to me. The solution is often to not do the problem at all. Doing by not doing. Math without math. And yet, at the same time, it gets straight to the matter of what the math is trying to express, and in the processes, helps you develop an intuition for it.
And if none of that made sense, here is an example. Consider the expression, for a positive value of x,
In general, we want to understand the behavior of that function with respect to x. Now, as a mathematician, I jump in and think, hmmm. What does that summation converge to? Can we express it in a nicer formula? Things start to jump out, generating functions, possible Taylor summations, etc, any one of which could be used to turn that summation into something recognizable. Usable.
But then, staring at the summation a good long while … nothing forms. The usual power series formulas are no help, due to the quadratic nature of the exponent. And the factor of (2j+1) in front of the exponent certainly complicates things. The form of the summation fits to nothing I recall seeing, ever. Asking Mathematica, it can’t find any convenient expression for the sum either. General maths suggest no reasonable way of turning this summation into something tractable. And, if we wanted to do more interesting things to that summation, such as take the log, and then various derivatives, by keeping the entire summation, formulas would rapidly turn quite nasty.
So, what is there to be done? How best to do that summation? How can we really appreciate what this function is? The answer is, to not do the summation.
(more…)
