I've always enjoyed finding out about crackpots whose odd ideas are years ahead of their time. Their ideas languish in obscurity for decades until some other nut comes along and points out the utility and novelty of the idea. Here are some of my favorites.

#### Quaternions

William Hamilton invented quaternions in 1843. Although there was some initial excitement about quaternions, they were overshadowed by vector algebra and vector calculus. Oliver Heaviside was openly hostile to the idea of quaternions and spent a fair amount of time pointing out the advantages of vector analysis. J. Willard Gibbs didn't quite ‘get’ quaternions (not that he didn't understand them, he just didn't see why you'd bother with the complexity of quaternions when you could just use a 4-vector). Quaternions generally fell by the wayside, but in recent years they have been adopted by the computer gaming industry as the standard way of dealing with rotations of objects in 3-dimensional space.

#### Reversible Computing

Rolf Landauer of IBM pointed out that the second law of thermodynamics implies that erasing information generates heat (that is, if you clear a register in a processor, the bits that you erase have to go somewhere, and that somewhere is the heat sink). On the other hand, if the computer never erased information, it could presumably run without generating heat. (In computers these days, each bit is represented by huge piles of electrons, and the vast bulk of the heat generated is because we are trying to move all these electrons around at very high speed. The heat from the logical information content is minuscule.) Although the connection between information theory and thermodynamics seems clear and obvious to me, it seems that many chemists and physicists consider it to be an analogy that has been stretched beyond its limits. Shannon's entropy from information theory has an identical formulation as Boltzmann's entropy from thermodynamics (modulo the conversion constant). The question is whether this is because they are analogous quantities or whether they are the same quantities.

#### Bayesian Statistics

Ronald Fisher, the father of modern statistics, was a staunch opponent of the Bayesian school of thought. The Bayesian approach to statistics has only recently started to gain traction. One important difference between standard statistics and Bayesian statistics is in the integrals over the likelihood function. The standard approach is to integrate over the sample space, but the Bayesian approach is to integrate over the hypothesis space. It is generally impossible to find a closed-form solution to the Bayesian integral, and attempting a numeric approximation is intractable. The standard integrals over the sample space are much easier, and if they cannot be solved analytically, they easily succumb to numeric techniques. But the Bayesian approach can give much better answers (when you can get them!). Modern computers allow a brute-force approach to numeric integration, and Bayesian statistics has been gaining in popularity.