Out of curiosity, I've been reading up a bit on the field of Machine Learning, and I'm surprised at the amount of computation and mathematics involved. One book I'm reading through uses advanced concepts such as Ring Theory and PDEs (note: the only thing I know about PDEs is that they use that funny looking character). This strikes me as odd considering that mathematics itself is a hard thing to "learn."

Are there any branches of Machine Learning that use different approaches?

I would think that a approaches relying more on logic, memory, construction of unfounded assumptions, and over-generalizations would be a better way to go, since that seems more like the way animals think. Animals don't (explicitly) calculate probabilities and statistics; at least as far as I know.

The behaviour of the neurons in our brains is very complex, and requires some heavy duty math to model. So, yes we do calculate extremely complex math, but it's done in a way that we don't perceive.

I don't know whether the math you typically find in A.I. research is entirely due to the complexity of the natural neural systems being modelled. It may also be due, in part, to heuristic techniques that don't even attempt to model the mind (e.g., using convolution filters to recognise shapes).

If you want to avoid the math but do AI like stuff, you can always stick to simpler models. In 90% of the time, the simpler models will be good enough for real world problems.

I don't know of a track of AI that is completely decoupled from math though. Probability theory is the tool for handling uncertainty which plays a major role in AI. So even if there was not-so-mathematical subfield, math techniques would most be a way to improve on those methods. And thus the mathematics would be back in game. Even simple techniques like the naive Bayes and decision trees can be used without a lot of math, but the real understanding comes only through it.

Machine learning is very math heavy. It is sometimes said to be close to "computational statistics", with a little more focus on the computational side. You might want to check out "Collective Intelligence" by O'Reilly, though. I hear they have a good collection of ML techniques without math too hard.