I don't think you need to know all the math behind NNs in order to successfully learn how to use one. I certainly don't and I've been using the encog library (by Heaton) for a while with great success. And you definitely do not need to be a mathematician to need one.

Yes, there are some general principles and concepts you need to know/understand, but nothing particularly deep.

Edit: Actually I'd highly recommend encog: it's very easy to use, it's extremely well documented and for those of us that do not have a formal education in NNs it's a great place to get started. http://www.heatonresearch.com/encog