<p>I am not a mathematician. I enjoy a good math puzzle, but I admit my weaknesses whole heartedly. That said, I've always had an interest in Neural Networks, and while I understand them enough to implement them from scratch, I hit a wall when I need to understand any concept that I can only find mathematic proofs for. Where is the programmer's guide to neural networks, using code instead of formula to explain the practical reasonings?</p>

Unfortunately, I don't know if there's a good single "programmers source" that will give you all of the concepts.  I liked
Neural and Adaptive Systems: Fundamentals through Simulations
.
The best way to have a "programmer's understanding" of neural networks is not so much by examining the code, but in the problem and the correct results.  So, if you don't want to look at math, I recommend you look at a given problem.  For example, consider the XOR problem as an example of why you need non-linear activation functions, look at the number of variables and their possible values for understanding why a neural network needs to be of a certain size and toplogy to be effective, and split your data into train/test regimes and do studies to see why overfitting is dangerous.  Examine the code with the data.
I also recommend not getting too hung up, but reading further.  Certain practices in feed-forward networks become more clear once you see their generalization in recurrent and constructive neural networks.  I also recommend going wider:  Bayesian networks, fuzzy cognitive maps, SOM, Boltzman machines, simulated annealing, and reinforcement learning all have intuitions.
Does this go towards answering your question?