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So what if you used more than one neuron?
The first neuron looks at the raw inputs and makes a simple call. Maybe it notices something about the shape of the ears. It doesn't know what a cat is. It just fires or doesn't based on what it sees.
The second neuron doesn't look at the raw inputs at all. It looks at what the first neuron decided. It takes that result, combines it with other results from other neurons in the first layer, and makes its own call. Maybe it's picking up on something about the overall shape of the face.
The third neuron looks at what the second found. And so on.
Each layer is asking a slightly more specific question than the one before, but no single neuron ever knows what a cat is. Each one just adds up its inputs, checks its threshold, and passes a signal forward.
Nobody writes the concept of "cat" down anywhere. It emerges from the layers, the same way the dog's decision to bark emerged from smell and sound and memory in the previous chapter. Once again, the understanding lives in the whole thing, not in any of the parts.