S-Shaped Signals: The Sigmoid Function from History to AI

Join us on a journey from ancient ideas to modern neural networks as we dissect the sigmoid function—the unmistakable S-curve. We’ll unpack its key mathematical properties (bounded outputs, differentiability, a single inflection point) and explain why they matter for training neural networks via backpropagation. Along the way we’ll trace its cross-disciplinary history—from psychology and engineering to statistics and biology—exploring prominent variants like the logistic function and tanh, and how the sigmoid shows up in real-world applications across fields. A concise look at how a simple curve informs prediction, learning, and complex systems.

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