Exercise 2: The Two New Architectural Ideas, and Why They're Architectural Not Mathematical — Possible Solution
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THE TWO NEW IDEAS, PER THIS CHAPTER
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Per this chapter: "Many neurons, side by side... a layer runs several
neurons in parallel against the same inputs — each with its own
independently learned weights and bias, each producing its own output,"
and "Layers, stacked... one layer's outputs become the next layer's
inputs."

WHY NEITHER IDEA CHANGES THE UNDERLYING MATH OF ONE NEURON
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Per Exercise 1's own reasoning, a single neuron's own computation — a
weighted sum plus bias, squashed through an activation function — is
exactly ml1-5's own formula, unchanged. Adding more neurons side by
side doesn't alter that formula at all; it simply runs the IDENTICAL
formula multiple times in parallel, once per neuron, each with its own
independent set of weights and bias. Stacking layers likewise doesn't
introduce any new kind of arithmetic operation — it just takes the
outputs already produced by one full application of that same formula
and feeds them in as the inputs to another application of the exact
same formula.

WHY THIS JUSTIFIES CALLING THEM "ARCHITECTURAL" CHANGES
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"Architectural" describes how existing pieces are arranged and
connected to one another, as opposed to "mathematical," which would
describe a genuinely new kind of computation being introduced. Both of
this chapter's own new ideas are entirely about ARRANGEMENT: how many
copies of the same neuron formula run side by side, and how the outputs
of one group of copies get routed as inputs into another group. Neither
idea modifies what a single neuron itself actually computes — it's the
same building block (already established as identical to ml1-5's
formula in Exercise 1), simply used in greater quantity and connected
in a new configuration.

WHY THIS DISTINCTION IS WORTH MAKING EXPLICIT
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If a reader assumed neural networks involved some fundamentally
different, more exotic mathematics than ml1-5's own logistic
regression, they would be missing the actual source of a network's own
power — which, per this chapter's own framing, comes entirely from
arrangement (many neurons, stacked in layers) rather than from any
single neuron computing something ml1-5 didn't already compute. Making
this explicit sets up exactly why nn1-2 and nn1-3's own resolution of
the XOR problem (previewed at this chapter's own close) will be an
architectural story — about adding a hidden layer — rather than a story
about inventing some new kind of formula for a single neuron to use.

WHY THIS WORKS AS AN ANSWER
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It restates the chapter's own two new ideas precisely, explains why
neither one alters a single neuron's own underlying formula (already
established as identical to ml1-5's own logistic regression), and
explains why this makes "architectural" — arrangement and connection,
not new arithmetic — the accurate description of what's actually new
in this course.