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Max Hodak - How do we know anything?

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This strikes us as obviously true and so we accept it, at least at the undergraduate level, as “proof” of the existence of the minimum. But really this is a feeling. Where does it come from? Some process in our brains is telling us to see something here, but when you try and pin it down our initial sense of clarity dissolves into a feeling of slipperiness. Mathematics has a long history of producing results that seem obviously true but which turn out later to be at least incomplete. More simply, everyone has had the experience of being struck by an idea clearly being right only to realize later something important was missed. Thinking about this, there is a fork in the road. On one side are interesting questions of how the brain produces the overall feeling of correctness and its reliability. On the other are the even deeper questions about, if this feeling is usually correct, what kinds of physics support the brain doing this in the first place. I’m not particularly concerned with the

Max Hodak - How do we know anything? Max Hodak - Writings How do we know anything? August 2022 Consider the following mathematical idea: A function $f(x)$ is smooth and continuous (i.e., it is differentiable everywhere) At some point $x_1$ the slope of $f$ is negative, $f’(x_1) < 0$ At some other point $x_2 > x_1$ the slope is positive, $f’(x_2) > 0$ Therefore by the intermediate value theorem there must be a minimum where $f’(x) = 0$ somewhere between $x_1 < x < x_2$ This strikes us as obviously true and so we accept it, at least at the undergraduate level, as “proof” of the existence of the

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