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I am currently reading "Science and Hypothesis" written by Poincaré. In chapter 2, "Mathematical Magnitudes and Experiments", I found the following sentence.

"We can show that there are curves which have no tangent, if we define such a curve as an analytical continuum of the second order."

What is this referring to? I would be grateful if you could tell me.

KReiser
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Atsu
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  • The Weierstrass function "is an example of a real-valued function that is continuous everywhere but differentiable nowhere" – Henry Jul 25 '23 at 16:44
  • @Henry This was also what came to mind when I first saw the question, but I think the Poincare quote (which is more the topic of discussion) is referring to something else than just the existence of nowhere differentiable curves. Here is the full quote. – Jam Jul 25 '23 at 16:47
  • @Jam Another version here on page 30. I do not totally understand Poincaré's language, but I think his "continuum of the second order" is essentially the real numbers. – Henry Jul 25 '23 at 16:53
  • @Henry Yes, I'm thinking the same. I'm beginning to think that Poincare is indeed referring to Weierstrass' function, but I'm finding it difficult to match the analogies with the mathematical objects they are supposed to symbolize. It's all very reminiscent of the hyperreal numbers. But, to Poincare's credit, the theory of real numbers wasn't yet complete at the time this was written (pardon the pun). – Jam Jul 25 '23 at 17:00

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