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There are meaningful questions we can ask about Euclidean geometry which could not have been posed in the time of Riemann or even of Hilbert, and which would have made no sense at all to Euclid. For example, does two-dimensional Euclidean geometry emerge as the large-scale limit of a quantum geometry? The fact that we are able to ask this question today demonstrates that the relevant constellation of absolute presuppositions, scene of inquiry, disciplinary matrix, or however you wish to phrase it, has simply changed.

pg-7, Towards a Philosophy of Real Mathematics

I'm trying to understand the highlighted line. What exactly is Quantum Geometry supposed in this context? Better, if the whole bolded question could be explained.

What I found so far:

  1. This wiki confuses me more since explanations assumes physics context I think.

  2. This MSE thread gives conflicting opinions, in particular, of one user saying this maybe an ambiguous question to ask.

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    $\begingroup$ My reading of the highlighted line is that the author is asking us to (1) invent a quantum geometry and then (2) prove that its large-scale limit is Euclidean geometry. (Maybe also invent a notion of large-scale limit.) $\endgroup$
    – Andreas Blass
    Commented Jun 12, 2023 at 21:04
  • $\begingroup$ So, you're saying those things don't exist already?? umm @AndreasBlass $\endgroup$
    – Brian
    Commented Jun 12, 2023 at 21:22
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    $\begingroup$ Maybe they don't exist yet, or maybe they exist in such great profusion that inventing one more won't do any harm. $\endgroup$
    – Andreas Blass
    Commented Jun 12, 2023 at 21:26

1 Answer 1

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I think it basically goes like this:

  1. Take an algebra with a product and a skew product, for example a suitable algebra of functions with a Poisson bracket $\{f,g\}=-\{g,f\}$. You can start with a complex manifold, or mechanical system, or a Lie group, and get a naturally skew-symmetric bracket.

  2. Replace multiplication that used to be commutative $fg-gf=0$ with a non-commutative one, with the deviation from commutativity being defined by said Poisson bracket. Define multiplication as $fg-gf=ih\{f,g\}$, where $h$ may be real or rational or imaginary, depending on what you want to study. See Moyal product. The "quantum" here alludes to the $h$ sometimes being interpreted as the Planck's constant.

  3. Study that non-commutative algebra and see what cool mathematics would fall out.

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