infinitesimals to atiyah singer


i learned some nonstandard analysis and clifford algebra, mostly through david hestenes’s ‘geometric algebra’.

it’s freaky because it seems to give a bridge between discrete and continuous in terms of cutting stuff into boxes.

all of these things suddenly came together, with pictures rather than algebra. it feels like what hodge theory and algebraic geometry touch on, but without nearly as much category theory/ homological algebra:

  • differential is difference, derivative is quotient of differences, and integral is sum
  • the tangent space as an infinitesimal vector space (smooth manifolds)
    • each tiny arrow is now literally an infinitesimal vector (the geometric arrow kind). the differential d as an affine displacement becomes much clearer.
  • these vector spaces and their dual space can be used to define an algebra of boxes and their orthogonal complements.
    • this is pretty much differential forms
    • i think it describes a bunch of hodge theory
    • by adding points at infinity, you get a unification of circles and lines and points. the order is point → circle → line. a point is a circle of infinitesimal radius. a circle is just a circle. a line is a circle of infinitesimal radius. circle inversion becomes really useful and natural.
    • homology becomes micro-triangulation. you can cut space up into triangles of different dimensions.
  • the idea of blowing up points into infinitesimal lines/boxes/circles/polygons takes you into what it means to gain and lose smoothness. hard (nonsmooth) contact in physics is measuring levels of contact to ensure that there’s at least 1 degree of smoothness left. you can locally add smoothness by using dirac deltas, which are now just plain functions (infinitely large value in an infinitely small interval (~1 point), so its size (measure/integral) is still finite).
  • atiyah’s work becomes really natural.
  • the end run feels like an infinite arithmetic/nice theory for differential equations.

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