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I recently had a bit of an epiphany around GA the other day (and Dual numbers, and Screw Theory with Twists and Wrenches): I realized that I had it all backward

I recently had a bit of an epiphany around GA the other day (and Dual numbers, and Screw Theory with Twists and Wrenches): I realized that I had it all backwards. I’d seen GA as an alternative formulation for projective geometry and for Maxwell’s equations, Dual Numbers for auto-differentiation, and Screw Theory for rigid-body dynamics; the question that had always bothered me was “how in the hell did they manage to take these engineering problems and derive these crazy linear algebra theories to solve them?” The realization was that it went the other way around! Clifford Algebras are an interesting linear algebra formulation. As it turns out, by appropriately defining the e^2=0, e^2=1, and e^2=-1 axes of the general Clifford Algebra framework, a bunch of these engineering problems map into the algebra and then adopt a super compact notation for solving them.

leumassuehtam 2y ago
I like to think of an algebra the same as a language. You want to define and constrain your language such that you can describe elements of your universe is clear and consice. For example, if I am trying to describe a art piece in the museum, I need concepts and words that connect what I am seeing to reality. In the same way in electromagnetism, proper time is part of the reality and it is an experimental fact, so it must be included in the algebra. Another example is a functional programming language where you identify high-level functions as the building concepts of many computer science problems. Defining an algebra tailored for the problem you're working with actually highly constrain the space, and this is what makes many problems trivial.
Sharlin 2y ago
As it usually goes. See also Wigner's classic essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

leumassuehtam 2y ago

I like to think of an algebra the same as a language. You want to define and constrain your language such that you can describe elements of your universe is clear and consice. For example, if I am trying to describe a art piece in the museum, I need concepts and words that connect what I am seeing to reality. In the same way in electromagnetism, proper time is part of the reality and it is an experimental fact, so it must be included in the algebra. Another example is a functional programming language where you identify high-level functions as the building concepts of many computer science problems. Defining an algebra tailored for the problem you're working with actually highly constrain the space, and this is what makes many problems trivial.

Sharlin 2y ago

As it usually goes. See also Wigner's classic essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...