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Hestenes’s tireless promotion is necessary because most everyone else, having grown up with a different formalism, is too entrenched / busy to try to make a bas

Hestenes’s tireless promotion is necessary because most everyone else, having grown up with a different formalism, is too entrenched / busy to try to make a basic change to what we teach and use, even when that change would provide tremendous benefits. In physics there was some uptake where appropriate, but mathematicians for the most part scoff (“this is just a different notation for structures that were studied in the 19th century, there’s nothing new here” kind of thing). But being able to multiply and divide vectors is a big deal, and GA is packed with incredibly useful identities. Working coordinate-free in a rich enough language of geometric relationships makes it (a) a lot easier to solve gnarly geometric problems, and (b) a whole heck of a lot easier to demonstrate and explain those solutions. The change of perspective has been most impactful in fields where the details can’t be hand-waved away because they are getting interpreted by a computer, so streamlining them helps a lot. In particular, computer graphics, computer vision, robotics, physical simulation, and the like.

JadeNB 4y ago
I think it can be very good for someone who is steeped in other perspectives to see Hestenes's take on things and so broaden their existing understanding, but I disagree strongly that Hestenes should be one's first read on the subject. His ideology might lead one to conclude, wrongly, that geometric algebra is the one true answer and that one need know no other; and this is as bad as not knowing about the geometric-algebra formalism at all. (Or, if you wish not to grant that it is as bad, at least it is also bad.)
- jacobolus 4y ago
  My own personal experience was that I read about geometric algebra and thought it was neat but didn’t really become anything like fluent in it for years (frankly I still don’t feel entirely fluent). But I like to work on geometry problems of various sorts (related to computer graphics, cartography, computational geometry, ...). Over and over again, spread out over maybe a decade, I would try solving some problem in the languages I learned in school – Gibbs-style vectors, matrices, differential forms, complex numbers, trigonometric functions, synthetic geometry, etc. – and fill pages of scratch paper with equations that balloon in size making it hard to spot patterns or avoid mistakes, and eventually I would get entirely stuck somewhere. Then I would slap my head, try rewriting the problem in GA terms, and end up replacing like 2 pages of completely incoherent scratch work with about 3–5 lines of concise and easily geometrically interpretable GA manipulations, yielding both a clear answer to my problem and a clear and intuitive demonstration of why it should be right. In particular, dividing by vectors is an unbelievably underrated idea. I don’t know about “one true language” as some kind of crusade, and I am not a physicist or mathematician, but in my opinion every engineer, scientist, and mathematician would benefit tremendously from becoming substantially fluent with GA,† ideally starting with the basic ideas in high school. It is a very clear and expressive language, substantially better for many purposes than the tools currently taught to students in their technical coursework. † Including you, if you ever solve geometric problems. Give it a serious try sometime.

JadeNB 4y ago

I think it can be very good for someone who is steeped in other perspectives to see Hestenes's take on things and so broaden their existing understanding, but I disagree strongly that Hestenes should be one's first read on the subject. His ideology might lead one to conclude, wrongly, that geometric algebra is the one true answer and that one need know no other; and this is as bad as not knowing about the geometric-algebra formalism at all. (Or, if you wish not to grant that it is as bad, at least it is also bad.)

jacobolus 4y ago
My own personal experience was that I read about geometric algebra and thought it was neat but didn’t really become anything like fluent in it for years (frankly I still don’t feel entirely fluent). But I like to work on geometry problems of various sorts (related to computer graphics, cartography, computational geometry, ...). Over and over again, spread out over maybe a decade, I would try solving some problem in the languages I learned in school – Gibbs-style vectors, matrices, differential forms, complex numbers, trigonometric functions, synthetic geometry, etc. – and fill pages of scratch paper with equations that balloon in size making it hard to spot patterns or avoid mistakes, and eventually I would get entirely stuck somewhere. Then I would slap my head, try rewriting the problem in GA terms, and end up replacing like 2 pages of completely incoherent scratch work with about 3–5 lines of concise and easily geometrically interpretable GA manipulations, yielding both a clear answer to my problem and a clear and intuitive demonstration of why it should be right. In particular, dividing by vectors is an unbelievably underrated idea. I don’t know about “one true language” as some kind of crusade, and I am not a physicist or mathematician, but in my opinion every engineer, scientist, and mathematician would benefit tremendously from becoming substantially fluent with GA,† ideally starting with the basic ideas in high school. It is a very clear and expressive language, substantially better for many purposes than the tools currently taught to students in their technical coursework. † Including you, if you ever solve geometric problems. Give it a serious try sometime.