6 ms·

The article begins: >In this post we will re-invent a form of math that is far superior to the one you learned in school. The ideas herein are nothing short of

The article begins: >In this post we will re-invent a form of math that is far superior to the one you learned in school. The ideas herein are nothing short of revolutionary. and concludes: > I firmly believe that in 100 years, Geometric Algebra will be the dominant way of introducing students to mathematical physics. In the same way that Newton's notation for Calculus is no longer the dominant one, or that Maxwell's actual equations for Electromagnetism have been replaced by Heaviside's, textbooks will change because a better system has come along. These claims are wrong. There are three standard notation methods in physics: vectors, tensors, and differential forms. Geometric algebra is, as the article points out, a more powerful version of the usual vector notation. But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free). [I'm oversimplifying a bit, but a full discussion is too long for a comment here.] Anyway, geometric algebra is not some esoteric secret. People know about it and have decided not to teach it, because the stuff that's already taught is better. [I picked up this specific phrasing from another user here, knzhou, which I think is a particularly good way of explaining it.]

topaz0 5y ago
This may be true in some places, but my undergraduate physics education spent a lot of time on standard Gibbs-style vector calculus. Taylor Classical Mechanics and Griffiths Electrodynamics especially depend on them. Maybe there is a case to be made that first years should start with differential forms, but until that happens I think geometric algebra could be a big improvement.
csdvrx 5y ago
> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation. But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free). [I'm oversimplifying a bit, but a full discussion is too long for a comment here.] I have no opinion about the claims, but I loved the article, as I quickly saw the gains from this algebra for my very basic needs. After checking out your 2 suggested alternatives, I'm not so convinced they are easier to understand.
forgotpwd16 5y ago
Maybe those are better for calculations but are they better for learning?
- alisonkisk 5y ago
  Most people need calculations to help them learn...
LotusFunctor 5y ago
>Geometric algebra is, as the article points out, a more powerful version of the usual vector notation That's not just a gross oversimplification, this is also flat out wrong if what you meant was that it only has vectors. It has more general objects called multivectors through pretty much the same process you get one, two, etc. forms from the wedge product. In fact, both GA and differential forms build from the exterior algebra, and you can go from the former to the latter through geometric calculus (one key difference e.g. would be the method of reciprocal bases to compute inner products with non-orthonormal bases, rather than explicitly working out a basis and then its dual). So I'm confused about your remark regarding its alleged deficiency vs. differential forms if you pretty much reconstruct it within the GA/GC system (especially regarding working basis-free). With regards to tensor notation in terms of calculations, if you mean all that index gymnastics, well GC still openly provides that way of computing things out from what you're used to. What I like about geometric algebra/geometric calculus is precisely the way in which it's nothing new: it's putting everything people use in one system by clarifying the connections between these seemingly disparate systems. Even lie groups/lie algebras can be constructed rather efficiently in the algebra. Another appealing feature of GA is its ability to make pretty transparent an old theorem from Cartan and Dieudonne that says you can view geometric transformations like rotations, and even translations (in projective geometry) as compositions of reflections. There's other appealing features like this in terms of classifying and relating different geometries together that harken back to the Erlangen program, but my point is even in terms of concrete calculations, it's not quite right to say it's just a "more powerful version of the usual vector notation" as it includes more general objects than vectors, and still includes a lot of very similar ways of doing calculations (almost a kind of "backwards-compatibility?") you're used to with tensor index calculations, just with the added bonus of making the transition to the tensors used from vector calculus seamless, alongside other added relation to other systems made more transparent.
- 6gvONxR4sf7o 5y ago
  > ... by clarifying the connections between these seemingly disparate systems. This is the big reason I like it. I remember learning bits and pieces of linear algebra whose rules seemed so entirely random (vectors vs pseudo-vectors, curl, quaternions, some spin calculations I can't even remember anymore, etc), that turned out to have more unified geometric interpretations once I learned GA. It definitely slowed me down as a student who hated 'just memorize it' pedagogy.
chriswarbo 5y ago
> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation > the stuff that's already taught is better These two statements seem contradictory. > But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free) The author made no claims about tensor notation or differential forms; perhaps those might displace all vector-like notation (GA or otherwise) in 100 years, in which case the author's claim can be weakened to "in 100 years, GA will be the dominant form of vector notation". As for "the stuff that's already taught is better", notice that such stuff includes: - Complex numbers - Pseudovectors - Cross-products - Matrix algebra - Dirac notation You seem to agree that GA is "more powerful" than the Gibbs-style vector algebra normally taught. The article is arguing that GA is also a simpler and more consistent approach, which I agree with (complex numbers are certainly simpler on their own, but are a little redundant if we're using GA for the rest). From my own experience in formal education (UK high school and undergraduate physics), I never encountered tensors or differential forms. I've since learned a little about tensors (for general relativity), but that's been due to my own curiosity; I've learned a little about GA for the same reason. I never used quaternions (hence why I left them out of the above list), although I'm aware of them and that they're used e.g. in computer graphics. I used vectors, pseudovectors, and matrices a lot, and I'm certain those topics would have been easier to learn and comprehend if they'd used GA instead.
- Cybiote 5y ago
  spekcular is correct. One could draw the following graph very roughly capturing how these concepts developed: Grassman’s exterior algebra -> Differential forms Cartan’s Exterior Derivatives -> Differential forms Quaternions -> Clifford Algebra exterior algebra -> Clifford Algebra -> geometric algebra Differential forms are a particular kind of tensor and tensors can be defined in terms of multilinear maps. As spekcular says, the standard curriculum covers differential forms, tensors and vectors. This entails becoming familiar with multivectors, the wedge product and multilinear algebra, making geometric algebra a relatively small delta to pick up. On the other hand, the standard course will also prepare you for mathematical topics like lie derivatives, differential geometry and de Rham cohomology. Other than physics, the standard approach equips you with the mathematical machinery underlying many topics in machine learning and statistics like Hamiltonian monte carlo, automatic differentiation, information geometry and geometric deep learning. The central advantage of geometric algebra over the standard approach isn't that it's better or more general, it's that pedagogical material for it is generally leagues and magnitudes better than those for the standard course.
captainmuon 5y ago
I wish I would have been introduced to Geometric Algebra or calculus of forms or whatever it is called during my physics studies. We learned all the conventional things you need for classical mechanics and electromagnetism, like div and curl and BAC-CAB. But there were a couple of things that we were not tought well, which caused problems later. One thing is that at first, a vector was just an N-tuple. But in physics, something is only a vector or a tensor if it behaves under a very specific way during transformations. (The infamous "a tensor is an object that transforms like a tensor"...) The other thing is that I hit a wall when trying to read theory papers, because nobody ever explained what a wedge operator or a two-form and so on is. I was able to mechanically follow calculations, but never got a real good intuition. We learned math together with the mathematicians in a very axiomatic way. Without a geometric intuition, I had a hard time understanding why you need dual vectors like one-forms or covariant vectors. They seem just like a convoluted way to write the scalar product.
- deleted 5y ago
  [deleted]
- spekcular 5y ago
  If you want an "intuitive" approach, you might check out this new book: https://www.amazon.com/Visual-Differential-Geometry-Forms-Mathematical/dp/0691203709/ https://www.amazon.com/Visual-Differential-Geometry-Forms-Ma.... Disclaimer: I haven't read it (but I have heard good things).
  - jacobolus 5y ago
    I have read (part of) it, and it is excellent. I highly recommend it (alongside Needham’s VCA book). But the lessons learned from there are largely orthogonal to the lessons you will learn from working with GA for a while. To the grandparent, I would also recommend the very pretty book http://matrixeditions.com/5thUnifiedApproach.html http://matrixeditions.com/5thUnifiedApproach.html
matt-noonan 5y ago
> it is deficient in various ways when compared to [...] differential forms (e.g. if you want to work basis-free) There is nothing basis-dependent in Geometric Algebra. This presentation started from a basis, but then again so do many presentations of differential forms, leading to 2-forms like dx \wedge dy and so on. The actual difference is that Geometric Algebra requires a choice of inner product (actually, you can get away with any bilinear form), while differential forms do not. However, some of the important operations on differential forms in physics do require an inner product (e.g. the hodge star operator and the codifferential), so you end up back on equal footing with GA again.
- spekcular 5y ago
  I worded that in an unclear way. My main beef is that GA is just way clunkier than differential forms, which are clearly the "right" approach if you want to approach the subject from a theoretical perspective. I see no advantage over the usual treatment, and many disadvantages.
  - ogogmad 5y ago
    I'm a skeptic too, but I might not be the intended user for the GA formalism. Please explain your reasons. My skepticism of the supposedly superior pedagogy of Geometric Algebra is the following: - 3D vector algebra with the cross product operation and the dot product operation is fairly easy and intuitive. Its replacement by GA might not be so easy. So maybe GA should be introduced after the vector formalism. - An arbitrary element of a Geometric Algebra might not have a geometric meaning. For instance, some elements of a GA are vectors, while some are scalars, but there are also these exotic mixed quantities which are scalars plus vectors. This is pretty hard for me to understand intuitively. - An arbitrary matrix has a geometric meaning. It's essentially just a linear transformation. By contrast, I don't feel that an arbitrary element of a geometric algebra has a geometric meaning. - Consider those elements of a Geometric Algebra which represent rotations -- they are called rotors. Observe that if "z" is a rotor then the element "-z" is also a rotor which stands for the same rotation as "z". So there is more to a rotor than whatever rotation it describes. This seems very unintuitive and advanced. (I know that this behaviour has applications for the study of spin-1/2 particles in quantum physics). I also have trouble understanding where the rule for multiplying two elements of a geometric algebra comes from. It's an operation, introduced from seemingly nowhere, which happens to have some applications in some areas. But I'm not comfortable with a multiplication rule being introduced out of nowhere without being derived out of something. The claim that it has a consistent geometric meaning from which it can be derived is never justified. My criticisms are therefore largely pedagogical.
sebastos 5y ago
If deficiency to tensors and differential forms is a reason not to be taught, then why do we learn about vector notation? Oh right, because it's a natural language for talking about geometry and mechanics, by far the most common and important type of reasoning that the average student will need to do. And geometric algebra is demonstrably superior for that domain. So your comment is pointless.
jacobolus 5y ago
> People know about it and have decided not to teach it Generally teachers don’t (can’t) individually decide this. Decisions about what to teach have incredible historical inertia, and are largely decided based on what the teacher learned when they themself went to school decades ago, what everyone else is teaching, what materials are easily available, what notations are used in past literature, etc. Substantial transitions in the teaching of existing material take generations. In 2020 our basic math/science curriculum and pedagogy in high schools and universities has all been pretty well statically fixed for 50+ years (many parts are unchanged in 200+ years), except in computer science where some of the basic ideas are newer than that, and in graduate-level courses that get closer to the cutting edge. * * * The place where geometric algebra has seen most rapid adoption is in computer programming, where code actually has to work, and a more effective formalism makes correct code easier to write and reason about, saving a ton of time and effort even for basic examples. Even in physics, where a better formalism leads to improved physical intuition and deeper conceptual understanding, a transition is an uphill struggle, because symbolic fluency with geometric algebra takes years of practice. (Some) pure mathematicians on their high horses scoff at anything that doesn’t advance their own obscure abstract research, which is unconcerned with conceptual obstacles faced by undergraduate students, scientists, or engineers. They can hand-wave a better formalism away with “this is isomorphic to X and Y other structures, so there’s no value in it”.
- spekcular 5y ago
  > Substantial transitions in the teaching of existing material take generations. > In 2020 our basic math/science curriculum and pedagogy in high schools and universities has all been pretty well statically fixed for 50+ years (many parts are unchanged in 200+ years), except in computer science where some of the basic ideas are newer than that, and in graduate-level courses that get closer to the cutting edge. This is wildly incorrect. Even in the past ~20 years we've seen a sea change in our understanding of science pedagogy. Look up the work of Carl Wieman on active learning, or https://www.pnas.org/content/111/23/8410 https://www.pnas.org/content/111/23/8410. Inclusive classroom practices are another thing that's come into fashion in the last ~10 years. The curriculum has also evolved; the most obvious thing to point to is the new emphasis on connections to data science in math/stats courses. If you're someone who doesn't stay up-to-date on pedagogy, then yes, it takes your retirement to bring about a change. But a lot of people, especially those teaching at small liberal acts colleges, have continually evolving teaching practices. There are entire conferences where people get together to talk about college teaching. > Even in physics, where a better formalism leads to improved physical intuition and deeper conceptual understanding, a transition is an uphill struggle, because symbolic fluency with geometric algebra takes years of practice. Do you really believe this? To anyone to recognizes that it's the standard stuff in (a clunkier) disguise, it shouldn't take years. Appealing to these two frictions does not offer a convincing theory of why GA has not been adopted despite being around for, what, 50+ years? The fact that it's worse than existing notation does.

topaz0 5y ago

This may be true in some places, but my undergraduate physics education spent a lot of time on standard Gibbs-style vector calculus. Taylor Classical Mechanics and Griffiths Electrodynamics especially depend on them. Maybe there is a case to be made that first years should start with differential forms, but until that happens I think geometric algebra could be a big improvement.

csdvrx 5y ago

> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation. But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free). [I'm oversimplifying a bit, but a full discussion is too long for a comment here.] I have no opinion about the claims, but I loved the article, as I quickly saw the gains from this algebra for my very basic needs. After checking out your 2 suggested alternatives, I'm not so convinced they are easier to understand.

forgotpwd16 5y ago

Maybe those are better for calculations but are they better for learning?

alisonkisk 5y ago
Most people need calculations to help them learn...

LotusFunctor 5y ago

>Geometric algebra is, as the article points out, a more powerful version of the usual vector notation That's not just a gross oversimplification, this is also flat out wrong if what you meant was that it only has vectors. It has more general objects called multivectors through pretty much the same process you get one, two, etc. forms from the wedge product. In fact, both GA and differential forms build from the exterior algebra, and you can go from the former to the latter through geometric calculus (one key difference e.g. would be the method of reciprocal bases to compute inner products with non-orthonormal bases, rather than explicitly working out a basis and then its dual). So I'm confused about your remark regarding its alleged deficiency vs. differential forms if you pretty much reconstruct it within the GA/GC system (especially regarding working basis-free). With regards to tensor notation in terms of calculations, if you mean all that index gymnastics, well GC still openly provides that way of computing things out from what you're used to. What I like about geometric algebra/geometric calculus is precisely the way in which it's nothing new: it's putting everything people use in one system by clarifying the connections between these seemingly disparate systems. Even lie groups/lie algebras can be constructed rather efficiently in the algebra. Another appealing feature of GA is its ability to make pretty transparent an old theorem from Cartan and Dieudonne that says you can view geometric transformations like rotations, and even translations (in projective geometry) as compositions of reflections. There's other appealing features like this in terms of classifying and relating different geometries together that harken back to the Erlangen program, but my point is even in terms of concrete calculations, it's not quite right to say it's just a "more powerful version of the usual vector notation" as it includes more general objects than vectors, and still includes a lot of very similar ways of doing calculations (almost a kind of "backwards-compatibility?") you're used to with tensor index calculations, just with the added bonus of making the transition to the tensors used from vector calculus seamless, alongside other added relation to other systems made more transparent.

6gvONxR4sf7o 5y ago
> ... by clarifying the connections between these seemingly disparate systems. This is the big reason I like it. I remember learning bits and pieces of linear algebra whose rules seemed so entirely random (vectors vs pseudo-vectors, curl, quaternions, some spin calculations I can't even remember anymore, etc), that turned out to have more unified geometric interpretations once I learned GA. It definitely slowed me down as a student who hated 'just memorize it' pedagogy.

chriswarbo 5y ago

> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation > the stuff that's already taught is better These two statements seem contradictory. > But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free) The author made no claims about tensor notation or differential forms; perhaps those might displace all vector-like notation (GA or otherwise) in 100 years, in which case the author's claim can be weakened to "in 100 years, GA will be the dominant form of vector notation". As for "the stuff that's already taught is better", notice that such stuff includes: - Complex numbers - Pseudovectors - Cross-products - Matrix algebra - Dirac notation You seem to agree that GA is "more powerful" than the Gibbs-style vector algebra normally taught. The article is arguing that GA is also a simpler and more consistent approach, which I agree with (complex numbers are certainly simpler on their own, but are a little redundant if we're using GA for the rest). From my own experience in formal education (UK high school and undergraduate physics), I never encountered tensors or differential forms. I've since learned a little about tensors (for general relativity), but that's been due to my own curiosity; I've learned a little about GA for the same reason. I never used quaternions (hence why I left them out of the above list), although I'm aware of them and that they're used e.g. in computer graphics. I used vectors, pseudovectors, and matrices a lot, and I'm certain those topics would have been easier to learn and comprehend if they'd used GA instead.

Cybiote 5y ago
spekcular is correct. One could draw the following graph very roughly capturing how these concepts developed: Grassman’s exterior algebra -> Differential forms Cartan’s Exterior Derivatives -> Differential forms Quaternions -> Clifford Algebra exterior algebra -> Clifford Algebra -> geometric algebra Differential forms are a particular kind of tensor and tensors can be defined in terms of multilinear maps. As spekcular says, the standard curriculum covers differential forms, tensors and vectors. This entails becoming familiar with multivectors, the wedge product and multilinear algebra, making geometric algebra a relatively small delta to pick up. On the other hand, the standard course will also prepare you for mathematical topics like lie derivatives, differential geometry and de Rham cohomology. Other than physics, the standard approach equips you with the mathematical machinery underlying many topics in machine learning and statistics like Hamiltonian monte carlo, automatic differentiation, information geometry and geometric deep learning. The central advantage of geometric algebra over the standard approach isn't that it's better or more general, it's that pedagogical material for it is generally leagues and magnitudes better than those for the standard course.

captainmuon 5y ago

I wish I would have been introduced to Geometric Algebra or calculus of forms or whatever it is called during my physics studies. We learned all the conventional things you need for classical mechanics and electromagnetism, like div and curl and BAC-CAB. But there were a couple of things that we were not tought well, which caused problems later. One thing is that at first, a vector was just an N-tuple. But in physics, something is only a vector or a tensor if it behaves under a very specific way during transformations. (The infamous "a tensor is an object that transforms like a tensor"...) The other thing is that I hit a wall when trying to read theory papers, because nobody ever explained what a wedge operator or a two-form and so on is. I was able to mechanically follow calculations, but never got a real good intuition. We learned math together with the mathematicians in a very axiomatic way. Without a geometric intuition, I had a hard time understanding why you need dual vectors like one-forms or covariant vectors. They seem just like a convoluted way to write the scalar product.

deleted 5y ago
[deleted]
spekcular 5y ago
If you want an "intuitive" approach, you might check out this new book: https://www.amazon.com/Visual-Differential-Geometry-Forms-Mathematical/dp/0691203709/ https://www.amazon.com/Visual-Differential-Geometry-Forms-Ma.... Disclaimer: I haven't read it (but I have heard good things).
- jacobolus 5y ago
  I have read (part of) it, and it is excellent. I highly recommend it (alongside Needham’s VCA book). But the lessons learned from there are largely orthogonal to the lessons you will learn from working with GA for a while. To the grandparent, I would also recommend the very pretty book http://matrixeditions.com/5thUnifiedApproach.html http://matrixeditions.com/5thUnifiedApproach.html

matt-noonan 5y ago

> it is deficient in various ways when compared to [...] differential forms (e.g. if you want to work basis-free) There is nothing basis-dependent in Geometric Algebra. This presentation started from a basis, but then again so do many presentations of differential forms, leading to 2-forms like dx \wedge dy and so on. The actual difference is that Geometric Algebra requires a choice of inner product (actually, you can get away with any bilinear form), while differential forms do not. However, some of the important operations on differential forms in physics do require an inner product (e.g. the hodge star operator and the codifferential), so you end up back on equal footing with GA again.

spekcular 5y ago
I worded that in an unclear way. My main beef is that GA is just way clunkier than differential forms, which are clearly the "right" approach if you want to approach the subject from a theoretical perspective. I see no advantage over the usual treatment, and many disadvantages.
- ogogmad 5y ago
  I'm a skeptic too, but I might not be the intended user for the GA formalism. Please explain your reasons. My skepticism of the supposedly superior pedagogy of Geometric Algebra is the following: - 3D vector algebra with the cross product operation and the dot product operation is fairly easy and intuitive. Its replacement by GA might not be so easy. So maybe GA should be introduced after the vector formalism. - An arbitrary element of a Geometric Algebra might not have a geometric meaning. For instance, some elements of a GA are vectors, while some are scalars, but there are also these exotic mixed quantities which are scalars plus vectors. This is pretty hard for me to understand intuitively. - An arbitrary matrix has a geometric meaning. It's essentially just a linear transformation. By contrast, I don't feel that an arbitrary element of a geometric algebra has a geometric meaning. - Consider those elements of a Geometric Algebra which represent rotations -- they are called rotors. Observe that if "z" is a rotor then the element "-z" is also a rotor which stands for the same rotation as "z". So there is more to a rotor than whatever rotation it describes. This seems very unintuitive and advanced. (I know that this behaviour has applications for the study of spin-1/2 particles in quantum physics). I also have trouble understanding where the rule for multiplying two elements of a geometric algebra comes from. It's an operation, introduced from seemingly nowhere, which happens to have some applications in some areas. But I'm not comfortable with a multiplication rule being introduced out of nowhere without being derived out of something. The claim that it has a consistent geometric meaning from which it can be derived is never justified. My criticisms are therefore largely pedagogical.

sebastos 5y ago

If deficiency to tensors and differential forms is a reason not to be taught, then why do we learn about vector notation? Oh right, because it's a natural language for talking about geometry and mechanics, by far the most common and important type of reasoning that the average student will need to do. And geometric algebra is demonstrably superior for that domain. So your comment is pointless.

jacobolus 5y ago

> People know about it and have decided not to teach it Generally teachers don’t (can’t) individually decide this. Decisions about what to teach have incredible historical inertia, and are largely decided based on what the teacher learned when they themself went to school decades ago, what everyone else is teaching, what materials are easily available, what notations are used in past literature, etc. Substantial transitions in the teaching of existing material take generations. In 2020 our basic math/science curriculum and pedagogy in high schools and universities has all been pretty well statically fixed for 50+ years (many parts are unchanged in 200+ years), except in computer science where some of the basic ideas are newer than that, and in graduate-level courses that get closer to the cutting edge. * * * The place where geometric algebra has seen most rapid adoption is in computer programming, where code actually has to work, and a more effective formalism makes correct code easier to write and reason about, saving a ton of time and effort even for basic examples. Even in physics, where a better formalism leads to improved physical intuition and deeper conceptual understanding, a transition is an uphill struggle, because symbolic fluency with geometric algebra takes years of practice. (Some) pure mathematicians on their high horses scoff at anything that doesn’t advance their own obscure abstract research, which is unconcerned with conceptual obstacles faced by undergraduate students, scientists, or engineers. They can hand-wave a better formalism away with “this is isomorphic to X and Y other structures, so there’s no value in it”.

spekcular 5y ago
> Substantial transitions in the teaching of existing material take generations. > In 2020 our basic math/science curriculum and pedagogy in high schools and universities has all been pretty well statically fixed for 50+ years (many parts are unchanged in 200+ years), except in computer science where some of the basic ideas are newer than that, and in graduate-level courses that get closer to the cutting edge. This is wildly incorrect. Even in the past ~20 years we've seen a sea change in our understanding of science pedagogy. Look up the work of Carl Wieman on active learning, or https://www.pnas.org/content/111/23/8410 https://www.pnas.org/content/111/23/8410. Inclusive classroom practices are another thing that's come into fashion in the last ~10 years. The curriculum has also evolved; the most obvious thing to point to is the new emphasis on connections to data science in math/stats courses. If you're someone who doesn't stay up-to-date on pedagogy, then yes, it takes your retirement to bring about a change. But a lot of people, especially those teaching at small liberal acts colleges, have continually evolving teaching practices. There are entire conferences where people get together to talk about college teaching. > Even in physics, where a better formalism leads to improved physical intuition and deeper conceptual understanding, a transition is an uphill struggle, because symbolic fluency with geometric algebra takes years of practice. Do you really believe this? To anyone to recognizes that it's the standard stuff in (a clunkier) disguise, it shouldn't take years. Appealing to these two frictions does not offer a convincing theory of why GA has not been adopted despite being around for, what, 50+ years? The fact that it's worse than existing notation does.