81–90 of 198 posts
Earlier quoted context omitted.
> It seems that this is written out of a perspective of some missing knowledge. Well, the author talks about what "we learned in school ", not university , so that checks out but only because you two have different audiences in mind.
In the US, it’s common to refer to university as school.
As a programmer it seems to me that the number one problem of math notation is that it's weakly typed. There's abuse and reuse of notation everywhere, which makes learning it needlessly difficult. I want a strongly typed fork of math notation. 90% of existing math notation would just be laughed at if it had to go through code review.
The situation that leads to weak typing in computer algorithms -- when you get data from a file or another process and don't know in advance what type it's going to be -- is basically non-existent in blackboard mathematics. Rigorous mathematical papers always tell you what set a variable belongs to when it is introduced, as well as the domain and co-domain of any functions that are defined. This is the blackboard equivalent of strong typing.
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…
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.
As a programmer it seems to me that the number one problem of math notation is that it's weakly typed. There's abuse and reuse of notation everywhere, which makes learning it needlessly difficult. I want a strongly typed fork of math notation. 90% of existing math notation would just be laughed at if it had to go through code review.
Ironically, this post is an abuse of the concept of "weak typing". There's nothing "weakly typed" about, say, the plus sign being used to add both numbers and sets, or a dot being used for both multiplication of numbers and the dot product of vectors. It just means those symbols dispatch based on the types of their arguments, which is perfectly consistent with strong typing (cf. the Julia language). The situation tha…
Take for example the law of total expectation, usually written as E( E(X|Y) ) = E(X). It's totally non obvious (so much that it's harmful IMO) that the outer E is a function of Y. Hiding the summation parameter of E does nothing but hurt math learners here.
As a programmer it seems to me that the number one problem of math notation is that it's weakly typed. There's abuse and reuse of notation everywhere, which makes learning it needlessly difficult. I want a strongly typed fork of math notation. 90% of existing math notation would just be laughed at if it had to go through code review.
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…
> 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 t…
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.
While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge. The basic object that the author seems to be interested in is that of an "algebra over a field" ( https://en.wikipedia.org/wiki/Algebra_over_a_field ). Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied f…
No, this is wrong. Geometric algebras aren't division algebras in general: they usually have zero divisors. Objects that live in a single grade are invertible, but composite objects don't always have multiplicative inverses. As a concrete example, consider the elements 1 + x and 1 - x. Their product is 1 + x - x - xx = 1 + x - x - 1 = 0. So certainly 1 + x doesn't have an inverse, either.
Geometric algebra (Clifford Algebra) unfortunately came late historically. It's a shame, because the whole theory is a very useful (eg for engineering / applied math) superset of linear algebra. I really wish I had learned this first in my undergrad years, would have made a whole bunch of things way clearer from the get go: differential forms tensor calculus linear algebra etc From zero to geo is a very good video in…
Its initial discovery was not too late, but both Clifford and Maxwell died too young in 1879 and after their too early death there was no one left who could finalize the applications of this theory to physics.
In their absence, the geometric algebra theory was ignored and both the theory of vectors and the theory of electromagnetic field were simplified to forms which are good enough for restricted contexts, but which are nonetheless inconsistent and fail in more general cases (the so-called Maxwell differential equations are valid only in much more restricted conditions than the original integral equations of Maxwell).
As a child I have also learned the theory of vectors in the incorrect way, e.g. including the so-called "vectorial product", so I lost time later until understanding that it is not a vector, and then I lost more time until understanding that the so-called pseudovectors a.k.a. axial vectors and the so-called pseudoscalars are not independent entities that come from nowhere, but their existence is just the natural consequence of the properties of the vectors.
It would have been much more efficient if the complete theory of geometric algebra would have been taught from the beginning.
English/American style of explanation fascinates me. First, they show some algebra formulas and mention dot product and cross product. But then they start introducing a definition of a vector! With images! Why, oh why do you need to waste yours and reader's time to introduce basic definitions, if any reader of the article definitely knows that? If they haven't, they wouldn't be able to read the first paragraph at all…
Do you have an example of "Russian style explanation"?
Ah, another geometric algebra evangelist? I can't figure out if GA actually adds anything substantial, or if it merely lets us write some equations in a more succinct fashion. But it certainly looks cool. As to vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse, but if you define new operations on them, well then that operation can hav…
It's just some of the more advanced theory you'd get from studying modules repackaged a bit. Basically the extra stuff that is usually skipped in first year linear algebra courses are the symmetric and asymmetric (often called exterior) products. These form algebras, of course. The exterior product, or wedge product, has a natural interpretation in terms of signed areas (or volumes) and from this you get the determin…