Dear All, I cannot find the following reference mentioned in PGA4CS. Can you help me, please 
Thanks.
L. Dorst, “Notational correspondence between Browne’s “Grassmann
algebra” and “PGA: Plane-based geometric algebra”,” 2020, appendical
note on bivector.net.
Dont know this paper, but basically Dorst has made several mistakes and Stephen De Kenninck are spreading around several inconsistent definitions, which are not compatible with differentjal geometry and not compatible with the original Grassmann algebra, and also not compatible with Hodge stuff either. I’ve explained this several times before in other places.
Unfortunately, I’m banned from the discord where I posted most of my proofs of the inconsistencies. I don’t recall if these specific inconsistencies are due to Dorst or Stephen De Kenninck alone, somebody sent a screenshot so I need to rely on recalling it with other people’s screenshots:
The following are inconsistent results from ganja.js, which demonstrate that their definitions are inconsistent, which makes them incompatible with original Grassmann.
The first two results here are supposed to be equal to each other (-e13), while the third result is supposed to be positive (e123), demonstrating there are issues with their definitions and is not compatible with Original Grassmann.
Here is the result from my Grassmann.jl package, which is carefully based on the OG original Grassmann:
julia> using Grassmann; basis"3"
(⟨×××⟩, v, v₁, v₂, v₃, v₁₂, v₁₃, v₂₃, v₁₂₃)
julia> (!v12)∧(!v23)
-1v₁₃
julia> !(v12&v23)
-1v₁₃
On the other hand, I do recall that one of the inconsistencies perpetuated by Dorst specifically is related to his definition of inner product. Dorst uses what I call the “conventional definition”, which is incompatible with the OG (original Grassmann) and the Hodge stuff.
This table is taken from my documentation, I am the first person to fully disambiguate these inner products and figured out why Dorst definitions are wrong as far as being compatible with the Original Grassmann goes. Not aware of anyone else talking about this.
I do have a paper I wrote where I had this table in it, but I never officially published the article yet. Guess I should get around to publishing it.
Since I am not a prestigous univeristy professor and not in academia, people aren’t paying attention to what I have to say about Dorst’s and DeKenninck’s stuff. Maybe it embarrasses them that I found out these issues.
Recently I made a video describing some of the details
@chakravala thank you for providing Grassmann.jl It is the reason, why I actually try to learn a bit of Julia (although I am more a Python guy ). I have some question concerning your posts above. In my own calculations I use “scalar” products between multivectors were I used the formulas from Doran et al. from GA4Ph. As far as I understood at least for vectors and k-vectors they work like the interior product \iota_X \omega for some vector X and some k-form \omega.
So from your talk, the posts above, and the foundational paper draft from 2021 (which is by far above my horizon, I have to admit) I learned that you suggest to use the form \langle \tilde{\eta_r} \omega_s \rangle_{r-s} as a “scalar” product. Is this correct? I have some questions concerning this:
Grassmann.jl?)At the end of the day, I am interested in a consistent framework which is compatible with most of the other mathematical frameworks (like e.g. Cartan differential calculus), but I am mostly interested in PGA stuff which is difficult due to its degenerate metric. So no need for me for the differential geometric stuff Therefore, may I ask, what would be your suggestion for a “scalar” product with the desired (anti-)symmetries for PGA stuff?
Thanks for your help!
Btw: English is not my mother tongue, so if something sounds harsh or impolite, this is due to my missing experience with the fine details and nuances of the English language. No offense intended.
This is essentially where you are still confused and you do not yet even grasp what differential geometric algebra achieves over so called “PGA”.
Differential geometric algebra correctly defines what an element squaring to 0 really means in the context of differential geometry, while the so called “PGA” which uses a degenerate metric tensor actually introduces incorrect minus signs which both theoretically and practically make it a mathematically inconsistent with the foundations of differential geometry. This is because my own definitions in differential geometric algebra allow the construction of elements squaring to zero without defining a degenerate metric tensor, thus avoiding the introduction of problematic alternating minus signs where they should not appear in a differential geometry context. The alternating algebra of Grassmann is needed in differential geometry, but unfortunately so called “PGA” does not have a refined enough perspective on the foundations and logical consistency with differential geometry like my approach has. In fact, their “PGA” is inconsistent with it due to this subtley regarding how elements squaring to zero are introduced in my system vs theirs. My system is more general than theirs and includes their system as a special case, so technically you can still practice their system within my system. The other way around is not true, their “PGA” system is not as general as mine and does not encompass the other ways of doing things which I consider more correct and consistent with differential geometry.
Therefore, no matter what the argument, my system can encompass both my way and their way… while their so called “PGA” way is not as general and does not encompass all the concepts of differential geometric algebra I defined in my writings.
It can vanish if the grade of the element on the right side is greater than that of the left side (hence you can also use the symbol > for this product in Grassmann.jl). Likewise, the opposite is true of the left contraction, which you can also ger with < to denote that it can be zero if the element on the left has greater grade than the right element).
When the grades of the elements are equal, then it does not vanish in such an asymmetric way, which is consistent with traditional vectors you are familiar with.
It is theoretically possible to have a more symmetric product, which for example Dorst has discussed in the context of his definitions, and I believe many others including myself have thought about it as well. People have an additional name for this concept, it’s called scalar product, I believe. However, the notion of having these left and right contractions is not at all new and has been in the differential geometry literature for a long time and can also be arrived at due to the definitions I discuss in differential geometric algebra.
That is correct, I suggest utilizing the original Grassmann complement, which was thought of in the context of an identity metric tensor. If one defines the original Grassmann complement to be the Hodge complement evaluated with an identity metric tensor, then this will give a complement which does not become zero if there is an otherwise degenerate metric tensor in the algebra definitions.
What I mean by “original Grassmann complement” or OG complement for short, refers to the fact that Grassmann was not really working with a generalized metric tensor, so naming the definition in this way is consistent with the historical development of that. In a more general way, what I mean by original Grassmann is that many of these newer Hestenes based strains of geometric algebra are lacking some of the original Grassmann point of view. This is not an insult, but a fact even stated by Hestenes himself, who acknowledges that he did not access to original Grassmann translations until later on in his research. Therefore, it is historically correct to understand that when observing the Hestenes strain (including for example Doran, Lasenby, etc), their work was built up perhaps somewhat negligent of the original Grassmann principles of thought. It is difficult for me to say or pinpoint what the source of the Hestenes-et-al strain of thought is and related people. All I can say is that the 2014 book on Grassmann from John Browne really helped me with making sure that my original Grassmann foundations are more rigorous, although I had to look at countless other sources as well.
Finally, I would like to mention that I put so called “PGA” into quotes here, because the general term PGA need not refer to the specific approach of Dorst, DeKenninck, etc, it is possible to use the term to not refer to their specific system… but to refer to the general concepts of applying projective geometry to geometric algebra. Apparrently, they have tried to brand the term PGA in such a way that it refers to their personal approach. Similarly, I brand my style “differential geometric algebra” to distinguish myself and to emphasize my intention of logical consistency with differential geometry. As far as I am aware, the terms “differential geometric algebra” have not been used much before in that exact order, so I feel it is open and relevant to apply this term of words to my system.
Thanks for posting this and the other videos.
@chakravala thank you for your immediate and detailed response! I found the definition of Original Grassmann complement and some of the other information I requested within your foundational paper, so my bad indeed This brought further clarification in addition to your answer for me. The hint about the book from John Browne was also very valuable. I already read the first chapter, which he provides on his website. For me this chapter alone brought very useful insights in how to think about the regressive product and the introduction of a metric and inner products from it. So thank you very much for this! Now I’m having a look into the online version of his book. I think I need time to digest this stuff and I think I will come back to you with some further questions.
Best wishes
Johannes
A few years ago I talked to Chris Doran on zoom about this stuff. It turns out he’s not a mathematician, he’s a physicist. He’s not willing to be mathematically rigorous like I am. Of course, it’s not easy to convince someone who has written a book on a topic that the mathematical definitions he used are logically inconsistent with actual Grassmann algebra and differential geometry. It works for him because he’s just doing physics and not trying to find a logically consistent mathematical system.
This is just an example, I have had discussions like this with several other people as well, talked with David Hestenes on the phone too. I’m not interested in bothering David Hestenes with these nuances, as there is no point in it, as he’s retired and he’s earned the right to not be bothered.
However, to the younger people and the next generation, those are the target audience who are worth impacting.
People like Stephen DeKenninck, Leo Dorst, Chris Doran, etc, they all seem like a lost cause to me. They have their “prestigious” academic bureaucrat positions, and are therefore incentivized to deny that there is any issue or logical inconsistency with their track record of papers/books in regards to Grassmann laws and the foundations of differential geometry.
They cannot be convinced because it is against their financial and academic bureaucrat interests to acknowledge mathematical inconsistencies I point out.
I used to think that it is important to make sure that people get exposed to the correct and logically consistent mathematics, which is why I tried occasionally to get these people to understand the inconsistencies in their definitions. This is futile, as a grown man, I can’t go running after other grown men and tell them their math is logically inconsistent.
All I can really do is work on my own things and make sure I am actually mathematically rigorous. These people are not interested in mathematical rigor, so I don’t take them very seriously as mathematicians. There is stuff you can learn from them, but you always have to beware that whatever they wrote is based on inconsistent logic, so the form of their equations has to always be scrutinized and picked apart.
To me, it’s a bit cringe when I think about how these people go out of their way to be popular and try to get so much attention, but they are spreading inconsistent mathematics to these people … it is what it is, meanwhile I quietly build correct and rigorous mathematics software. I tried telling people, I gave up on telling people, and now I just do my own thing and chuckle to myself when I find more and more people spreading the logically inconsistent version of this math.
There will be some people like @joha2 who are willing to look into what I say, and those few people will reap the rewards of it, while the academic bureaucrats bury themselves in an ever deeper pile of inconsistent logic because they are too “prestigious” to listen.
chakravala: What is mathematical rigor?
ChatGPT:
Mathematical rigor refers to the level of precision, clarity, and logical validity that is applied to the development and presentation of mathematical concepts, theorems, and proofs. It is a fundamental aspect of mathematics that ensures that mathematical arguments are sound, conclusions are reliable, and mathematical ideas are communicated effectively.
In a mathematically rigorous argument or proof:
Mathematical rigor is essential for several reasons:
Overall, mathematical rigor is the backbone of the mathematical discipline, allowing for the development of new theories, applications, and insights while maintaining the highest level of precision and validity.
chakravala: What if a non-academic researcher is more mathematically rigorous than some prestigious academic professors on a certain topic?
ChatGPT:
The rigor of mathematical work is not solely determined by one’s academic status or affiliation. It is possible for a non-academic researcher to produce more mathematically rigorous work than some prestigious academic professors on a certain topic. Here are a few reasons why:
Dedication and Passion: Non-academic researchers might be deeply passionate about their topic and devote significant time and effort to ensure the rigor of their work, as they may not have the same institutional pressures or distractions that some academic professors might face.
Independent Perspective: Non-academic researchers often have more flexibility to explore unconventional or less mainstream approaches. This freedom can lead to fresh insights and more thorough explorations of a topic.
Niche Expertise: Some non-academic researchers may specialize in very specific niches or topics that aren’t extensively covered in mainstream academic circles. As a result, they might become experts in those areas and produce work that’s more rigorous due to their in-depth understanding.
Lack of Publish-or-Perish Pressure: Academic professors often face pressure to publish a certain quantity of papers to maintain their reputation and secure funding. This can sometimes lead to less rigorous work being published for the sake of quantity. Non-academic researchers might not be bound by this pressure and can focus more on quality.
Thorough Review Process: Non-academic researchers might spend more time on self-review or seeking feedback from peers since they’re not constrained by the rapid publication cycles of academic journals.
Freedom from Traditional Constraints: Academic institutions might have certain expectations or paradigms that researchers feel compelled to adhere to. Non-academic researchers might not feel bound by these norms, allowing them to pursue rigor in alternative ways.
Resource Constraints: Non-academic researchers might not have access to the same resources and support as academic professors. To compensate, they might need to be more thorough in their approach to ensure the quality of their work.
It’s important to note that rigor isn’t an inherent trait of an individual; rather, it’s a characteristic of the work itself. Rigorous work is characterized by clear definitions, logical arguments, meticulous proof-writing, and adherence to established mathematical principles.
While some non-academic researchers might excel in rigor, it’s also important to recognize that academia has its own rigorous standards and processes for peer review, collaboration, and dissemination of knowledge. Academic professors have access to institutional resources, peer networks, and ongoing discussions that can contribute to the overall advancement of knowledge in their field.
In the end, the quality and rigor of mathematical work should be evaluated based on the work itself, its adherence to established mathematical principles, and its contributions to the field, rather than the academic status of the individual producing it.
chakravala: If a non-academic researcher is more mathematically rigorous than some prestigious academic professors on a certain topic, it is difficult to get the field to advance due to the academic bureaucrat being incentivized to avoid confrontation / acknowledgement of an inconsistency.
ChatGPT:
You raise a valid concern. In academia, there can be various factors that lead to resistance in acknowledging or addressing inconsistencies, even when presented by non-academic researchers who may have more mathematically rigorous insights. Some of these factors include:
It’s important to approach this issue with an understanding of the complexities inherent in academia. Change, even in the face of rigorous evidence, can be gradual. However, the pursuit of truth and rigor remains a fundamental aspect of scientific and mathematical progress.