Many people may not realize that while Grassmann.jl by default uses the original Grassmann interior products, it also allows you to use the incorrect definitions used by many people and papers, such as Hestenes, Dorst, Doran, etc.
By default the interior products are < and >, yet I still defined the incorrect interior products too: << and >>.
So it’s not like you can’t use Grassmann.jl if you disagree with me on what the definitions are.
If you disagree with my definition of complement, you can easily define mycomplement(x) = x*I or whatever you prefer and define it whatever way you wish.
Grassmann.jl never stopped people from using incorrect definitions found in various papers, although I severely disagree with those.
Grassmann.jl truly is designed with the correct original Grassmann definitions.
Yet it doesn’t prevent users from using the incorrect definitions, if they want to insist on making mistakes.
It has to do with defining a consistent axiomatic system as a foundation for mathematics. When an alternative definition is used that is inconsistent with the definitions of the original Grassmann interior product, then the result is an inconsistent mathematical system with statements and expressions which cannot be evaluated consistently.
Originally, Grassmann had defined how interior products should work, so when later at some time in history there was a different choice of definition (such as the people using the Hestene-Dorst stuff), then those people are introducing inconsistent expressions into the mathematical formalism.
I have studied and analyzed the inconsistencies introduced into mathematics as a consequence of this situation and came to the conclusion that Grassmann’s original way was wise, which is why I always make sure to stay consistent with it.
My work in the foundations of geometric algebra has provided the proper and correct way to do geometric algebra fully consistent with the original Grassmann.
If Grassmann were to review with me these other developments of what I consider incorrect, then Grassmann himself would surely agree with me.
Merely the fact that the definitions are not the same means that they contradict each other in general. Evaluating them has different results, which is why they can be recognized as separate definitions.
Have you tested if ganja.js satisfies the Grassmann laws of Grassmann algebra? I have checked it and found it to be entirely inconsistent with the axioms of Grassmann algebra… this is completely unacceptable for someone like me, as I require rigorous formulas founded in mathematical formalisms such as those developed on the Grassmann algebra.
If you do not take the laws of Grassmann algebra seriously, then I also do not consider you a serious mathematician. That’s all fine, not everyone needs to be a rigorous mathematician, but please do NOT call yourself a mathematician if you don’t care about the details of the mathematical formalism.
The people who use these sloppy incorrect definitions cannot be considered rigorous mathematicians, they are non-math people like physicists or computer scientists, they are not expert mathematicians.
You have said that Hestenes’s definition introduces the inconsistencies into the mathematical formalism. Is there a specific example that can be tested?
Here is an example with ganja.js, I do not know if this example is representative of all authors, this is just one particular example I have discussed before.