Geometric algebra and the scientific method

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@WhiteEyes here is the problem:

The purpose of religion in human society is entirely different from the purpose of science for us.

First of all, you have to consider what is the purpose of language is in its context. We’re using language to express something to other humans in a context.

Purpose of religious language is to construct a context within other people. What is the purpose of the linguistic context? The purpose of it is to deal with the inquiry of how human individuals and society behave in the greater context of the world.

Religious inquiry constructs a context within which to ask and answer questions about how humans behave in the context of the world and other beings.

Scientific inquiry is an entirely different linguistic context with a different purpose. It’s to communicate and construct a context within which to ask and answer empirical questions about objects in world.

In science, we do not inquire about what is the correct moral behavior for a human being, because these things are dependent on cultural context. In some cultures it is socially acceptable to behave this way or that way, in other cultures inappropriate.

Science is not like this. when science is communicated, it should always be interpreted in the same way by every interpreter. It is independent of cultural context or morality.

You are confusing science and religion. By combining science and religion, you will muddy the waters of science and muddy religion also.

By inserting cultural dependent references and terms into science, you are making science context dependent on being in a particular culture. This means that your science might be inappropriate in another different culture, due to conflicting context on what is socially acceptable in that culture.

Furthermore, neither scientists nor religious people will take you seriously, because your science is mixed up with culturally dependent contexts, and the cultural context of some other religion is now incompatible with your idea of religion.

So you’re not going to get very far with this, among other reasons. There’s a lot of reasons why confusing scientific contexts with cultural religion context is going to lead to getting nowhere.

23

I am not really concerned with what is “socially acceptable” in science and “religion”,
My project is about rephrasing, highlighting and reviving what others has said before me to the best of my ability. I’ve quoted grassmann, and proclus, and referred to many books. I dont feel this discussion is leading anywhere. So we end it here.

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I am also not concerned with what is socially acceptable.

I’m just letting you know that you will get nowhere with this.

You are of course welcome to do something which will get you nowhere, just as I am doing.

25

Blockquote
“…i was wondering if there is any possibility of opening study programs based geometric algebra from the ground up. If not at a full university level maybe a shorter application focused s.t.e.m. stem program, and if there are any ideas on what can be done about it, finance-wise or in any other way.”

It is now 2024 and biVectorDOTnet is not alone in doing something about it by generating resources for Geometric Algebra study programs.

I have not yet seen a Brilliant.org interactive course on Geometric Algebra but STEMCstudio works with Geometric Algebra and David Geo Holmes has YouTube videos
about Geometric Algebra using STEMCstudio.

I found this recently: https://geometricalgebratutorial.com/

It’s interactive like W3schools live examples which can be a bit passive but it’s a pretty good start. I need to go through it more thoroughly but it looks a bit old-school in terms of the linear presentation of reading material. I would like to see an ASK™ hypermedia system and GeoGebra or Brilliant.org-type interactivity.

jimsmithinchiapas has GeoGebra material on Geometric Algebra.

I’m currently looking into using CoCalc. (Jupyter notebook Geometric Algebra Research.ipynb, How to use the galgebra (Geometric Algebra) package at CoCalc.)

There are several people like sudgylacmoe that are creating high quality Geometric Algebra educational short videos on YouTube.

There are many old-school lectures on the biVectorDOTnet site that can be followed using the Gangajs Coffeeshop.

There are several YouTube channels providing educational Geometric Algebra videos of varying length.

  • David Geo Holmes YouTube channel, Geometric Algebra First Course and STEMCstudio

  • Alan Macdonald YouTube channel and Luther College website

  • Pre-University Geometric Algebra Videos - Geometric Algebra for High-School Students and Teachers

Websites with lecture notes etc.
University of Cambridge Geometric Algebra - Introduction to GA, Geometric Algebra 2016 lecture course

The Geometric Calculus R&D website is now back online on the David Hestenes archive.

NOTE: MIT OpenCourseWare has a 1976 Applied Geometric Algebra Course by László Tisza online but it does not use the unified notation developed by David Hestenes and others. David Hestenes tried to appropriate the name “Geometric Algebra” for his unified formalism (that incorporates spinors, tensors and differential forms) to distinguish it from Clifford Algebra. However, for many people Geometric Algebra is just another name for Clifford Algebra and there are even books out now with the title “Geometric Algebra” that have nothing to do with Clifford Algebra so it can be confusing.

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@Paul_Gowan many of those sources are people barely know what they’re talking about, but decided to try and be popular on the internet, so watch out.

Even people like Chris Doran and Leo Dorst barely know what they’re talking about, given their intentional ignorance when confronted with proofs.

Personally, I have gotten 100% confident in my Grassmann-Clifford-Hodge foundation because I very thoroughly implemented it and verified it with proofs. And if I’m wrong, You can confront me and either I will disprove you or fix my mistakes.

Many people making videos or articles on the internet have not spent the necessary 10 years of mathematical preparation to be a proper teacher on nuanced definitions in this topic, which are easily confused or mistaken due to the high amount of incorrect or sloppy papers / books out there.

Currently, the various teaching materials, especially from people trying to be “popular” … are not all correct and contain fundamental errors.

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29

Since I am a bit new to this and you seem to be an expert I have a couple of questions.

First let’s examine the geometric product which seems weird to me.

ab = ab + a^b Now let’s suppose that the units of a are meters per second. Then just exactly what would be the unit of ab, ab and a^b?

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Provided that a and b are vectors where all coefficients have the same unit, such as meters, then both the interior product and also the exterior product would produce those units squared. For example, lengths would become areas. This is consistent with the notion that bivectors are areas and that the absolute value of a vector has the same unit. This isn’t very mysterious to figure out, once you check that multiplying two units results in the square unit.

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You seem to be suggesting that the units of all vectors are meters. But your argument fails even there. Area requires that the two dimensions be orthogonal. If b is broken down into orthogonal and parallels segments then your argument sort of holds for the pieces that are orthogonal (i.e. equivalent to the bivector) but not the pieces that are parallel. I mentioned this to David Hestenes and he agrees that GA does not include units. And I suspect it doesn’t apply to anything in this world. In a similar way, regular algebra doesn’t apply to anything in this world as well. The distributive principle states a(b + c) = ab + ac. There is nothing in this world that has the property that you can multiply them together and also add them together and have both operations make sense.
So can we agree that mathematics does not apply to anything in this world? It is just an imaginary game.

32

You are completely mistaken @macduff

The units are in fact scalar values which can be factored out of the vectors a and b.

For example [1m, 2m, 3m] = [1,2,3]m, where the unit of meters can be facted out as with any scalar.

Let’s take the geometric product [1m,2m,3m]*[3m,2m,1m]

The result is clearly [1,2,3]*[3,2,1]*m^2 =
10m^2 - 4m^2*v_{12} - 8m^2*v_{13} - 4m^2*v_{23}

Which is equivalent to
(10 - 4v_{12} - 8v_{13} - 4v_{23})m^2.

Units are in fact nothing but scalars, they are no different than multiplying any other scalar, and squaring a unit in fact makes a squared unit.

If you have trouble accepting this, then you are obviously not capable of basic scalar multiplication, and perhaps you need to go back and re-learn some implications of basic scalar multiplication.

33

Will you agree that in your 10m^2 the m^2 is composed of an xx, yy and zz. In other words the products of unit vectors. Can you explain to me why you are assuming these to be the same and why you can claim them to be equal to the numeral 1? And also why you are assuming that the xy, xz and yz are somehow different? And just exactly how are they different?
Are you also claiming that the units can’t be things like meters/sec or that these products only works with like units?

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@macduff you clearly have no idea what you are talking about when it comes to physical units

No, m^2 is a scalar, it’s a physical unit, all physical units are scalars quantities proportional to 1.

Yes, in fact area units are equivalent to 1 in Natural units, in Natural physics units, all units are 1 scalars, note the name “unit” and the number “one,” both refer to a unit of one.

The only reason why humans don’t use Natural units is because we prefer space and time to be scaled in such a way that the speed of light is 299792458 m/s.

However, if you actually know anything about physical units, you’d know that in Natural units, the speed of light is the scalar 1. In fact, all units are naturally the number 1, it’s only for the sake of human scale that we change the metric so that meters and seconds get involved as units.

I don’t owe you a basic lesson in physics, so if you want me to lecture you on how physical units work, maybe you should pay me for my time. I can certainly teach this stuff at the deepest most expert level, but i dont owe anyone my time and expertise for free.

People like you expect some kind of free lesson in math and physics, as if I have endless time to explain basic fundamental concepts to you, while you give me nothing in return, except drain my time.

This isn’t a fruitful discussion for me, because you expect me to clear up endless confusion you have about math and physics, while you give nothing to me in return for this, only draining my energy.

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chakravala, I really do appreciate you helping me out here, you have to understand I am just a lowly physicists.
“we prefer space and time to be scaled in such a way that the speed of light is 299792458 m/s.” How in advance did they manage to come up with that number? I was under the belief that they had determined the units first.
However, going back to the question of geometric products. Let’s suppose that a, b, c, d are all vectors with units of meters.
So a = (bc)d now if the units of bc are as you say, meters squared, then shouldn’t “a” now be meters cubed rather than meters.
Where have I gone wrong here? Or is David Hestenes correct on this?

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chakravala, I think you need to know that David Hestenes is the author of geometric algebra. Your lack of understanding of units leads me to believe your background is in mathematics. In the formula a =(bc)d, bc is an operator that converts d into a. Thus the product of bc has no units. In this case a and d have the same units. Currently this is an odd formalism when it comes to units and it creates huge amounts of confusion. What is happening is that bc is composed of four pieces and these pieces act on d in such a way as they use d to create a sequence of right triangles that form a.
The current formalism is a bit awkward but if you like I can show you or anybody else a much more easily understood version of GA that eliminates the confusion.
I might be wrong on this but, chakravala, I think your efforts in attempting to deal with this issue has led you to the idea that “length = volume”.

I believe that problem starts with the standard algebra that is taught in schools. It is nothing more than a game that has little or nothing to do with reality. Consider the distributive principle a(b + c) = ab + ac. There is nothing, absolutely nothing in this world that has the property that you can add them together and also multiply them together and the result make sense. Zero, nada, zilch. The destructiveness to human kind of this mathematics is beyond cruel. GA has a small step in the right direction when it defines n(a + b) = na + nb.

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I talked to David Hestenes on the phone, and asked him why he’s using incorrect definitions for geometric algebra. It’s because he didn’t have access to the original Grassmann algebra translation or something. I’m well aware of who he is and what mistakes are in his literature. He doesn’t know everything about geometric algebra, and is not always the best reference for geometric algebra. I respect his contribution, but he does make mistakes and uses some incorrect ideas.

No, you are misunderstanding me. I am in fact completely correct that length can equal volume, my UnitSystems.jl software and Similitude.jl can demonstrate this calculation:

julia> using Similitude

julia> meter(Metric)
𝟏 = 1.0 [m] Metric

julia> meter(Natural)
R∞⋅α⁻²τ⋅2 = 2.589605074835788e12 [𝟙] Natural

julia> length(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> volume(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> length(Metric) == volume(Metric)
false

julia> length(Natural) == volume(Natural)
true

The value of a Natural unit of length is exactly equal to 1 which is exactly equal to 1^3 which is the Natural unit of volume, 1=1^3 is easy to verify.

In the Metric unit system, length and volume units are not equal to each other, but in the Natural unit system length and volume are in fact equal units.

I do not lack any understanding of physical units, it is YOU who has no clue about physical units. It is a fact that length = volume when expressed in Natural units, and you lack understanding of physics units, so you are unable to understand this.

There is no mistake in the conception that length equals volume in Natural units, this is perhaps not obvious to uninformed people such as yourself, but it is in fact how the mathematics of units works.

When using Natural physics units, length and volume cannot be distinguished, they are equal. In Metric units, length and volume are not equal. This is all fact.

You didn’t explain your notation before and I don’t really care about engaging with you in this discussion anymore. I don’t have any issue with these topics, and I dont care about discussing it further. Good luck with your struggle and confusion about units.

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julia> length(Natural) == volume(Natural) true Here you are just defining this to be true, how would your software know its not really true?

“The value of a Natural unit of length is exactly equal to 1 which is exactly equal to 1^3 which is the Natural unit of volume, 1=1^3 is easy to verify.”

Suppose that your 1 is 1 m/s, prove to me that this is identical to 1m^3/s^3. Now I have absolutely no idea as to what 1 m^3/s^3 is but I would be delight for you to show me that it is nothing more than just 1 m/s. In fact, it is just another way of describing a regular ordinary everyday velocity.

Could you also explain to me as to why you would want to do this? The advantage would be?

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My software calculates physical units, how? Because I am capable of programming it into existence, I do not owe you an explanation of how my software works.


julia> using Similitude

julia> ms = Quantity(speed,Metric,1)
1 [m⋅s⁻¹] Metric

julia> ms*ms*ms
1 [m³s⁻³] Metric

julia> ms == ms*ms*ms
false

julia> ns = Quantity(speed,Natural,1)
1 [𝟙] Natural

julia> ns*ns*ns
1 [𝟙] Natural

julia> ns == ns*ns*ns
true

Using Natural units, speed and speed cubed are equal to each other, but not with Metric units.

Anyway, I don’t owe you an explanation, but I will say that Natural units are in fact valid physics units used by physicists and first pioneered by Max Planck, although he didn’t develop the most general form as I have implemented in software.

Physics unit dimensions are not at all like the dimensions of geometric algebra, the dimensions of physics can in fact be “non-dimensionalized” so that length = volume, this is what physicists call “Natural” units. Natural physics units are quite “alien” to human experience of physics, but they are in fact a valid physical unit formalism.

Why would people do this? Because some people have a deeper understanding of physical units than what you know.

When wikipedia or myself says that Natural units have certain physical constants = 1, that’s not a joke, that’s how physics actually works.

For example, mass = energy, that’s how the atomic bomb works, by equating mass and energy.

julia> mass(Metric) == energy(Metric)
false

julia> mass(Natural) == energy(Natural)
true

David Hestenes and yourself clearly don’t understand how physical units are actually defined, but the nuclear explosions of the past prove my perspective on physical units are correct. Do we really need to set off another nuclear bomb to prove that mass = energy once again? Or can you accept the mathematics?

My UnitSystems.jl software and Similitude.jl can demonstrate these calculations, it’s not a joke, I am a proven expert in this topic. You now owe me $500 for pressuring me into lecturing you on this topic.

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chakravala, Let me pay you back in this way. By introducing you to a Vector Algebra that incorporates vectors along with their units as you would like. And YOU get to tear it apart and demonstrate all its flaws.
Axioms a=1a = a1, n(a + b) = na + nb, 1=bb/bb, if x,y,z are unit vectors then xy=-yx as xy is a means of encoding a rotation. Note vector products only have meaning in the context of ratios where they form complex numbers.

Ok, let a = a1 = a (bb/bb) = (ab/bb)b ab/bb is an operator that rotates and dilates b into a. How does it do this? It is clearly unites so it passes that hurtle. It also has four components, one is a scalar that modifies the length of b so that it forms one leg of a right triangle. The complex part rotates b 90 degrees and scales it so that you create a. Thus it creates a sequence of right triangles that can be constructed visually so it becomes clear to exactly how the math actually works.
Check it out, first in two dimensions then three. Enjoy it, if you can!

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Sorry, I don’t really understand your formalism or how it relates to units, so I am unable to comment on it.

What I can say is that b/b = 1 for any vector b, so it is unnecessary to include the term b/b in your formulas.

Sinplifying, you have 1=bb, and a=a(bb)=(ab)b ab.

However, this second formula is incorrect, since (ab)b = a(bb) = a, then (ab)b ab = aab does not equal a.

Either the 2nd formula is incorrect, or I don’t understand what your notation or axioms are.

I tried understanding it, but it doesn’t make sense to me, perhaps I don’t understand your notation, or how you use parenthesis, or what it means.

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There is this current belief that with vectors b/b=1. So a = (a/b)b. let a = 6x + 7y and b = 3x - 2y. a/b = what? Note: x and y are unit vectors. In standard vector algebra xx=yy=zz=1 if vectors have units as you are insisting then shouldn’t the answer be 1 unit^2. So let’s let xx=yy=zz=1 unit^2. Squaring removes the orientation.

What I am suggesting is that the magnitude of b compared to the magnitude of b is equal to 1. Thus (ab/bb) will have units xx/xx and xy/xx or 1 and xy/xx. The xy/xx in standard literature is often referred to as the complex number i. In other words, this shows complex algebra is nothing but a variation of a vector algebra. The nice thing it gives the imaginary number an actual physical reference. However replacing xy/xx by i is not a good idea.