An Argument Against Geometric Algebra (2024)

Every so often, the digital sphere becomes saturated with discussions regarding Geometric Algebra (GA). The narrative usually presents GA as a groundbreaking mathematical framework capable of correcting every perceived deficiency in multivariable calculus and linear algebra.

Whenever I encounter this rhetoric, my instinctive reaction is:

"Hold on a second—that's not actually true! While GA is certainly onto something, it is riddled with issues. You likely just want the wedge product and the concept of multivectors!"

Unfortunately, this response is rarely effective. Explaining the why takes far too long, and the cycle simply repeats itself without any real progress. While many peers share my sentiment, most choose to simply ignore GA. I, however, actually want the project to succeed because I appreciate its core ambitions.

Therefore, I am writing this comprehensive piece to serve as a permanent reference for my critiques.

The Core Problem: A Flawed Foundation

The fundamental issue is that GA is structurally flawed and requires significant refinement. More concerningly, the community surrounding it seems either oblivious to these flaws or uninterested in fixing them.

The Obsession with the Geometric Product

Specifically, I believe that Hestenes’ Geometric Product is a subpar operation. Despite this, GA is obsessed with it, attempting to rebuild the entirety of geometry around it.

The "religious" fervor with which the geometric product is promoted is problematic. Proponents treat it as a canonical, self-evident truth, suggesting that the rest of the mathematical world is simply "blind" to its brilliance. This approach is:

Mathematically unsound: Treating specific models as universal defaults is a mistake.
Socially alienating: The zealotry pushes potential adopters away.

While the geometric product might have a place in a comprehensive theory of geometry, it should not be the centerpiece. By centering the theory on it, GA becomes less compelling than it could be.

The Path to Success

For GA to actually gain traction, it must:

Address the shortcomings of "establishment" mathematics more effectively.
Develop a framework that is universally acceptable.
Cultivate self-awareness regarding its current limitations.

Currently, GA's relationship with mainstream mathematics is fragile. Because the movement lacks self-awareness, it attracts a specific brand of $\text{zealous individual}$ who views the lack of mainstream adoption as a conspiracy—as if they are being $\text{persecuted by close-minded traditionalists}$ .

~~The mainstream is suppressing GA.~~ $\rightarrow$ Reality: GA is simply not compelling enough in its current state.

The Cycle of Mediocrity

GA continues to attract new enthusiasts because it does solve some real problems. However, a predictable pattern emerges:

This dynamic traps GA in a state of perpetual mediocrity. People defend the surface-level philosophy because they feel they are part of a revolution, but no one is doing the hard work of criticizing or improving the actual structure.

Comparison of Perspectives

Perspective	View of GA	View of Mainstream Math	Result
The Zealot	A perfect, revolutionary tool	Oppressive and outdated	Isolation/Echo Chambers
The Academic	A niche tool with no new proofs	The standard for a reason	Indifference/Eye-rolling
The Author	A promising but flawed project	Useful, but needs evolution	Desire for structural reform

Final Disclaimers and Intent

I want to be clear: I am not a professional mathematician. A formal mathematical critique would likely argue that GA doesn't prove anything new, so it's irrelevant. However, the goals of academic research differ from the goals of those using math for practical application. Furthermore, because I support the general philosophy of GA, I admit I am slightly on the "crank" side of the fence.

My goal is to push the field forward. I believe the philosophical direction of GA is correct—math should evolve this way—but we must first identify what is broken.

Mathematical Context

To illustrate the operation in question, the geometric product of two vectors $a$ and $b$ is often defined as: $ab = a \cdot b + a \wedge b$ Where:

$a \cdot b$ is the inner product (scalar).
$a \wedge b$ is the outer product (bivector).

While this looks elegant in code or on paper, the insistence that this specific combination be the primary lens for all geometry is where the theory falters.

Moving Forward: I will now describe my understanding of what GA is, its historical trajectory, and its position relative to physics and mathematics to ensure we are disagreeing about the same things. (Note: I am no historian; I welcome corrections on the following background).

Summary of GA as a Movement

Geometric Algebra is simultaneously a mathematical branch and a social movement. As a social movement, it consists of individuals who believe that:

Pedagogy should be reformulated for utility.
Research should prioritize the needs of non-mathematicians.