A primer on the Universe

Galileo Galilei was a key precursor to Isaac Newton and modern science. In Pisa, he famously conducted (or at least proposed) experiments by dropping objects of different masses from the Leaning Tower to demonstrate that their fall speed is independent of mass, directly challenging Aristotle’s longstanding views on gravity. This foundational work paved the way for Newton’s later formulation of the laws of motion and universal gravitation.

Galileo Galilei was a key precursor to Isaac Newton and modern science.

Isaac Newton created the modern view of the universe by conceiving space as a vast, three-dimensional arena and time as a uniform, linear progression—an independent backdrop against which all physical events unfold. He used mathematics to give precise form to the law of universal gravitation, expressing it as a rigorous equation: every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them. In this way, Newton unified terrestrial and celestial mechanics under a single mathematical framework.

Albert Einstein, together with Hermann Minkowski, transformed the Newtonian paradigm by fusing space and time into a single, four-dimensional continuum called spacetime. In this new setting, the geometry of events is governed by Lorentz transformations, revealing that space and time are interdependent and relative to the observer. Their work showed that simultaneity, length, and the passage of time are not absolute but depend on the observer's motion. Physical laws remain invariant under transformations that mix space and time coordinates—a cornerstone of special relativity. For example, two events that appear simultaneous in one reference frame may not be simultaneous in another moving at a different velocity.

Mainstream physics rapidly adopted this paradigm, and today the standard picture still rests on these principles—even though quantum theory and recent astronomical discoveries highlight persistent mismatches with the classical framework. Increasingly, there is a sense that space and time may not be fundamental entities. This shift motivates newer programs such as “process physics” and “relational physics,” in which the basic constituents of reality are not extended in space or time, but are instead conceived as Leibniz-like monads, processes, or relations. In such approaches, our experience of space and time emerges statistically from the collective behavior of countless underlying primary entities, and from the web of connections among them. As a concrete example, some relational models postulate a discrete network of interactions—rather than a smooth spacetime continuum—out of which familiar geometry only appears at large scales.

For further reading on the relational approach, you may refer to the review "Relational Paradigm in Asks and Answers" by M. I. Suslova and A. A. Sidorova-Biryukova here, and "On Explanations of Magnetic Fields of Astrophysical Objects in the Geometric and Relational Approaches" by Y.S. Vladimirov, S.V. Bolokhov, and I.A. Babenko here.

Perhaps physics requires a complete paradigm shift—perhaps it even needs entirely new mathematics, with “circular arrows” as invoked in some relational approaches. Personally, however, I favor a middle path, one based on algebra and geometry. I believe that algebra and geometry have not yet been exploited to their full potential.

Scientists usually believe that there exist fundamental laws or principles, and that their task is to discover and study them. There are at least two possible strategies for this pursuit: we may either collect data, analyze it, and try to deduce the laws by making educated guesses, or we may rely on our intuition or flashes of insight, boldly postulating fundamental principles (such as “duality” or “triality”), and then deducing more complex laws from these "prime principles." In practice, it is usually best to blend these approaches—combining empiricism and intuition—if we are to make significant progress within any reasonable time frame. I witness such a healthy attitude among participants in Yu.S. Vladimirov’s weekly seminar “Foundations of Fundamental Physics” at Peoples’ Friendship University of Russia, which I have been attending regularly for several months. The Vladimirov school is developing a mathematics of complex binary relations and, on this basis, constructing a worldview that incorporates the ideas of Leibniz and Mach, including the key notion of action at a distance.

“Foundations of Fundamental Physics” seminar 2025-11-06.

Y.S. Vladimirov on the far right.

Personally, I am a syncretist, blending elements from different traditions. Building the universe out of primitive yes-no alternatives, even with circular arrows and complex numbers, is a formidable task and may take an extremely long time—even following clever shortcuts. Instead of focusing on elementary monads or "atoms of existence," I prefer the attitude of a chemist, who works with molecules rather than elementary particles. Thus, I take algebras as already given, and geometry—which naturally follows from these algebras—as sufficiently stable to serve as the building blocks of our knowledge. These structures are the prototypes I use for modeling and describing our reality, both material and immaterial.

In my previous post, I mentioned a particular paper, written by five mathematicians, that describes the synthetic geometry of a family of such models. The title of the paper is "A Primer on the (2 + 1) Einstein Universe," and it is intended to introduce the “synthetic geometry of the Einstein universe.” Synthetic geometry is, in principle, based on axioms—like those found in Euclidean geometry. Fortunately, the paper does not pursue this formal path. Personally, I do not care for axiom systems, since they can often be replaced by others that express the opposite or yield incompatible structures. I prefer “constructions.” But constructions, in turn, often come with cumbersome formulas and calculations, making it easy to lose sight of the underlying essence. The paper I refer to, fortunately, takes a balanced approach: it introduces the main constructions, but then focuses on the “objects” involved and the web of relations (mostly incidence relations) between these objects—this is the part of synthetic geometry that I truly appreciate! Let us now delve into the paper’s details.

• The math part will follow in the next post
• A Substack version of this post is here:

Afternotes:

12-11-25 12:02 A passing Reader inquired in a comment to the previous post if I will discuss the Penrose diagram and twistors. Yes, this is related to the content of "A Primer", and we will talk about these concepts. There will be a separate post dealing with the Penrose diagram. BTW Y.S Vladimirov likes to quote the Penrose program and expands it into new areas using new methods,

12-11-25 18:47 Somewhat related new post  on Substack by Laura:
Mind, Matter, and the Epistemic Asymmetry: A Close Reading of Kastrup’s Argument.

The Spin Chronicles (Part 12) - Geometry, Kant, and the Limits of Physics

 Welcome back to the odyssey of geometric algebra, where the math gets deep and the philosophy… well, it occasionally dives off the deep end. This post picks up from "The Spin Chronicles (Part 11)" and continues our foray into the natural bilinear forms that emerge within the Clifford algebra of our 3D space. Don't worry if you’re still wondering what that is; just keep in mind that it's an essential way to understand how space itself “behaves.” And no, we won’t be referencing the cosmic dance between the stars and planets—but this will get philosophical enough to warrant a helmet.

Immanuel Kant, who preferred his mornings with a side of metaphysics, once said:

"Space is not something objective and real, nor a substance, nor an accident, nor a relation; instead, it is subjective and ideal, originating from the mind’s nature in accord with a stable law as a scheme, as it were, for coordinating everything sensed externally."

Which is to say, Kant thought of space not as some grand arena laid out by a divine hand, but as a sort of mental wallpaper, perfectly crafted to make sense of the world without needing to be “real” in the way rocks or trees are real. Then he doubles down:

"Space is not an empirical concept derived from outer experiences. In order for sensations to refer to something outside me, and for them to be represented as outside and alongside one another in different places, the concept of space must already exist within us."

Basically, Kant argues that space isn’t something we pick up by wandering around the world and mapping it in our heads; rather, it’s a kind of mental scaffolding that’s there from the get-go, allowing us to experience everything else in relation to it.

Now, whether or not you buy into Kant’s perspective here is a different story. Space, after all, feels pretty solid when you bump into a coffee table in the dark. We intuitively “know” space, and our DNA may even hold clues to why this intuitive knowledge works so well. Our spatial instincts aren’t simply musings; they come from somewhere deeply ingrained, refined by eons of needing to dodge pointy sticks and hungry animals.

But just as we start to think space is on our side, the physicists arrive, white coats flapping, to tell us: “Well, space (and time) aren’t absolute. They’re relative, dependent on the observer’s inertial frame of reference.” Einstein’s Special Relativity was a real buzzkill for those who thought they’d finally “figured out” space. Yet, as bold as these physics claims are, they’re always at the mercy of the next scientific revolution. Mathematics, by contrast, is a steady old friend—unchanging, consistent, and reliably grounded in the Euclidean spaces we’re exploring here.

Well, space (and time) aren’t absolute...

So, let's put the physics opinions aside for now and dive back into mathematics, where truths stay put and constants stay constant. Today, our quest involves understanding the natural bilinear forms within the geometric Clifford algebra of our classic 3D Euclidean space. Yes, we’re talking about the world where the Pythagorean theorem reigns supreme and the ratio of a circle's circumference to its diameter is that oh-so-familiar π ≈ 3.1415926...

So, let’s roll up our sleeves and get back to numbers, forms, and figures that don’t play hide-and-seek depending on who’s looking at them.

We have already discussed the bilinear forms B₀ and B_τ. Next in order is B_ν(u,v) = t_n(ν(u)v),

where 

ν(u) = (p₀,-p).

We are using a real basis E_A (A=0,1,...,7) in Cl(V):

E₀ = 1, E₁ = e₁, E₂ = e₂, E₃ = e₃, E₄ = ie₁, E₅ = ie₂, E₆ = ie₃, E7 = i.

In this basis the matrix of the real and imaginary parts of B_ν are given by:

Re(B_ν)

Im(B_ν)

Both are of neutral signature (++++----).

Finally B_π, with π given by 

π(u) = (p₀,-p)*, where "*" stands for the complex conjugation:

Re( B_π)

Im( B_π)

The real part is symmetric, with eigenvalues (+1,-1,-1,-1,-1,-1,-1,+1). The imaginary part is anti-symmetric.

Since Cl(V) carries a natural complex structure, it is even more instructive to consider our bilinear forms as complex valued. For this it is convenient to use the matrix representation with Pauli matrices. Then the complex basis consists of matrices (I, σ₁, σ₂, σ₃), as discussed in The Spin Chronicles (Part 9): Matrix representation of Cl(V) and The Spin Chronicles (Part 10) - Dressing up the three involutions. In this basis we calculate the quadratic forms B₀(u,u), B_τ(u,u), B_ν(u,u) and B_π(u,u), for u = (p₀,p). Here p₀ is a complex scalar, p is a complex vector. We can easily obtain:

B₀(u,u) = (p₀)² + p²,

B_τ(u,u) = |p₀|² + |p|²,

B_ν(u,u) = (p₀)² - p²,

B_π(u,u) = |p₀|² - |p|².

It is somewhat surprising that  for B_ν and B_π(u,u) we are getting a form resembling the 4D Minkowski metric of special relativity, but that's what mathematics leads to, for its own strange reasons. We will return to this issue in the future posts. Stay tuned...