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.