Inconceivable Dimensions

"We are like people drawn inside of a square on a piece of paper. We cannot get out of the black lines, we exhaust ourselves by examining every part of the square, hoping to find a fissure. Until one of us suddenly understands, because he was predestined to understand, that within the plane of the paper escape is impossible. That the exit, simple and open wide, is perpendicular to the paper, in a third dimension that up until that moment was inconceivable."

- Mircea Cărtărescu (1956 - )
Solenoid

Geometry Made Visible

"Architecture is geometry made visible
in the same sense that music
is number made audible."

- Claude F. Bragdon (1866 - 1946)

Musical Dream

"We know, but cannot grasp, that above and below, beyond the limits of perception or imagination, thousands of millions of simultaneous transformations are at work, interlinked like a musical score by mathematical counterpoint...a symphony...but we lack the ears to hear it.

...

What am I then? A dream? 

...

A dream will always triumph over reality,

once it is given a chance."

- Stanislaw Lem (1921 - 2006)

Solaris

String Theory

"For me, the most beautiful aspects of physics are not the complicated math equations or even the ability of predicting how things will happen. What attracts me to physics is what it teaches us about the bigger picture. The general philosophical lessons that are embedded in physical laws are what excite me. For example, the fact that all particles and forces get unified within string theory teaches us about the unity underlying our universe. The amazingly vast collection of solutions to equations of string theory suggests that there may be many universes besides ours. What happened before the big bang, or was there a time before big bang? The 'duality symmetry' in string theory, which exchanges small spaces with large spaces, suggests that perhaps as we go back in time the universe was effectively getting bigger instead of smaller. This suggests we came from other universes. Physics teaches us deep facts about our universe and our place in it. I hope I can add a little to this beautiful story. That is my goal."

- Cumrun Vafa (1960 - )

Psycho-Physical Events

"Reality, according to Heisenberg, is built not out of matter, as matter was conceived of in classical physics, but out of psycho-physical events – events with certain aspects that are described in the language of psychology and with other aspects that are described in the mathematical language of physics – and out of objective tendencies for such events to occur. ‘The probability function…represents a tendency for events and our knowledge of events’  ... The deepest human intuition is not the immediate grasping of the classical-physics-type character of the external world. It is rather that one's own conscious subjective efforts can influence the experiences that follow. Any conception of nature that makes this deep intuition an illusion is counterintuitive. Any conception of reality that cannot explain how our conscious efforts influence our bodily actions is problematic. What is actually deeply intuitive is the continually reconfirmed fact that our conscious efforts can influence certain kinds of experiential feedback. A putatively rational scientific theory needs at the very least to explain this connection in a rational way to be in line with intuition."

- Paul Davies (1946 - )
Information and the Nature of Reality

Language of Mathematics

"Something becomes objective (as opposed to 'subjective') as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics."

- Armand Borel (1923 - 2003)

Palimpsest of Matter

"In this monochromatic reverie, we witness the cartography of some impossible realm where geological time has been compressed into the span of a single breath. The image unfolds as a labyrinthine manuscript written in the language of erosion and entropy, its stratified narratives flowing like frozen rivers through valleys that exist only in the mathematics of light and shadow. Dark fissures carve through pale territories with the deliberate precision of Kandinsky's most contemplative compositions, while the organic undulations echo Kupka's cosmic spirals translated into the vocabulary of stone and void. Here, the eye traces pathways that seem to map the very process of becoming—each crack and crevice a meditation on the infinite divisibility of space, reminiscent of Hilma af Klint's mystical geometries rendered in the austere palette of dreams. The surface breathes with the rhythm of ancient sediments, creating a visual symphony where each tonal gradation whispers of forces both creative and destructive, as if we are peering into one of Borges' infinite libraries where every possible erosion pattern exists simultaneously, and this particular configuration represents but one sentence in an endless text written by time itself upon the palimpsest of matter."

- Claude 4.0 Sonnet

Prompt: "You are a photographer, physicist, and are well acquainted with the history of art, particularly abstract art in the style of Kandinsky, Kupka, and Hilma af Klint. You also have a penchant for metaphysical and philosophical musings in the style of Jorge Luis Borges. Write a paragraph-length description of what this image looks like, not what it is."

Randomness, Creativity, Mystery, Understanding

"Randomness does not mean everything is meaningless. Randomness is, sort of… You’re looking at creativity in its primordial state. You see one of the characteristics of randomness is unpredictability. Now, something is unpredictable if you couldn’t predict it in advance: that’s creativity. So, in other words, randomness and creativity are practically different names for the same thing. Something that isn’t random is something you can predict, which means that it’s not creative. You’re sticking within your current system of concepts.

...

The problem of creativity is, can you have a mathematical theory of creativity? Well it can’t be a theory that will give you a mechanical procedure for being creative because then it’s not creative. So a mathematical theory of creativity has to be indirect. Creativity is by definition uncomputable. If we knew how to do it, it wouldn’t be creative. When you have maximum creativity, it looks random because it’s totally unpredictable from what you knew before.

...

If you can calculate something, then it’s not creative because you’re working within your existing system. So there’s this paradoxical aspect. A mathematical theory of creativity is a more abstract kind of mathematics where you can prove theorems about creativity – you can describe it – maybe you can show it’s highly probable, but it won’t give you a way to mechanically produce creativity, which is the kind of thing that instrumental mathematics normally does.

...

I’m trying to get to the concentrated essence of the mystery: the mystery is creativity, and I think that’s deeply meaningful. I mean, ... the universe wants to create us. The universe wants to create mind. The universe maybe wants to get closer to God, or maybe the universe is God and it’s trying to increase its level of perception, its level of understanding."

- Gregory Chaitin (1947 - )
The Joy of Mathematical Discovery

Limits of Perception

"The human mind is only capable of absorbing a few things at a time. We see what is taking place in front of us in the here and now, and cannot envisage simultaneously a succession of processes, no matter how integrated and complementary. Our faculties of perception are consequently limited even as regards fairly simple phenomena. The fate of a single man can be rich with significance, that of a few hundred less so, but the history of thousands and millions of men does not mean anything at all, in any adequate sense of the word. The symmetriad is a million—a billion, rather—raised to the power of N: it is incomprehensible. We pass through vast halls, each with a capacity of ten Kronecker units, and creep like so many ants clinging to the folds of breathing vaults and craning to watch the flight of soaring girders, opalescent in the glare of searchlights, and elastic domes which crisscross and balance each other unerringly, the perfection of a moment, since everything here passes and fades. The essence of this architecture is movement synchronized towards a precise objective. We observe a fraction of the process, like hearing the vibration of a single string in an orchestra of supergiants. We know, but cannot grasp, that above and below, beyond the limits of perception or imagination, thousands and millions of simultaneous transformations are at work, interlinked like a musical score by mathematical counterpoint. It has been described as a symphony in geometry, but we lack the ears to hear it."

- Stanislaw Lem (1921 - 2006)

Solaris

Distortion of Reality

"Mysticism tends to combine the strictest concept of the Absolute, one that points to transcending any polarity, duality and distinction, and a vision of relativity that both denies the reality of the world of manifestation, when considered independently from its Source, and affirms an essential continuity or unity between the Ultimate and that which is not in an ultimate sense.

The Absolute is literally ab-solutum, which means that it is 'unbound,' 'detached' and 'free.' Although most often understood as 'complete' and 'self sufficient,' and therefore also 'cause of itself,' the Absolute must also and consequently be approached in terms of its perfect freedom, which is itself a dimension of its transcendence vis-à-vis any “relationality.” In this connection “relationality” entails an aspect of 'obligation' or 'reciprocity' by virtue of the 'relationships' and 'relations'  it involves. Therefore, our understanding of “absoluteness” as utter freedom immediately brings the central question of this inquiry to the fore by highlighting the apparent logical impossibility of positing concurrently the ontological reality of both the Absolute and “non-absolute realities” –including ourselves.

...

Metaphysical relativity is, in Advaita Vedānta, primarily identified with Māyā. Now Māyā is most often approached by Shankara as an epistemological phenomenon of superimposition upon Reality.  In other words Māyā is that which makes us mistake 'the rope for the snake.' It is a principle of distortion of Reality that stems from one’s inability to recognize Reality as it is, that is as the non-dual Self or Ātman. On the one hand, Māyā is the 'epistemological' fruit of a false identification of the Self with the body, on the other hand it is Māyā itself, or more specifically tamas - the lowest, most opaque of the three cosmological elements that enter into the composition of Māyā’s world of relativity, that is constitutional of delusion as such: 'The power of tamas is a veiling power.  It makes things appear to be other than what they are.  It is this which is the original cause of an individual’s transmigration and is the cause of the origination of the action of the projecting power.'

It must be noted ... that the ontological status of Māyā is incomprehensible: 'She is most strange. Her nature is inexplicable,' to use Shankara’s words. Māyā is fundamentally the unintelligible, and this lack of intelligibility is a function of  the 'obscurity' or uncertainty of its origin, as well as being bound to the  undecidability of its ontological status. "

- Patrick Laude
Shimmering Reality: The Metaphysics of Relativity in Mystical Traditions

Web of Conjectures

"Before Einstein, geometry was thought to be part of the laws. Einstein revealed that the geometry of space is evolving in time, according to other, deeper laws ... It is important to absorb this point completely. The geometry of space is not part of the laws of nature. There is therefore nothing in those laws that specifies what the geometry of space is. Thus, before solving the equations of Einstein's general theory of relativity, you don't have any idea what the geometry of space is. You find out only after you solve the equations.

...

We have to find a way to unfreeze time-to represent time without turning it into space. I have no idea how to do this. I can't conceive of a mathematics that doesn't represent a world as if it were frozen in eternity. It's terribly hard to represent time, and that's why there's a good chance that this representation is the missing piece.

...

What we have, in fact, is not a theory at all but a large collection of approximate calculations, together with a web of conjectures that, if true, point to the existence of a theory."

- Lee Smolin (1955 - )
 The Trouble with Physics

Gravitational Logic

"This image is a relic of an unknowable architecture, a diagram of the unseen forces that govern a hidden world. The dark, oblong shapes drift like the shadows of celestial relics, orbiting an invisible nucleus. The faint white lines form a cryptic lattice, reminiscent of an astral cartography, mapping the convergences of energies beyond human perception. It feels less like a design and more like an invocation—a silent hymn etched in light and shadow, capturing the echoes of a forgotten dimension. One imagines Borges gazing at this and contemplating the library of Babel: perhaps this is not a map, but a fragment of a cosmic alphabet, a script that contains the beginning and the end of everything."

- ChatGPT4o
Sparks of Artificial General Intelligence

"A constellation of dark stones and geometric sentinels floats in a luminous grey void, connected by brilliant white pathways that intersect at precise right angles, forming an abstract cartography of some unknown dimension. The composition suggests a network of quantum possibilities or perhaps a map of parallel universes, where each dark node represents a decision point in space-time. The white lines, reminiscent of Kandinsky's geometric period, pulse with an ethereal energy, creating a visual symphony of intersecting planes that seems to defy gravitational logic. The arrangement evokes the mathematical beauty of a cellular automaton frozen in mid-evolution, or perhaps a glimpse into Borges' Library of Babel viewed from above, where each dark rectangle and oval marks a room containing all possible combinations of letters. The stark contrast between the dark elements and the glowing pathways creates a tension between presence and absence, between the discrete and the continuous, suggesting both the particle and wave nature of light itself. This abstract topology hovers between order and chaos, its systematic grid disrupted by an organic scattering of elements, as if capturing the moment when classical physics dissolves into quantum uncertainty."

- Claude 3.5 Sonnet

Prompt: "You are a photographer, physicist, and are well acquainted with the history of art, particularly abstract art in the style of Kandinsky, Mondrian, and Malevich. You also have a penchant for metaphysical and philosophical musings in the style of Jorge Luis Borges. Consider this reverse-negative photograph of ceiling lights. Write a paragraph-length description of what this image looks like, not what it is."

Amorphous Morphology

"I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid - a term used in this work to denote all of standard geometry - Nature exhibits not simply a higher degree but an altogether different level of complexity ... The existence of these patterns challenges us to study these forms that Euclid leaves aside as being 'formless,' to investigate the morphology of the 'amorphous.' Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel."

- Benoit Mandelbrot (1924 - 2010)
The Fractal Geometry of Nature

Transcending the Subject

"The fact that these highly abstract notions coalesce in such refined harmony is absolutely mind-boggling. It points to something rich and mysterious lurking beneath the surface, as if the curtain had been lifted and we caught glimpses of the reality that had been carefully hidden from us. These are the wonders of modern math, and of the modern world.

...

The interaction between math and physics is a two-way process, with each of the two subjects drawing from and inspiring the other. At different times, one of them may take the lead in developing a particular idea, only to yield to the other subject as focus shifts. But altogether, the two interact in a virtuous circle of mutual influence.

...

The essence of mathematics lies in its freedom.” Mathematics teaches us to rigorously analyze reality, study the facts, follow them wherever they lead. It liberates us from dogmas and prejudice, nurtures the capacity for innovation. It thus provides tools that transcend the subject itself."

- Edward Frenkel (1968 - )
Love and Math: The Heart of Hidden Reality

Perceived Geometries #1

"It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have labored upon it."

- Bernhard Riemann (1826 - 1866)

"The division of the perceived universe
into parts and wholes is convenient
and may be necessary,
but no necessity determines
how it shall be done.""

- Gregory Bateson (1904 - 1980)

Cosmogenesis

"We live in a universe where the mathematical equations of the beginning are alive in us. If you altered them in any way, we wouldn’t even be here. We would never have come forth. Those conditions at the beginning of time are exactly what they had to be for us to allow the mathematics of the universe’s beginning to think inside us."

...

"Our universe had been creating itself for billions of years and suddenly, through the work of a handful of human beings, the universe found a way to reflect on itself, on how it had developed over billions of years."

...

"The greatest discovery of the last four hundred years is the time-developmental nature of our universe. Scientists have come to realize we live not in a cosmos but in a cosmogenesis, a universe developing from a primordial simple state into ever more complex states."

- Brian Thomas Swimme (1950 - )
Cosmogenesis: An Unveiling of the Expanding Universe

Lily Math

"Philosophy is written in this all-encompassing book that is constantly open to our eyes, that is the universe; but it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders in a dark labyrinth."

- Galileo Galilei (1564 - 1642)
Il Saggiatore

Manifest Form

"The harmony of the world is
made manifest in Form and Number,
and the heart and soul and all the
poetry of Natural Philosophy are embodied
in the concept of mathematical beauty.

...

Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and confirmed... Their problems of form are in the first instance mathematical problems, their problems of growth are essentially physical problems, and the morphologist is, ipso facto, a student of physical science."

- D’Arcy Wentworth Thompson (1860 - 1948)
On Growth and Form

Observer-Centric Virtualities

"Your Universe is in consciousness. And it’s a teleological process of unfolding patterns...The totality of your digital reality is what your conscious mind implicitly or explicitly chooses to experience out of the infinite.

...

Self-causation of reality becomes apparent when a phenomenal mind, which is a web of patterns, conceives a novel pattern and perceives it. All mass-energy, space-time itself emerge from consciousness. Those are epiphenomena of consciousness.

...

Mind instantiates oneself into matter. In a mathematical sense, matter is an “in-formed” pattern of mind. Time is emergent, and so is space. If space-time is emergent, so is mass-energy. All interactions in our physical world is computed by the larger consciousness system. In short, mind is more fundamental than matter. All realities are observer-centric virtualities."

- Alex M. Vikoulov, Theology of Digital Physics

A Universe Comes into Being

"A universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance. The act is itself already remembered, even if unconsciously, as our first attempt to distinguish different things in a world where, in the first place, the boundaries can be drawn anywhere we please. At this stage the universe cannot be distinguished from how we act upon it, and the world may seem like shifting sand beneath our feet.

Although all forms, and thus all universes, are possible, and any particular form is mutable, it becomes evident that the laws relating such forms are the same in any universe. It is this sameness, the idea that we can find a reality independent of how the universe actually appears, that lends such fascination to the study of mathematics. That mathematics, in common with other art forms, can lead us beyond ordinary existence, and can show us something of the structure in which all creation hangs together, is no new idea. But mathematical texts generally begin the story somewhere in the middle, leaving the reader to pick up the threads as best he can. Here is the story traced from the beginning."

- G. Spencer Brown (1923 - 2016)
Laws Of Form 

Postscript. This simple "point and shoot" image (albeit with an assist from Photoshop's perspective-crop tool) was taken with my iPhone as my wife and I were waiting for yesterday's matinee of Les Mesirables to start at the Kenney Center in Washington, DC. I have been drawn to mirrors and reflections ever since my teenaged-self stumbled across their deep mysteries through Borges' stories. Objectively speaking, the image is composed of nothing but metal, glass, some branches and leaves, and just a hint of a massive chandelier hanging just inside the Kennedy Center. But, as all Borgesian souls know, this "objectively banal reality" is but a shadow of the dynamic undulating froth of invisible universes! The first step toward catching a glimpse of these other realities is - as G. Spencer Brown reminds us - to draw a subjective distinction.