How is an electron like a cube?

Reading | Ontology

Brian Fang, BSc | 2026-07-24

Think of a billiard ball. Your brain automatically splits it into two distinct categories: the “stuff” it is made of, and the properties attached to that stuff. The ball is red, glossy, smooth, and heavy. It’s intuitive to assume that if you could somehow scrape away the redness and the gloss, a naked lump of pure “stuff” would remain underneath. For centuries, our definition of reality has rested on this basic assumption: substance comes first, and qualities come second.

But look closer at a simple wooden cube sitting on the desk.

You never actually see a three-dimensional cube all at once. At any given moment, you see a flat, trapezoidal patch of color from one specific angle. Move your head, and that patch shifts. Walk around the desk, and new faces slide into view while others vanish. Over a century ago, the philosopher Edmund Husserl pointed out a strange fact about perception: if we only ever encounter a shifting sequence of partial profiles, why are we so certain we are looking at one solid cube? [1]

Husserl’s answer was that the cube is not simply a hidden object concealed behind its facets. In experience, the cube is disclosed through the lawful unity of those facets. The changing profiles are not random; they follow strict geometric constraints. Each angle implies the others. Each visible face points beyond itself to the sides currently unseen. When you point to the desk and say “that is a cube,” you are not merely pointing to an unseen lump of underlying material; you are recognizing a pattern of invariant structure that remains stable across changing perspectives.

What makes the cube the object it is, then, is not only what it is made of. It is also the geometry that persists through transformation. We could construct the same cube from wood, glass, ice, or steel. The material changes, but the cubical structure remains. Its identity does not depend entirely on the substance. It depends on the relations that hold its form together.

This is not just a quirk of human psychology or perception. Something remarkably similar appears at the foundations of modern physics.

Take the electron. We naturally picture it as a microscopic billiard ball: a tiny piece of “electron-stuff” carrying mass, charge, and spin. But quantum mechanics does not support that picture. Every electron in the universe is identical. Not merely very similar, like pennies fresh from the mint, but indistinguishable even in principle [2]. If two electrons in a helium atom are exchanged, the universe does not merely lose track of which is which; according to quantum theory, there is and was not any fact of the matter regarding which electron was which. Quantum statistics, which determine the behavior of matter and light at the foundational level, depend on this indistinguishability [2].

Separate lumps of substance should, at least in principle, be individually trackable. But in quantum theory, there is no hidden label that the universe keeps tucked away.

In the 1930s, the physicist Eugene Wigner approached this issue from a different angle. He asked what fixes the identity of an elementary particle within the theory. His answer was technical, but its conceptual meaning is profound: a particle is defined by how it transforms [3].

In modern physics, one way to identify a particle is to ask what remains unchanged when we transform the mathematical frame around it. Rotate the system. Shift it in space. Change its state of motion. Apply the symmetries the theory allows. What survives those transformations tells us what kind of thing we are dealing with. In Wigner’s work, mass and spin appear as defining features tied to spacetime symmetry. In later quantum field theory, electric charge is tied to a different kind of symmetry: the way the electron field responds to internal phase changes [3,4].

This sounds abstract, but the basic idea is simple: the electron is not best understood as a tiny object that first exists and then happens to possess mass, charge, and spin; within the theory, those features are bound up with the electron’s lawful pattern of transformation. What the electron is cannot be cleanly separated from how it behaves under symmetry.

There is no obvious extra layer beneath those characterizations. The equations describe fields, states, and their lawful transformations. Introducing some deeper “electron-stuff” that carries the properties adds no explanatory power if it makes no difference to the theory.

Taking this perspective, the electron is less like a noun, like a chair, than a stable pattern of behavior. We sometimes say that quantum fields “vibrate,” but unlike water waves or sound waves, these are not vibrations of an independently identifiable material medium. An electron is not a little bead riding on a wave; it is an excitation of a field: a structured mode within the theory’s mathematical framework [4].

On this view, asking why all electrons are identical is less like asking why all pebbles are the same and more like asking why all instances of the same pattern have the same structure. The symmetries do not merely decorate the electron after the fact; they help define what kind of thing the electron is.

The parallel with the cube is striking. In both cases, identity is connected to what remains invariant under transformation. For the cube, it is invariance across changes of physical perspective. For the electron, it is invariance across mathematical transformations. Different domains, same basic insight: what a thing is cannot always be separated from the structure that remains stable as everything else changes.

In mathematics, these transformations form a symmetry group, and the way something responds to them is called a representation. To understand an object is, in part, to understand its representation: its specific pattern of invariance.

There is, however, an important difference between the cube and the electron. For the cube, you can still say that the pattern is realized in wood, metal, glass, or ice. There is indeed a lump of stuff sitting on the table, bearing qualities so and so. But for the electron, that classical escape route is much harder to maintain. The theory contains no clear role for a tiny underlying substrate to play. Any supposed subatomic “substance” behind the electron would be explanatorily idle: a ghost present in language, borne from our deepest intuitions from childhood, but totally unnecessary for these theories—theories which explain the basis of the periodic table and enable technologies such as semiconductors.

In 1989, the philosopher John Worrall argued for a related lesson in the history of physics. When major scientific theories are overturned, the “stuff” we thought existed often disappears, while important mathematical structures survive. For generations, physicists believed that light traveled through a physical, elastic medium called the ether. The ether was eventually discarded, but much of the mathematical structure used to describe optical phenomena carried over into later theories. Worrall’s lesson was not that science discovers nothing real; it was that what survives theory change is often not the imagined substance, but the structure [5].

Physics may not be revealing a hidden inventory of tiny things in quite the way common sense expects; it may be uncovering objective patterns, relations, and invariances that remain stable beneath changing descriptions.

This dependence on structure becomes visually explicit in bistable percepts like the Necker cube: a simple line drawing of twelve lines on a flat page. When you look at it, the cube flips. What appeared to be the front face suddenly becomes the back, and then reverses again. The ink on the page remains exactly the same, yet the perceived object changes its orientation. The physical marks do not change, but the organization does.

The Necker cube does not prove that physical reality is created by perception; but it does reveal something important: identity is not always dictated by material substrate alone. Organization matters. Structure matters. The same underlying marks can support different experienced objects depending on the relational pattern through which they are interpreted.

We do not need to invoke mysticism, or claim that consciousness creates reality out of thin air, to appreciate the shift happening here. If we look closely at what things actually are, they are far removed from our classical intuition of mere blocks of “stuff.” The physical universe may not be a collection of self-contained objects waiting to be counted; it may be better understood as a web of stable relations, symmetries, and transformations waiting to be expressed.

Reality may not be a theater of hidden substances wearing visible properties; it may be better understood through structure: identity as what remains the same despite change.

 

References

1. Husserl, Edmund. Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy, First Book: General Introduction to a Pure Phenomenology. Translated by F. Kersten. The Hague: Martinus Nijhoff, 1983. Original work published 1913. https://www.finophd.eu/wp-content/uploads/2018/01/Husserl-Ideas-First-Book.pdf
2. French, Steven. “Identity and Individuality in Quantum Theory.” The Stanford Encyclopedia of Philosophy. First published February 15, 2000; substantive revision February 29, 2024. https://plato.stanford.edu/entries/qt-idind/
3. Wigner, Eugene P. “On Unitary Representations of the Inhomogeneous Lorentz Group.” Annals of Mathematics 40, no. 1 (1939): 149–204.  https://www.math.utoronto.ca/mgualt/courses/25-QM/docs/Wigner-1939.pdf
4. Kuhlmann, Meinard. “Quantum Field Theory.” The Stanford Encyclopedia of Philosophy. First published June 22, 2006; substantive revision August 10, 2022. https://plato.stanford.edu/entries/quantum-field-theory/
5. Worrall, John. “Structural Realism: The Best of Both Worlds?” Dialectica 43, no. 1–2 (1989): 99–124. https://doi.org/10.1111/j.1746-8361.1989.tb00933.x.  https://joelvelasco.net/teaching/3330/Worrall 1989 Structural Realism .pdf