Summary

A three-year-old cannot put her shoes on before her socks. Not because she lacks motor skills — because the operations do not commute.

The same structural constraint, dressed in the language of operators on a Hilbert space, is why Heisenberg’s uncertainty principle holds. This post is about that connection: the accidental algebra lesson built into getting dressed, and why the physicists of 1925 had to abandon one of arithmetic’s most taken-for-granted assumptions.

Getting Dressed Is a Non-Abelian Problem

Start with the mundane. Your morning routine imposes a strict partial order on operations: underwear before trousers, socks before shoes, cap before chin-strap if you cycle. Try reversing any pair and the sequence fails — physically, not just socially. You cannot pull a sock over a shoe.

The operation “put on socks” followed by “put on shoes” produces a wearable human; the reverse produces neither, and no amount of wishing commutativity into existence will help.

In the language of abstract algebra, two operations \(A\) and \(B\) commute if \(AB = BA\) — if doing them in either order yields the same result. Everyday life is full of operations that do not commute: rotate a book 90° around its vertical axis then 90° around its horizontal axis; now reverse the order. The final orientations differ. Turn right then turn left while driving; left then right. Different positions.

The intuition is not hard to build. What is surprising is how rarely we note it, and what it costs us when we finally hit a domain — quantum mechanics — where non-commutativity is not an inconvenient edge case but the central fact.

Piaget Said Seven; Toddlers Disagreed

Jean Piaget argued that children do not acquire operational thinking — the ability to mentally perform and reverse sequences of actions — until the concrete operational stage, roughly ages seven to eleven (Inhelder & Piaget, 1958). Before that, he claimed, children lack the understanding that an operation can be undone or reordered.

Post-Piagetian research pushed back hard. Patricia Bauer and Jean Mandler tested infants aged sixteen and twenty months on novel, multi-step action sequences (Bauer & Mandler, 1989). For causally structured sequences — where step A physically enables step B — infants reproduced the correct order after a two-week delay. They were not told the order was important. They had no language to encode it. They just knew, implicitly, that the operations had a necessary direction.

A 2020 study by Klemfuss and colleagues tested 100 children aged roughly two-and-a-half to five on temporal ordering questions (Klemfuss et al., 2020). Children answered “what happened first?” questions correctly 82% of the time. The errors that did appear followed an encoding-order bias — children defaulted to reporting the next event in the sequence as originally experienced, regardless of what was asked. The ordering knowledge was intact. What children lack, for Piaget’s full seven years, is the formal recursive conception of reversibility. The procedural knowledge — that some sequences must be done in the right order and cannot be freely rearranged — is there from the second year of life.

Which means: learning that \(AB \neq BA\) is not learning something exotic. It is articulating something the nervous system already knows.

The Mathematician’s Commutator

Abstract algebra formalized this intuition in the nineteenth century. A group is abelian (commutative) if every pair of elements satisfies \(ab = ba\). Integers under addition: abelian. Rotations in three dimensions: not.

Arthur Cayley’s 1858 memoir established matrix algebra as a formal theory (Cayley, 1858). Multiply two \(2 \times 2\) matrices:

$$ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$$$ AB = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}, \quad BA = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix} $$

\(AB \neq BA\). Non-commutativity is not a curiosity; it is the generic condition for matrix products. Commutativity is the special case — and requiring justification.

William Rowan Hamilton had already gone further. On 16 October 1843, walking along the Royal Canal in Dublin, he discovered the quaternions and carved their multiplication rule into the stone of Broom Bridge:

$$ i^2 = j^2 = k^2 = ijk = -1 $$

From this it follows immediately that \(ij = k\) but \(ji = -k\). Hamilton’s four-dimensional number system — the first algebraic structure beyond the complex numbers — was non-commutative by construction. He did not apologize for it. He celebrated it.

The Lie algebra structure underlying these commutator relations is the same skeleton that governs Messiaen’s modes of limited transposition, which I traced in a previous post on group theory and music — a very different physical domain, but identical algebraic machinery.

Born, Jordan, and the Physicist’s Shock

Classical mechanics treats position \(x\) and momentum \(p\) as ordinary real numbers. Real numbers commute: \(xp = px\). The Poisson bracket \(\{x, p\} = 1\) encodes a classical relationship, but the underlying quantities are scalars, and scalars commute.

In July 1925, Werner Heisenberg published a paper that could not quite bring itself to say what it was doing (Heisenberg, 1925). He replaced classical dynamical variables with arrays of numbers — what we would now call matrices — and found, uncomfortably, that the resulting quantum condition required order to matter.

While Heisenberg was on vacation, Max Born and Pascual Jordan finished the translation into matrix language (Born & Jordan, 1925). They wrote the commutation relation explicitly, recognized it as the fundamental law, and showed that it reproduced the known quantum results:

$$ [\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i\hbar $$

Non-commutativity of position and momentum was not a mathematical accident. It was the theory.

The uncertainty principle followed four years later as a theorem, not an additional postulate. Howard Robertson proved in 1929 that for any two observables \(\hat{A}\) and \(\hat{B}\), the Cauchy–Schwarz inequality on Hilbert space yields (Robertson, 1929):

$$ \Delta A \cdot \Delta B \geq \frac{1}{2} \left| \langle [\hat{A}, \hat{B}] \rangle \right| $$

Substituting \(\hat{A} = \hat{x}\), \(\hat{B} = \hat{p}\), \([\hat{x}, \hat{p}] = i\hbar\):

$$ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} $$

This is the uncertainty principle. It does not say nature is fuzzy or that measurement disturbs systems in some vague intuitive sense. It says: position and momentum are operators that do not commute, and the Robertson inequality then constrains their joint variance. Non-commutativity is the uncertainty principle. Put the shoes on before the socks and the state is not defined.

The same logic applies to angular momentum. The three components satisfy:

$$ [\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z, \quad [\hat{L}_y, \hat{L}_z] = i\hbar \hat{L}_x, \quad [\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y $$

This is the Lie algebra \(\mathfrak{su}(2)\). You cannot simultaneously determine two components of angular momentum to arbitrary precision — not because the measurement apparatus is noisy, but because the operations of measuring them do not commute.

The fiber bundle language that underlies these rotation groups also appears, in different physical dress, in the problem of the falling cat and geometric phases — another case where the order of rotations has non-trivial physical consequences (see that post).

Connes and Non-Commutative Space

Alain Connes asked what happens if we allow the coordinates of space itself to be non-commutative. In ordinary geometry, the algebra of coordinate functions on a manifold is commutative: \(f(x) \cdot g(x) = g(x) \cdot f(x)\). Connes’ non-commutative geometry replaces this with a spectral triple \((\mathcal{A}, \mathcal{H}, D)\): an algebra \(\mathcal{A}\) of operators (possibly non-commutative) acting on a Hilbert space \(\mathcal{H}\), with a generalized Dirac operator \(D\) encoding the geometry (Connes, 1994).

The payoff was remarkable. With Ali Chamseddine, Connes showed that if \(\mathcal{A}\) is chosen as a specific non-commutative product of the real numbers, complex numbers, quaternions, and matrix algebras, the spectral action principle reproduces the full Lagrangian of the Standard Model coupled to general relativity from a single geometric principle (Chamseddine & Connes, 1996). The Higgs field, the gauge bosons, the graviton: all from the geometry of a non-commutative space.

Classical geometry is the special case where the coordinate algebra is commutative. Drop that assumption and you open up a vastly richer landscape. Quantum mechanics lives in that landscape. Possibly, so does the structure of spacetime at the Planck scale.

The Lesson Pre-Schoolers Already Know

There is an irony here that I cannot quite leave alone. Students learning linear algebra for the first time consistently make the same mistake. Anna Sierpinska documented it carefully: they assume \(AB = BA\) for matrices because they have spent years in arithmetic and scalar algebra where multiplication commutes (Sierpinska, 2000). The commutativity of ordinary multiplication is so deeply internalized that abandoning it feels like breaking a rule.

But the pre-schooler in the sock-and-shoe scenario never had that problem. Her procedural memory, documented in infants as young as sixteen months by Bauer and Mandler, encoded the correct asymmetry directly. The order of operations is the first thing a developing mind learns about actions in the world, before the arithmetic of school teaches it the convenient fiction that order is irrelevant.

Arithmetic is the outlier. \(3 + 5 = 5 + 3\) because counting does not depend on where you start. But putting on clothes, multiplying matrices, rotating rigid bodies, measuring quantum observables: these operations carry memory of order, and they repay the attention a child already brings to them before she can name a number.

The universe is non-abelian. We are born knowing it. School briefly convinces us otherwise. Physics eventually agrees with the pre-schooler.

References

  • Inhelder, B., & Piaget, J. (1958). The Growth of Logical Thinking from Childhood to Adolescence. Basic Books.
  • Bauer, P. J., & Mandler, J. M. (1989). One thing follows another: Effects of temporal structure on 1- to 2-year-olds’ recall of events. Developmental Psychology, 25, 197–206.
  • Klemfuss, J. Z., McWilliams, K., Henderson, H. M., Olaguez, A. P., & Lyon, T. D. (2020). Order of encoding predicts young children’s responses to sequencing questions. Cognitive Development, 55, 100927. DOI: 10.1016/j.cogdev.2020.100927
  • Cayley, A. (1858). A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London, 148, 17–37. DOI: 10.1098/rstl.1858.0002
  • Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik, 33, 879–893.
  • Born, M., & Jordan, P. (1925). Zur Quantenmechanik. Zeitschrift für Physik, 34, 858–888.
  • Robertson, H. P. (1929). The uncertainty principle. Physical Review, 34, 163–164. DOI: 10.1103/PhysRev.34.163
  • Connes, A. (1994). Noncommutative Geometry. Academic Press. ISBN 0-12-185860-X.
  • Chamseddine, A. H., & Connes, A. (1996). Universal formula for noncommutative geometry actions: Unification of gravity and the standard model. Physical Review Letters, 77, 4868–4871. DOI: 10.1103/PhysRevLett.77.4868
  • Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the Teaching of Linear Algebra (pp. 209–246). Kluwer Academic Publishers. DOI: 10.1007/0-306-47224-4_8

Changelog

  • 2026-02-03: Corrected the age range for the Klemfuss et al. (2020) study from “two to four” to “roughly two-and-a-half to five” — the actual participants were aged 30–61 months.
  • 2026-02-03: Updated the characterisation of Klemfuss et al. (2020) findings to reflect the paper’s central result: errors follow an encoding-order bias (children default to the next event in encoding sequence). The paper’s title — “Order of encoding predicts young children’s responses” — names the mechanism.