Eighteen centuries of equations that arrived early

Deep Dive · Philosophy of Mathematics

Eighteen centuries of equations that arrived early

Why abstract mathematics keeps describing a physical world it was never built for — the eight strongest cases, the three best naturalist replies, and where the argument actually leads.

Around 20 minutes · 20 references · white.org.nz

Part one

The question underneath physics

Physics answers questions of the form how does this behave? It does this superbly. What it does not do — what it is not designed to do — is explain why the answer takes the form of an equation in the first place.

The point is easy to miss because the success is so total. A physicist writes down a Lagrangian, turns a mathematical crank, and out comes a number that matches an experiment to eleven or twelve significant figures. The anomalous magnetic moment of the electron is the standard example: theory and measurement agree to roughly one part in a trillion. Nothing else in human intellectual life works like this.

But notice what has been assumed rather than explained. Why is there a Lagrangian? Why does turning a mathematical crank on a symbolic expression tell you anything at all about an electron in a laboratory in Illinois? These are not questions physics can answer using physics, because any answer would itself be an equation, and the question is about why equations work.

This is not a gap that better instruments will close. It is a structural feature of the enterprise. Science presupposes the mathematical intelligibility of nature; it cannot derive it.

Part two

Three answers to “what is mathematics?”

Before asking why mathematics fits the world, it is worth being clear about what kind of thing mathematics is taken to be. Three positions have serious defenders.

Platonism: mathematical objects exist independently

On this view, numbers, sets and functions are real, abstract, non-physical, and would exist whether or not any mind ever contemplated them. Mathematicians discover rather than invent, in the same sense that explorers discover continents. Kurt Gödel held this openly, comparing mathematical intuition to sense perception. Roger Penrose defends a version of it at length in The Road to Reality. Hardy stated it as plainly as anyone has.

The case for Platonism is not mystical; it is mostly the observation that mathematics behaves like a subject matter and not like a game. Mathematicians are surprised by results. They get things wrong. Whole programmes fail. Fermat’s Last Theorem was not settled by a vote or a convention — it resisted for 358 years, and then it did not, and nobody involved thought they had changed the truth by proving it.

Nominalism and fictionalism: there are no mathematical objects

The strongest opposing view says that talk of numbers is a useful fiction. Hartry Field’s Science Without Numbers (1980) is the landmark attempt to make this work: Field tried to show that Newtonian gravitational theory can be reformulated without quantifying over mathematical entities at all, using only spacetime regions and relations among them.

It is serious philosophy and it is not obviously wrong. But two things should be noted. First, the programme has never been extended successfully to quantum mechanics, where the mathematics is far less dispensable. Second, and more importantly for our purposes, fictionalism does not dissolve the puzzle we are dealing with. If mathematics is a fiction, it becomes more baffling, not less, that the fiction predicts the positron.

Theistic conceptualism: mathematics as thought

A third position holds that mathematical truths are necessary thoughts in a necessary mind. Numbers are not free-floating abstract objects, and they are not fictions; they are the contents of an eternal intellect. Greg Welty defends this as “theistic conceptual realism,” and William Lane Craig — who explicitly rejects Platonism — develops a related case in God Over All.

We will come back to why this matters in Part eight. For now, note only that all three positions agree on one thing: whatever mathematics is, it is not a human invention in the way that chess or the metric system is.

A test case

Five hundred million years ago, one trilobite crossed the seabed of a Cambrian sea with no other trilobite anywhere near it. There were no humans, no numerals, no counting. Was it true that there was exactly one? If yes, then at least some mathematical facts did not wait for human beings. The numeral 1 is ours. The quantity is not.

Part three

Eight times the mathematics came first

The following are ordered by when the mathematics was done. In every case, the structure was worked out for internal mathematical reasons — beauty, curiosity, consistency — with no physical application in view. In every case, the physical world turned out to be using it.

Conic sections and the shape of orbits

c. 1,800 years

Around 200 BC, Apollonius of Perga worked out the geometry of curves produced by slicing a cone: circles, ellipses, parabolas, hyperbolas. It was pure Greek geometry, motivated by nothing but the elegance of the problem. There was no reason to expect the sky to care.

In 1609, after six years of failing to fit Mars’s orbit to a circle, Johannes Kepler found that it fitted an ellipse — one of Apollonius’s curves — with the Sun at a focus. Eighteen centuries of abstract geometry turned out to be a description of planetary motion.

Complex numbers and quantum mechanics

c. 350 years

Sixteenth-century Italian algebraists solving cubic equations kept running into square roots of negative numbers. Cardano called them useless; Bombelli, in 1572, worked out arithmetic for them anyway. For centuries they were regarded as a convenient fiction — the name “imaginary” was intended as an insult.

In 1926 Schrödinger wrote down the equation governing quantum systems, and the imaginary unit i was sitting in it, structurally, unremovably. Quantum mechanics cannot be formulated over the real numbers alone without losing content. A fiction invented to tidy up cubics turned out to be the grammar of matter.

Neptune, found on paper

Same night

Uranus was not where Newtonian mechanics said it should be. In 1845–46, Urbain Le Verrier in Paris and John Couch Adams in Cambridge independently asked what unseen body could produce the discrepancy, and calculated where it would have to be.

Le Verrier posted his coordinates to Johann Galle at the Berlin Observatory. Galle received the letter on 23 September 1846, looked that same evening, and found Neptune within about one degree of the predicted position. No new observations of the planet had informed the prediction — only the existing record of Uranus’s orbit and the mathematics of perturbation theory.

Riemannian geometry and general relativity

61 years

In 1854, Bernhard Riemann delivered a lecture on the foundations of geometry in which he described curved spaces of arbitrary dimension. It was regarded as beautiful and entirely abstract. Riemann had no physical application in mind, and none existed.

In 1915, Einstein needed exactly this machinery. General relativity is Riemannian geometry applied to four-dimensional spacetime; without Riemann’s abstract work, the theory could not have been written. Einstein did not invent the mathematics he needed. It was already there, sixty years old, waiting.

Gravitational waves

99 years

In 1916 Einstein derived, from the field equations, that accelerating masses should radiate ripples in spacetime itself. The predicted effect was so small that he doubted for decades whether it was physically real or a mathematical artefact.

On 14 September 2015, the two LIGO detectors recorded a signal from a pair of black holes that had merged more than a billion years earlier. The measured distortion was a change in a four-kilometre arm length of about one part in 10²¹ — roughly a thousandth of the width of a proton. The waveform matched the equations.

Dirac’s equation and antimatter

4 years

Dirac’s 1928 relativistic equation for the electron had solutions with negative energy. They appeared to be a fault in the theory. Dirac first tried to identify them with the proton, then in 1931 concluded they described something genuinely new: a particle identical to the electron but oppositely charged.

Carl Anderson photographed exactly such a track in a cloud chamber and published it in 1933. The positron had been deduced from the internal demands of an equation before anyone had reason to think antimatter existed.

Group theory and the omega-minus

3 years

By 1961 particle physics was drowning in newly discovered particles with no evident organisation. Murray Gell-Mann and Yuval Ne’eman independently noticed that they could be sorted using the symmetry group SU(3) — an object from nineteenth-century pure algebra with no connection to physics.

The scheme left one slot empty. Gell-Mann said a particle must exist to fill it, and specified its charge, strangeness and mass in advance. In February 1964 the omega-minus was found at Brookhaven, with the properties the symmetry required. This is perhaps the cleanest case of all: the prediction came not from a physical model but from the shape of an abstract group.

The Higgs mechanism

48 years

In 1964 three independent groups — Robert Brout and François Englert; Peter Higgs; Gerald Guralnik, Carl Hagen and Tom Kibble — published within months of each other on how a field could give mass to particles that gauge symmetry required to be massless.

The Large Hadron Collider was built, at a cost of billions, substantially to test a mathematical consistency requirement from 1964. On 4 July 2012, ATLAS and CMS announced a new boson at about 125 GeV. Two of the theorists were still alive to see it.

There is a pattern here that ought to be uncomfortable for anyone. In none of these cases did mathematicians tailor their work to physics. In several, the mathematics was explicitly prized for being useless. And in every case the physical world was found to be running on it.

Part four

Wigner’s problem, stated precisely

Eugene Wigner, Nobel laureate in physics, published the classic statement of this puzzle in 1960 under the title The Unreasonable Effectiveness of Mathematics in the Natural Sciences. It is worth being exact about what he claimed, because the paper is often cited loosely.

Wigner’s argument has two stages. The first is that mathematical concepts are chosen for their aesthetic and formal properties, not for their usefulness in describing nature. The second is that these concepts nevertheless describe nature with accuracy far beyond anything the original evidence could justify — and, crucially, that they keep doing so when extrapolated into regimes no one had tested.

His conclusion was that the appropriateness of mathematical language to physics is a gift we neither understand nor deserve, and he added that we should be grateful for it and hope it continues to hold. He offered no explanation, and said so.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

Albert Einstein, Geometry and Experience, address to the Prussian Academy of Sciences, 27 January 1921

Mark Steiner extended the argument considerably in The Applicability of Mathematics as a Philosophical Problem (1998). Steiner’s move is important: he distinguishes the general fact that mathematics is useful — which a naturalist can perhaps explain — from the specific fact that physicists have repeatedly succeeded by making purely formal analogies, reasoning from mathematical structure alone to novel physical conclusions. Steiner argues this second phenomenon looks anthropocentric in a way naturalism should find hard to accommodate: the universe appears to be arranged so that human aesthetic preferences in mathematics are a reliable guide to physical truth.

Part five

Mathematics in nature: what survives scrutiny

This is the part of the argument most often overstated, so it is worth separating the claims that hold from the ones that do not. We will do the discarding first.

Claims that should be retired

Popular claim	Why it fails
The nautilus shell is a golden spiral	It is a logarithmic spiral, but measured growth ratios cluster around 1.31–1.33 per revolution, not 1.618. Clement Falbo measured a large sample and found none matching φ.
DNA’s dimensions encode φ (34 Å / 21 Å)	The 34 Å helical pitch is right. The width is standardly given as about 20 Å. The ratio only yields φ if you select the figure that produces φ.
The Parthenon was designed on the golden ratio	No ancient source mentions it. The proportion appears only if you choose which edges to measure from. George Markowsky catalogued this and several related errors in 1992.

These claims circulate widely in apologetics, and they are damaging. A reader who checks one of them and finds it false will reasonably assume the rest of the argument was assembled the same way. The genuine examples do not need the help.

Claims that hold

Fibonacci phyllotaxis. Seed and leaf spirals in sunflowers, pine cones, pineapples and daisies count out consecutive Fibonacci numbers with high reliability — 8 and 13, 34 and 55, 55 and 89. This is measured, not asserted.

There is an honest complication, and it should be stated. In 1992 Stéphane Douady and Yves Couder showed that Fibonacci spirals emerge spontaneously from a simple physical process: drops of magnetic fluid repelling one another as they are added at a centre at regular intervals settle into exactly these arrangements. There is a mechanism, and it is not mysterious.

Why a mechanism does not close the question

Finding the mechanism relocates the puzzle rather than removing it. Douady and Couder showed that a system minimising repulsion converges on the golden angle. But that is a mathematical fact about optimisation — it is true because φ is the most irrational number, the hardest to approximate by ratios, and therefore the packing angle that avoids overlap longest. The plant is not obeying an arbitrary rule. It is landing on a theorem. The question was never how living things reach the optimum; it is why an optimum specified by number theory is what physical growth converges on.

Honeycomb. The hexagonal grid is the least-perimeter way to divide a plane into equal-area cells. Bees have built it for tens of millions of years. Mathematicians conjectured it for roughly two thousand and only proved it in 1999, when Thomas Hales published a demonstration of the honeycomb conjecture. Here again the physics is understandable — surface tension and packing — and here again the endpoint is a theorem that took human beings two millennia to establish.

Periodic cicadas. North American Magicicada emerge on 13-year and 17-year cycles. Both are prime, and no non-prime cycles occur. The standard explanation is selective: prime cycles minimise the frequency of coincidence with predator population cycles and reduce hybridisation between broods, as Yoshimura argued in 1997.

We flag this one as weaker for our purposes, because a straightforward Darwinian account is available. It belongs in the file as an example of number theory appearing in biology, not as evidence that selection cannot produce it.

Scaling laws. Metabolic rate across organisms scales with body mass to the power of three-quarters — Kleiber’s law, holding across roughly twenty-one orders of magnitude from bacteria to whales. A quarter-power exponent is a strange thing to find in biology, and West, Brown and Enquist derived it in 1997 from the geometry of branching transport networks. Once more: a mathematical structure sitting underneath living systems that nobody put there deliberately.

Part six

Euler’s identity, framed correctly

e^iπ + 1 = 0 Five constants from five unrelated branches of mathematics

This identity is regularly presented in apologetics as though five numbers had improbably lined up, and someone must have arranged it. That framing is a mistake, and a mathematically literate reader will spot it immediately.

Euler’s identity is a theorem. It is a special case of Euler’s formula, e^iθ = cos θ + i sin θ, which Feynman called the most remarkable formula in mathematics in his Lectures on Physics. Given the definitions of e, of π, and of complex exponentiation, the identity could not have been otherwise. It is necessary, not contingent. Nobody arranged it, because there was nothing to arrange.

But now notice what has actually been established, because it is stronger than the version being discarded. What Euler’s formula shows is that exponential growth, circular rotation and the imaginary unit are not three subjects. They are one subject seen from three directions. The unity was there before anyone noticed it, and it was not put there by the noticing.

And then this necessary, self-contained structure — developed entirely without reference to the physical world — turns out to be the mathematics of every oscillation in nature: alternating current, wave optics, signal processing, and the quantum wavefunction. The mystery is not internal to mathematics. The mystery is that the physical universe is built on the same necessities.

Part seven

The three strongest naturalist replies

An argument is only worth as much as the objections it can survive. Here are the three best, stated as strongly as we can put them.

Objection one · The selection effect

Mathematics is the study of all possible abstract structure. The physical world has some structure. Of course some branch of mathematics fits it — you are astonished that a key found in an infinite key shop opens your door.

This is the most common reply and it has real force. But it does not fit the record. If mathematics were merely a vast catalogue from which physicists select after the fact, we would expect physics to be fitted to mathematics retrospectively. What actually happens repeatedly is the reverse: the mathematics is chosen for internal reasons, and it then predicts things that had not been observed.

The catalogue reply cannot account for the omega-minus. Gell-Mann did not survey SU(3) after the particle was found. The symmetry left a hole, he specified what would fill it, and three years later it was there with the stated mass. A catalogue does not do that. Neither does it explain why physics keeps needing the mathematically elegant option rather than an arbitrary one from the shelf.

Objection two · Evolutionary epistemology

Our brains were shaped by selection to model the world. Mathematics is what that modelling capacity looks like when it runs off-leash. No wonder it fits — it was built to fit.

Persuasive for arithmetic and Euclidean geometry, which plausibly track features of the mid-sized environment our ancestors navigated. It becomes very weak very quickly beyond that.

Selection pressure on the African savannah did not include Hilbert spaces, non-commutative operator algebras, eleven-dimensional manifolds or SU(3) symmetry. These are exactly the regions where mathematics has been most spectacularly predictive, and they are exactly the regions where no ancestor ever had to survive anything. An evolutionary account explains why we can count sabre-tooth cats. It does not explain why a mind tuned for counting cats can derive antimatter.

Objection three · The mathematical universe hypothesis

Mathematics describes physical reality perfectly because physical reality is a mathematical structure. Max Tegmark’s proposal: all mathematical structures exist, and ours is one of them.

This is the most interesting response, and it should be taken seriously. Note first what it concedes: Tegmark agrees the fit is too good to be an accident and needs explaining at the deepest level. He has abandoned the idea that mathematics is a human construct entirely.

What it costs is considerable. To avoid a mind behind the mathematics, Tegmark posits the actual existence of an infinity of universes, most of them unobservable in principle. He must also explain why we find ourselves in an unusually orderly one, which requires anthropic reasoning of a kind many physicists regard as unfalsifiable. And he must answer a prior question he cannot answer within his own framework: why is there a concrete world instantiating the structure rather than nothing at all? Abstract structures do not have causal powers. A description of a universe does not produce one.

Set beside the alternative, the ledger is worth reading honestly. One explanation requires an infinite unobservable multiverse plus a bare brute fact about instantiation. The other requires a mind. Neither is directly testable. Only one has explanatory economy on its side.

Part eight

Why Platonism alone is not enough

This is the point at which a great deal of Christian apologetics on this topic stops too early, and it is worth being candid about it.

Suppose the argument so far succeeds and mathematical realism is established: mathematical truths are objective, mind-independent, necessary, and hold prior to and independently of the physical universe. A thoughtful atheist can accept every word of that and remain an atheist. Hardy did. Gödel was a theist, but his Platonism was not what made him one. Platonism about mathematics is not theism, and pretending otherwise is not an argument.

Worse, a straightforward Platonism creates a problem for theology. If numbers, sets and propositions exist necessarily, eternally, and independently of God, then God is not the source of all reality. There is a second eternal realm he did not make and cannot alter. This is why Craig — who is not usually accused of conceding ground to naturalists — rejects Platonism outright in God Over All: he regards it as incompatible with divine aseity, the doctrine that God alone exists a se, from himself.

The stronger position

Theistic conceptual realism holds that mathematical truths are necessary thoughts in a necessary mind. They are objective and mind-independent with respect to human minds — which is everything the arguments above required — while not constituting an eternal realm outside God. Numbers are not creatures, and they are not rivals. They are what an infinite intellect thinks.

On this account the puzzle we began with dissolves rather than deepens. If the physical universe was thought before it was made, by a mind whose thinking is what mathematical truth consists in, then of course the universe is mathematically describable, and of course a rational creature made in the image of that mind can partly read it. Wigner’s unreasonable effectiveness stops being unreasonable. It becomes exactly what you would predict.

This is what makes the argument evidential rather than merely rhetorical. Theism predicted a mathematically intelligible universe accessible to human reason before anyone knew whether it was. Naturalism did not, and has spent sixty-five years since Wigner offering accounts of why it should not have been surprised.

Part nine

Where this leaves us

Nothing here is a proof, and it should not be presented as one. What it is, is an explanatory comparison, and those are how rational people settle most questions that matter.

The facts to be explained are these. Structures developed for their internal elegance, with no physical motivation, describe the physical world to twelve decimal places. They do so predictively, across gaps of four to eighteen hundred years. The same small set of mathematical solutions recurs in plants, in insects, and in galaxies. Mathematics invented as an admitted fiction turned out to be the grammar of matter. And the human mind, which had no evolutionary reason to be able to do any of this, can.

You can hold that this is all a very large coincidence which requires no account. You can pay for it with an infinite multiverse. Or you can conclude that the universe is written in mathematics because it was thought by a mind for which mathematics is not an invention but a native language.

Nobody is obliged to find the third option compelling. But it is not the unreasonable one, and it is worth noticing that the alternative is not really an explanation — it is a decision to stop asking.

An author who speaks the language

No one who does not know English writes an English novel. Output requires the competence behind it. When archaeologists find geometric figures cut into stone, they do not conclude that wind produced them. If a radio telescope received the first hundred primes in sequence, no astronomer would file it as background noise — SETI has been built around that inference for sixty years.

An unguided expansion of matter and energy has no mathematical competence and no capacity to prefer one structure over another. The universe it is said to have produced is fluent in a mathematics so far ahead of us that its conclusions routinely predate the physics they describe.

When he prepared the heavens, I was there: when he set a compass upon the face of the depth… then I was by him, as one brought up with him. Proverbs 8:27, 30, KJV

References

Hardy, G. H. A Mathematician’s Apology. Cambridge University Press, 1940, §22.
Wigner, E. P. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications in Pure and Applied Mathematics 13, no. 1 (1960): 1–14.
Einstein, A. Geometrie und Erfahrung. Address to the Prussian Academy of Sciences, Berlin, 27 January 1921.
Steiner, M. The Applicability of Mathematics as a Philosophical Problem. Harvard University Press, 1998.
Gödel, K. “What Is Cantor’s Continuum Problem?” Revised version in Benacerraf & Putnam, eds., Philosophy of Mathematics: Selected Readings. 1964.
Penrose, R. The Road to Reality: A Complete Guide to the Laws of the Universe. Jonathan Cape, 2004.
Field, H. Science Without Numbers: A Defence of Nominalism. Princeton University Press, 1980.
Balaguer, M. Platonism and Anti-Platonism in Mathematics. Oxford University Press, 1998.
Tegmark, M. “The Mathematical Universe.” Foundations of Physics 38, no. 2 (2008): 101–150.
Craig, W. L. God Over All: Divine Aseity and the Challenge of Platonism. Oxford University Press, 2016.
Welty, G. “Theistic Conceptual Realism.” In P. M. Gould, ed., Beyond the Control of God? Six Views on the Problem of God and Abstract Objects. Bloomsbury, 2014.
Dirac, P. A. M. “Quantised Singularities in the Electromagnetic Field.” Proceedings of the Royal Society A 133 (1931): 60–72.
Anderson, C. D. “The Positive Electron.” Physical Review 43 (1933): 491–494.
Abbott, B. P., et al. (LIGO Scientific Collaboration and Virgo Collaboration). “Observation of Gravitational Waves from a Binary Black Hole Merger.” Physical Review Letters 116 (2016): 061102.
Higgs, P. W. “Broken Symmetries and the Masses of Gauge Bosons.” Physical Review Letters 13 (1964): 508–509. See also Englert & Brout, PRL 13 (1964): 321–323, and Guralnik, Hagen & Kibble, PRL 13 (1964): 585–587.
Gell-Mann, M. The Eightfold Way: A Theory of Strong Interaction Symmetry. California Institute of Technology Report CTSL-20, 1961.
Hales, T. C. “The Honeycomb Conjecture.” Discrete & Computational Geometry 25 (2001): 1–22.
Douady, S., and Y. Couder. “Phyllotaxis as a Physical Self-Organized Growth Process.” Physical Review Letters 68 (1992): 2098–2101.
Markowsky, G. “Misconceptions about the Golden Ratio.” The College Mathematics Journal 23, no. 1 (1992): 2–19. See also Falbo, C. “The Golden Ratio — A Contrary Viewpoint.” The College Mathematics Journal 36, no. 2 (2005): 123–134.
West, G. B., J. H. Brown, and B. J. Enquist. “A General Model for the Origin of Allometric Scaling Laws in Biology.” Science 276 (1997): 122–126. See also Yoshimura, J. “The Evolutionary Origins of Periodical Cicadas During Ice Ages.” The American Naturalist 149, no. 1 (1997): 112–124.

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