Chance, Necessity and Convention
Seven essays on whether mathematics, physics and biology are found or made turn out to have been circling the oldest pair in natural philosophy: chance and necessity. They also turn out to need a third term the pair does not have — and the third term is where the disciplines differ.
In 1970 Jacques Monod, who had shared a Nobel Prize five years earlier for working out how bacteria switch their genes on and off, published a short book with a long argument and a two-word title. Le Hasard et la Nécessité opens with an epigraph attributed to Democritus — everything existing in the universe is the fruit of chance and necessity — and spends two hundred pages claiming that biology, properly understood, has nothing else in it.[1] Necessity is the physics and chemistry an organism obeys without choosing to. Chance is the mutation, which arrives with no relation to what will later prove useful and is only afterwards sorted by selection. There is no third thing. There is no purpose in the molecules, no direction in the history, and the appearance of both is what you get when a blind process runs long enough through a rigid medium.
The seven essays that precede this one asked a different-sounding question — is mathematics found or made, and physics, and life — and arrived, seven times, at a division of labour: the structure is found, the vocabulary is made. It is worth asking, at the end, whether that was Monod's pair under other names. The answer is that it was, for biology, almost exactly; that it was not, for mathematics, in a way that matters; and that the reason it was not is that Monod's pair has two places and the sciences need three. The third place is the one this afterword is about.
Two kinds of having-to-be
Monod's necessity is physical. The bacterium's regulatory switch works because of the chemistry of a protein binding to DNA, and the chemistry does not consult the bacterium. Everything the biology essay in this series filed under “found” is necessity of that kind: Turing's stripes come from two reaction rates and two diffusion rates; Kleiber's three-quarters comes, if West and his colleagues are right, from the geometry of a branching network in three dimensions; the eye has been invented forty times because there are few ways to focus light. These are laws of the world, and the world could in principle have had others.
The necessity the mathematics essays met is not that. When Frobenius proved in 1877 that there are exactly three finite-dimensional associative division algebras over the reals, he was not reporting a fact about this universe.[2] The constraint holds in any universe, or in none; it is not a law that matter obeys but a limit on what can be consistently said. Goodstein's sequences terminate whether or not anything exists to compute them. The large cardinals line up by consistency strength in a hierarchy no one designed and no physics could disturb. Hales's honeycomb theorem — that the hexagon is the least-perimeter way to partition a plane — is why the bees are right, but the bees could be abolished and the theorem would stand.[3]
So the “found” column of the series holds two things Monod's word runs together: necessity of law, which the world happens to have, and necessity of structure, which it could not lack. The distinction is old — it is roughly Leibniz's contingent and necessary truths — and the series did not need it until now. It needs it now because the second kind of necessity is what makes mathematics applicable at all: the world instantiates structures, and the structures it instantiates are constrained by what structures there can be. Monod's necessity explains why the stripe has a wavelength. The other kind explains why there are only so many kinds of stripe.
Chance, and the thing that is not chance
Monod's chance is the mutation, and the biology essay's “made” column is full of it: the population in Lenski's flask that found the citrate because of an earlier mutation that did nothing visible on its own, the inverted vertebrate retina that no engineer would draw, the ribosome's fifty proteins accreted around a core from a world before proteins.[4] None of these was chosen. Each is what a blind process left behind, and Monod's word for it is the right one.
Now look at the “made” column of the mathematics essays. Bombelli extended the number system. Hilbert chose axioms. Cohen built universes in which the continuum hypothesis fails and universes in which it holds. Lakatos's mathematicians renegotiated the definition of a polyhedron, four times, to keep a theorem true. Feynman drew lines for electrons and wavy lines for photons and could have drawn them the other way round. Not one of these was a mutation. Each was a decision, made by someone, for reasons, and each could have gone otherwise for other reasons. That is not chance. It is the thing Monod's pair has no word for, and the word the tradition gives it is convention.
The doctrine has an author. Henri Poincaré argued in 1902 that the axioms of geometry are neither synthetic a priori truths, as Kant had said, nor facts of experience, as Mill had said: they are conventions — definitions in disguise, adopted for convenience, which experience can guide but cannot refute.[5] To ask whether Euclidean geometry is true, he said, has no more sense than to ask whether the metric system is true. One geometry can only be more convenient than another. The second essay in this series told the story of the parallel postulate, Beltrami's proof of its independence, and Einstein's physics choosing the non-Euclidean option, without naming the philosophy that had been built on exactly those events; it should have, because Poincaré's conventionalism is the “made” pole of the whole series stated as a thesis.
Rudolf Carnap pushed it further in 1934 with his principle of tolerance: in logic there are no morals, everyone is free to build their own system, and the only obligation is to say clearly what the rules are.[6] Quine replied in 1936 that convention cannot be the whole story, because to derive anything from conventions you already need logic, which cannot therefore itself be a convention.[7] Einstein had made the analogous point about geometry in 1921: Poincaré is right that geometry alone is conventional, but geometry plus physics is not, and the pair together can be tested.[8] The series arrived at the same place by its own route. We choose the rules; what follows from them is not up to us. Convention names the choosing. Necessity names what it lets loose.
So “made” is two things that look alike from a distance and are opposites up close. Chance is what history does without anyone deciding. Convention is what someone decides, with reasons. Both produce contingency — things that could have been otherwise — but one is blind and one is not, and the difference between a blind contingency and a chosen one is the difference between biology and mathematics.
Three places, three sciences
Put the three terms at the corners of a triangle and each science lands somewhere inside it.
The table is a claim, not a survey, and two of its cells deserve a note. Mathematics has almost no chance in it because its objects do not have histories in the relevant sense: the complex numbers were discovered in a particular century by particular people, but nothing about the complex numbers records that. Biology had almost no convention in it until, in 2013 and 2019, laboratories rewrote the genetic code and the bacteria grew.[9] Those experiments are philosophically stranger than they are usually described, because they are the first time the convention column has been opened in a science that had only ever had the other two. Jason Chin chose a sixty-one-codon code the way Hamilton chose the quaternions.
Two remarks and a stalemate
The triad settles a puzzle the first essay raised and the biology essay repeated. Eugene Wigner asked in 1960 why mathematics is unreasonably effective in physics;[10] Israel Gelfand is reported to have said that the only thing more unreasonable is its ineffectiveness in biology.[11] Both remarks have the same explanation. Mathematics describes the necessity column and nothing else. It cannot describe convention, because convention is what one does before the mathematics starts, and it cannot describe chance, because chance is precisely what has no structure to describe. Physics is mostly necessity and some convention, so mathematics fits it nearly everywhere. Biology is mostly chance and some necessity, so mathematics fits it exactly where the biology essay found it fitting — in the forms, not the parts — and nowhere else. Neither effectiveness nor ineffectiveness is unreasonable. They are the widths of a column.
It also names the stalemate at the centre of the second essay. Gödel and Woodin hold that the axioms of set theory are, in the end, necessity: that the right ones will force themselves on us as true. Hamkins holds that they are convention: that the multiverse of set-theoretic universes is the whole story and the choice among them is ours. Eighty years after Cohen the question is open, and the triad shows why it is hard: it is a question about which corner of the triangle a whole discipline sits nearest, and there is no experiment that measures that. What there is, in mathematics, is the observation that conventions chosen for convenience keep turning out to have consequences nobody chose — Frobenius again — which is the evidence that convention has been tracking necessity all along.
And it relocates a question from physics. The first essay ended on four decades of fundamental physics as an unresolved experiment in whether beauty is evidence. The Feynman essay found convention under the diagrams and necessity under the convention. The remaining case is the constants of nature: the fine-structure constant, the masses, the cosmological constant that Steven Weinberg argued in 1987 could be as large as it is only in a universe where galaxies had managed to form.[12] If that argument is right, the constants are chance — Monod's mutation, at the scale of universes — and physics has a third column after all. If a final theory fixes them, they are necessity. Nobody knows which, and the question is Monod's question one floor up.
Convention is a bet on necessity
The three corners are not independent, and the way they depend on each other is the last thing the series has to say.
Convention is choice with reasons, and the reasons the mathematicians and physicists give are the ones the first and fifth essays examined: convenience, elegance, economy, the sense that a definition is right. Poincaré said conventions are adopted because they are convenient; Dirac said equations should be beautiful before they are true.[13] Both are describing the same act, and the act is a wager: that a rule chosen for its shape will turn out to have consequences that fit. The history of the complex numbers, of non-Euclidean geometry, of the diagrams, is the history of that wager paying. The history of Kepler's solids and SU(5) is the history of it failing. Convention, in other words, is how a finite mind gets at necessity: it cannot survey the space of all consistent rules, so it picks by beauty and then discovers what it picked. When the picking is good the convention disappears into the necessity and looks, in retrospect, found.
Chance has no such relation. It does not track anything; it is what is left when neither necessity nor choice has fixed an outcome. That is why biology's made things — the retina, the ribosome, the citrate — do not look found in retrospect and never will. They look like what they are, which is history. The synthetic biologists are interesting precisely because they have started replacing chance with convention in the one science that had none: choosing a code instead of inheriting one, and then discovering, as Bombelli did, what the choice entails.
So the division of labour the series proposed in its first essay was right, and incomplete. The structure is found; that is necessity, and in mathematics it is necessity of the deeper kind. The vocabulary is made; but made is two things, and which of the two it is decides what kind of science one is doing. Where the vocabulary is chosen, the science can hope that its choices will be absorbed into what it finds, and mathematics and physics have spent their histories watching that happen. Where the vocabulary is inherited from a blind process, the science can only catalogue it, and the catalogue is most of biology.
Democritus, if the fragment is his, had two words for what there is. The sciences have needed three. The third is the one that lets a mind that did not make the world make a start on it.
Where this could go
The constants, decided. Whether the cosmological constant and the rest are necessity, convention or chance is the most consequential open cell in the table, and there is a real literature on each reading: final-theory programmes, natural-unit arguments, anthropic and multiverse arguments from Weinberg onward. An essay that sorted the constants one by one into the three columns, with the evidence for each, would be the physics companion this afterword lacks.
Poincaré, Einstein, Carnap, Quine. The philosophical dispute over convention is a century old and largely settled in Quine's favour among philosophers, while working mathematicians go on behaving like Carnap. The gap between what the philosophy concluded and what the practice does is itself a subject, and the second essay's Hamkins–Woodin stalemate is where it is being fought now.
Measuring the columns. Lenski's replays are the only experiment that puts a number on chance against necessity in a living system: how often does a rerun find the same solution? The same question can be asked of the history of mathematics — how often were the same objects discovered independently, and how alike were they? — and of physics, where the double copy and the amplituhedron suggest that different conventions have been finding the same structure. A comparative essay on convergence across the three sciences would test the table.
The convention column of biology. Recoded genomes, xenobiology with unnatural base pairs, minimal synthetic cells: biology is acquiring the column it never had, and nobody has asked what that does to Monod's argument. The frozen accident thaws; the question is what freezes next.
Beauty as a wager. If convention is a bet on necessity and beauty is how the bet is placed, then the first essay's unmeasured base rate — how often aesthetic choices have paid off — is the number that would tell us how good the mind's access to necessity actually is. It is still nobody's project.
This piece was written collaboratively with Claude Fable 5.1 (Anthropic): human specification, editorial direction and critical review; machine synthesis, drafting and figure generation. It closes the Found or Made series and was produced the same way as its parts.
The figure is an argument drawn as a diagram: the positions of the three sciences inside the triangle are the essay's claim, not coordinates derived from anything, and the caption says so. The table is likewise a proposal.
The Democritus epigraph is quoted as Monod quotes it; whether the sentence is genuinely Democritean is doubted by classicists, and the text hedges accordingly. Gelfand's remark is reported, not published. Weinberg's 1987 anthropic bound is presented as one reading of the constants among several, and the essay takes no position on the multiverse. The reading of Poincaré, Carnap and Quine is compressed to a paragraph each and should be read as a pointer to the primary texts, not a summary of them.
The claim that “made” splits into chance and convention, and that this split is what separates the sciences, is the authors' synthesis. None of the cited writers put it that way.
Authored by: Luis Matos Ferreira — Physicist, Developer, Writer
- Invented, Then Unavoidable — imaginary numbers, rigidity, and why physicists trust beauty.
- The Limits of the Rules — Gödel, Goodstein, the continuum, and what happens when the axioms run out.
- What a Proof Is Now — Lakatos, the four colour theorem, Lean, and proofs no one has read.
- Where Number Comes From — the number sense, cultures without counting, and whether aliens would share our arithmetic.
- The Bus at Coutances — how mathematicians and physicists actually think: images, incubation, birds and frogs, and minds without pictures.
- The Diagram That Did the Sum — companion: Feynman's diagrams, and the shape underneath them.
- The Frozen Accident — companion: biology, made by history and found by physics.
- Chance, Necessity and Convention — afterword. This essay.
- Monod, Le Hasard et la Nécessité, 1970 (English: Chance and Necessity, 1971); the epigraph is attributed to Democritus and is not found among the surviving fragments.
- Frobenius, real division algebras, 1877.
- Hales, “The honeycomb conjecture”, Discrete & Computational Geometry, 2001.
- Blount, Borland & Lenski, “Historical contingency and the evolution of a key innovation in an experimental population of Escherichia coli”, PNAS, 2008.
- Poincaré, La Science et l'Hypothèse, 1902, chs. III–V; Kant, Critique of Pure Reason, 1781; Mill, A System of Logic, 1843.
- Carnap, Logische Syntax der Sprache, 1934, §17 (the principle of tolerance).
- Quine, “Truth by Convention”, 1936.
- Einstein, “Geometrie und Erfahrung”, 1921.
- Lajoie et al., “Genomically recoded organisms expand biological functions”, Science, 2013; Fredens et al., “Total synthesis of Escherichia coli with a recoded genome”, Nature, 2019.
- Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, 1960.
- Gelfand, as reported by Borovik in Mathematics under the Microscope, 2010.
- Weinberg, “Anthropic bound on the cosmological constant”, Physical Review Letters, 1987.
- Dirac, “The Evolution of the Physicist's Picture of Nature”, Scientific American, 1963.
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