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Is water H₂O? (3): How chemists learned to count invisible atoms

16 September 2026 at 14:00

Author: José Luis Granados Mateo is a postdoctoral researcher in the Department of Philosophy at the University of the Basque Country (EHU) and a member of the Integrated History and Philosophy of Science (iHPS) research group. His work focuses on history and philosophy of science, science and values, and the epistemology of scientific practices.

 

Water is H₂O. Few scientific formulas look more familiar, more compact, or more final. Two atoms of hydrogen, one of oxygen: the phrase seems to say what water really is beneath the shifting appearances of ice, vapour, rain, rivers and the liquid in a glass.

Yet that little subscript was not easy to earn.

By the early nineteenth century, chemists had good reasons to treat water as a compound of hydrogen and oxygen. The old idea of water as a simple element had lost its authority, as we saw in Is Water H₂O? (1) From Element to Compound. Electrolysis then gave that claim a new experimental force, making hydrogen and oxygen appear under electrical action, although not without raising puzzles of its own, as discussed in Is Water H₂O? (2) The Puzzle of Water Electrolysis. What remained unsettled was a more delicate question. Not whether water had constituents, but how many of each entered one molecule.

Was water HO? H₂O? Something else?

Today the answer feels almost automatic. In the first decades of atomic chemistry, it was not. Dalton’s atomic theory made HO the natural formula: one atom of hydrogen, one atom of oxygen. Avogadro proposed something much closer to the modern answer very early, but his proposal did not settle the matter. For more than half a century, some of the best chemists in Europe disagreed over the formula of one of the most ordinary substances in the world.

The confusion was public enough to become comic. In the 1860s, when agreement was finally beginning to form, an author identifying himself as a certified lunatic at Hanwell Asylum wrote to Chemical News mocking the chemists’ inability to agree even about water. One could contemplate the bewildering variety of formulas, he suggested, until the brain first became confused, then swam, and finally softened.

The joke had bite because the embarrassment was real. Chemical tables were full of combining weights, volumes, radicals, equivalents and atomic symbols. Water had long ceased to be elementary. Yet the old difficulty remained: to know the formula, chemists had to count atoms; to count atoms, they would have had to see them. And they could not.

Liebig put the matter painfully in 1851. Chemists possessed no direct means of ascertaining the number of atoms even in the simplest compound, because that would require seeing and counting them. Atomic chemistry needed numbers for entities no one could inspect.

So the problem was not that the answer lay beyond all reach. It lay beyond direct reach. Chemists had to learn to count invisible things by making them answer to operations that could be performed.

 

A number, and two worlds

The difficulty begins with a ratio.

Hydrogen and oxygen combine in fixed proportions by weight. In round modern numbers, water contains one part hydrogen for eight parts oxygen. Chemists could establish the ratio; the difficulty began when that ratio had to be translated into atoms.

Suppose water is HO. One atom of hydrogen combines with one atom of oxygen. Since the oxygen contribution weighs eight times as much as the hydrogen contribution, the oxygen atom must weigh eight times as much as the hydrogen atom.

Now suppose water is H₂O. Two atoms of hydrogen combine with one atom of oxygen. The same measured ratio now yields a different atomic weight: oxygen must weigh sixteen times as much as hydrogen.

The laboratory ratio has not changed. The invisible architecture has.

That was the circular trap. Molecular formulas required atomic weights; atomic weights required molecular formulas. Measurement gave combining proportions, not little labelled particles waiting to be counted. A consistent chemistry could be built with HO and oxygen as 8. Another could be built with H₂O and oxygen as 16. Once the difficulty spread across many compounds, chemists were no longer facing a local uncertainty about water. They were choosing among possible atomic worlds.

The difficulty appeared at the worst possible place: the point where atomic chemistry had to begin producing reliable numbers.

 

Dalton’s necessary guess

Dalton knew that atomic theory needed a starting point.

He approached water from within a broader problem. Chemical substances combined in fixed proportions; in some cases, the same elements combined again in different but still simple ratios. Atomic theory offered a machinery for these regularities. If matter consisted of particles with definite weights, then chemical combination could be understood as their joining in determinate numbers, and the arithmetic of chemistry began to point towards an unseen order. What it still failed to provide was the count itself: the number of atoms entering each compound.

Dalton therefore introduced rules of simplicity. They were not decorative assumptions added to an otherwise complete theory. They were what allowed the theory to produce numbers at all. When only one compound of two elements was known, Dalton assumed the simplest possible atomic constitution: one atom of each. Since water was the known compound of hydrogen and oxygen, its formula became HO. Hydrogen could be assigned atomic weight 1, oxygen 8, and the system could begin to move.

Judged from later chemistry, Dalton’s rule has an air of innocence. It was a methodological expedient rather than an experimental result, a way of allowing atomic theory to proceed when direct counting was unavailable. Dalton was not looking into water molecules and counting badly; he was working with weights, tables and regularities, in a period before spectroscopy, quantum theory or a settled molecular chemistry had made atoms answerable in other ways.

HO was wrong. It was not random. It belonged to a method that could bring order to combining weights before chemistry had better ways of counting atoms.

H₂O
Figure 1. Dalton’s atomic symbols, from A New System of Chemical Philosophy (1808), Plate IV. Dalton’s table did not merely decorate atomic theory; it made atoms calculable through symbols, weights and simple combinations. In this scheme, water or steam appears as a binary compound of one atom of oxygen and one atom of hydrogen. Source: Wikimedia Commons, public domain.

 

What gas volumes seemed to promise

Another route opened through gases.

Volumes could be measured, and gases combined with a striking regularity. Gay-Lussac showed that gases entered reactions in simple volume ratios. Two volumes of hydrogen combine with one volume of oxygen to form water vapour; other gases display similarly neat relations. To anyone looking for a way to count invisible particles, this was tempting.

If equal volumes of gases contained equal numbers of particles, volume could become a counting device. It would not show atoms individually, but it might give chemists a way to infer how many particles were involved.

Avogadro made that move. Equal volumes of gases, under the same conditions, contain equal numbers of molecules. Grant that, and the formation of water vapour changes shape. Two volumes of hydrogen and one volume of oxygen produce two volumes of water vapour. The arithmetic works if hydrogen and oxygen gases are themselves molecular, H₂ and O₂, and if water is H₂O:

2H₂ + O₂ → 2H₂O

H₂O
Figure 2. Opening page of Avogadro’s 1811 essay on the relative masses of elementary molecules. Avogadro’s hypothesis made gas volume into a possible way of counting molecules: equal volumes of gases, under the same conditions, could be treated as containing equal numbers of molecules. The proposal pointed towards H₂O, but it did not immediately command the field. Source: Amedeo Avogadro, ‘Essai d’une manière de déterminer les masses relatives des molécules élémentaires des corps…’, Journal de Physique, 73, 1811; Wellcome Collection / Classic Chemistry.

There it is, almost modern chemistry.

Yet the hypothesis did not conquer the field. The reasons were not merely psychological, nor reducible to conservatism. Avogadro solved one problem while creating others that looked serious at the time. Dalton saw no obvious reason why atoms of the same element should join into stable pairs. In his picture, atoms were surrounded by repulsive atmospheres of caloric; piling like atoms together was hardly the natural assumption. For electrochemical dualists, the difficulty sharpened further. Like atoms ought to have like electrical characters, and like electrical characters ought to repel. Why should oxygen atoms pair with oxygen atoms? Why should hydrogen atoms pair with hydrogen atoms? Why stop at pairs?

The hypothesis also asked chemists to treat elementary gases as molecules containing several atoms of the same element, and to imagine those molecules splitting during chemical combination. To those who trusted weights more readily than invisible gas particles, this was a large demand.

Even sympathetic chemists found the route uneven. Dumas tried, for a time, to develop Avogadro’s approach, but vapour-density results did not behave with perfect neatness. Mercury, phosphorus and sulphur made it hard to treat every elementary vapour on the model of hydrogen or oxygen.

Avogadro had offered a powerful clue. It was not yet the common language of chemistry. Water could have been H₂O in 1811; historically, that was not enough.

 

Atoms with handles

Behind the disagreement over water lay a larger problem. Nineteenth-century chemists were learning how to handle atoms at all: as weights, as volumes, as equivalents, as electrical units, as radicals, as centres of substitution.

For some, atoms were bearers of weight. For others, units in an electrochemical opposition. In Avogadro’s programme, gases offered a way of counting particles by volume. In organic chemistry, atoms and radicals became things that could be moved, exchanged and followed through reactions. Later structural chemistry would give them positions and patterns of connection.

Atoms entered chemistry through these practices unevenly: as things to be weighed, counted, exchanged, classified or connected.

A formula could be useful in one setting and awkward in another. Atomic weights could seem natural when derived from one class of operations and arbitrary when forced into another. Equivalents worked well for neutralisation and replacement. Volumes worked well for gases, though not without trouble. Substitution worked powerfully in organic chemistry. Electrochemical dualism arranged substances by polarity and affinity.

No single practice simply revealed atoms as they were. Each gave atoms a handle.

That is why the history of H₂O is not just a story about choosing the correct formula. The formula became convincing when atoms could be made to enter operations: weighed in one context, counted in another, substituted in a third, arranged elsewhere by their capacities for connection. Seeing was unavailable; work had to do the job of sight.

 

The caution of equivalents

One response to the circular trap was to avoid stepping fully into it.

Many chemists were cautious about atoms. Atomic language sounded metaphysical, or at least more speculative than the discipline required. Equivalent weights appeared safer. One substance could neutralise, replace or combine with another in a definite proportion, and those relations could be measured without deciding what atoms really were or how many of them sat inside a molecule.

Equivalents allowed calculation while holding invisible architecture at arm’s length. A chemist could say how much of one substance answered to another in reaction, without committing to a microscopic count.

This caution was productive. It helped chemists calculate, compare and classify. It also preserved ambiguity. In practice, an equivalent could behave rather like an atomic weight, even when chemists insisted that they were avoiding speculative atomism.

Water once again exposed the difficulty. If oxygen’s equivalent was 8 relative to hydrogen, HO looked natural. If oxygen’s atomic weight was 16, H₂O became natural. The issue depended on the passage from observable combining relations to invisible atomic units.

The appeal to equivalents allowed chemists to postpone bolder claims about atoms, while leaving a theoretical judgment embedded in the passage from measurable reactions to invisible units.

 

The detour through organic chemistry

The decisive work did not come from staring harder at water.

That is one of the surprises in this history. H₂O gained its strength elsewhere: in the movement of radicals, in substitutions, in families of organic compounds. Organic chemistry gave chemists a way to manipulate groups of atoms and to follow their replacements through reactions.

The key idea was the water type.

If water is H₂O, oxygen can be treated as holding two units together. In water, the two units are hydrogens. In alcohols and ethers, one or both hydrogens can be replaced by organic radicals. Water, alcohols and ethers could therefore be seen as members of a family, built around oxygen’s capacity to connect two branches.

At first this sounds like a tidy diagram. It became much more than that.

Williamson’s work on etherification gave the water type experimental substance. The older Liebig-Dumas view treated etherification largely as a removal of water from alcohol under the dehydrating action of sulphuric acid. Williamson rewrote the process as a traffic of groups. Sulphuric acid took an ethyl group from alcohol, leaving water behind; the resulting sulphovinic acid then handed that ethyl group to another alcohol molecule, taking hydrogen in return. Sulphuric acid was regenerated, and ether was produced.

The formula was doing work. It allowed chemists to follow the movement of groups through a sequence of transformations instead of merely recording elemental proportions at the beginning and the end.

Then came the sharper test. If the water-type picture was right, oxygen could hold two different radicals at once. Williamson therefore contrived ethers containing different combinations of methyl, ethyl and amyl groups. These were mixed ethers, not mere mixtures of symmetrical ethers. Under the rival Liebig-Dumas scheme, such mixed ethers should not have appeared as distinct compounds at all; at most, the experiments should have produced mixtures of symmetrical ethers.

Their production gave the water type operational force.

 

H₂O
Figure 3. A modern schematic representation of the Williamson ether synthesis. The diagram is not a reproduction of Williamson’s nineteenth-century notation, but it captures the substitutional logic that made his work so important: ethers could be produced by replacing hydrogen in alcohols with organic radicals. This helped make the “water type” experimentally powerful, since water, alcohols and ethers could be understood as related compounds organised around oxygen’s twofold combining capacity. Source: Wikimedia Commons, CC0.

H₂O gained strength because it travelled. It helped chemists understand not only water, but alcohols, ethers and families of organic compounds. It made substitutions intelligible. It made room for oxygen as an atom with a definite combining capacity.

The problem had widened. Water was no longer an isolated substance to be analysed, but the simplest expression of oxygen’s capacity to bind, exchange and organise chemical groups.

 

Counting by substitution

Substitution reactions gave atoms a kind of experimental visibility.

No one saw oxygen holding two hydrogens. No one watched radicals attach themselves like beads on a wire. But chemists could replace one group with another and see what sort of compound resulted. If an atom or radical could be substituted without the molecule falling apart, that said something about the architecture of the compound. If replacing a central atom broke the molecule into two products, that said something else.

Here the language of valency began to acquire force. Oxygen and sulphur behaved as if they could bind two units. Chlorine behaved differently, as if it had only one hand. The metaphor is crude, but useful; later it would be disciplined into valency.

Water became the simplest member of a larger argument. If oxygen could connect two radicals in ethers, then the two hydrogens in water no longer looked like an arbitrary doubling. They were the simplest expression of oxygen’s twofold combining capacity.

The older picture of a central binding atom would later lose its privileged status. What remained was more durable: a number attached to an element’s capacity for combination.

This was a different kind of counting from Dalton’s simplicity rule or Avogadro’s gas volumes. Dalton counted by assuming the simplest combination. Avogadro counted through equal volumes. Organic chemists counted through substitutions, replacements and families of compounds. None of these methods made atoms visible. Each made them usable in a different way.

H₂O became embedded in a network of operations in which oxygen’s twofold capacity could be tested across many compounds. The formula began to earn its keep.

 

Cannizzaro and the story we like to tell

There is a familiar heroic version of the story. Avogadro discovered the key; chemists forgot it; Cannizzaro revived it at Karlsruhe in 1860; the fog lifted; H₂O won.

There is some truth in this. Not enough.

Cannizzaro’s intervention mattered. His presentation of atomic and molecular weights was clear, forceful and timely. By using Avogadro’s ideas, he gave chemists a way to distinguish atoms from molecules, molecular weights from atomic weights, formulas from combining proportions. His pamphlet circulated after Karlsruhe and influenced younger chemists who were looking for order in a confusing field.

The achievement was real. It was not miraculous.

By the time of Karlsruhe, many of the crucial battles had already been fought elsewhere. Organic chemistry had done a great deal of the work. Substitution, types, radicals and valency had given chemists practical ways to count and sort atoms. Cannizzaro clarified and spread a system that was already becoming workable; he did not unlock the problem by himself.

Nor did his account contain the strongest organic side of the story. Williamson, types, radicals and substitution had already made H₂O useful in ways that a purely Avogadrian argument could not.

Consensus emerged because several practices began to reinforce one another. Volumes, atomic weights, substitution patterns, organic structures and valencies gradually pulled in the same direction. H₂O did not win because an old hypothesis was finally remembered. It won because chemists had learned to make the formula work across a widening range of operations.

A formula becomes harder to doubt when it begins to do work elsewhere.

When H₂O became workable

By the 1860s, H₂O was no longer an isolated claim about water. It belonged to a broader chemical world.

Hydrogen was monovalent. Oxygen was divalent. Water belonged to a family of compounds in which oxygen could connect two units. Gas volumes supported the distinction between atoms and molecules. Organic substitutions gave chemists ways to test combining capacities. Atomic weights and molecular formulas could now be adjusted together with less arbitrariness than before.

Agreement on H₂O did not mean agreement on the reality of formulas. Some chemists treated them as useful instruments; others took them as claims about molecular architecture; many occupied positions somewhere between those poles. Chemical atomism became operationally secure before philosophical realism about atoms became uncontested.

The settlement also did not turn chemistry into a single perfectly unified language. Equivalent weights, atomic weights, empirical formulas, structural formulas and physical atomism did not collapse into one view. The field was rearranged. H₂O became stable within that rearrangement, not because every older tension disappeared, but because enough practices now supported the same count.

Once that happened, HO no longer looked like a reasonable simplification. It looked wrong. But it looked wrong from inside a later practice.

 

What H₂O hides

H₂O looks almost weightless on the page. Three symbols, one subscript, a formula learnt so early that it seems to have no history at all.

It has a long one.

Before chemists could write that subscript with confidence, water had to lose its ancient simplicity, acquire hydrogen and oxygen as constituents, and then submit to a harder demand. Its invisible parts had to be counted.

That last demand was the strangest. Weights could be measured, gases collected, volumes compared, substances replaced, compounds arranged into families. Yet no one could open a molecule and inspect its furniture. The number two had to come from elsewhere.

It came from operations. From volumes that made gases countable; from equivalents that allowed calculation; from substitutions that let radicals change places; from families of compounds in which oxygen repeatedly behaved as a twofold centre of combination. A count that no eye could read became stable through chemical work.

This is what the formula now conceals. H₂O is neither a convention nor a snapshot of an invisible object. It is a compressed history of practices, a small sign in which balances, gases, paper formulas, ethers, substitutions and valencies have disappeared into familiarity.

Water is H₂O. The difficulty lies in remembering how much chemistry had to be invented before that sentence could become obvious.

 

References

Avogadro, A. (1811). ‘Essai d’une manière de déterminer les masses relatives des molécules élémentaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons’. Journal de physique, de chimie, d’histoire naturelle et des arts, 73, pp. 58–76.

Cannizzaro, S. ([1858] 1910). Sketch of a Course of Chemical Philosophy. Alembic Club Reprints, no. 18. Edinburgh: The Alembic Club.

Chang, H. (2012). Is Water H₂O? Evidence, Realism and Pluralism. Dordrecht: Springer. DOI 10.1007/978-94-007-3932-1.

Dalton, J. (1808). A New System of Chemical Philosophy, Part I. Manchester: Printed by S. Russell, for R. Bickerstaff, Strand, London.

Gay-Lussac, J. L. (1809). ‘Mémoire sur la combinaison des substances gazeuses, les unes avec les autres’. Mémoires de physique et de chimie de la Société d’Arcueil, 2, pp. 207–234.

Klein, U. (2003). Experiments, Models, Paper Tools: Cultures of Organic Chemistry in the Nineteenth Century. Stanford: Stanford University Press.

Liebig, J. von. (1851). Familiar Letters on Chemistry, in Its Relations to Physiology, Dietetics, Agriculture, Commerce, and Political Economy. 3rd ed. London: Taylor, Walton, & Maberly.

Rocke, A. J. (1984). Chemical Atomism in the Nineteenth Century: From Dalton to Cannizzaro. Columbus: Ohio State University Press.

Rocke, A. J. (2010). Image and Reality: Kekulé, Kopp, and the Scientific Imagination. Chicago: University of Chicago Press.

Williamson, A. W. (1850). ‘Theory of Ætherification’. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 3rd series, 37(251), pp. 350–356.

Williamson, A. W. (1852). ‘On Etherification’. Quarterly Journal of the Chemical Society of London, 4, pp. 229–239. DOI 10.1039/QJ8520400229.

 

The post Is water H₂O? (3): How chemists learned to count invisible atoms appeared first on Mapping Ignorance.

Is Water H₂O? (2): The puzzle of water electrolysis

9 September 2026 at 14:00

Author: José Luis Granados Mateo is a postdoctoral researcher in the Department of Philosophy at the University of the Basque Country (EHU) and a member of the Integrated History and Philosophy of Science (iHPS) research group. His work focuses on history and philosophy of science, science and values, and the epistemology of scientific practices.

By 1800, water had already changed once. It was no longer one of nature’s simple terms, but a compound of hydrogen and oxygen (see Is Water H₂O? (1) From Element to Compound). The new chemistry had a powerful account of that change. What it still wanted was a more direct experiment: water itself, giving up its constituents.

Then came Volta’s pile.

It was not a spectacular object. Discs of metal stacked in pairs, pieces of moist material between them. Yet it altered the experimental life of electricity. Static machines produced sparks, shocks and theatrical moments. The pile offered something steadier. Electricity could now be made to act for long enough to enter chemistry.

electrolysis
Figure 1 Voltaic pile, the first electric battery, invented by Alessandro Volta in 1800. It did not look spectacular, but it changed what experimenters could do with electricity: it provided a sustained electrical action, long enough for electricity to become a chemical instrument. Source: Science History Institute / Public Domain

Nicholson and Carlisle tried it on water almost at once. They placed wires from the pile in the liquid. Gas appeared, or rather the two sides of the reaction began to declare themselves. Hydrogen was released at one wire; oxygen, in the earliest arrangement, fixed itself to the other, nearly two inches away. Later, with gold or platinum electrodes, oxygen could be collected as gas. The awkward fact remained. The two supposed products of one decomposition appeared at opposite electrical ends of the apparatus.

At first glance, this looked like the experiment the new chemistry had been waiting for.

The trouble was that it looked a little too good.

 

A cleaner decomposition

The first article in this series followed water from element to compound. That change had not been a simple passage from error to fact. It involved a wider transformation of chemical practice: new gases, new names, balances, combustion experiments, recomposition, and a growing confidence in quantitative composition.

Electrolysis seemed to bypass much of that history. No flame, no combustion vessel, no burning of inflammable air with oxygen, no droplets collected on glass. The battery appeared to make water surrender its constituents directly. If water was composed of hydrogen and oxygen, here was synthesis in reverse: pass electrical action through the liquid and the two gases would come out.

Strictly speaking, electrical decomposition of water was not new. In 1789, Adriaan Paets van Troostwijk and Jan Rudolph Deiman had used repeated sparks of static electricity to obtain a mixture of hydrogen and oxygen from water, then recombined that mixture into water. Their experiment mattered. It was also awkward. The gases came together, in small quantities, and separating them for testing was difficult.

Nicholson and Carlisle’s version looked sharper. The Voltaic pile produced sustained action. The products could be made to appear separately. Hydrogen here, oxygen there. Each could be collected, examined and identified.

That was the improvement. It was also where the puzzle began.

 

When neatness becomes suspicious

Suppose water is decomposed. What, exactly, has been decomposed? A particle of water, perhaps. Yet if that particle is split into hydrogen and oxygen, why do the products not appear together? Why does hydrogen emerge at one wire and oxygen at the other? Why should the layout of the battery govern the place where each product appears, while the water between the wires shows no visible sign of being torn apart?

This was not an experiment going wrong. The gases appeared reliably. They appeared at the expected poles. That was precisely what made the situation uncomfortable.

Nicholson noticed the difficulty at once. Hydrogen appeared at one wire; oxygen was found at the other, nearly two inches away. The space between them did not look like a region where water was being visibly pulled into its parts. No trail of bubbles marked the route. The products simply appeared at opposite ends of the electrical arrangement, as if the battery had assigned them places.

Others saw the same embarrassment. Cuvier put it bluntly: if oxygen and hydrogen came from the same particle of water, why did they appear at two distant points, each always at its own wire? Haüy pressed the problem in molecular terms. If one molecule of water was decomposed, why were its products separated? If two molecules were decomposed, why did one give only hydrogen and the other only oxygen?

The question was almost childishly simple. That was why it would not go away.

Today we speak of ions, electrodes and charge transport. Those words make the old difficulty vanish before we have properly felt it. Around 1800, chemists had poles, attractions, affinities, particles, electrical fluids, acids, alkalis, metals, and a growing suspicion that electricity and chemistry belonged together. The experiment was clear enough. The mechanism was not.

Chang calls this the distance problem. The phrase is exact. The gases appeared with too much order for a simple story of local decomposition.

 

The distance problem

The obvious interpretation could still be defended. Perhaps electricity pulled the constituents of water apart. Oxygen went to one pole, hydrogen to the other. Many experiments already suggested that substances had different electrical tendencies, so the idea was hardly absurd.

It left the hardest part untouched.

What travelled through the water? Hydrogen? Oxygen? Something already separated? Something charged? If the constituents were detached somewhere between the electrodes, why did they not appear on the way? If they moved invisibly, why did they not recombine? And if decomposition happened only at the wires, how could it be the decomposition of the same portion of water?

Changes in apparatus did not remove the difficulty. Davy tried water held in separate gold cups, connected only by a moist bridge of fine asbestos, an arrangement he attributed to William Hyde Wollaston. The products still appeared at distant sites. The more carefully experimenters separated the places of action, the more pressing the question became: where, exactly, was water being decomposed?

electrolysis
Figure 2. The distance problem in water electrolysis. Hydrogen appeared at one electrical pole and oxygen at the other, even when the sites of action were separated. The experiment seemed to show that water was decomposed, yet it left open the question of how the products travelled, or whether they were instead produced locally at the electrodes. Source: Diagram generated by the author using AI, based on Chang (2012) and Davy’s 1807 experiments.

 

That was the oddity. Electrolysis was not messy. If anything, it was too orderly. Hydrogen went one way, oxygen another. The battery gave a geography to the products of decomposition.

A flame did nothing like this. When inflammable air burned with oxygen, water appeared as the product of combination. There were difficulties, of course, but not difficulties of distance. Electrolysis changed the spatial form of chemical evidence. Water did not answer in one place. It answered at two.

Under electrical action, water no longer seemed merely something to be composed or decomposed. It seemed to contain routes.

 

Ritter changes the verb

Johann Wilhelm Ritter changed the verb.

For Lavoisierian chemists, electrolysis was analysis. Electricity decomposed water into hydrogen and oxygen. Ritter proposed the reverse. Electrolysis was synthesis.

At the negative pole, water combined with negative electricity and yielded hydrogen. At the positive pole, water combined with positive electricity and yielded oxygen. No product needed to travel from a decomposed particle somewhere in the middle of the liquid. The gases were made where they appeared.

It was a radical move. It was not foolish. Ritter took the most striking feature of the experiment seriously: the gases emerged at different poles. He did not treat that separation as an inconvenience to be patched up later. He built his interpretation around it. If hydrogen appears here and oxygen there, perhaps the two gases are being formed locally by two different electrical processes.

The price was high. Water became elementary again. Hydrogen and oxygen became compounds of water and electricity.

To modern chemistry, this looks perverse. Seen from the apparatus, it had one plain advantage: the gases were produced where they were found. Electricity was still widely treated as something with material agency. If a pile drove something into a liquid, why could that something not combine chemically with the liquid? Why must electricity be only the knife, and never part of what is made?

The phlogistonist echo was clear enough. If inflammable air could be read as water combined with phlogiston, hydrogen could be read as water combined with negative electricity. Identify phlogiston with negative electricity, and Ritter’s scheme comes surprisingly close to older anti-Lavoisierian accounts of water.

Priestley, from America, also saw that the new experiment did not force the Lavoisierian reading. He followed the debate through Nicholson’s journal and argued that electrolysis need not be understood as the decomposition of compound water. His intervention matters because it shows how differently the same electrical process could be read. For Lavoisierians, the gases were constituents released from water. For Priestley, they could still be products of another chemical-electrical process.

Ritter’s view did not last. Chemistry was becoming more firmly Lavoisierian, and elementary water had little room left in the emerging discipline. Still, electrolysis itself had not supplied a neat refutation. Ritter had solved the distance problem rather well. His view lost ground as chemists built an electrochemical practice around compound water.

Keeping water a compound

The Lavoisierian problem was delicate. Chemists had to hold three claims together. Water was composed of hydrogen and oxygen. Electricity decomposed it. Yet the products appeared at different electrodes, with no visible decomposition in between.

So the theory needed help.

One option was invisible transfer: one constituent moved through the liquid to its proper pole. Another was a chain of decompositions and recompositions, in which neighbouring particles handed hydrogen and oxygen along until the products emerged at the ends. In related versions, the whole liquid was imagined as electrically ordered, so that the final gases appeared only where that order met the metal.

These were rescue hypotheses. The phrase can sound dismissive, although it need not be. They were attempts to make a powerful practice intelligible. Electrolysis worked. Its effects could be repeated, varied and extended. What was missing was a mechanism that made the spatial separation of the gases sit comfortably with compound water.

The chain idea had a certain elegance. No single particle of hydrogen or oxygen needed to travel the whole distance. A constituent could be passed along through a succession of exchanges, like a message moving through a crowd. The gas would appear only at the end of the chain.

But what exactly was being passed? How were the exchanges coordinated? Why did the process end at just the right pole? The explanation helped, though it left plenty in the dark.

The transfer idea had its own attraction. If chemical particles had electrical characters, perhaps they could be drawn through the liquid. Again, however, the path remained hidden. The water between the wires did not display the process one wanted to see. Without a developed theory of ions and charge transport, the mechanism remained speculative.

None of this made the hypotheses useless. They kept research going. They suggested new apparatuses, new distances, new electrodes, new liquids and new comparisons. An experiment need not be fully understood before it becomes useful.

That is more or less what happened.

 

When water stopped being negotiable

By the early nineteenth century, compound water was becoming part of the working furniture of chemistry. Not every detail had been explained. The distance problem remained. Yet fewer and fewer chemists were prepared to reopen the older question of whether water might be elementary.

Water as a compound was no longer simply a hypothesis under trial. Increasingly, it was a condition for further work.

That shift mattered. Once water was taken as compound, electrochemistry could ask other questions. What moved towards each pole? What kinds of substances were decomposed by the pile? How did acids, alkalis, salts and metals behave under electrical action? What changed when the electrode, distance or solution changed? How much chemical change corresponded to how much electrical action?

The material practice hardened. Trough batteries became easier to handle. Gold and platinum electrodes helped prevent the products from disappearing into reactions with the metal. Separate cups connected by wet asbestos became a standard arrangement. The theory remained unsettled, but the apparatus was becoming disciplined.

The unresolved puzzle did not stop electrochemistry. Experimenters varied the arrangement. They separated vessels, introduced bridges, changed electrodes, added acids and salts, and studied what appeared where. The field learnt to handle the phenomena before it possessed a settled account of what moved inside the liquid.

Later developments brought more discipline. Faraday gave electrolysis a quantitative discipline in the 1830s. By the end of the nineteenth century, Arrhenius’s theory of ionic dissociation would help make sense of processes that had long remained obscure. That later clarity, though, should not be read backwards into 1800. For decades, electrochemistry advanced with a problem at its centre.

 

What electricity did to water

Electrolysis changed what water could be made to do in the laboratory. It did not simply reveal a truth waiting inside the liquid. It altered the route by which that truth could be pursued.

The first article in this series followed water into the balance sheet of chemistry. Water became a substance that could be produced, analysed, recomposed, weighed and named through its constituents. Electrolysis added another sort of depth. Water could now be interrogated electrically. Its constitution became a problem of poles, currents, attractions, transport and hidden motion.

Lavoisierian chemistry had given water constituents. Electrochemistry gave it pathways.

The familiar classroom demonstration hides that difficulty. The bubbles are easy to see. Their meaning was not. Hydrogen here, oxygen there: that simple separation forced chemists to imagine invisible processes inside a liquid that otherwise looked continuous and unchanged.

Electrolysis made compound water harder to abandon, without making it transparent. It turned water into an electrochemical object: something that could conduct, polarise, transport, release and exchange. The question was no longer only whether hydrogen and oxygen belonged to water. It was how electrical action could make them appear at all.

Still, electrolysis could not settle the formula. It could make water yield hydrogen and oxygen. It could not tell chemists how those elements should be counted.

That was the next difficulty. Once water had constituents, chemistry still had to learn how to count them.

 

References

Chang, H. (2012). Is Water H2O? Evidence, Realism and Pluralism. Dordrecht: Springer. doi: 10.1007/978-94-007-3932-1.

Davy, H. (1807). ‘The Bakerian Lecture, on Some Chemical Agencies of Electricity’. Philosophical Transactions of the Royal Society of London, 97, pp. 1–56. doi: 10.1098/rstl.1807.0001.

Faraday, M. (1834). ‘Experimental Researches in Electricity.—Seventh Series’. Philosophical Transactions of the Royal Society of London, 124, pp. 77–122. doi: 10.1098/rstl.1834.0008.

Nicholson, W. (1800). ‘Account of the New Electrical or Galvanic Apparatus of Sig. Alex. Volta, and Experiments Performed with the Same’. Journal of Natural Philosophy, Chemistry, and the Arts, 4, pp. 179–187.

Singer, G. J. (1814). Elements of Electricity and Electro-Chemistry. London: Printed for Longman, Hurst, Rees, Orme, A. Brown, and R. Triphook.

Volta, A. (1800). ‘On the Electricity Excited by the Mere Contact of Conducting Substances of Different Kinds’. Philosophical Transactions of the Royal Society of London, 90, pp. 403–431. doi: 10.1098/rstl.1800.0018.

Wilkinson, C. H. (1804). Elements of Galvanism, in Theory and Practice: With a Comprehensive View of Its History, from the First Experiments of Galvani to the Present Time. 2 vols. London: John Murray.

 

The post Is Water H₂O? (2): The puzzle of water electrolysis appeared first on Mapping Ignorance.

Is water H₂O? (1): From element to compound

31 August 2026 at 14:00

Author: José Luis Granados Mateo is a postdoctoral researcher in the Department of Philosophy at the University of the Basque Country (EHU) and a member of the Integrated History and Philosophy of Science (iHPS) research group. His work focuses on history and philosophy of science, science and values, and the epistemology of scientific practices.

 

Take a glass of water. Nothing about it seems to ask for a theory. It has no colour worth naming, no dramatic smell, no visible architecture. It runs, freezes, boils, evaporates, dissolves, erodes, circulates and nourishes. It belongs to weather, geology, physiology, cooking and plumbing before it belongs to chemistry.

Then science gives us the famous abbreviation: H₂O.

There is a curious finality in those three symbols. They seem to do what ordinary experience cannot. Rain, steam, ice, rivers, blood and laboratory wash bottles are all brought under a clean chemical statement. Two atoms of hydrogen, one atom of oxygen. For many people, H₂O is more than a formula. It is almost an emblem of scientific knowledge, the moment when science gets behind appearances and tells us what something really is.

Yet the formula stands at the end of a long story. Before water could be written as H₂O, it had to acquire a chemical interior. Chemists first had to learn to treat the most familiar liquid in the world as something that could be taken apart, recomposed and assigned constituents. The first transformation was not from one formula to another. It was from element to compound.

H₂O
Figure 1. Experiment on the decomposition of water, from Antoine-Laurent Lavoisier’s Traité élémentaire de chimie (1789), Plate IV; engraving by Marie-Anne Pierrette Paulze Lavoisier. Before water could be written as H₂O, it had to become an experimental object, handled through gases, vessels, combustion and weighing. Source: Science History Institute, Public Domain Mark 1.0.

 

This is the starting point of Hasok Chang’s Is Water H₂O?, a book that does for water something close to what Inventing Temperature did for thermometry. Temperature now looks like a number waiting to be read from a scale. Chang showed that the scale had to earn that appearance of simplicity. Fixed points had to be stabilised, instruments compared, extrapolations justified. Measurement became routine only after it had first been constructed as a reliable practice.

Water presented another kind of difficulty. The problem was not how to make a phenomenon answer numerically, but how to make a familiar substance answer analytically. Chemists had to learn to look at water as more than something that flowed, froze, boiled, dissolved, condensed or could be purified. They had to ask whether it had an inside: whether it could be taken apart, made again, and understood through its constituents.

To us, that question feels almost unavoidable. In the eighteenth century, it was anything but.

 

When water had no parts

For much of European natural philosophy, water was an element. That claim can sound childish to us, though only because we hear it through the categories that later displaced it.

The idea was older than Aristotle. Thales had imagined water as the origin of things; Empedocles later placed it among the four basic “roots” of the world, alongside earth, air and fire. Aristotle gave that inherited scheme a more systematic form. But these were not elements in the modern chemical sense. No one was classifying them by atomic number, nuclear charge or molecular structure. They belonged to a physics of qualities and transformations, of hot and cold, wet and dry. Water was simple because, in that world, it occupied a basic place in the order of natural change.

By the eighteenth century, that older grammar had been heavily rewritten. Chemistry had become crowded with acids, alkalis, precipitates, metallic calxes, distillations and newly isolated “airs”. Water was everywhere in this world. It served as solvent, medium, reagent, product and experimental environment. It could be purified, frozen, boiled, condensed and recovered.

Still, none of that made it automatically compound. A substance can be experimentally familiar and still have no recognised chemical interior.

The difficulty is easy to miss because we already speak the language that solved it. We say that hydrogen burns in oxygen to form water. But eighteenth-century chemists did not begin with hydrogen and oxygen. They had inflammable air, dephlogisticated air, fixed air, nitrous air and other gaseous bodies whose identities were still unsettled. The experiment did not come with modern subtitles. The vocabulary in which its meaning now appears self-evident was itself one of the products of the revolution.

Water became a compound when certain operations began to count as analysis and synthesis: taking water apart, producing it again, and treating the substances involved as its constituents. This was more than a taxonomic adjustment. It changed the standard of chemical knowledge. To know a substance now meant, increasingly, to know what it was made of.

 

The wrong theory that worked

The defeated rival was phlogiston theory. Posterity has enjoyed defeating it again and again.

In the usual caricature, phlogiston was a ghostly fire-stuff. Combustible bodies contained it and released it when they burned. Metals contained it too. When heated in air, they lost phlogiston and became calxes. Heat the calx with charcoal, and the metal returned because phlogiston had been restored. Then Lavoisier introduced oxygen, and the ghost vanished.

What makes this story so satisfying? Precisely what makes it misleading. It tells the Chemical Revolution from the winner’s seat, where oxygen looks inevitable and phlogiston appears only to make its exit.

From inside the eighteenth-century laboratory, the story looks less like a comedy of error. Phlogiston was not a fantasy floating above the experiments. It belonged to things chemists and metallurgists handled every day. Metals dulled and lost their malleability when calcined. Calxes could be brought back to metal with charcoal. Ores became usable metals in furnaces. Fire, air, metals, respiration and reduction seemed to be tied together by a common traffic of activity. Phlogiston named that traffic. It was a mistaken name for a real pattern.

No one shows this better than Joseph Priestley. He was not a dim survivor of pre-modern chemistry, nor an experimental genius trapped in mere confusion, but one of the great investigators of the eighteenth century, a dissenting minister, political radical and virtuoso of pneumatic chemistry. In Leeds, living beside a brewery, he studied the “fixed air” rising from fermentation, now carbon dioxide. Later, by heating the red calx of mercury, he obtained a far more remarkable gas. Flames burned in it with unusual brilliance; animals could breathe it for longer than ordinary air; Priestley even tried it himself, noticing what he described as a peculiar lightness in his chest.

We call that gas oxygen. Priestley did not. To him, it was dephlogisticated air.

 

H₂O
Figure 2. Gas-manipulating apparatus associated with Joseph Priestley’s pneumatic chemistry, 1775. Priestley’s “dephlogisticated air” was not an arbitrary name, but part of a working experimental culture built around the production, collection and manipulation of gases. Source: Wellcome Collection, M0018264: Illustration of gas manipulating apparatus, 1775. Public Domain.

 

The name sounds absurd only when detached from the operations that made it meaningful. In phlogiston chemistry, ordinary air could become loaded with phlogiston. Air deprived of phlogiston would therefore be especially ready to absorb it again. It should support combustion powerfully and sustain respiration unusually well. That is exactly what Priestley saw. His interpretation was wrong, but it was not arbitrary. It made sense of real experimental behaviour.

Nor was the winning term free of theory. Lavoisier called the gas oxygen, meaning “acid-generator”, because he believed oxygen was the universal principle of acidity. That too was false. The defeated word preserved one mistake; the victorious word preserved another. One became a fossil of discarded chemistry. The other became familiar enough to hide its own history.

 

A few drops on the glass

The next turn in the story came from a gas with a wonderfully plain eighteenth-century name: inflammable air. We now call it hydrogen.

Henry Cavendish had studied this light, combustible gas in the 1760s. It could be produced by reacting metals with acids, and it burned readily. When inflammable air was burned together with dephlogisticated air, something extraordinary happened. Water appeared. Two invisible airs had produced the most familiar liquid in the world.

It is hard, now, not to let modern chemistry finish the sentence for us. Hydrogen combines with oxygen to form water. Of course. But the droplets on the glass did not settle the matter by themselves. They had to be interpreted.

A phlogistonist could tell the story differently. Perhaps inflammable air was water combined with phlogiston. Perhaps dephlogisticated air was water deprived of it, or at least air peculiarly ready to receive it. On that reading, the experiment did not reveal that water was composed of two gases. It showed that ordinary water could be restored through the right exchange of phlogiston.

That may sound evasive from our side of the Chemical Revolution. At the time, it was not an absurd reply. A flame, a gas, a misted vessel, a change in weight. None of these comes with its meaning attached. Evidence becomes evidence inside a working system of concepts, operations and standards. What counts as a substance? What counts as a component? What counts as analysis rather than transformation? Which variable should be trusted most, appearance, activity, weight, or reproducibility?

The new chemistry changed the setting in which the experiment was read. The appearance of water was no longer merely a striking event at the end of combustion. It became part of a quantitative account. How much inflammable air had disappeared? How much oxygen had been involved? How much water had been produced? Could water be decomposed and then recomposed, with the numbers matching closely enough to make the same substance appear on both sides of the operation?

At that point, water was no longer simply appearing on the glass. It was entering the balance sheet of chemistry.

 

The balance makes composition visible

A metal is heated in air. It dulls, loses its shine, gives up its malleability and turns into a calx. In phlogiston chemistry, this looked like a loss. The metal had given up the principle that made it metallic.

Then came the awkward fact on the bench. The calx was heavier.

Heavier, after a supposed loss.

That did not destroy phlogiston theory in a single blow. Good chemists could still defend it, and did. Yet the weight gain gave oxygen chemistry an opening. Perhaps calcination was better understood as combination with something from the air. Follow the weights carefully and chemical change begins to look different. Appearances still matter. Activity still matters. So do colour, heat, flame, residue and reversibility. But weight gives the whole operation a new discipline. It turns transformation into material accounting.

 

H₂O
Figure 3. Lavoisier’s laboratory balance and gas-measuring apparatus, from Antoine-Laurent Lavoisier’s Traité élémentaire de chimie (1789), Plate VIII; engraving by Marie-Anne Pierrette Paulze Lavoisier. In the new chemistry, weighing was not merely a technical aid. It helped make composition visible by turning chemical reactions into material accounts. Source: Science History Institute. Public Domain.

 

The same habit of attention changed the status of water. A substance that can be broken down into measured quantities of other substances, and then produced again from them, starts to lose its old simplicity. It has become analysable. It has constituents.

This was the strongest hand of the new chemistry. Combustion, calcination, gases and water could now be drawn into the same quantitative order. Chemical names changed too. A name was expected to do more than preserve origin, appearance, use or inherited convention. Ideally, it should tell chemists what a substance contained.

The point needs care. This was never a struggle between careful measurers and careless speculators. Cavendish was one of the most exact experimentalists of the century. Richard Kirwan, among the serious defenders of phlogiston, treated weight as a genuine difficulty. The disagreement concerned the authority of weight. Should it count as one experimental constraint among others, or should it become the privileged route to chemical identity?

That question slowly shifted the terrain. Phlogiston chemistry was most at home in a language of principles, powers and transformations. What gives a body combustibility? Why does a metal lose its metallic character? How can a calx be revived? What passes from one substance to another when reaction occurs?

The new chemistry pressed elsewhere. What does this substance contain? In what proportions? Can those constituents be separated and brought together again while the material account remains intact?

Once that style of reasoning became dominant, water’s old simplicity became harder to defend. Water could still be boiled, condensed, purified and recovered, just as before. Now it could also be weighed into a different story. It could be decomposed, recomposed, renamed and treated as a body with an interior.

 

No single knockout blow

The usual story of the Chemical Revolution likes a clean scene. Priestley isolates oxygen and fails to recognise it. Lavoisier understands what Priestley could not. Phlogiston collapses. Modern chemistry begins.

History was less obliging.

Priestley knew the new chemistry and resisted it. Cavendish remained cautious. Other competent chemists hesitated, adopted parts of the oxygen system, or kept some version of phlogiston alive. Their reluctance was not simple blindness. Phlogiston theory still connected combustion, respiration, calcination, reduction and the behaviour of airs. It had also helped produce many of the experimental facts that later counted in favour of oxygen chemistry.

Lavoisier’s system, for all its power, carried its own burdens. His theory of acidity was wrong. The very word oxygen preserved that mistake, since it named the gas as the supposed generator of acids. His chemistry also relied on caloric, an imponderable fluid of heat. Seen from the present, caloric does not look obviously more respectable than phlogiston. Lavoisierian chemistry was elegant, disciplined and transformative. It was not modern chemistry arriving whole and immaculate.

So the real historical question is more interesting than the textbook one. Why did one way of doing chemistry become more convincing, more portable, more teachable and more productive than another?

Much of the answer lies in the rise of compositionism. Oxygen chemistry made substances legible as combinations of measurable constituents. Reactions became material accounts. Chemical names began to function as compressed statements of composition. Laboratory operations, weights, gases and nomenclature could now be made to speak the same language.

Phlogiston still made sense of many transformations, but it sat uneasily in that emerging world. It named a principle of activity, a way of thinking about combustibility, metallic character, reduction and chemical power. The new chemistry increasingly wanted components. It asked what substances were made of, how much of each constituent they contained, and whether those constituents could be separated and brought together again.

As compositionism gained ground, phlogiston began to lose its natural habitat. It could still speak about combustion, reduction and chemical activity, but chemistry was moving towards another language: constituents, proportions, balances, material accounts. The revolution changed the answer, certainly. More importantly, it changed the kind of answer chemistry wanted.

The first metamorphosis of water

When water became a compound, it did more than shift from one scientific category to another. Its identity for chemistry changed.

Before this transition, water could pass through laboratory practice while keeping its apparent simplicity. It could be boiled, frozen, distilled, condensed, purified and recovered. It appeared everywhere, as liquid, vapour, solvent, medium, residue or product. Familiarity gave it stability. It did not yet give it an interior.

After the Chemical Revolution, the same substance began to occupy a different role. Water could be produced from other bodies, analysed through them, and assigned constituents. The liquid that returned from vapour or appeared after combustion now carried a hidden depth. It was no longer merely recovered; it could be accounted for.

Nobody saw that hidden constitution. There were no water molecules under a microscope, no direct image of hydrogen and oxygen joined together. The evidence came through laboratory work: gases prepared and collected, vessels sealed, mixtures burned, droplets condensed, weights compared, names revised. Compound water became credible through practice before it became imaginable as a molecular structure.

This is why the story should not run too quickly towards H₂O. Lavoisierian chemistry helped establish water as a compound of hydrogen and oxygen. It did not settle the formula. It was one thing to say that water was made from hydrogen and oxygen; it was another to decide how many atoms of each entered the compound. That question would take nineteenth-century chemistry into a different set of difficulties.

First came the more basic metamorphosis. Water ceased to be one of nature’s simple terms. It became a substance that could be decomposed, recomposed and written in the language of other substances.

 

The cost of clarity

The new chemistry clarified immensely. It disciplined measurement, stabilised nomenclature and helped make possible the modern science of composition. But clarity selects. It illuminates some things by pushing others into shadow.

Phlogiston theory had been concerned with chemical activity. Why do substances burn, reduce, revive, calcine, combine and transform? Early compositional chemistry was much better at saying what entered and left a reaction than at explaining why the reaction happened at all. It provided an excellent account book, though not yet a full theory of chemical agency.

Some questions once handled by phlogiston later returned in different forms. Chemical potential energy, oxidation-reduction and electrons all came to occupy parts of the conceptual space that phlogiston had once held. That does not vindicate phlogiston as a literal substance. It suggests something subtler and more useful. A false theory can preserve real questions. A victorious theory can be right and still leave work unfinished.

This is a more realistic picture of scientific progress. Oxygen chemistry won for good reasons. But those reasons were not contained in one devastating experiment before which only fools could hesitate. They belonged to a wider reorganisation of chemical practice.

Water’s first great chemical transformation was therefore not the discovery of a formula. It was the making of a new kind of object. The familiar liquid became something that could be analysed, recomposed, weighed and named through its constituents. Only after that could the later question become pressing: not whether water was compound, but what exactly its composition was.

Electricity seemed about to settle the matter. It made water yield hydrogen and oxygen directly, as if nature had finally supplied the decisive experiment.

Instead, the proof produced a new puzzle.

 

References

Chang, H. (2012). Is Water H₂O? Evidence, Realism and Pluralism. Springer. doi 10.1007/978-94-007-3932-1.

Chang, H. (2004). Inventing Temperature. Measurement and Scientific Progress. Oxford University Press.

Cavendish, H. (1784). ‘Experiments on Air’. Philosophical Transactions of the Royal Society of London, 74, 119–153.

Kirwan, R. (1789). An Essay on Phlogiston and the Constitution of Acids. London, J. Johnson.

Lavoisier, A.-L. ([1789] 1965). Elements of Chemistry. Dover.

Priestley, J. ([1796] 1969). Considerations on the Doctrine of Phlogiston, and the Decomposition of Water. Kraus Reprint.

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Generative AI has changed mathematics forever

12 August 2026 at 14:00

Author: Melissa Lee, Senior Lecturer, School of Mathematics, Monash University

Some weeks ago, artificial intelligence company OpenAI announced ten advances in mathematics and computer science made with their as-yet unreleased model Astra. The discoveries cover a wide range of mathematical fields, including geometry, cryptography and coding theory.

These ten are just the latest in a rapidly growing list of mathematical breakthroughs made by generative artificial intelligence (AI) systems.

OpenAI accompanied the announcement with a statement on “responsibility to the mathematical community”. Here, the company acknowledges concerns about attribution, accountability for the correctness of AI-assisted results, and the changing nature of mathematical discovery.

The statement reflects a growing understanding that the remarkable technical advances due to AI are only part of a bigger story about what mathematics means, and what its future will look like.

mathematics
Image created by ChatGPT when prompted this: illustration for an article titled “Generative AI has changed mathematics forever”. Source: OpenAI / ChatGPT

AI is raising big questions

Large language models (LLMs) such as ChatGPT and Claude are creating disruptions in all parts of the mathematical community.

University departments are grappling with the ethical implications of these tools in their research and teaching. Editorial boards of many peer-reviewed journals are being inundated with AI-written papers and must decide how to assess and disclose the use of generative AI. The arXiv preprint repository has seen a sharp increase in mathematical submissions in recent months.

Math papers uploaded to @arxiv per month. Jan 1992 to Jul 2026. Source: https://t.co/nrqjhJBtij pic.twitter.com/pmHvtbYK2X

— Pablo Groisman 🎲 (@pgroisma) August 3, 2026

Funding agencies are starting to develop policies governing acceptable use of AI in applications and assessment.

Broader philosophical questions about how and why mathematics is done in the first place are hard to avoid.

If an LLM can discover a proof, construct an example, or formulate a new question, is there anything uniquely valuable about human mathematical creativity? Is mathematics mostly about producing new theorems, or is it about developing understanding? When a machine contributes to a discovery, who should get the credit?

Growing concern – but little consensus

Among working mathematicians, there is little consensus about the answers. Some have genuine anxiety about the future of the discipline. One mathematician wrote that the advance of AI-assisted mathematics had triggered a “profound spiritual crisis”.

Others have tried to articulate principles for integrating AI into research in a responsible way. The recent Leiden Declaration, signed by thousands of mathematicians from around the world, argues AI should augment rather than replace human mathematical creativity. At the same time, it emphasises transparency, accountability, and proper attribution.

Two weeks ago at the International Congress of Mathematicians, the largest and most prestigious meeting in the field, Terence Tao spoke about “the age of AI”. The famous Fields medallist urged mathematicians to think ahead. In his view, the question is how AI systems might be incorporated into research to strengthen the culture and values of mathematics.

Two attitudes to AI

Two collaborations from my own research over the past few weeks show just how varied these attitudes have become.

My colleague Saul Freedman and I announced an answer to the “semiregularity problem” about highly symmetric networks. This had been the subject of dozens of peer-reviewed papers over several decades.

From the outset, Saul made it clear he did not want AI used in any way during our collaboration due to his concerns about the environmental and social impacts of the technology. Therefore, the work proceeded entirely without the use of these tools.

After we announced our result, two colleagues told me they had independently tried to use LLMs to solve the same problem without success.

AI-powered discovery – with human connection

In contrast, a couple of weeks ago another colleague, Aluna Rizzoli, emailed to tell me he had found an object that collaborators and I had spent over two years searching for. We had used considerable computing resources, but Aluna had used an OpenAI model and a supercomputing cluster. His computation took a mere 43 hours.

The method ChatGPT had generated, working within the intriguingly named “Monster group”, relied on a much more intricate version of an existing algorithm. It would have taken my collaborators and me months to develop this generalisation.

Perhaps the most striking aspect is what happened next. Rather than “scooping” our ongoing work, Aluna invited us to write a joint paper based on the discovery, which appeared on arXiv last week.

Working out where AI fits

Each of these collaborations produced important mathematical advances, yet they embody starkly contrasting attitudes to generative AI.

One deliberately excluded the technology on ethical grounds, while the other embraced it as an indispensable research partner, without abandoning the etiquette of mathematical research. Neither approach appears inherently incompatible with doing excellent mathematics.

That is perhaps the defining feature of the current moment. Rather than converging on a single view, mathematicians are still working out where AI fits within the research process.

The conversation is no longer about whether LLMs are capable of contributing to mathematics research. Instead, we are talking about how their use can be reconciled with the values of collaboration, transparency, and intellectual integrity that form the foundation of our discipline.The Conversation

 

This article is republished from The Conversation under a Creative Commons license.  Original article.

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International scientific institutions between war and peace. One hundred years of IUPAP (1)

6 July 2026 at 14:00

Author: Jaume Navarro is an Ikerbasque Research Professor at the University of the Basque Country

A few weeks after the invasion of Ukraine by the Russian army, many scientific institutions felt the need to issue public statements against that war. At the time, I was president of the Commission for the History of Physics within the International Union of Pure and Applied Physics (IUPAP) and, as such, I took part in the discussions leading to the final declaration. The timing could not be better for historians of science to make a significant, albeit small, contribution: my commission was working on a project to seriously study the origins and development of IUPAP, which was about to celebrate its centenary. Paying attention to history was important so as to articulate a statement that would avoid past mistakes and incorporate positive experiences of the Union in its one-hundred-year history. Because, like other scientific international unions, IUPAP was born with an idea of peace in mind, yet it had to maneuver through times of war, both hot and cold.

 

It was after the tragedy of the Great War (1914-1918) that a new wave of internationalism in science took shape. If excessive nationalism had drawn Europe to its worse carnage in history, so the argument went, it was time for international institutions to collaborate in the construction of peace. And scientific research, understood as a cultural enterprise, was one such area in which collaboration rather than competition was to play a key role. In hindsight, one of the ironies of this search for peace was that the losers (Germany, Austria, Hungary, Bulgaria, and what remained of a now highly unstable Ottoman Empire) both collectively and individually, were banned from taking part in the new international order. That meant not only that their national academies could not join the new international unions, but also that individual scientists from those countries were excluded from meetings and collaborations sponsored by the unions. This was particularly poignant in the case of physics.

 

Indeed, international collaboration had a somewhat long tradition in fields like astronomy or geodesy due to the global scale of some of their research projects. But physics, while the “king of the sciences” in the nineteenth century, had not seen the international cooperation of national academies in the same way. That partly explains why, with the creation of the International Research Council (IRC) by delegates of twelve nations in July 1919, the International Unions of Astronomy (IAU), Geodesy and Geophysics (IUGG) and Pure and Applied Chemistry (IUPAC) immediately approved their initial statutes, soon to be followed by the Mathematical Union (IMU) and the Union for Scientific Radio Telegraphy (URSI); yet, a union of physics would have to wait, both in form and, especially, in practice.

 

In 1922, a number of physicists from France, Belgium, Britain, the USA and a few other countries met in Paris and came up with a first draft of statutes for an International Union of Physics. It was the young American experimentalist, Robert A. Millikan, who pushed for a name that included both pure and applied physics. The main goals of the Union were the promotion of up-to-date scientific bibliographies, the coordination of fundamental units, and the organization or sponsorship of international conferences. Those present at the meeting to celebrate the fiftieth anniversary of the Société française de physique (SFP) in Paris in December 1923 finally approved the statutes of IUPAP nominating the British and Nobel prize winner William H. Bragg as president, and Henri Abraham, professor of physics at the École Normale Supérieure, as secretary general. But it was informally agreed that the Union would remain basically dormant until it could be fully international; i.e., include German physicists.

IUPAP
Henri Abraham (1868-1943) in 1935. Source: Estudio Harcourt / Public Domain / Wikimedia Commons

By 1924, the vindictive philosophy of the IRC against the Central Powers was being challenged. The Council had grown and now included national academies from up to twenty countries, mostly neutral during the war and who did not see the need for such exclusionary policy. Also, an increasing number of individual scientists were pushing towards a more reconciliatory and inclusive internationalism. French and Belgian representatives were the most reticent to amend the statutes of the IRC and lift the foundational ban. So much so that real changes only came in 1931 when the IRC dissolved and the new International Council of Scientific Unions (ICSU) took its place. From then on, not only was internationalism a real possibility but also each individual scientific union was freer to organize itself as they saw fit.

 

In the 1920s, however, internationalism in physics materialized in other forms. Two are the best-known and most influential institutions on this aspect: the Solvay Conferences and Niels Bohr’s Institute for theoretical physics in Copenhagen. The former were the initiative of the Belgian industrialist Ernest Solvay to promote relatively small, truly international, by-invitation-only conferences to address pressing issues in both physics and chemistry. After the success of the first two meetings in physics in 1911 and 1913, the third and fourth Solvay meetings in 1921 and 1924 took part in the boycott against the Germans. But, by 1927, the mood had changed and the fifth Solvay conference in Brussels, which eventually became one of the most influential events in shaping the transformation of the early quantum physics into the new quantum mechanics, saw the presence of five German physicists: Max Planck, Albert Einstein, Wolfgang Pauli, Werner Heisenberg and Max Born. In a way, that success was partly due to the tradition that Bohr had managed to create in his institute in Copenhagen. Indeed, Denmark had been a neutral country during the war and the Carlsberg Foundation had generously funded a research center for theoretical physics to keep Bohr in the country. Thanks to both elements, Copenhagen became the central point for physicists and mathematicians to meet and freely exchange ideas on the new quantum physics.

 

But let us go back to the early history of IUPAP because, while Bohr had played a significant role in bringing together physicists from both sides of the war, he was also partly responsible for the failure of IUPAP to take off. In 1931, Bragg resigned and Millikan was appointed president. His plan was to use his tenure as a springboard for American physicists to become more and more internationally relevant, as well as to materialize the inclusion of the German physical society into IUPAP. He expected to achieve both goals with the organization of a major international physics conference in Chicago in 1933, year in which the wind city was celebrating its centenary. With the presence of a large number of invited physicists IUPAP might finally formalize the German membership.

IUPAP
Source: Weimer Pursell, silkscreen print by Neely Printing Co., Chicago (1933) Source: Public Domain / Wikimedia Commons

Millikan’s grand plan did not succeed. First, the recession after the crash of 1929 affected the organization of events during the “Century of Progress” celebrations and money to invite scientists from abroad was short. Second, IUPAP and all other unions had one national scientific institution representing the country. But Germany had two such societies, the Deutsche Physikalischer Gesellschaft (DPG) and Gesellschaft für Technische Physik (GTP), and the attempts by Millikan and, especially Abraham, who was still serving as secretary, in correspondence with Planck and other German representatives could not settle the question as to which German society was to represent the country in the Union. And, third, the political climate in Germany was changing and the nazi ideology taking hold of the country was certainly not keen on internationalism.

 

Chicago did finally organize a minor scientific meeting, but it was far from being the major event Millikan had envisaged to expand and consolidate IUPAP. Promoters of the Union did not, however, give up and decided to use a major physics conference organized by the British Institute of Physics and the Royal Society in both London and Cambridge in October 1934 as the place to formalize German membership. And as a sign to underline the true international character of the Union, they offered the presidency to Bohr as icon of internationalism. The correspondence between Millikan, Abraham, Bohr, Schrödinger and a few other physicists show an amusing yet sad chain of misunderstandings: Abraham and Bohr thought Millikan had accepted the presidency by misreading a telegram of the latter; but Bohr did not want to get involved unless the German membership was fully signed, not simply promised. And that never happened.

 

So, Millikan having resigned and Bohr having rejected the presidency, the soul of the dormant union was its long-serving secretary, Abraham, who worked hard to find a candidate to act as president while things were clarified. In 1937, the Swedish physicist and Nobel Prize winner Manne Siegbahn agreed to be the president of IUPAP, but the Union was still short of activities, cohesion and a clear goal. It was also the indefatigable Abraham who continued his search for ways to keep the Union alive. He proposed a conference in Copenhagen in 1938, with the support of Bohr, or in Stockholm, organized by Siegbahn or, even as late as late as May 1939 he was thinking of a conference in Paris for 1940. None of them ever materialized and World War II not only destroyed the hopes and dreams of having a fully functional and international Union of Pure and Applied Physics, but even more poignantly, ended with the life of its most ardent defendant: Abraham was murdered in Auschwitz in 1943. As we shall see in the next installment, the Union reinvented itself after the war and had to learn how to navigate the new peace of the Cold War.

 

 

Further reading:

 

Lalli, R. and Navarro, J. (2024). Globalizing Physics: One Hundred Years of the International Union of Pure and Applied Physics. Oxford University Press. [Available in open access: https://academic.oup.com/book/58182]

 

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