When René Descartes appended his Géométrie to the Discours de la méthode in 1637, he performed what has since been canonized as one of the great conceptual ruptures in the history of mathematics: the systematic identification of geometric curves with polynomial equations relative to a fixed coordinate frame. The standard narrative treats this as an act of invention — the creation, from philosophical first principles, of a new mathematical ontology in which geometry and algebra become one. It is a compelling story, and it is not exactly wrong. But it obscures a more uncomfortable possibility: that Descartes’s principal achievement was not the discovery of the coordinate method but its codification, and that the difference between these two acts has been doing quiet ideological work for nearly four centuries.
The argument is not that Descartes plagiarized anyone, or that priority claims in mathematics are zero-sum. It is, rather, that the very machinery by which mathematical knowledge acquires a name, an author, and a place in the canon is systematically biased toward epistemic cultures that prize formal articulation over operational fluency — and that this bias generates a persistent illusion: that knowledge begins at the moment it is written down in the approved form.
Consider the Jiuzhang Suanshu, the Nine Chapters on the Mathematical Art, compiled in its canonical form around the first century of the common era but drawing on material considerably older. Its eighth chapter, titled Fangcheng (方程, roughly “rectangular arrays”), presents eighteen problems solved by a procedure that is, in every computationally meaningful sense, Gaussian elimination. Coefficients are arranged in columns on a counting board; cross-multiplication and subtraction reduce the system to triangular form; back-substitution yields the solution. The procedure handles systems of up to six equations in six unknowns. Problem 13 is indeterminate — five equations in six unknowns — making it the earliest known treatment of underdetermined linear systems in any civilization’s mathematical record. Roger Hart’s meticulous reconstruction in The Chinese Roots of Linear Algebra (2011) leaves no serious room for doubt: the operational content of what we call matrix algebra and Gaussian elimination was present in Chinese mathematical practice at least sixteen centuries before Gauss and seventeen before the formalization of matrix theory in the nineteenth-century West.
But the word “operational” is doing essential work in that sentence, and the nature of that work is precisely what is at stake. Hart emphasizes a remarkable sociological feature of the fangcheng tradition: the practitioners who actually manipulated the counting rods — who possessed the motor-learned, visually mediated, two-dimensional intuition for cross-multiplying entries across an array — were most likely illiterate. The written records we possess are translations of their practices into literary Chinese, compiled by literati-officials seeking imperial patronage. Hart’s devastating observation is that “the translation of fangcheng calculations into classical Chinese, modern English prose, or modern mathematical terminology renders these practices almost incomprehensible.” The knowledge was not propositional. It was not encoded in theorems. It lived in the hands, in the spatial grammar of rod placement on a grid, in an embodied competence that resisted textualization. It was, in the fullest sense of Michael Polanyi’s term, tacit.
And here the argument acquires its philosophical edge. In Polanyi’s framework, tacit knowledge is not merely knowledge that happens not to have been written down; it is knowledge that cannot be fully articulated without loss of essential content. The fangcheng practitioner’s fluency with the counting board was not a deficient version of Gauss’s explicit method awaiting its proper formulation. It was a different mode of knowing — one in which the two-dimensional spatial structure of the problem was apprehended directly, through manipulation, rather than mediated through symbolic notation. The counting board was itself a coordinate system: a physical grid in which numerical quantities occupied positions and were transformed by positional operations. To say that the Chinese “didn’t have” coordinate geometry because they didn’t write 
This confusion is not innocent. It is, in fact, the central mechanism by which certain mathematical cultures accumulate credit at the expense of others. The pattern is worth stating explicitly. First, a body of operational mathematical knowledge develops within a practice-oriented tradition: the fangcheng rod-calculators, Liu Hui’s algorithmic geometry, Li Zhi’s conversion of circle-triangle problems into polynomial equations via the tianyuan shu in the thirteenth century. This knowledge is transmitted through apprenticeship, through embodied practice, through a pedagogical mode that privileges doing over stating. Second, centuries later, a scholar in a different tradition performs the codification event: the knowledge is named, formalized, situated within a deductive architecture, and published in a form optimized for textual transmission. Third, the codification event is retroactively identified as the moment of invention, and the codifier is credited as the inventor. The prior operational tradition is either unknown, dismissed as “merely computational,” or acknowledged only as a vague anticipation — never as the real thing.
Liu Hui’s commentary on the Nine Chapters, completed in 263 CE, is an extraordinary case study in the costs of this pattern. Karine Chemla’s three decades of scholarship — culminating in her critical edition with Guo Shuchun (2004) and her edited volume The History of Mathematical Proof in Ancient Traditions (2012) — has demonstrated beyond reasonable dispute that Liu Hui’s work contains genuine mathematical proofs: rigorous demonstrations of algorithmic correctness using what Chemla calls “algebraic proofs in an algorithmic context.” These are not heuristic sketches or lucky guesses. They are systematic, they are general, and they deploy a mode of mathematical reasoning that Chemla identifies as “attested to nowhere so far in ancient traditions except in ancient China.” Yet for generations, Western historians dismissed Chinese mathematics as lacking proof altogether, because the proofs did not take the form of the Euclidean deductive chain — axiom, definition, proposition, QED. The absence of a particular form of articulation was mistaken for the absence of the substance of rigorous thought. This is the cartographer’s trick: if it does not appear on my map, it does not exist.
The trick operates with particular force in the case of Li Zhi (also known as Li Ye, 1192–1279), whose Ceyuan Haijing (Sea-Mirror of Circle Measurements) deploys the tianyuan shu to convert intricate geometric configurations — circles inscribed in triangles, intersecting figures, surveying problems — into polynomial equations of degree up to six, which are then solved by Horner-type methods on the counting board. The tianyuan (“celestial element”) is explicitly an unknown quantity: the functional equivalent of Descartes’s 
Yet this entire tradition was lost. When the Ming dynasty overthrew the Mongol Yuan, the new regime’s suspicion of Yuan-era knowledge — compounded by the inherent fragility of tacit transmission — rendered Zhu Shijie’s and Li Zhi’s texts incomprehensible. Without the living practice of rod calculation to anchor the written instructions, the texts became opaque, dismissed by Ming literati as numerological arcana. They were rediscovered and decoded only after Western algebra entered China in the nineteenth century, at which point the bitter irony became apparent: the Chinese had possessed, and lost, methods equivalent to what the West was now presenting as its own invention. The knowledge had been real, had been operational, had solved hard problems — and had vanished because it was encoded in hands rather than in books.
Dan Wang’s analysis of contemporary Chinese manufacturing provides the economic and sociological framework for understanding why this pattern recurs. Wang argues that the West — and particularly the United States — has developed a systematic overvaluation of codified knowledge (patents, papers, formal intellectual property) relative to tacit knowledge (process engineering, supply-chain expertise, the “feel” of a production line). This overvaluation is not merely an analytical error; it is a structural feature of institutions designed to allocate credit, funding, and prestige. Universities reward publication. Patent offices reward formal claims. Historiographies reward named discoverers. In each case, the system selects for a particular form of knowledge expression, and this selection is then mistaken for a judgment about the content of the knowledge itself.
The mathematical case reveals that Wang’s framework is not merely about contemporary industrial policy; it describes a deep epistemological pattern that has been shaping the distribution of intellectual credit for millennia. The fangcheng practitioner, Liu Hui, Li Zhi, and the anonymous rod-calculators of imperial China were not doing “proto-mathematics” or “approximate” versions of what Gauss and Descartes would later do properly. They were doing mathematics — fully, rigorously, and often with greater generality than their later Western counterparts. What they were not doing was codifying their mathematics in the particular textual form that the Western canon recognizes as the signature of genuine mathematical achievement. And so the credit went elsewhere.
Minhyong Kim’s recent essay in the Notices of the American Mathematical Society — “History, Identity, and Ownership in Mathematics,” published in September 2025 and expanded from lectures delivered between 2022 and 2024 — addresses this dynamic with the authority of a leading arithmetic geometer working at the intersection of anabelian geometry and mathematical public engagement. Kim argues for what he calls the “unity of the global heritage of mathematics,” contending that while genuine diversity exists in mathematical ideas and methods across cultures and periods, “any attempt to divide up the contributions into distinct zones or clusters is ultimately artificial.” He draws an analogy to population genetics: just as the attempt to partition human biological variation into discrete races fails to capture the continuous, clinal nature of actual genetic diversity, the attempt to assign mathematical innovations to discrete civilizations — “Greek geometry,” “Chinese algebra,” “Western analysis” — imposes artificial boundaries on what is in reality a single, continuous, globally distributed intellectual endeavor.
Kim’s analogy is more than rhetorical. It identifies the precise logical structure of the error. The codification event — Euclid’s Elements, Descartes’s Géométrie, Gauss’s Disquisitiones — functions like a bottleneck in a population-genetic model: it concentrates credit in a single named lineage while obscuring the wider genetic (or, in this case, intellectual) diversity that preceded it. The codifier becomes the “founder” of a tradition that in reality had many sources, many contributors, and many prior instantiations in forms that the codifier’s own tradition was not equipped to recognize. The Elements did not invent deductive geometry; it packaged it. The Géométrie did not invent the algebraic treatment of geometric problems; it named it. The naming was genuinely valuable — it enabled transmission, generalization, pedagogy at scale. But it also generated a founder effect in the historiography, concentrating all subsequent citation and credit on the codifier and his intellectual descendants while rendering the prior distributed tradition invisible.
Hart’s Imagined Civilizations (2013) provides the most granular demonstration of this mechanism in action. When Matteo Ricci and Xu Guangqi translated Euclid’s Elements into Chinese in the early seventeenth century, the standard narrative — constructed almost entirely from Jesuit sources — presents this as the moment when China encountered, and recognized the superiority of, Western mathematics. Hart, working from Chinese primary sources, inverts the story. He shows that Xu Guangqi was operating from a position of considerable political power relative to Ricci, that Chinese mathematical treatises of the period contained methods equivalent to or more sophisticated than what the Jesuits were offering, and that the extravagant claims of Western mathematical superiority made by certain Chinese literati were “bids for patronage through memorials in which they fashioned themselves as statesmen with novel solutions to late-Ming crises.” The Jesuits codified; the Chinese literati recognized the political utility of the codification; and Western historians, reading the Jesuits’ own account of the encounter, mistook a political transaction for an intellectual revelation.
The irony is recursive. The very act of writing about these dynamics is itself a codification event — one that risks reproducing the pattern it describes, by translating the tacit, practice-embedded knowledge of Chinese rod-calculators into the explicit, propositional, citational form that Western academic discourse recognizes as legitimate. There is no way to escape this recursion entirely. But there is value in naming it, in making the mechanism visible, in insisting that the map is not the territory and the name is not the thing. When we say “Cartesian coordinates,” we are not describing a fact about the history of mathematics. We are performing a speech act — one that attributes ownership, assigns credit, and renders invisible the centuries of distributed, tacit, embodied mathematical practice that preceded and enabled the codification event we have chosen to celebrate.
The counting board was a coordinate system. The rod was the unknown. The array was the matrix. They were all there, unnamed, for seventeen hundred years before anyone thought to write a treatise about them. The treatise was important. But the knowledge was older than the name.
References
Chemla, Karine. 1997. “What is at Stake in Mathematical Proofs from Third Century China?” Science in Context 10 (2): 227–251.
Chemla, Karine. 2005. “The Interplay Between Proof and Algorithm in 3rd Century China: The Operation as Prescription of Computation and the Operation as Argument.” In Visualization, Explanation and Reasoning Styles in Mathematics, edited by Paolo Mancosu, Klaus Frovin Jørgensen, and Stig Andur Pedersen, 123–145. Synthese Library, vol. 327. Dordrecht: Springer.
Chemla, Karine, ed. 2012. The History of Mathematical Proof in Ancient Traditions. Cambridge: Cambridge University Press.
Chemla, Karine, and Guo Shuchun. 2004. Les Neuf Chapitres: Le Classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.
Cullen, Christopher. 2002. “Learning from Liu Hui? A Different Way to Do Mathematics.” Notices of the American Mathematical Society 49 (7): 783–790.
Descartes, René. (1637) 1954. The Geometry of René Descartes. Translated by David Eugene Smith and Marcia L. Latham. New York: Dover.
Hart, Roger. 2011. The Chinese Roots of Linear Algebra. Baltimore: Johns Hopkins University Press.
Hart, Roger. 2013. Imagined Civilizations: China, the West, and Their First Encounter. Baltimore: Johns Hopkins University Press.
Kim, Minhyong. 2025. “History, Identity, and Ownership in Mathematics.” Notices of the American Mathematical Society 72 (8): 846–853.
Polanyi, Michael. 1966. The Tacit Dimension. Garden City, NY: Doubleday.
Shen, Kangshen, John N. Crossley, and Anthony W.-C. Lun. 1999. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford: Oxford University Press.
Swetz, Frank J. 1977. Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China. University Park: Pennsylvania State University Press.
Swetz, Frank J. 1992. The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China. University Park: Pennsylvania State University Press.
Wang, Dan. 2018. “How Technology Grows (A Restatement of Definite Optimism).” July 2018. https://danwang.co/how-technology-grows/.
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