The Man Who Made Madhava Legible

First in the Yukti series. The series argues that across five very different domains, the surface output is observable but the reasoning behind it has never been written down in a form anyone can build on. This one is about Madhava and Jyesthadeva: the result, and the man who wrote down the reasoning two generations later.

1. The Asymmetry

Most people who have heard of the Kerala School know one name. Madhava. The man who wrote down an infinite series for π three hundred years before Newton.

Fewer can name the person who made Madhava legible.

Jyesthadeva wrote his Yuktibhasha around 1530, in Malayalam.¹ It is the book where the proofs live. Without Jyesthadeva’s work, Madhava survives as a claim. With it, Madhava survives as mathematics.

I have been reading the English translation of the Yuktibhasha on breaks. It is patient writing. The kind of prose a teacher uses when the student is sitting across the table and cannot be expected to infer. That patience is itself a discovery. Most Indian mathematical tradition before this was composed in Sanskrit verse, tight enough to memorise, too tight to explain. Jyesthadeva refused that compression.

He made a choice, and nobody outside a small scholarly circle in Kerala read the result for four hundred years.

That gap is what this essay is about.

2. What Jyesthadeva Did

The mathematics that Madhava and his successors worked out between the 14th and 16th centuries is not in dispute. They derived infinite series expansions for π, sine, and cosine. They built planetary models that predicted observations more accurately than the Aryabhatan tradition they inherited. They were, by any measure, doing calculus before Newton.

What they did not have was a stable way to transmit the why.

Indian mathematical tradition compressed results into sutras and verse. A number could be encoded into a Sanskrit syllable using the katapayadi system, so that a planetary constant would arrive as a memorable line of poetry. This was beautiful and fragile. The verse told you what. The meaning lived in the teacher. When the teacher-disciple chain broke, the verse was still there and the proof was gone.

K.V. Sarma, who spent decades translating the Yuktibhasha, described the decline this way: as teacher-pupil and father-son traditions waned, commentaries stopped giving inner significance and reduced themselves to word meanings.¹ The formula survived. The reasoning did not.

Jyesthadeva sat inside this collapse and did the opposite. He wrote in Malayalam prose. He walked through the proofs step by step, the way a teacher does when the student is in the room. Kim Plofker calls it an informal exposition.² P.P. Divakaran calls it unambiguous Malayalam prose, a far cry from the enigmatic sutras of earlier masters.³

Madhava’s infinite series is the result. Jyesthadeva’s Yuktibhasha is the yukti, the reasoned proof of why the result holds. One is a discovery. The other is a decision about how discovery should be transmitted.

In any civilisation that ends up mattering, both have to survive. In this one, only one of them did.

3. The Elitism That Killed It

Why didn’t it survive?

The usual answer is colonialism. Portuguese arrival in 1498, the disruption of Malabar trade, then British imperial suppression of indigenous knowledge. All of this is real. None of it is the primary cause.

The primary cause is that Kerala’s mathematical tradition was locked inside a small caste. The Malayalam professor Litty Chacko, speaking on Manu Pillai’s podcast, put it cleanly: this mathematical tradition was not available to everyone due to caste norms, which is also why this mathematical tradition was never popular or widely studied.⁴

Temples got built, and stone does not cut itself. The sthapatis who raised Brihadeeswarar and Padmanabhaswamy had their own mathematics, transmitted through the Vishwakarma craftsmen-caste and the Shilpa Shastra texts. They knew proportion, load, and astronomical orientation. What they did not have was a shared discourse with the Namboothiri scholars writing infinite series across town. India did not have one knowledge elite that failed to distribute. It had several elites, all casted, all parallel, each holding its own mathematics as a private inheritance.

How porous those boundaries were in everyday practice, we cannot now reconstruct. Private conversations leave no manuscripts. What we can observe is the pattern the system produces, then and now. Each lineage keeps its own mathematics. Each lineage thins over time. When a lineage breaks, the knowledge inside it breaks with it. That is the failure mode we can still watch operating in India today and in living memory. Kerala’s calculus was lost the same way knowledge is still lost in this country: by being inherited rather than shared.

You can see the consequence when the British arrived. Charles Whish, the East India Company officer who first collected Kerala School manuscripts, died young and left them in London, where they sat unread. In 1863, the British scholar-administrator Charles P. Brown dismissed the bequest with a single sentence: being in the Tamil, Malayalam and Telugu character, they probably will be useless in England.⁵ When Kerala calculus did surface in later Western scholarship, it was treated as a clever hoax, or assumed to have been copied from Europeans.

Colonial bias is real. But colonial bias cannot destroy a knowledge tradition that has already been widely transmitted. It can only delete one that was never widely held in the first place.

The evidence is close at hand. Malayalam survived. Tamil survived. Bengali, Marathi, Sanskrit, Hindi, Urdu all survived. A billion people still read, argue, and pray in these languages after five centuries of invasion and colonisation. What did not survive was a mathematics kept inside a handful of families. The civilisation resisted where the threat met a distributed base. Where the base was narrow, it fell.

4. “We Knew Everything”: The Same Elitism in Defence Mode

There is a modern reaction to this loss that I want to name clearly, because it is the wrong response and it is growing.

The reaction takes the form of a claim: ancient India knew everything. Airplanes in the Vedas. Quantum physics in the Upanishads. Plastic surgery in Sushruta. Stem cells in the Mahabharata.

This reaction is not scientific temper but scientific nostalgia, and it is the same elitism that killed the Kerala School tradition, wearing defensive clothing.

Madhava and Jyesthadeva would not have tolerated it. Their standard was yukti, reasoned proof. Not shruti, revealed knowledge. The whole point of the Yuktibhasha is that a claim without a proof is not mathematics. You work the derivation. You expose the logic. You let the reader check.

When a contemporary movement says we already had all of it, it is making a claim without yukti. It is also, quietly, making the same move that killed the original tradition: treating knowledge as hereditary entitlement rather than as a working standard that has to be earned each generation.

If Madhava and Jyesthadeva were alive today, they would not be defending their ancestors. They would be working the problem. They would be irritated by the poster that says India had calculus first, and asking why the poster is not a textbook.

The most honest homage we can pay them is to behave like them. Find what is not yet known. Write it down so the next person can follow it. Refuse the temptation to confuse ancestry with achievement.

5. Scientific Temper, Forward

The scientific temper the Kerala School practised was specific, and it translates to the present without any adjustment.

First, no result without a proof. Results are cheap. Proofs are expensive. The discipline sits in not accepting a claim until the derivation can be walked. This is as true for a planetary model as it is for a pricing decision, a policy recommendation, or a machine-learning output. When we accept an answer we cannot reconstruct, we are no longer in the yukti tradition. We are back in the late sutra period where the verse survives and the meaning does not.

Second, no knowledge is valid if it is intentionally hidden. This is the Jyesthadeva standard. He refused the compression that kept mathematics inside a scholarly circle and wrote in Malayalam. That refusal is the test. If your knowledge is hoarded by platform, paywall, coaching fee, language, or caste, it is not knowledge in the civilisational sense. It is inventory.

I noticed this mapping late. Before I started reading the Yuktibhasha, I had been designing systems at ChainAlign against a principle I call anti-atrophy: no decision should survive in a system without the reasoning that produced it. When the reasoning is lost, the decision decays into a record that cannot be explained or corrected. The principle came from watching enterprise architectures fail in familiar ways, not from reading any ancient text. The Jyesthadeva standard turns out to be the same principle, named five centuries earlier, in Malayalam, a language I grew up reading. The heritage had been there all along. Nobody had pointed me to it.

Third, the point of working rigorously is to keep going, not to stop and celebrate. The Kerala School found errors in the Aryabhatan models they inherited, and they did not file the errors under divine will. They filed them under mathematics to be done. The posture is forward. The past is a shoulder to stand on.

A civilisation that cannot stand on its shoulders cannot stay on its feet.

6. AI as the New Malayalam, and the Shruti Trap

We are now living through something that looks very much like a second printing press.

AI can translate a graduate-level derivation into a language a child understands. It can take a paywalled paper and explain it in a tongue that no academic journal publishes in. A student in a village with a phone and an internet connection has access to a tutor who never sleeps and speaks every language.

In the structural sense, this is a Jyesthadeva moment. Joel Mokyr, studying why the Industrial Revolution happened where it did, pointed to a specific precondition. The cost of acquiring knowledge fell, and the pipeline between the theorist and the practitioner, between what Mokyr called the savants and the fabricants, opened up.⁶ Elizabeth Eisenstein made a related argument about the press itself. Print introduced fixity, exactly repeatable pictorial statements that did not drift with each copy.⁷ Both arguments are debated. Some historians, Gregory Clark and John R. Harris among them, argue the shift was more gradual and more driven by tacit artisanal skill than by books.⁸ The debate is real. The structural claim is what matters here: cumulative civilisational learning needs distribution, and distribution needs a cheap, reliable pipeline between the ones who know and the ones who can build.

AI is the newest such pipeline. It is cheaper than any printing press the world has seen.

The danger is that we throw the chance away by using AI the wrong way.

There is a version of AI use where the model gives you the answer and you stop there. You memorise the output, you paste it into the next conversation, you never reconstruct the reasoning. This is the late sutra tradition, industrialised. The verse arrives. The yukti does not. And when the context shifts, as it always does, we have a formula and no way to fix it.

There is a subtler version of the same trap. AI produces the illusion of understanding in domains where we have none. Fluency arrives before yukti. When the model gives a polished explanation of a proof, the reading feels like understanding, though what we have absorbed is the verse, not the reasoning. The surface is indistinguishable. The difference shows up only when the problem shifts slightly, or when someone asks a follow-up the output did not anticipate. Sutra fluency imitates yukti well enough to fool us. It fools us hardest when the yukti matters most.

Jyesthadeva wrote proofs because he knew the verse alone would not survive the next generation. We are choosing, now, whether to use AI to build proofs or to build verses.

If we choose verses, we lose the civilisational compounding that this moment is quietly offering.

If we choose proofs, we may finally honour the standard Jyesthadeva set.

Three Principles

There is no need to prove we were great in the 14th century. The evidence is in the record, for anyone who cares to read. The question that matters is whether the standard that made us great then is a standard we are willing to meet now.

Three principles to hold on to.

The Yukti Standard. No result is accepted without a proof that can be walked.

The Jyesthadeva Mandate. No knowledge is valid if it is intentionally hidden.

The AI Obligation. Use the tools of this moment to break the barriers of language, cost, caste, and paywall that killed the tradition last time.

Madhava is the memory. Jyesthadeva is the method. We can choose, today, which of the two we are interested in continuing.

Sources

1. Jyesthadeva, Ganita-Yukti-Bhasha (Rationales in Mathematical Astronomy), 2 vols., edited and translated by K.V. Sarma with explanatory notes by K. Ramasubramanian, M.D. Srinivas, and M.S. Sriram. Hindustan Book Agency, New Delhi, 2008. Also the quotation on the waning of teacher-pupil and father-son traditions is drawn from Sarma’s editorial commentary in this edition.

2. Kim Plofker, Mathematics in India. Princeton University Press, 2009.

3. P.P. Divakaran, The Mathematics of India: Concepts, Methods, Culture. Hindustan Book Agency / Springer, 2018.

4. Litty Chacko, interviewed by Manu S Pillai on SAGA with Manu S Pillai, Episode 4: Sangamagrama Madhava. ARPO, 2024.

5. Charles P. Brown’s 1863 dismissal of Charles Whish’s manuscript bequest, quoted in George Gheverghese Joseph, A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact. Sage Publications, 2009.

6. Joel Mokyr, The Gifts of Athena: Historical Origins of the Knowledge Economy. Princeton University Press, 2002. And The Enlightened Economy: An Economic History of Britain 1700-1850. Yale University Press, 2009.

7. Elizabeth L. Eisenstein, The Printing Press as an Agent of Change: Communications and Cultural Transformations in Early Modern Europe. Cambridge University Press, 1979.

8. Counter-views on the printing-press and knowledge-economy theses. Gregory Clark, A Farewell to Alms. Princeton University Press, 2007. John R. Harris, Industrial Espionage and Technology Transfer: Britain and France in the Eighteenth Century. Ashgate, 1998. Paul Saenger, Space Between Words: The Origins of Silent Reading. Stanford University Press, 1997.