A. Piṅgala and computation
Historical evidence Modern interpretation
Piṅgala’s Chandaḥśāstra, a treatise on Sanskrit metre usually dated to the last centuries BCE (the date is debated), classifies syllables as laghu (light) and guru (heavy) and gives procedures for listing and counting the patterns they form.
- Topics
- Sanskrit prosody; laghu and guru patterns; prastāra; enumeration; binary representations; recursive pattern generation; combinatorial algorithms; historical and modern interpretations.
- Questions
- How can the source procedures for listing (prastāra) and counting be reconstructed as explicit algorithms? In what exact sense are the patterns “binary”, and what is gained or lost by saying so? What do variant readings and commentaries change?
- Method
- Close reading of editions and translations, formal reconstruction, and implementation in code to confirm that the reconstruction produces what the source describes.
Binary-like pattern representations can be studied mathematically without assuming that their historical users had digital computers. The question concerns structure, not machines.
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B. Meru Prastāra
Historical evidence Modern interpretation
The meru-prastāra, often translated “staircase of Mount Meru”, is a triangular array that records how many patterns of a given length contain a given number of laghu. Its entries are binomial coefficients. Commentators on Piṅgala describe it, with Halāyudha’s account in the 10th century CE being well known. The same array appears in other traditions and is called Pascal’s triangle in Europe.
- Topics
- Combinatorial structures; recursive counting; binomial coefficients; pattern enumeration; mathematical reconstruction; connections with contemporary algorithmic methods.
- Questions
- By what rule do the sources build the array? How does their recursive counting relate to modern combinatorics? How do their methods compare with modern algorithms for the same counts?
- Method
- Derive and verify counts by direct enumeration and by recursion; compare descriptions across sources.
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C. Indian numerical methods
Historical evidence Modern interpretation
Indian mathematical texts developed place-value arithmetic, rational approximations, sine tables and series. Examples include the sine approximation attributed to Bhāskara I (7th century CE) and the infinite-series methods associated with Mādhava and the Kerala school (14th to 16th centuries CE).
- Topics
- Arithmetic and representation; approximation; exact computation; numerical algorithms; error analysis; alternative computational representations.
- Questions
- How accurate are these methods, and how did their authors control error? Which procedures are exact and which approximate? What alternative representations of number do they use?
- Method
- Reimplementation of the procedures, error analysis against modern reference values, and interval bounds where possible.
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D. Information theory and computational mechanics
Modern interpretation Working hypothesis
Modern information theory measures how much uncertainty a pattern source carries. Computational mechanics finds the smallest model that predicts a source as well as possible, using causal states. Metrical patterns are a clean testbed for both.
- Topics
- Information representation; causal states; pattern structure; recursive computation; adaptive numerical methods; AI-assisted mathematical investigation.
- Questions
- How much information do metrical constraints carry? What are the minimal predictive models of constrained pattern sets? How can AI tools help explore such structures without replacing proof?
- Method
- Compute entropies exactly, build and compare automata, and record for each finding whether it was proved or only found by search.
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E. Geometry and physical science
Historical evidence Modern interpretation
The Śulbasūtras (attributed to Baudhāyana, Āpastamba and Kātyāyana, the oldest usually dated to the first millennium BCE) give rope-based geometric constructions for ritual altars, including rules for the diagonal of a rectangle and approximations of √2.
- Topics
- Śulba mathematical traditions; geometry and measurement; physical modelling; fluid dynamics; historical hydraulic structures; mathematical approaches to engineering systems.
- Questions
- How do the construction procedures work and how accurate are they? Can historical water-management structures, such as stepwells and reservoirs, be modelled with modern fluid dynamics to understand how they performed?
- Method
- Reconstruct constructions, build numerical models, and compare with published surveys where available.
Related publications
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F. Adaptive computation
Working hypothesis
A computation can store a number in several ways: as an exact ratio, as a floating-point value, as a logarithm, or as an interval that is guaranteed to contain it. Each is better in some situations. This programme asks when and how to switch between them.
- Topics
- Representation switching; exact ratios; interval arithmetic; logarithmic representations; adaptive algorithms; computational efficiency; connections to intelligent physical systems.
- Questions
- When does switching representation improve accuracy or cost? Can the choice be made automatically? How does this connect to VIVO, where decisions must be made from uncertain measurements?
- Method
- Algorithm design, benchmarks and error bounds, reported with the cases where the idea fails.
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