The Ontology and Epistemology of Mathematics in Indian Tradition: From Ganita to Modern Abstraction

1. Introduction

What is mathematics? Is it a body of abstract truths, a language of the universe, or a sacred mirror of cosmic principles? This question, while seemingly modern, has animated the intellectual landscapes of multiple civilizations. In ancient India, mathematics—gaṇita—was not a secular discipline abstracted from the world but was intrinsically woven into the ritual, cosmological, architectural, and linguistic fabric of life.

This paper is a meditation on the ontological and epistemological dimensions of mathematics across two major streams of thought: the Indian classical tradition and Western formalism. It examines foundational notions such as the definition of number, the conceptual emergence of zero and infinity, and the logical infrastructure of inference in the Nyāya school compared to Western deduction. Further, it explores how mathematical ideas emerge within linguistic structures, sacred geometry, and cosmology.

The Indian tradition, with its recursive temporal cycles (kalpa, yuga), algorithmic grammar (as in Pāṇini’s Aṣṭādhyāyī), and intricate altar designs (Śulba Sūtras), prefigured many principles now formalized in modern logic, automata theory, set theory, and topology. The concept of śūnyatā (emptiness), deeply rooted in Buddhist metaphysics, parallels the mathematical void represented by zero and the set-theoretic empty set ∅.

Through rigorous comparative analysis and dense mathematical formulations—including recursive equations, symmetry groups, combinatorial functions, and topological invariants—this paper aims to reframe mathematics not merely as computation but as a form of metaphysical inquiry. Mathematics is revealed to be a system of signification and structure, one that bridges the seen and the unseen, the finite and the infinite, the ritual and the rational.

2. The Concept of Zero in Ancient Indian Mathematics

The origin of the concept of zero is deeply rooted in Indian mathematical traditions. The symbol for zero and the idea of using it as a number with its own properties were developed in India by mathematicians such as Brahmagupta in the 7th century CE. In his seminal work, the Brahmasphutasiddhanta, Brahmagupta provided the earliest known rules for arithmetic operations involving zero. He stated:

$$a+0=a,a-0=a,a\times 0=0$$

(1)

However, his rules for division involving zero were not entirely accurate, as he stated:

$${\displaystyle \frac{a}{0}}=\infty$$

(2)

which contradicts the modern understanding that division by zero is undefined.

The Bakshali manuscript, dated between the 3rd and 4th century CE through radiocarbon dating, contains a dot-like symbol representing zero as a placeholder. The mathematical notation seen in this manuscript confirms the Indian origin of the positional decimal system that uses powers of ten. The influence of Indian zero reached the Islamic world, where scholars such as Al-Khwarizmi and Al-Uqlidisi adopted it, eventually transmitting it to Europe. This transmission laid the foundation for the modern numerical systems and computational algorithms.

The arithmetic involving zero became critical in developing algebra and calculus, especially with limits and continuity, which were later formalized in European mathematics. Indian texts such as the Sulbasutras, although focused on geometric principles, hinted at abstract mathematical thinking that eventually matured in Brahmagupta’s works [1,2].

3. Subtle Differences in the Concept of Zero Across Number Systems

The concept of zero has evolved significantly as mathematical structures matured from natural numbers to reals and beyond. Though symbolized identically across systems, its interpretation, properties, and algebraic behavior depend on the broader context of the number system in which it resides. As such, zero as a whole number differs subtly, but critically, from zero as an integer, and again from zero in the real number system.

In the domain of whole numbers, typically denoted as $\mathbb{W}=\{0,1,2,3,\dots \}$, zero represents the beginning of the counting process and is the cardinality of the empty set. In this set-theoretic definition, zero fulfills the role of an identity element under addition:

$$a+0=0+a=a,\forall a\in \mathbb{W}$$

(3)

This identity axiom is fundamental in Peano arithmetic, which defines whole numbers inductively with zero as the base. However, zero in this context lacks an additive inverse, since subtraction such as $0-5$ is undefined in $\mathbb{W}$. Moreover, the set $\mathbb{W}$ is not closed under subtraction.

When we transition to the set of integers, $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$, zero acquires more algebraic functionality. Not only does it remain the additive identity, but it becomes the neutral element between the positive and negative integers. It satisfies:

$$-0=0$$

(4)

and serves as a unique number whose sign is undefined in typical number theory contexts. Zero also facilitates solution sets to linear equations. For instance, the solution to $ax=0$ where $a\ne 0$ is uniquely:

$$x=0$$

(5)

Additionally, integer domains support the existence of additive inverses. For each $a\in \mathbb{Z}$, there exists $-a\in \mathbb{Z}$ such that:

$$a+(-a)=0$$

(6)

which is a crucial property for the construction of abelian groups under addition.

The transition from $\mathbb{Z}$ to the real numbers $\mathbb{R}$ reveals even deeper conceptual subtleties. In $\mathbb{R}$, zero is not merely a number or boundary; it is a critical limit point and a reference for continuity, differentiability, and analytic behavior. In calculus, zero plays a central role in the definition of derivatives. Consider:

$$\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f\left(x\right)}{h}}$$

(7)

This formulation hinges on the ability to approach zero arbitrarily closely without attaining it. The limiting behavior near zero is pivotal in defining smoothness and local linearity of functions. Moreover, in real analysis, zero is treated as an accumulation point. Given any $ϵ>0$, there exists a $\mathrm{\delta }>0$ such that for all x satisfying $\left|x\right|<\mathrm{\delta }$, x lies within the $ϵ$-neighborhood of zero:

$$\forall ϵ>0,\exists \mathrm{\delta }>0:\left|x\right|<\mathrm{\delta }\Rightarrow x\in (-ϵ,ϵ)$$

(8)

This property underlies the $ϵ$-$\mathrm{\delta }$ definitions of limit and continuity in $\mathbb{R}$ [3].

Additionally, zero in the real number system supports more nuanced structures such as signed zeros in floating-point arithmetic. According to the IEEE 754 standard, computers distinguish between $+0$ and $-0$ during certain calculations. Although $+0=-0$ in mathematical terms, these distinct representations affect numerical outcomes in limits and directional derivatives, especially in complex and applied numerical algorithms [5].

Topologically, zero is a non-isolated point in $\mathbb{R}$; between any two distinct reals, however close, there exist infinitely many other numbers. This is in contrast to the integers or whole numbers, where zero is discrete and no other number lies strictly between 0 and 1.

In symbolic logic and foundational mathematics, zero was treated differently depending on the philosophical stance. Bertrand Russell, in his logicist framework, defined zero as the class of all classes with no members [3]. In formal logic, zero denotes falsity, whereas one denotes truth. This is reversed in Boolean algebra, where:

$$a\wedge 0=0,a\vee 0=a$$

(9)

In abstract algebra, the zero element plays a role in ring and field structures. In any field $\mathbb{F}$, zero is the unique additive identity satisfying:

$$a+0=a,a·0=0,{\displaystyle \frac{a}{0}}\mathrm{is}\mathrm{undefined}\mathrm{for}a\ne 0$$

(10)

Furthermore, in topology and metric spaces, the zero vector forms the kernel of linear transformations, and acts as the base point in vector spaces over $\mathbb{R}$. The zero function $f\left(x\right)=0$ is the neutral element in the space of continuous real-valued functions under pointwise addition.

Hence, across $\mathbb{W}$, $\mathbb{Z}$, and $\mathbb{R}$, the role of zero transitions from a counting origin to an algebraic pivot to an analytic anchor. These distinctions are not merely semantic but foundational, affecting how equations are solved, how limits are defined, and how systems are modeled. Zero’s evolution reflects the deepening of mathematical abstraction across centuries of intellectual history [4,5].

4. The Emergence and Formalization of the Concept of Infinity in Scientific History

The concept of infinity has intrigued mathematicians, philosophers, and scientists for over three millennia. Unlike quantities grounded in sensory experience, infinity represents the limitless, the unbounded, and the non-finite, which makes it philosophically challenging and mathematically abstract. The treatment of infinity has evolved across distinct epochs: ancient metaphysical speculation, early mathematical intuition, Renaissance symbolism, and finally, modern set-theoretic rigor. This section presents a detailed account of the conceptual emergence and rigorous formalization of infinity in mathematical history.

In ancient India, the concept of infinity appeared under the term “Ananta,” meaning boundless or endless. The Upanishadic texts associated Ananta with Brahman, the infinite and formless cosmic principle. In Jain mathematical philosophy, developed around 500 BCE, a classification of infinity was proposed that surpassed other ancient cultures in sophistication. Jain thinkers distinguished between different types of infinities: infinite in one direction, infinite in two directions, infinite in area, and infinite everywhere [7]. These were not metaphorical assertions, but reflected a formal attempt to categorize magnitudes beyond finite measurement.

The Greek philosophical tradition engaged with infinity primarily through paradoxes. Anaximander introduced the notion of the “apeiron” (boundless), while Zeno of Elea’s paradoxes posed serious challenges to the coherence of infinite divisibility. For example, in the Achilles and the Tortoise paradox, Zeno argued that Achilles could never overtake a tortoise if space could be divided infinitely. The paradox hinges on the infinite geometric series:

$$\sum _{n=0}^{\infty }{\left({\displaystyle \frac{1}{2}}\right)}^n=1$$

(11)

Although this result is accepted in modern analysis, it was philosophically problematic in Zeno’s time, leading Aristotle to introduce a critical distinction between potential and actual infinity. Aristotle, in his Physics, allowed processes to be infinite (e.g., counting forward without end), but denied that a completed infinite quantity could exist [3].

In the medieval period, theological and metaphysical uses of infinity became dominant. Christian, Islamic, and Jewish thinkers invoked the infinite nature of God, but largely avoided mathematical applications. Thomas Aquinas referred to God as the “act of being itself” and thus inherently infinite. Nonetheless, such conceptualizations remained philosophical and did not influence the formal development of mathematics [4].

The Renaissance marked a turning point. Galileo Galilei observed a paradox of infinite sets, noting that the set of perfect squares $\{1,4,9,16,\dots \}$ can be put into one-to-one correspondence with the natural numbers $\{1,2,3,4,\dots \}$. That is:

$$f\left(n\right)=n^2\mathrm{is}\mathrm{a}\mathrm{bijection}\mathrm{from}\mathbb{N}\to \{n^2:n\in \mathbb{N}\}$$

(12)

This led Galileo to conclude that concepts such as “greater than” and “equal to” are not meaningful when comparing infinite sets, a paradox that would remain unresolved until the late 19th century.

John Wallis, in 1655, introduced the modern symbol for infinity $(\infty )$ in his Arithmetica Infinitorum, treating infinity as an extreme value that grows beyond any finite bounds. Calculus emerged soon after with Newton and Leibniz, whose use of infinitesimals implicitly invoked infinity. The derivative:

$${\displaystyle \frac{dy}{dx}}=\underset{\Delta x\to 0}{lim}{\displaystyle \frac{\Delta y}{\Delta x}}$$

(13)

relies on approaching zero through an infinite process, although the concept of infinity remained intuitive rather than formal during their time [5].

The foundational breakthrough came in the 1870s with Georg Cantor, who rigorously defined actual infinities using set theory. Cantor showed that infinite sets can have different cardinalities. He demonstrated that the set of natural numbers $\mathbb{N}$ is countably infinite, denoted:

$$\left|\mathbb{N}\right|={\aleph }_0$$

(14)

whereas the real numbers $\mathbb{R}$ form an uncountable set, satisfying:

$$\left|\mathbb{R}\right|>{\aleph }_0$$

(15)

Cantor’s diagonal argument proved that no bijection exists between $\mathbb{N}$ and $\mathbb{R}$, thereby establishing a hierarchy of infinities [8]. Moreover, he introduced transfinite numbers and developed ordinal and cardinal arithmetic, laying the foundation for modern set theory.

Cantor also showed that the power set of $\mathbb{N}$, denoted $\mathcal{P}\left(\mathbb{N}\right)$, has a cardinality strictly greater than that of $\mathbb{N}$:

$$\left|\mathcal{P}\left(\mathbb{N}\right)\right|=2^{{\aleph }_0}$$

(16)

This result suggests an infinite sequence of larger and larger infinities:

$${\aleph }_0<2^{{\aleph }_0}<2^{2^{{\aleph }_0}}<\dots$$

(17)

which revolutionized mathematical thinking and resolved the ancient ambiguities of infinite comparison.

Despite its elegance, Cantor’s work was initially controversial. Leopold Kronecker rejected actual infinity, stating that "God made the integers; all else is the work of man." Cantor’s ideas were later incorporated into axiomatic set theory, most notably the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC), which underpins much of modern mathematics [4].

In summary, infinity transitioned from a metaphysical abstraction to a mathematically rigorous concept only in the late 19th century. Its emergence marks one of the most profound conceptual shifts in human thought, reconciling ancient paradoxes with modern formalism.

5. Projective Duality of Zero and Infinity Through Conformal Mappings

In classical Euclidean geometry, zero and infinity are positioned as conceptual and numerical extremes, often lacking a formal connection. However, in the frameworks of projective geometry and conformal mappings, especially within complex analysis, these two concepts attain a duality that reveals a unified geometric and analytic structure. This section explores the interplay between zero and infinity via projective compactification, Möbius transformations, and the conformal Riemann sphere.

We begin with the definition of the complex projective line, denoted ${\mathbb{P}}^1\left(\mathbb{C}\right)$, which compactifies the complex plane by adding a point at infinity. Formally, ${\mathbb{P}}^1\left(\mathbb{C}\right)$ is defined as the space of equivalence classes:

$${\mathbb{P}}^1\left(\mathbb{C}\right)=\left\{[z:w]\in {\mathbb{C}}^2\setminus \left\{(0,0)\right\}|[z:w]\sim [\lambda z:\lambda w],\lambda \in {\mathbb{C}}^{*}\right\}$$

(18)

This definition introduces a coordinate-free treatment of points on the complex line, allowing for a natural representation of infinity. The affine complex plane $\mathbb{C}$ is embedded in ${\mathbb{P}}^1\left(\mathbb{C}\right)$ via the map:

$$z\mapsto [z:1]$$

(19)

with the additional point at infinity represented by:

$$\infty \equiv [1:0]$$

(20)

The projective line is topologically equivalent to the Riemann sphere $S^2$, obtained through stereographic projection. Every meromorphic function on $\mathbb{C}$ can be extended to a continuous function on the Riemann sphere by assigning values at infinity. This allows infinity to be treated on par with any finite complex number [9].

An important structure preserving the conformal geometry of ${\mathbb{P}}^1\left(\mathbb{C}\right)$ is the group of Möbius transformations, or fractional linear transformations, defined by:

$$f\left(z\right)={\displaystyle \frac{az+b}{cz+d}},ad-bc\ne 0$$

(21)

These transformations form the group $\mathrm{PSL}(2,\mathbb{C})$, acting transitively on the Riemann sphere. Of particular interest is their behavior under inversion. Consider the transformation:

$$f\left(z\right)={\displaystyle \frac{1}{z}}$$

(22)

This transformation exchanges zero and infinity:

$$f\left(0\right)=\infty ,f(\infty )=0$$

(23)

revealing a fundamental symmetry in projective geometry. This symmetry is preserved in the homogeneous matrix representation:

$$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

(24)

which represents the map $z\mapsto {\displaystyle \frac{1}{z}}$. This involution maps circles to circles, and in particular, interchanges the extended real line and the unit circle under stereographic projection [10].

The symmetry between zero and infinity becomes essential in the analysis of rational functions. Consider a rational function:

$$R\left(z\right)={\displaystyle \frac{P\left(z\right)}{Q\left(z\right)}},\mathrm{with}degP=m,degQ=n$$

(25)

Its behavior at zero and at infinity is analyzed by examining:

$$\underset{z\to 0}{lim}R\left(z\right),\underset{z\to \infty }{lim}R\left(z\right)$$

(26)

For instance, the transformation $z\mapsto {\displaystyle \frac{1}{z}}$ allows us to study poles at infinity by converting them into poles at zero:

$$\underset{z\to \infty }{lim}R\left(z\right)=\underset{w\to 0}{lim}R\left({\displaystyle \frac{1}{w}}\right)$$

(27)

This duality is fundamental in residue theory and Laurent series expansions, where the principal part of a function near $z=\infty$ corresponds to the negative power terms in the Laurent expansion around zero.

The behavior of analytic functions under inversion further connects zero and infinity through functional symmetries. For example, the logarithm satisfies:

$$log\left(z\right)+log\left({\displaystyle \frac{1}{z}}\right)=log\left(1\right)=0$$

(28)

and the exponential function behaves reciprocally:

$$exp(-z)={\displaystyle \frac{1}{exp\left(z\right)}}$$

(29)

From a geometric perspective, the Riemann sphere allows every meromorphic function $f:{\mathbb{P}}^1\to {\mathbb{P}}^1$ to be expressed as a rational function. The total number of zeros and poles (counting multiplicities) of such a function equals its degree:

$$\mathrm{Number}\mathrm{of}\mathrm{zeros}+\mathrm{Number}\mathrm{of}\mathrm{poles}=deg\left(f\right)$$

(30)

This balance reflects a projective symmetry that extends the classical Fundamental Theorem of Algebra to compactified domains [11].

In algebraic geometry, the dual role of zero and infinity is intrinsic to divisors and line bundles on ${\mathbb{P}}^1$. The divisor of a rational function is defined as:

$$\mathrm{div}\left(f\right)=\sum _{p\in {\mathbb{P}}^1}{\mathrm{ord}}_p\left(f\right)·p$$

(31)

where ${\mathrm{ord}}_p\left(f\right)$ is the order of vanishing (or pole) at p. The condition:

$$deg\left(\mathrm{div}\right(f\left)\right)=0$$

(32)

forces zeros and poles, including those at infinity, to balance in degree — again enforcing duality between 0 and ∞.

In mathematical physics, especially in conformal field theory (CFT), this duality is exploited in the operator product expansion (OPE) near singularities, and in the construction of vertex operator algebras where points at zero and infinity correspond to insertion and extraction of states from the vacuum [12].

Thus, the notions of zero and infinity, though numerically distant, are geometrically and algebraically intertwined in the structure of projective and conformal mappings. Their duality under inversion, their symmetry in Möbius transformations, and their balance in rational function theory reflect a unified framework in which both can be analyzed not as boundaries, but as interdependent entities within a compactified mathematical universe.

6. The Semantics and Philosophical Differences Between Gaṇita and Mathematics

The term “Mathematics” in modern English and its equivalent “गणित” (Gaṇita) in Hindi and Sanskrit are often used interchangeably in educational and technical discourse. However, a deeper etymological, historical, and philosophical investigation reveals that they encapsulate fundamentally distinct epistemic traditions. While both encompass number, magnitude, and structure, their origins, methods, and worldviews diverge significantly.

The Sanskrit term “गणित” originates from the verbal root “गण्” (gaṇ), which means “to count”, “to enumerate”, or “to reckon”. This etymology places computation and practical enumeration at the heart of Gaṇita. The term is found extensively in ancient Indian mathematical treatises such as the Āryabhaṭīya of Āryabhaṭa (c. 499 CE), the Līlāvatī of Bhāskarācārya (c. 1150 CE), and earlier references in Vedic literature related to Vedā$\dot{\mathrm{n}}$ga Jyotiṣa [7,13].

In contrast, the word “Mathematics” stems from the Greek “μάθημα” (*mathema*), meaning “that which is learnt”. The related term “μαθηματικός” (*mathematikos*) referred to one who is fond of learning. The emphasis in the Greek tradition, particularly after Euclid, shifted toward deductive logic, abstract axiomatization, and idealized forms [4]. Thus, while Gaṇita foregrounds computation and approximation, Mathematics in the Greek-European tradition emphasizes abstract proof.

In terms of structure, classical Indian mathematical texts categorize Gaṇita into several branches. For example, Bhāskara’s Līlāvatī delineates topics such as:

Arithmetic (पाटीगणित), Algebra (बीजगणित), Geometry (रेखागणित), Mensuration (परिमाणगणित)

(33)

These subfields often involved recursive methods, empirical approximations, and rule-based computations. Consider, for instance, Bhāskara’s rule for approximating square roots:

$$\sqrt{N}\approx x+{\displaystyle \frac{N-x^2}{2x}}$$

(34)

This formula closely resembles Newton-Raphson’s iteration in numerical analysis. Unlike Euclidean geometry, which prioritizes deductive axioms, Gaṇita validates methods through pragmatic utility, iterative verification, and empirical accuracy.

The philosophical foundation of Gaṇita also integrates with metaphysical systems. The Śulba Sūtras (c. 800 BCE) include geometric constructions for altars, where the diagonal of a square is described in terms of the Pythagorean theorem:

$$\mathrm{If}a=b,\mathrm{then}c=\sqrt{2}a$$

(35)

Here, $\sqrt{2}$ is approximated with high precision, such as:

$$\sqrt{2}\approx {\displaystyle \frac{577}{408}}$$

(36)

as seen in Baudhāyana’s and Āpastamba’s Sulbasūtras. These approximations were not derived axiomatically but were refined through recursive arithmetic and geometric reasoning [14].

The notion of number in Gaṇita diverges ontologically from that in Western Mathematics. In Gaṇita, numbers are computational entities. In contrast, in Platonist mathematics, numbers are often viewed as ideal forms. Consider the modern formulation of natural numbers via the Peano axioms, where:

$$0\mathrm{is}\mathrm{a}\mathrm{natural}\mathrm{number},\mathrm{If}n\in \mathbb{N},\mathrm{then}S\left(n\right)\in \mathbb{N}$$

(37)

This axiomatic approach abstracts numbers from computational activity. Gaṇita, however, emphasizes techniques such as the "Kuttaka" (pulverizer) algorithm for solving indeterminate equations:

$$ax+by=c$$

(38)

used famously by Āryabhaṭa to find multiplicative inverses modulo n, centuries before the extended Euclidean algorithm appeared in Europe [13].

Moreover, Gaṇita integrates epistemology with pramāṇa theory (means of knowledge). Valid knowledge arises from pratyakṣa (perception), anumāna (inference), and upamāna (comparison), among others. Mathematics in India was not just an abstract system but a practical, valid means of acquiring knowledge about the world, including celestial cycles, architecture, and rituals [15].

Even terminologically, Gaṇita was often aligned with Jyotiṣa (astronomy/astrology), unlike the Western division between pure and applied mathematics. The Sanskrit word for number, “सङ्ख्या” (sa$\dot{\mathrm{n}}$khyā), and related words like “बीज” (bīja, seed) and “रेखा” (rekhā, line) emphasize generative and geometric origins of computation. Algebra as “bījagaṇita” implies a connection to roots and growth, emphasizing intuition over strict logical derivation.

In summary, Gaṇita and Mathematics, while both dealing with quantitative reasoning, emerge from vastly different intellectual and cultural contexts. The former values computation, approximation, and utility, whereas the latter idealizes deduction, abstraction, and internal consistency. This divergence affects pedagogy, epistemology, and the framing of mathematical truths across civilizations.

7. Analogues of Abstract Mathematical Concepts in Ancient Indian Mathematics

Modern mathematical developments in abstract algebra, set theory, and group theory represent the pinnacle of formalization and abstraction in the mathematical sciences. Concepts such as Cantor’s transfinite cardinals, the axiomatic structure of groups, rings, and fields, and the formulation of infinity as a mathematical object, are hallmarks of nineteenth and twentieth-century mathematics. This section investigates whether similar structural insights—albeit in different forms—can be identified in Ancient Indian Mathematics.

One of the most striking developments in modern mathematics is Cantor’s formulation of cardinalities of infinite sets. Cantor introduced transfinite numbers and defined the cardinality of countably infinite sets such as the natural numbers as:

$$\left|\mathbb{N}\right|={\aleph }_0$$

(39)

and established that the cardinality of the real numbers satisfies:

$$\left|\mathbb{R}\right|=2^{{\aleph }_0}>{\aleph }_0$$

(40)

While ancient Indian mathematics did not employ formal set-theoretic definitions, the Jain tradition (circa 500 BCE) introduced a classification of different types of infinities. As recorded in the Tattvārtha Sūtra, Jain scholars posited that infinities could be of various kinds: infinite in one dimension, two dimensions, or omnidirectionally infinite. This conceptual richness anticipates the comparative study of infinities without appealing to cardinal arithmetic [18].

Modern group theory studies sets equipped with binary operations that satisfy associativity, identity, and invertibility. A group is formally defined as a pair $(G,·)$ where:

$$\forall a,b,c\in G,(a·b)·c=a·(b·c)$$

(41)

$$\exists e\in G,\forall a\in G,a·e=e·a=a$$

(42)

$$\forall a\in G,\exists a^{-1}\in G,a·a^{-1}=e$$

(43)

While there is no evidence of such an explicit structure in ancient Indian texts, functional analogs exist in grammar and astronomy. Panini’s grammar (Ashtadhyayi, 5th century BCE) consists of transformation rules that can be represented as rewriting systems with invertible transitions and closure under composition. Scholars such as Staal and Bronk argue that Panini’s rules can be modeled using formal automata and transformational algebra [23].

Likewise, astronomical computations in Āryabhaṭa’s and Bhāskara’s works utilize modular arithmetic to compute planetary positions. These cyclic processes resemble the properties of finite cyclic groups. For instance, the Kuttaka algorithm solves equations of the form:

$$ax+by=c$$

(44)

which is functionally equivalent to the extended Euclidean algorithm used to find modular inverses, a core operation in group theory over ${\mathbb{Z}}_n^{*}$ [13].

The algebraic work of Bhāskara II in Bījagaṇita reveals a deep understanding of polynomial manipulation and symbolic representation. Quadratic and cubic equations were solved using algorithmic processes, and expressions involving roots and surds were manipulated according to rules such as:

$$\sqrt{a}·\sqrt{b}=\sqrt{ab}$$

(45)

$${(a+b)}^2=a^2+2ab+b^2$$

(46)

These manipulations mirror the algebraic identities preserved in ring structures, albeit without an explicit definition of algebraic systems as seen in modern axiomatic algebra [14].

Furthermore, Indian mathematicians showed a profound understanding of series and limits. The Kerala school, particularly Mādhava of Sangamagrama (circa 14th century), derived infinite series expansions for trigonometric functions. For example, the power series for $arctan\left(x\right)$ was given as:

$$arctan\left(x\right)=x-{\displaystyle \frac{x^3}{3}}+{\displaystyle \frac{x^5}{5}}-{\displaystyle \frac{x^7}{7}}+\dots$$

(47)

Setting $x=1$ yields:

$${\displaystyle \frac{\pi }{4}}=1-{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{5}}-{\displaystyle \frac{1}{7}}+\dots$$

(48)

This series, now known as the Gregory–Leibniz series, was computed in India centuries before its rediscovery in Europe. It demonstrates an early conceptual grasp of convergence and infinite processes [19].

Combinatorics and recursion were extensively developed in works such as Pingala’s Chandaḥśāstra. The binary representation of poetic meters (with laghu and guru syllables) leads to a recursive enumeration of combinations:

$$C(n,r)=C(n-1,r-1)+C(n-1,r)$$

(49)

This recursion, functionally equivalent to Pascal’s Triangle, shows a deep understanding of algorithmic generation of combinatorial coefficients [24].

In conclusion, although ancient Indian mathematicians did not formulate abstract structures like sets, groups, or fields in a modern axiomatic sense, they demonstrated an exceptional operational grasp of these ideas. They explored different types of infinity, manipulated algebraic expressions symbolically, solved congruences algorithmically, and computed infinite series and combinatorial identities. These achievements illustrate a profound engagement with abstract patterns, symmetry, and structure—revealing Gaṇita as not just a practical discipline but one with deep philosophical and proto-theoretical dimensions.

8. On the Ontological and Structural Definition of Number

The notion of “number” is foundational to all of mathematics, yet its definition is deceptively non-trivial. In everyday language, a number is something that tells us “how many” of something there are. However, from a philosophical and formal mathematical point of view, this naive idea quickly deepens into a complex interplay of logic, structure, and abstraction. One concise and conceptually rich formulation is the following:

“Number is the only non-trivial property of an arbitrary collection.”

This statement encapsulates a structuralist perspective. An “arbitrary collection” may be interpreted formally as a set, as defined in Zermelo-Fraenkel set theory (ZF). A “non-trivial property” is one that remains invariant under isomorphisms — in this case, bijections between sets. If two collections can be placed in a one-to-one correspondence, then they are said to have the same number. This leads us to the formal definition of cardinality:

$$\left|A\right|=\left|B\right|\iff \exists f:A\to B\mathrm{such}\mathrm{that}f\mathrm{is}\mathrm{bijective}$$

(50)

Thus, the number of a set is not dependent on what elements it contains, but solely on how many elements it contains. All singleton sets have cardinality one; all ten-element sets have cardinality ten, regardless of the nature of the objects. This aligns with Frege’s logicist program, where numbers are defined as classes of equinumerous sets [16].

To build this formally, we consider the equivalence classes of finite sets under the equivalence relation:

$$A\sim B\iff \exists f:A\to B\mathrm{such}\mathrm{that}f\mathrm{is}\mathrm{bijective}$$

(51)

Then, the cardinal number associated to A is defined as the equivalence class $\left[A\right]$ under ∼. For example:

$$\left|\right\{x,y,z\left\}\right|=3,\left|\right\{1,2,3\left\}\right|=3,\mathrm{because}\{x,y,z\}\sim \{1,2,3\}$$

(52)

The Peano axioms provide a recursive, logical basis for the natural numbers:

$$0\in \mathbb{N},\forall n\in \mathbb{N},S\left(n\right)\in \mathbb{N}$$

(53)

$$\forall n\in \mathbb{N},S\left(n\right)\ne 0,\mathrm{and}S\left(n\right)=S\left(m\right)\Rightarrow n=m$$

(54)

This construction treats numbers as arising from a primitive zero and the application of a “successor” operation, without reference to any collection. The tension between these views — one rooted in set-theoretic cardinality, the other in logical recursion — illustrates the multifaceted nature of number theory.

In category theory, the natural numbers object (NNO) in a cartesian closed category can be defined via universal properties. Let 1 be the terminal object and $\mathbb{N}$ the object such that:

$$\mathrm{There}\mathrm{exist}\mathrm{maps}z:1\to \mathbb{N}\mathrm{and}s:\mathbb{N}\to \mathbb{N}$$

(55)

such that for any object X, and morphisms $f:1\to X$, $g:X\to X$, there exists a unique morphism $h:\mathbb{N}\to X$ making the diagrams commute. This formalizes recursion as a universal construction [17].

Turning to Indian traditions, the Sanskrit term for number is सङ्ख्या (sa$\dot{\mathrm{n}}$khyā), derived from the root “sa$\dot{\mathrm{n}}$kh” meaning “to enumerate” or “to reckon.” The word गणित (gaṇita) emphasizes computation, counting, and numerical operations. In works such as Bhāskara II’s Līlāvatī, numbers are introduced contextually, as measurements, counts, and magnitudes, without abstraction into sets. Nevertheless, the operational definition implicitly assumes cardinality.

$$25÷5=5$$

(56)

which reflects a set-theoretic partition into five disjoint subsets of equal cardinality, again supporting the intuition that number measures equivalence of collections.

In Jain metaphysics, numbers are not merely tools of arithmetic but ontological categories. Infinite types of numbers are discussed — countably infinite, uncountably infinite, and infinitesimals. The Jaina doctrine classified infinities into three main types: enumerable infinite, non-enumerable infinite, and absolute infinite [18]. This implies an early awareness that even among arbitrary collections, “number” can have qualitative gradations.

In abstract algebra, numbers evolve into elements of structures. For example, in modular arithmetic:

$${\mathbb{Z}}_n=\{0,1,2,\dots ,n-1\}$$

(57)

numbers are equivalence classes under the relation:

$$a\equiv bmodn\iff n\mid (a-b)$$

(58)

This generalization shifts focus from cardinality to congruence and algebraic structure. Similarly, in group theory, numbers can be thought of as labels of group elements, such as:

$${\mathbb{Z}}_n\cong (\mathbb{Z},+)/n\mathbb{Z}$$

(59)

Thus, in different contexts, “number” may be cardinal, ordinal, nominal, or structural. Yet your proposed definition retains its integrity across all these views: a number abstracts all qualities except quantity.

This philosophical clarity allows one to generalize number beyond finiteness. Cantor’s transfinite numbers define:

$${\aleph }_0=\left|\mathbb{N}\right|,{\aleph }_1=\left|\mathbb{R}\right|,\mathrm{if}\mathrm{the}\mathrm{Continuum}\mathrm{Hypothesis}\mathrm{holds}$$

(60)

Even here, a number denotes the only invariant under bijections among infinite collections.

In conclusion, defining number as “the only non-trivial property of an arbitrary collection” is not only intuitively elegant but aligns with deep mathematical traditions across set theory, logic, algebra, and Indian mathematical philosophy. It highlights number as a measure of invariance, a relational essence unperturbed by the identity of constituents, and a bridge between the concrete and the abstract.

9. Logic and Proof: Nyāya Reasoning versus Western Deductive Systems

The notion of logical inference lies at the core of mathematical proof and epistemology across cultures. While the Western tradition, originating in Aristotle’s syllogism and formalized in Frege-Russell logic, centers on deduction from axioms using formal operators, the Indian tradition of Nyāya provides an equally rich yet distinct approach based on structured inference, empirical grounding, and multi-valued cognition. This section undertakes a comparative study of these systems.

The classical form of a deductive argument in Western logic is the modus ponens rule:

$$P\to Q,P,\therefore Q$$

(61)

This can be interpreted as: “If P implies Q, and P is true, then Q is necessarily true.” This binary structure is truth-functional and operates under bivalent semantics. The roots of this system trace to Aristotle’s syllogism, which takes the form:

$$\mathrm{All}M\mathrm{are}P,\mathrm{All}S\mathrm{are}M,\therefore \mathrm{All}S\mathrm{are}P$$

(62)

In contrast, the Nyāya school, as expounded in Gautama’s Nyāya Sūtras (circa 2nd century BCE), defines inference (anumāna) through a five-membered syllogism:

$$1.\mathrm{Pratij}ñā\left(\mathrm{Proposition}\right):P\mathrm{is}\mathrm{true}$$

(63)

$$2.\mathrm{Hetu}\left(\mathrm{Reason}\right):\mathrm{Because}\mathrm{Q}$$

(64)

$$3.\mathrm{D}ṛṣṭā\mathrm{nta}\left(\mathrm{Example}\right):\mathrm{As}\mathrm{in}\mathrm{case}\mathrm{R}$$

(65)

$$4.\mathrm{Upanaya}\left(\mathrm{Application}\right):\mathrm{This}\mathrm{case}\mathrm{is}\mathrm{like}\mathrm{R}$$

(66)

$$5.\mathrm{Nigamana}\left(\mathrm{Conclusion}\right):\mathrm{Therefore}\mathrm{P}$$

(67)

A concrete example follows: Pratijñā: There is fire on the hill. Hetu: Because there is smoke. Dṛṣṭānta: Wherever there is smoke, there is fire, as in a kitchen. Upanaya: The hill has smoke, which is like the kitchen. Nigamana: Therefore, the hill has fire.

The epistemological structure is significantly different. Nyāya insists on the necessity of citing a general rule (vyāpti) and an observed instance (dṛṣṭānta) to validate inference, emphasizing causal and empirical justification over formal deduction [25]. The general form of inference in Nyāya can be written as:

$$\forall x\left(S\right(x)\to H(x\left)\right),S\left(a\right),\therefore H\left(a\right)$$

(68)

where $S\left(x\right)$ represents smoke, and $H\left(x\right)$ represents fire. However, the inclusion of a concrete example (as in the kitchen) introduces a third-order quantification that is not present in classical syllogistic logic.

Modern propositional logic utilizes truth tables and logical connectives such as:

$$\neg P,P\wedge Q,P\vee Q,P\to Q,P\leftrightarrow Q$$

(69)

Indian logic, especially in Buddhist logic (Dignāga and Dharmakīrti), extended inference into a trivalent logic including true, false, and indeterminate, anticipating developments in paraconsistent and many-valued logic [26]. This aligns with the modern fuzzy logic systems and non-classical logical frameworks.

Moreover, Nyāya epistemology is grounded in four pramāṇas (means of valid knowledge):

$$1.\mathrm{Pratyak}ṣ\mathrm{a}\left(\mathrm{Perception}\right),2.\mathrm{Anum}ā\mathrm{na}\left(\mathrm{Inference}\right),3.\mathrm{Upam}ā\mathrm{na}\left(\mathrm{Comparison}\right),4.Ś\mathrm{abda}(\mathrm{Verbal}\mathrm{Testimony})$$

(70)

This broadens the scope of logic beyond deductive syntax to encompass the sources and conditions of cognitive validity. In comparison, Western logic isolates inference from sensory or testimonial sources, relying instead on formal systems of rules and symbol manipulation.

The Nyāya method assumes the possibility of cognitive error (viparyaya) and introduces the role of tarka (suppositional reasoning) as a method of eliminating false alternatives. In terms of formal reasoning, this resembles the reductio ad absurdum (RAA):

$$\neg Q\Rightarrow \mathrm{Contradiction},\therefore Q$$

(71)

This is present in Nyāya as well, but within a broader hermeneutic and dialogic framework, often in debate or dialectical settings.

In Western symbolic logic, the system is formalized using propositional and predicate calculus. For instance, the first-order formula:

$$\forall x\left(P\right(x)\to Q(x\left)\right),P\left(a\right),\therefore Q\left(a\right)$$

(72)

has a direct syntactic proof. In contrast, the Nyāya system places semantic emphasis on the known regularities and empirical observation supporting the rule $P\left(x\right)\to Q\left(x\right)$. The presence of the Dṛṣṭānta ensures that the generalization is grounded, not merely stipulated [27].

Another important distinction is that in Nyāya, knowledge is validated via pramāṇa, and not simply by syntactic derivability. A valid inference must be accompanied by a reliable causal connection (vyāpti), where:

$$\forall x,S\left(x\right)\Rightarrow H\left(x\right),\mathrm{and}S\left(x\right)\mathrm{is}\mathrm{invariably}\mathrm{connected}\mathrm{to}H\left(x\right)$$

(73)

The reliability of inference is thus epistemically weighted, unlike the purely formal derivability of Western logic.

In conclusion, the Nyāya system of logic, with its five-step inferential model, provides a rich and layered alternative to Western binary syllogism and modern symbolic logic. While Western logic emphasizes syntactic deduction from axioms using binary connectives, Nyāya integrates inference with ontology, epistemology, and semantics. The presence of empirical grounding through Dṛṣṭānta, the emphasis on pramāṇas, and the subtle inclusion of cognitive states render it a powerful and sophisticated system.

10. Mathematics of Language: Pāṇini’s Aṣṭādhyāyī and Formal Systems

The Aṣṭādhyāyī, composed by the grammarian Pāṇini around the 5th century BCE, stands as one of the most rigorous formal systems in the history of human thought. Consisting of nearly 4,000 concise rules (sūtras), it encodes the morphology and syntax of classical Sanskrit in a generative, algorithmic structure. In recent decades, scholars have increasingly recognized that Pāṇini’s grammar anticipates key features of modern formal systems, including finite state automata, generative grammars, rule rewriting.

At its core, Pāṇini’s system employs a metalanguage that defines how phonemes, morphemes, and syntactic units interact through a rule-based derivation process. The rules are arranged to be both generative and transformational. A typical derivation involves multiple stages of substitution, affixation, and morphophonemic adjustment, executed according to clearly defined precedence constraints.

The formal structure of Pāṇini’s grammar is highly compatible with the concept of a finite-state transducer (FST). A finite automaton is defined by the quintuple:

$$M=(Q,\mathrm{\Sigma },\mathrm{\delta },q_0,F)$$

(74)

where Q is a finite set of states, $\mathrm{\Sigma }$ is the input alphabet, $\mathrm{\delta }:Q\times \mathrm{\Sigma }\to Q$ is the transition function, $q_0\in Q$ is the initial state, and $F\subseteq Q$ is the set of final states. In a finite-state transducer, the output is generated alongside the transitions:

$$T=(Q,\mathrm{\Sigma },\mathrm{\Gamma },\mathrm{\delta },\omega ,q_0,F)$$

(75)

where $\mathrm{\Gamma }$ is the output alphabet, and $\omega :Q\times \mathrm{\Sigma }\to {\mathrm{\Gamma }}^{*}$ defines the output function. In Pāṇini’s system, the phonological transformations, particularly in the Pratyāhāra-sūtras and Sandhi-rules, can be modeled using such transducers [20].

Each grammatical rule functions as a production in a generative system, comparable to a rewriting rule in formal language theory. In Chomsky’s classification, a context-sensitive rule takes the form:

$$\alpha A\beta \to \alpha \gamma \beta$$

(76)

where A is a non-terminal, and $\gamma$ is a string possibly containing terminals and non-terminals. Many of Pāṇini’s rules follow this format, transforming substrings conditionally based on their contexts. For example, in the rule:

$$a+i\to e$$

(77)

the vowel coalescence can be encoded as a rule with left and right contexts. This is equivalent to:

$$\mathrm{If}x=a,\mathrm{and}y=i,\mathrm{then}x\circ y=e$$

(78)

such rules are deterministic and local, making them suitable for implementation in computational parsers and phonological analyzers [21].

Moreover, Pāṇini’s paribhāṣā-sūtras (meta-rules) define how the rule system itself behaves. These are rules about rules, akin to higher-order functions in the lambda calculus. Lambda calculus, formalized by Church, uses expressions like:

$$\lambda x.f\left(x\right)$$

(79)

to denote anonymous functions. Similarly, Pāṇini uses variables and substitution operators to create templates for grammatical operations. For instance, a paribhāṣā rule might govern how optionality is interpreted across multiple derivational stages:

$$R_i\mathrm{applies}\mathrm{only}\mathrm{if}{R}_j\mathrm{does}\mathrm{not}\mathrm{apply}$$

(80)

This is analogous to function application precedence or rule ordering in term-rewriting systems. The precedence system in Aṣṭādhyāyī is known as vipratiṣedha (rule conflict resolution), which states:

$$\mathrm{In}\mathrm{case}\mathrm{of}\mathrm{conflict},\mathrm{the}\mathrm{later}\mathrm{rule}\mathrm{prevails}:R_j>R_i\mathrm{if}j>i$$

(81)

This rule ordering mimics priority queues or ordered rewriting in computational logic systems.

Pāṇini’s metalanguage even anticipates aspects of Backus-Naur Form (BNF), used to define the syntax of programming languages. In BNF, a production like:

$$<\mathrm{sentence}>::=<\mathrm{noun}-\mathrm{phrase}><\mathrm{verb}-\mathrm{phrase}>$$

(82)

has its equivalent in Pāṇini’s use of technical terms and definitions (saṃjñā-sūtras), which define categories of grammatical elements. These definitions function as macros or data types, recursively instantiated in rule execution.

Some scholars have gone further to encode the entire Aṣṭādhyāyī in Prolog and other formal languages, confirming its computational expressiveness [22]. The key insight is that Pāṇini’s system is not just a catalog of linguistic observations but a formal grammar capable of generating an infinite number of valid Sanskrit utterances from a finite rule base.

The use of recursion, optionality, context-sensitivity, and rule ordering in the Aṣṭādhyāyī anticipates features of both procedural and declarative programming paradigms. Consider the recursion in affixation rules:

$$X\to X+affi{x}_n\to X+affi{x}_n+affi{x}_m$$

(83)

where $affi{x}_n$ triggers another affixation depending on morphosyntactic features. Such chaining is computationally similar to nested function calls or recursive descent parsing in compilers.

In conclusion, Pāṇini’s Aṣṭādhyāyī exhibits a level of abstraction and formal rigor comparable to that of modern computational theories. Its equivalence to finite-state systems, compatibility with rewriting logic, and inclusion of meta-rules underscore its sophistication. When viewed through the lens of mathematical linguistics and computer science, Pāṇini’s grammar is not merely a linguistic artifact but a formal system with deep computational properties.

11. Geometry and Measurement in the Śulba Sūtras

The Śulba Sūtras, a subset of the Kalpa Sūtras within the corpus of Vedic literature, are among the earliest known mathematical texts, dating from circa 800–500 BCE. Their primary concern is the construction of Vedic altars (vedi) with exact shapes and areas, which required sophisticated geometric reasoning. While the ritualistic motivation is theological, the mathematical content is rigorously developed and offers striking parallels to later Greek and modern mathematical developments.

The Śulba Sūtras frequently use the result known in later times as the Pythagorean theorem. For a square of side a, the diagonal c is given in the Baudhāyana Śulba Sūtra as:

$$c=a\sqrt{2}$$

(84)

Although this is not presented with symbolic algebra, the verbal form in Sanskrit describes that “the diagonal of a square produces double the area,” which is functionally equivalent to the geometric identity:

$$a^2+a^2=c^2\Rightarrow c=\sqrt{2}a$$

(85)

This identity is repeatedly used in constructing squares of equal area and converting square-based layouts into rectangles or trapeziums. The Baudhāyana and Āpastamba Śulba Sūtras describe explicit constructions for transforming one shape into another while maintaining area invariance, such as:

$$\mathrm{Area}\left(Square\right)=\mathrm{Area}\left(Rectangle\right),\mathrm{via}\mathrm{geometric}\mathrm{dissection}$$

(86)

These constructions anticipate the later geometric techniques of Euclid, particularly in Book II of the Elements, which deals with geometric algebra. For example, the transformation of a square into a rectangle of equal area employs a compass-straightedge dissection method that is both precise and repeatable [28].

A fundamental problem addressed in these texts is the construction of fire altars of equivalent area but differing shapes, such as squares, rectangles, trapezoids, and even complex bird-shaped figures (śyenaciti). These require solving equations of the type:

$$A_1=A_2,\mathrm{where}{A}_1,A_2\mathrm{are}\mathrm{functions}\mathrm{of}\mathrm{length}\mathrm{variables}$$

(87)

In many cases, approximate values for irrational numbers are provided. For example, the value of $\sqrt{2}$ given by the Baudhāyana Śulba Sūtra is:

$$\sqrt{2}\approx 1+{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{3\times 4}}-{\displaystyle \frac{1}{3\times 4\times 34}}={\displaystyle \frac{577}{408}}$$

(88)

This is remarkably close to the actual value $\sqrt{2}\approx 1.41421356$, showing a high level of numerical precision [29].

The Śulba Sūtras also provide constructions for squaring the circle and vice versa, though the latter is approximate. Letting $A_C$ and $A_S$ denote the areas of a circle and a square respectively, the problem is to find s such that:

$$A_C=\pi r^2=s^2\Rightarrow s=r\sqrt{\pi }$$

(89)

Given that $\pi$ was not explicitly known, the approximations employed imply values around 3.2, used to construct squares of equal area as circles. This introduces an early version of what modern analysis classifies under measure-preserving transformations.

The altar designs often employ geometric constructions involving bisection, trisection, and complex transformations. For instance, to construct a square equal in area to a given rectangle $ab$, one constructs the mean proportional x such that:

$${\displaystyle \frac{a}{x}}={\displaystyle \frac{x}{b}}\Rightarrow x=\sqrt{ab}$$

(90)

This is a classic example of a geometric mean, used frequently in Greek and Indian geometry alike. Such constructions relate directly to Euclidean problems solvable with compass and straightedge.

The comparison with modern measure theory arises when we note that these constructions preserve area across transformations. Given a measure space $(X,\mathrm{\Sigma },\mu )$, a transformation $T:X\to X$ is measure-preserving if:

$$\mu \left(T^{-1}\left(A\right)\right)=\mu \left(A\right)\forall A\in \mathrm{\Sigma }$$

(91)

In the Śulba Sūtra constructions, the measure is the geometric area, and each construction step ensures that the resulting figure retains the same area as the original, hence implementing a kind of geometric measure-preserving map.

Furthermore, these geometric ideas also relate to integral geometry and calculus. Given that many altar constructions rely on assembling or disassembling parts of shapes, it resembles the principle of integration via summation of infinitesimal units, a core idea in Riemannian measure:

$$A={\int }_a^{b}f\left(x\right)dx$$

(92)

While not formalized as such in Vedic literature, the underlying intuition aligns closely with these concepts.

In conclusion, the Śulba Sūtras represent a remarkably sophisticated body of geometric knowledge, encompassing concepts of irrational numbers, constructive geometry, area preservation, and even elements of what modern mathematics calls measure theory. They provide a unique, independently developed foundation of geometry, rooted in religious ritual but expressed in precise mathematical logic.

12. Mathematical Cosmology and Cyclic Time in Indian Thought

Ancient Indian cosmology is distinguished by its profoundly mathematical conception of time, space, and recurrence. Rather than a linear chronology, the cosmological framework in texts like the Purāṇas, Sūrya Siddhānta, and Mahābhārata espouses a cyclic and recursive model of temporal evolution, spanning billions of years per cycle. This model is defined by numerically precise time units such as the kalpa, manvantara, and yuga, which are embedded within a hierarchical and fractal manner.

The kalpa, or a "day of Brahmā", is defined as:

$$1\mathrm{kalpa}=4.32\times {10}^9\mathrm{years}$$

(93)

This is followed by an equally long night of Brahmā, making one full cycle of day and night:

$$1\mathrm{day}-\mathrm{night}\mathrm{cycle}\mathrm{of}\mathrm{Brahm}ā=8.64\times {10}^9\mathrm{years}$$

(94)

A manvantara is a subdivision of a kalpa and is given by:

$$1\mathrm{manvantara}=71\mathrm{mah}ā\mathrm{yugas}+1.728\times {10}^6\mathrm{years}\left(\mathrm{junctions}\right)=308.448\times {10}^6\mathrm{years}$$

(95)

There are 14 manvantaras in a kalpa, leading to the equation:

$$14\times 308.448\times {10}^6=4.318272\times {10}^9\mathrm{years}\approx 1\mathrm{kalpa}$$

(96)

This recursive structure of kalpa → manvantara → mahāyuga → yuga embodies a fractal view of time. Each mahāyuga comprises four yugas in decreasing moral and temporal order:

$$\mathrm{K}ṛ\mathrm{ta}\left(\mathrm{Satya}\right)=1.728\times {10}^6,\mathrm{Tret}ā=1.296\times {10}^6,\mathrm{Dv}ā\mathrm{para}=864\times {10}^3,\mathrm{Kali}=432\times {10}^3$$

(97)

Adding up gives:

$$1\mathrm{mah}ā\mathrm{yuga}=4.32\times {10}^6\mathrm{years}$$

(98)

The harmonic and geometric progression of these yugas—where each subsequent yuga is shorter by a factor of $3/4$—suggests a highly structured numerical cosmology, possibly to encode entropy or decay in dharma over time [30].

In calendrical calculations, Indian astronomy made use of precise combinatorics. For instance, the number of tithis (lunar days) in a synodic lunar month is 30, and the solar year is calculated to be approximately:

$$365.2587565\mathrm{days}$$

(99)

Thus, the alignment of solar and lunar calendars requires intercalation, achieved via cycles such as the Metonic 19-year cycle, which approximately aligns:

$$19\mathrm{solar}\mathrm{years}\approx 235\mathrm{lunar}\mathrm{months}$$

(100)

This synchronization required precise tracking of tithis and nakṣatras, and astronomers such as Āryabhaṭa and Bhāskara II developed combinatorial methods to track conjunctions and periodicities [31].

The Sūrya Siddhānta further provides mean revolution numbers of celestial bodies over a mahāyuga (4.32 million years). For instance, the number of synodic lunar months (lunations) in a mahāyuga is given as:

$$53,433,336\mathrm{synodic}\mathrm{months}$$

(101)

This leads to the average length of the synodic month:

$$\mathrm{Mean}\mathrm{synodic}\mathrm{month}={\displaystyle \frac{4.32\times {10}^6\mathrm{years}\times 365.2587565}{53,433,336}}\approx 29.5306\mathrm{days}$$

(102)

which is remarkably close to the modern value. This confirms that ancient Indian cosmology combined theology with rigorous observational astronomy and arithmetic.

The recurrence of yugas and kalpas bears structural similarity to fractals, in that the same pattern recurs at varying scales. This recursive definition can be expressed using self-similarity functions:

$$T_{n+1}=k·T_n,\mathrm{where}k=\mathrm{scaling}\mathrm{factor}$$

(103)

This scaling factor is often ${\displaystyle \frac{3}{4}}$ across the yugas and ${\displaystyle \frac{1}{2}}$ across night and day cycles. These nested cycles produce a cosmology where the universe oscillates, expands, and contracts periodically, aligning loosely with cyclic cosmological models in modern theoretical physics [32].

In conclusion, the Indian tradition of mathematical cosmology is a unique fusion of metaphysical, astronomical, and arithmetic insight. By encoding vast and precise time cycles, recursive structures, and combinatorial calendrical systems, it laid a foundation for both ritual timing and astronomical prediction. It exemplifies how ancient cultures employed mathematical constructs not only for measurement but to express deep philosophical and cosmological ideas.

13. Combinatorics and Enumerative Techniques in Pre-modern Indian Mathematics

Indian mathematical texts dating from the Vedic period to the Jain canonical literature demonstrate a sophisticated grasp of combinatorics, long before the formalization of this branch in the West. Central to this tradition are the recursive enumeration methods of Pi$\dot{\mathrm{n}}$gala, the binary interpretations of prosodic patterns, and the Jain treatment of permutations and partitions. These contributions foreshadow ideas in information theory, algorithmic recursion, and integer partition theory.

The earliest formal combinatorial insights appear in the Chandaḥśāstra of Pi$\dot{\mathrm{n}}$gala (circa 200 BCE), a treatise on Sanskrit prosody. The fundamental problem is to count the number of metrical patterns of length n made from long (G) and short (L) syllables. This binary choice leads directly to the recursive relation:

$$M\left(n\right)=M(n-1)+M(n-2)$$

(104)

with initial conditions $M\left(0\right)=1,M\left(1\right)=1$, yielding the Fibonacci sequence:

$$1,1,2,3,5,8,13,21,\dots$$

(105)

The Fibonacci numbers $F_n$ also satisfy the closed-form expression known as Binet’s formula:

$$F_n={\displaystyle \frac{1}{\sqrt{5}}}\left({\left({\displaystyle \frac{1+\sqrt{5}}{2}}\right)}^n-{\left({\displaystyle \frac{1-\sqrt{5}}{2}}\right)}^n\right)$$

(106)

Pi$\dot{\mathrm{n}}$gala’s representation of sequences using binary code—where $G\equiv 1$, $L\equiv 0$—can be interpreted as the earliest known encoding of binary numbers. This mapping anticipates modern data encoding schemes and forms a conceptual precursor to information theory [33].

The Jain mathematicians of the 5th–10th centuries CE contributed significantly to permutation theory. In Tiloyapaṇṇatti and other canonical texts, the number of permutations of n objects is explicitly stated as:

$$P\left(n\right)=n!$$

(107)

where $n!$ denotes the factorial of n. For instance, the number of ways to arrange 4 different items is:

$$P\left(4\right)=4!=4\times 3\times 2\times 1=24$$

(108)

In the context of choosing r elements from n distinct items, the number of combinations is:

$$C(n,r)={\displaystyle \frac{n!}{r!(n-r)!}}$$

(109)

These formulas were used in Jain cosmological calculations and philosophical discussions on the finitude and infinitude of knowledge [34].

Partition theory, though not formalized until the works of Euler and Ramanujan, has antecedents in Indian texts. A partition of an integer n is defined as the number of distinct ways to express n as a sum of positive integers. The partition function is denoted as:

$$p\left(n\right)=\mathrm{number}\mathrm{of}\mathrm{integer}\mathrm{partitions}\mathrm{of}n$$

(110)

Although ancient Indian texts did not state generating functions explicitly, recursive enumeration strategies were developed. For instance, Bhāskara II, in the Līlāvatī, solves problems of combinations with constraints using near-partition logic [35].

The Jain mathematicians also introduced classification systems of sets based on the cardinality of their elements and developed taxonomies involving sets of sets (power sets). The number of subsets of a set with n elements is given by:

$$2^n$$

(111)

This directly relates to modern set theory and combinatorics. The recursive structure of subset enumeration is evident in Pi$\dot{\mathrm{n}}$gala’s binary matrix (prastāra), which is a table of all binary sequences of length n.

In algorithmic terms, these constructions correspond to depth-first traversal of a binary tree of height n, with each node branching into either G or L, and the paths representing metrical patterns.

The use of permutations and combinations in astronomical and calendrical calculations is also noteworthy. For example, the number of ways to align seven planets with twelve zodiac signs is given by:

$${12}^7=35,831,808$$

(112)

This reflects the scale at which combinatorial enumeration was used in astrological charting.

In conclusion, Indian combinatorics evolved not only as a mathematical curiosity but as an integral part of literary, philosophical, and cosmological systems. The recursive constructions of Pi$\dot{\mathrm{n}}$gala, the factorial formulations of Jain thinkers, and the implicit use of partition theory in classical texts show a continuity of enumerative reasoning that prefigures many modern mathematical frameworks.

14. Concept of Void and Śūnyatā in Mathematics and Metaphysics

The concept of emptiness or void occupies a unique position at the intersection of metaphysics and mathematics. In Indian philosophical traditions, particularly Buddhist Madhyamaka, the notion of śūnyatā (emptiness) signifies the absence of inherent existence or svabhāva in all phenomena. This idea has parallels in mathematical constructs such as the null set, zero as a placeholder and operator, and even modern ideas of vacua in physics. This section explores the philosophical concept of śūnyatā.

Mathematically, the empty set is denoted by:

$$\varnothing =\left\{\right\}$$

(113)

It is defined as the set containing no elements, and it serves as the foundational building block in set theory. Every set is constructed inductively from the empty set using operations like union, power set, and Cartesian product. In Zermelo–Fraenkel set theory (ZF), all natural numbers can be represented using nested sets starting from ∅. For example:

$$0=\varnothing ,1=\{\varnothing \},2=\{\varnothing ,\{\varnothing \left\}\right\},\mathrm{and}\mathrm{so}\mathrm{on}$$

(114)

In this formulation, zero does not just signify "nothing" but acts as an ontological seed from which the entire number system is generated [36].

In Buddhist metaphysics, particularly in Nāgārjuna’s Mūlamadhyamakakārikā, the assertion is that all phenomena are empty of essence:

$$\forall x,svabhāva\left(x\right)=\varnothing$$

(115)

This radical ontological emptiness denies the existence of independent entities. Instead, everything is contingent upon causes and conditions—a notion called pratītyasamutpāda (dependent origination). This is formally analogous to the mathematical axiom of foundation in set theory:

$$\forall x,\exists y\in x\Rightarrow y\ne x$$

(116)

This prevents self-referential sets, ensuring foundational dependence rather than ontological circularity. Śūnyatā and ∅ both represent minimal objects that are non-self-sufficient, serving as voids upon which structure is built.

René Descartes’ dualism posited a split between res cogitans (thinking substance) and res extensa (extended substance). This is often formalized in Cartesian coordinates, where spatial points exist independently of consciousness:

$$\mathrm{res}\mathrm{extensa}\sim {\mathbb{R}}^n,\mathrm{res}\mathrm{cogitans}\sim \mathbb{M}$$

(117)

where $\mathbb{M}$ denotes mental objects or subjective states. In contrast, Buddhist śūnyatā dissolves this duality entirely, proposing that mind and matter co-arise in emptiness and thus lack ontological distinction. This non-duality challenges the Cartesian foundation of mathematics as operating on fixed substances.

In terms of number neutrality, zero functions as both an additive identity:

$$\forall x\in \mathbb{R},x+0=x$$

(118)

and a null element for multiplication:

$$\forall x\in \mathbb{R},x·0=0$$

(119)

In this sense, zero is structurally passive yet ubiquitous, echoing the śūnya which pervades all but is independent of all.

Moreover, in algebraic structures like rings and fields, zero has a unique role:

$$0\in \mathbb{Z},\exists -x\mathrm{such}\mathrm{that}x+(-x)=0$$

(120)

The existence of additive inverses centered around zero reflects its balancing or neutralizing function, akin to the metaphysical balance śūnyatā provides to extremes of existence and non-existence [37].

From a logical standpoint, the empty set is the false proposition in set-theoretic logic. Given:

$$x\in \varnothing \Rightarrow \mathrm{false}$$

(121)

This makes ∅ a grounding point for constructing negation, as every membership assertion is false. In Buddhist thought, śūnyatā likewise serves as a negation of essence, forming the basis for logical and epistemological inference.

In conclusion, the mathematical concept of zero and the empty set reflect structural absence, while the metaphysical concept of śūnyatā reflects ontological absence. Despite arising in distinct intellectual traditions, both serve as foundational voids—neutral, generative, and paradoxically fertile. Their comparison reveals how mathematical and philosophical emptiness are not mere nihilisms, but deep frameworks for understanding existence, structure, and truth.

15. Visualizing Mathematical Cosmologies: Diagrams, Mandalas, and Topology

Indian cosmology has historically used visual, symbolic, and geometric models to represent complex metaphysical and mathematical ideas. Among the most profound of these are mandalas and yantras—structured diagrams that reflect the underlying principles of symmetry, recursion, spatial order, and cosmogenesis. These diagrams, often realized with mathematical precision, serve not only as ritual objects but also as implicit topological constructs, reflecting notions of spatial transformation, symmetry groups.

The geometric symmetry in mandalas and yantras can be analyzed using the formalism of group theory. A symmetry group G of a geometric object is the set of all transformations that preserve the object’s structure. For a typical circular mandala with n-fold rotational symmetry, the group of rotations forms the cyclic group $C_n$:

$$C_n=\{R_k\mid R_k\left(\theta \right)=\theta +{\displaystyle \frac{2\pi k}{n}},k=0,1,\dots ,n-1\}$$

(122)

When reflection symmetries are added, the symmetry group becomes the dihedral group $D_n$, defined as:

$$D_n=\{R_k,M_k\mid R_k\in C_n,M_k=\mathrm{reflection}\mathrm{across}\mathrm{axis}\theta ={\displaystyle \frac{\pi k}{n}}\}$$

(123)

These groups preserve geometric invariants and allow classification of sacred diagrams based on their transformation properties [38].

Tessellations and tilings form another structural element in sacred geometry. The repeated patterning found in yantras and temple floor designs adheres to tiling rules in Euclidean and non-Euclidean geometry. A regular tessellation is a partition of the plane using congruent regular polygons with identical vertices. There are exactly three such tessellations:

$$\mathrm{Triangles}:(3,6),\mathrm{Squares}:(4,4),\mathrm{Hexagons}:(6,3)$$

(124)

More generally, a tessellation pattern can be described using a symmetry group known as a wallpaper group. There are exactly 17 such groups in Euclidean 2-space, classified by their translational and reflectional symmetries. In yantric constructions such as the Śrī Yantra, nested triangles reflect these symmetry properties while converging to a central bindu, forming an infinite series of nested recursive polygons.

The Śrī Yantra itself is composed of nine interlocking triangles—five pointing down and four pointing up—yielding a total of 43 smaller triangles in the final composition. Each triangle obeys strict angular rules and is constructed with precise overlapping:

$$T_k=\mathrm{Intersection}(T_{up},T_{down})\Rightarrow \mathrm{Total}\mathrm{triangles}=43$$

(125)

From a topological perspective, the arrangement maps a surface into regions bounded by simple closed curves, each of which can be assigned an Euler characteristic:

$$\chi =V-E+F$$

(126)

In the planar projection of the Śrī Yantra, the Euler characteristic remains invariant under topological deformation, signifying its homeomorphism to a disk ($\chi =1$). Such invariance connects mandala construction to ideas in algebraic topology and graph theory.

The idea of centrality and periphery, so prominent in mandala architecture, maps to concepts in topology such as concentric neighborhoods and compactness. The nested levels of a mandala—like those seen in the Bhū-maṇḍala diagrams—can be modeled as concentric rings:

$$X=\bigcup _{i=1}^{n}B(x_i,r_i),\mathrm{where}B(x_i,r_i)\mathrm{is}\mathrm{an}\mathrm{open}\mathrm{ball}\mathrm{of}\mathrm{radius}{r}_i$$

(127)

Such constructions serve as intuitive models for layered universes and recursive cosmology.

Furthermore, the recursive patterns in yantras reflect self-similarity, a hallmark of fractal geometry. While classical mandalas are not fractals in the strict mathematical sense, the repeated scaling and pattern nesting resemble affine transformations in iterated function systems (IFS):

$$x_{n+1}=f_i\left(x_n\right),\mathrm{where}{f}_i\mathrm{are}\mathrm{affine}\mathrm{maps}$$

(128)

This recursive iteration echoes the cosmological repetition of kalpas and yugas in Indian time theory, thereby uniting the geometric and temporal domains.

Artists like M. C. Escher employed mathematical ideas of symmetry, tessellation, and infinite recursion in their artworks. Indian sacred geometry shares these principles, particularly in temple tiling (e.g., at Khajuraho and Belur) and mandala painting. The Escherian idea of a tiling that reflects across hyperbolic space finds a parallel in tantric yantras that symbolize cosmic contraction and expansion.

In conclusion, sacred diagrams such as mandalas and yantras are more than spiritual symbols. When viewed through a mathematical lens, they encode a rich variety of topological, combinatorial, and algebraic structures. They act as cosmological maps, group-theoretic visualizations, and recursive algorithms—making them ideal for exploring intersections between metaphysics, geometry, and mathematical topology.

16. Mathematics as Sacred Science in Ancient India

In ancient Indian thought, mathematics (Gaṇita) was not an isolated or secular domain. Rather, it functioned as an integral part of sacred knowledge systems, embedded within astronomy (jyotiṣa), ritual science (karma-kāṇḍa), and sacred architecture (vāstu-śāstra). This holistic approach treated numbers, shapes, and proportions not merely as utilitarian constructs but as intrinsic to cosmic harmony, spiritual design, and metaphysical symbolism. The mathematical structures are evident in temple construction.

In the context of temple geometry, the canonical text Śulba Sūtras provided rules for constructing fire altars and geometrical grounds that evolved into principles used in temple site planning. The most common sacred geometrical figure was the square, symbolizing the earth and stability. The central shrine or garbhagṛha was always positioned at the center of the temple mandala, reflecting axial symmetry. The basic layout adhered to the vāstu-puruṣa-maṇḍala, a grid of $n\times n$.

$$\mathrm{Area}\mathrm{of}\mathrm{each}\mathrm{padma}\left(\mathrm{module}\right)={\displaystyle \frac{A_{\mathrm{total}}}{n^2}}$$

(129)

The mandala of 64 or 81 squares structured the spatial dynamics and ensured proportional scaling. The vertical axis through the sanctum and the central spire (śikhara) corresponded to the cosmic axis (axis mundi). These proportions are rooted in mathematical ratios, many of which approximated what is now known as the golden ratio:

$$\varphi ={\displaystyle \frac{1+\sqrt{5}}{2}}\approx 1.6180339887$$

(130)

This ratio emerged in the height-to-base proportions of shrines and was used to maintain visual harmony across scales [41]. Consider a base length a, and the height $h=\varphi a$. The recursive property of the golden ratio ensures self-similarity in design:

$${\varphi }^2=\varphi +1$$

(131)

The Śrīcakra or Śrī Yantra is a prime example of sacred geometry grounded in mathematical construction. It consists of nine interlocking triangles producing 43 smaller triangles, with recursive symmetry and axial alignment. Its construction requires precise trigonometric calculations, including angle divisions such as:

$$\theta ={\displaystyle \frac{{360}^{\circ }}{43}}\approx 8.{372}^{\circ }$$

(132)

These triangles are nested to form concentric circles and lotus petals, adhering to fixed proportions based on multiples of 9 and 108, numbers of deep ritual significance.

In jyotiṣa, numerical cycles were central to calendrical computations and astrological models. The concept of the kālacakra (wheel of time) divides cosmic time into vast numerical units like yugas, kalpas, and manvantaras. Each cycle is mathematically articulated:

$$1\mathrm{mah}ā\mathrm{yuga}=4.32\times {10}^6\mathrm{years},1\mathrm{kalpa}=4.32\times {10}^9\mathrm{years}$$

(133)

These multiples of 432 appear frequently in temple design. For example, a temple’s perimeter or axis length might be made proportional to 108 or 360, referencing the number of tithis, days, or deities. The recursive use of these numbers turns architecture into a temporal and astronomical instrument.

In ritual mathematics, Vedic sacrifices were calibrated using complex arithmetic and geometric transformations. The agnicayana altar required specific brick counts arranged in layers, satisfying equations such as:

$$\sum _{i=1}^n{b}_i=B,\mathrm{where}B=\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{bricks}\mathrm{in}\mathrm{layer}$$

(134)

These configurations were not arbitrary but had cosmological mappings—number of bricks representing days of the year, layers representing heavens and earth.

The sacredness of number also extended into musical theory (sa$\dot{n}$gīta), where ratios of 2:1 (octave), 3:2 (perfect fifth), and 4:3 (perfect fourth) governed the tuning of strings and flutes. These are mathematically simple ratios producing harmonic consonance, thereby binding Gaṇita to sonic aesthetics and spiritual resonance.

In conclusion, ancient Indian mathematics was profoundly sacred. Numbers were mantric, proportions were cosmological, and constructions were metaphysical. Whether in the golden-ratio-aligned spires of temples, the numerical cycles of kālacakra, or the carefully calculated fire altars, mathematics was a science of divine correspondence. Far from secular abstraction, Gaṇita was the very syntax through which the sacred was measured, aligned, and made manifest.

17. Srinivasa Ramanujan: Mathematical Contributions

Srinivasa Ramanujan (1887–1920) made substantial contributions to number theory, particularly in the fields of partitions, modular forms, and continued fractions. His identity involving infinite series and irrational numbers remains celebrated to this day. For example, one of his famous approximations for $\pi$ is given by:

$${\displaystyle \frac{1}{\pi }}={\displaystyle \frac{2\sqrt{2}}{9801}}\sum _{k=0}^{\infty }{\displaystyle \frac{\left(4k\right)!(1103+26390k)}{{(k!)}^4{396}^{4k}}}$$

(135)

This rapidly converging series was used by modern computer algorithms for computing millions of digits of $\pi$.

His work on the partition function $p\left(n\right)$ led to an asymptotic formula with Hardy:

$$p\left(n\right)\sim {\displaystyle \frac{1}{4n\sqrt{3}}}exp\left(\pi \sqrt{{\displaystyle \frac{2n}{3}}}\right)$$

(136)

He also introduced the Ramanujan tau-function $\tau \left(n\right)$, appearing in the Fourier expansion of the discriminant modular form:

$$\Delta \left(q\right)=q\prod _{n=1}^{\infty }{(1-q^n)}^{24}=\sum _{n=1}^{\infty }\tau \left(n\right)q^n$$

(137)

Ramanujan’s intuition extended to mock theta functions, which remained mysterious until modern researchers linked them to harmonic Maass forms. His collaboration with G.H. Hardy at Cambridge led to a systematic publication of many of his theorems, many of which were stated without proof. Later work by mathematicians such as George Andrews and Bruce Berndt validated these results using modern methods [6,44,45].

18. Harish-Chandra and the Rise of Abstract Modern Indian Mathematics

Harish-Chandra (1923–1983), initially trained as a physicist under Paul Dirac, transitioned to mathematics and became a pioneering figure in representation theory and harmonic analysis on semisimple Lie groups. His most important contribution is the Harish-Chandra character formula, which provides an explicit expression for the characters of irreducible representations of real reductive Lie groups.

Let G be a semisimple Lie group and let $\mathfrak{g}$ be its Lie algebra. The Harish-Chandra isomorphism relates the center $\mathfrak{Z}\left(\mathfrak{g}\right)$ of the universal enveloping algebra $U\left(\mathfrak{g}\right)$ with invariant polynomials on a Cartan subalgebra $\mathfrak{h}$:

$$\mathfrak{Z}\left(\mathfrak{g}\right)\cong S{\left(\mathfrak{h}\right)}^W$$

(138)

where $S{\left(\mathfrak{h}\right)}^W$ denotes the symmetric algebra on $\mathfrak{h}$ invariant under the Weyl group W.

He also contributed to the Plancherel measure for semisimple Lie groups and developed the theory of discrete series representations. These have far-reaching consequences in number theory and automorphic forms, particularly within the Langlands program.

Harish-Chandra’s influence on Indian mathematics was institutional as well, contributing to the development of mathematics at the Tata Institute of Fundamental Research (TIFR) and the Harish-Chandra Research Institute in Prayagraj [46,47].

19. Āryabhaṭa and Foundational Contributions to Indian Mathematics

Among the earliest and most influential mathematicians of ancient India, Āryabhaṭa (476–550 CE) occupies a central position. His treatise, the Āryabhaṭīya, composed in 499 CE, represents a compact yet profound synthesis of astronomical and mathematical knowledge. One of his most remarkable contributions was his approximation of $\pi$. In Āryabhaṭīya (Ganitapāda, verse 10), he states a rule which translates to:

$$\pi \approx {\displaystyle \frac{62832}{20000}}=3.1416$$

(139)

This value of $\pi$ is correct to four decimal places and was unparalleled in accuracy during his time [7,13]. His method shows a deep understanding of geometric ratios and irrational quantities, long before such concepts were formalized in Europe.

Āryabhaṭa’s work in trigonometry is equally significant. He introduced the sine function, termed as ardha-jya, and constructed tables of values at intervals of $3.{75}^{\circ }$. Taking the radius of the circle as 3438 minutes (since $1^{\circ }=60$ minutes and the arc of a semicircle is $180\times 60=10800$ minutes), the sine of an angle $\theta$ can be given by:

$$\mathrm{Sin}\mathrm{e}\left(\theta \right)=r·sin\left(\theta \right),\mathrm{where}r=3438$$

(140)

The precision of these trigonometric tables had practical applications in astronomical observations and calendrical computations, and they were developed centuries before similar constructs emerged in Islamic and European traditions [48,49].

Another profound contribution of Āryabhaṭa is his method for solving linear indeterminate equations, especially those of the form:

$$ax\equiv b(modn)$$

(141)

This is equivalent to the modern linear congruence problem. Āryabhaṭa’s algorithm, known as the Kuttaka or pulverizer method, is functionally equivalent to the extended Euclidean algorithm. For instance, to solve:

$$17x\equiv 1(mod43)$$

(142)

one seeks integers x and k such that $17x-43k=1$. The Kuttaka method provides a systematic way to find such integers. The solution to this equation is:

$$x=38(\sin \mathrm{ce}17\times 38=646\equiv 1(mod43))$$

(143)

This illustrates the early Indian understanding of modular arithmetic and the existence of multiplicative inverses modulo n, which form the basis for group theory and modern cryptography [13].

Further, Āryabhaṭa provided an approximation for the length of the sidereal year as:

$${\mathrm{Year}}_{\mathrm{Aryabhata}}=365+{\displaystyle \frac{6·12+15}{60·60·24}}=365.25868\mathrm{days}$$

(144)

The current accepted modern value is approximately 365.25636 days, showing that Āryabhaṭa’s calculation differed by less than 3 minutes per year. Such precision had significant implications in the development of astronomical models and accurate calendars in India [49].

In addition to numerical work, Āryabhaṭa devised notational systems that encoded large numbers using Sanskrit consonants and vowels, thus allowing complex calculations to be preserved in verse form. His place-value system, although not using a symbol for zero explicitly, required a place-holding concept, which paved the way for the later symbolic zero introduced by Brahmagupta [7].

Finally, his awareness of the rotation of the Earth on its axis (stated clearly in astronomical verses of Āryabhaṭīya) marks a significant deviation from prevailing cosmological views of his time. This cosmological insight would influence later Indian astronomical schools and their computational techniques.

Thus, Āryabhaṭa not only laid the mathematical foundations of arithmetic and algebraic processes in ancient India, but also profoundly impacted astronomy, trigonometry, and algorithmic thinking. His methodologies were carried forward by later Indian mathematicians and eventually influenced the Arabic mathematical corpus during the Islamic Golden Age, which in turn informed the European Renaissance [13,48].

20. Other Ancient Indian Mathematicians and Their Contributions

The mathematical tradition of ancient India is not confined to a few individuals. Over centuries, a lineage of scholars emerged who systematically developed arithmetic, algebra, geometry, trigonometry, and astronomical computations. This section explores the works of Bhāskara I, Brahmagupta, Mahāvīra, Śrīdhara, Bhāskara II, Manjula, and Nīlakaṇṭha Somayāji.

Bhāskara I (c. 600–680 CE) was among the earliest commentators of Āryabhaṭa. He authored texts such as Mahābhāskarīya and Laghubhāskarīya, which elaborated trigonometric computations. He provided an accurate rational approximation for the sine function. The interpolation formula he used is:

$$sinx\approx {\displaystyle \frac{16x(\pi -x)}{5{\pi }^2-4x(\pi -x)}}$$

(145)

where x is in radians. This approximation holds with remarkable accuracy for small angles and demonstrates an understanding of function behavior well before calculus emerged in Europe [7].

Brahmagupta (598–668 CE), a seminal figure, defined arithmetic operations involving zero and negative numbers. In his text Brāhmasphuṭasiddhānta, he formulated rules such as:

$$a+0=a,a-0=a,a·0=0,a^0=1$$

(146)

But his rules for division by zero, such as:

$${\displaystyle \frac{a}{0}}=\infty \mathrm{and}{\displaystyle \frac{0}{0}}=0$$

(147)

were incorrect by modern standards. Nonetheless, this was a profound advancement in treating zero as a number [13].

In algebra, Brahmagupta addressed the general quadratic equation:

$$a{x}^2+bx=c$$

(148)

and provided solutions using verbal algorithms equivalent to the quadratic formula. He also dealt with indeterminate equations of the Pell’s type:

$$x^2-N{y}^2=1$$

(149)

Brahmagupta’s solutions to such equations influenced later Arabic mathematicians, including Al-Khwarizmi, and through them, European algebraic development [7,49].

Mahāvīra (9th century CE), in his work Gaṇitasārasan̄graha, extended algebraic and combinatorial thinking. He clearly defined permutations and combinations, and explored series summations. For instance, the formula for the sum of the first n natural numbers is presented as:

$$S_n={\displaystyle \frac{n(n+1)}{2}}$$

(150)

and for square numbers:

$$S_n^{\left(2\right)}={\displaystyle \frac{n(n+1)(2n+1)}{6}}$$

(151)

Śrīdhara (c. 850 CE) contributed to root extraction techniques and quadratic solutions. He gave explicit solutions to:

$$a{x}^2+bx+c=0$$

(152)

His method corresponds to the modern formula:

$$x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$$

(153)

He discussed irrational numbers, and presented systematic ways to handle them [13].

Bhāskara II (1114–1185 CE), often called Bhāskarāchārya, synthesized arithmetic, algebra, geometry, and astronomy. His works include Līlāvatī, Bījagaṇita, and Siddhānta Śiromaṇi. In Bījagaṇita, he solved Diophantine equations such as:

$$N{x}^2+1=y^2$$

(154)

using the Chakravāla method — a cyclic algorithm akin to continued fractions. For example, solving:

$$61x^2+1=y^2$$

(155)

yields $x=226153980$, $y=1766319049$, showcasing the power of the Chakravāla technique [7].

He also presented the laws of motion and concepts similar to differential calculus. For example, he described uniform acceleration using:

$$s=ut+{\displaystyle \frac{1}{2}}a{t}^2$$

(156)

hundreds of years before Newton. In Līlāvatī, he addressed practical problems in commerce, construction, and surveying using arithmetic and geometry.

Manjula (c. 10th century CE) advanced trigonometric methods through interpolation and refinement of sine tables. His refinements improved the accuracy of astronomical models that were in use for centuries [49].

Finally, Nīlakaṇṭha Somayāji (1444–1544 CE), part of the Kerala School of Astronomy and Mathematics, introduced infinite series expansions. He derived series like:

$$arctanx=x-{\displaystyle \frac{x^3}{3}}+{\displaystyle \frac{x^5}{5}}-\dots$$

(157)

centuries before Gregory and Leibniz. These results are recorded in Tantrasangraha and commentaries on Āryabhaṭa’s works, revealing early glimpses of calculus [48].

These scholars collectively built a comprehensive mathematical tradition that blended practical computation with deep theoretical insight. Their treatises were composed in Sanskrit verse, ensuring transmission through oral tradition and commentaries over generations.

21. Concluding Remarks and Legacy

The Indian contribution to mathematics spans millennia, from ancient numerical abstraction to deep algebraic structures of the modern age. The invention and formalization of zero by Brahmagupta established arithmetic frameworks that powered algebraic evolution. Centuries later, Ramanujan astonished the world with equations emerging from intuition rather than formal training. Harish-Chandra advanced Indian mathematics into the core of abstract algebra and representation theory, transforming the global mathematical landscape.

Understanding this continuum demonstrates not just isolated genius, but a sustained tradition of intellectual inquiry that continues to influence the evolution of mathematics. Institutions such as the Indian Statistical Institute, TIFR, and the Harish-Chandra Research Institute preserve and amplify this legacy.

This paper has traversed a wide spectrum of mathematical, philosophical, and cultural ideas rooted in the Indian tradition and compared them with developments in Western mathematical thought. By engaging deeply with concepts such as zero (śūnya), infinity, cyclic time, recursive grammar, sacred geometry, and combinatorics, we have aimed to reveal mathematics not merely as a technical tool but as a profound cognitive and spiritual structure embedded in metaphysical worldviews.

The Indian tradition does not isolate mathematics from ritual, cosmology, or language. Instead, gaṇita was always simultaneously practical, symbolic, and sacred. Whether in the precise combinatorics of Pi$\dot{\mathrm{n}}$gala’s metrical patterns, the recursive depths of Pāṇini’s grammar, the astronomical calculations of the Jyotiṣa, or the diagrammatic elegance of mandalas and yantras, mathematics appears as a framework for ordering both the cosmos and consciousness.

By contrasting Nyāya’s five-membered logical inference with Western syllogistic reasoning, by exploring the structural use of binary and factorial operations in Jain and Vedic literature, and by revisiting the geometry of the Śulba Sūtras, we see a picture of mathematics that is richly pluralistic. Moreover, the metaphysical resonance between the Buddhist śūnyatā and the mathematical empty set ∅, or between sacred proportions and the golden ratio $\varphi$, reveals a trans-cultural depth in mathematical abstraction.

Ultimately, this inquiry supports a view of mathematics not merely as a formal system but as an ontological interface between the intelligible and the ineffable. In ancient Indian thought, mathematics becomes a language not just for the world, but for that which transcends it. It is not only how we calculate, but how we contemplate. As such, recovering and reinterpreting these foundational perspectives is not merely of historical interest—it is essential to expanding the philosophical and cultural scope of modern MAthematics.