The Nyaya school of philosophy, known for its emphasis on logic and reasoning, played a crucial role in shaping Indian mathematical thought. Nyaya philosophers developed intricate systems of logical analysis and reasoning, which influenced the development of mathematical concepts and methods.

The concept of Brahman, representing the ultimate reality and unity of the universe, was closely linked to the study of geometry in ancient India. Geometric patterns and shapes were seen as manifestations of the underlying unity and order of the cosmos, reflecting the philosophical notion of Brahman.

Bhaskara II, a renowned mathematician and astronomer, is widely recognized for his contributions to the study of infinity. He developed the concept of 'ananta', which encompassed both the infinitely large and the infinitely small, and explored its implications in various mathematical contexts.

Brahmagupta, a prominent mathematician and astronomer, is credited with introducing the concept of 'shunya' or zero in Indian mathematics. He recognized the importance of zero as a placeholder and developed rules for arithmetic operations involving zero, significantly advancing the field of mathematics.

The Vaisheshika school of philosophy placed a strong emphasis on empirical observation and experimentation as means of acquiring knowledge. This approach influenced the development of mathematical methods that relied on observation and experimentation, contributing to the advancement of mathematical knowledge in ancient India.

The Samkhya school of philosophy incorporated the concept of 'lila' or playfulness into its understanding of the universe. This idea influenced the development of mathematical concepts and methods that were seen as playful explorations of the underlying order and beauty of the cosmos.

Bhaskara II, a renowned mathematician, is credited with developing the 'chakravala' method, a systematic approach for solving indeterminate equations. This method involved a series of transformations and iterations to find integer solutions to complex equations, demonstrating the advanced mathematical techniques developed in ancient India.

The Advaita Vedanta school of philosophy, propounded by Adi Shankara, emphasized the concept of 'maya' or illusion. This idea influenced the development of mathematical concepts and methods that were seen as tools for understanding the underlying reality beyond the veil of illusion.

Brahmagupta, a prominent mathematician and astronomer, is credited with developing a method for calculating the area of a triangle using half the product of its base and height. This formula, known as Brahmagupta's formula, is a fundamental result in geometry and demonstrates the advanced mathematical techniques developed in ancient India.

Aryabhata, a renowned mathematician and astronomer, developed a geometric approach for solving quadratic equations. This method involved constructing a rectangle with sides equal to the coefficients of the quadratic equation and then using geometric properties to find the roots of the equation. This approach demonstrates the ingenuity and creativity of ancient Indian mathematicians.

The concept of 'dharma' or righteousness played a significant role in Indian mathematical thought. It influenced the development of mathematical concepts and methods that were seen as tools for upholding righteousness and justice in society.

Bhaskara II, a renowned mathematician, developed a geometric approach for solving cubic equations. This method involved constructing a circle and a parabola with specific properties and then using geometric techniques to find the roots of the cubic equation. This approach demonstrates the advanced mathematical techniques developed in ancient India.

The concept of 'moksha' or liberation played a significant role in Indian mathematical thought. It influenced the development of mathematical concepts and methods that were seen as tools for achieving liberation from the cycle of birth and rebirth.

Bhaskara II, a renowned mathematician, developed a geometric approach for solving quartic equations. This method involved constructing a circle and a hyperbola with specific properties and then using geometric techniques to find the roots of the quartic equation. This approach demonstrates the advanced mathematical techniques developed in ancient India.

The concept of 'karma' or action played a significant role in Indian mathematical thought. It influenced the development of mathematical concepts and methods that were seen as tools for understanding the consequences of one's actions and for achieving liberation from the cycle of birth and rebirth.