Relevance of Mathematics in Science

(v) Arab Mathematics:- After the decline of Greeks and Romans, the works on mathematics were carried further by Indians and Arabs. Mathematics flourished for hundred of years in India and the Islamic world. Indian mathematics reached Baghdad, a major early center of Islam about 800 AD. Supported by the ruling Caliphs and richest people, scholars and translators in Baghdad, published Arabic version of Greece and Indian works on mathematics.

The need for translators was stimulated by mathematical research in the Arab world. Arab mathematics was used as a tool in dividing inheritances according to the Islamic laws; that is in predicting the time of the new moon; when the next moon began; in determining the direction of Mekkah for the orientation of mosques and fixing of time for daily prayers, which were divided facing Makkah.

The period from the 9^th  to 11^th  century was golden age of Arab Science. The great contribution of Muslim Scientists in the advancement of scientific knowledge in the fields of astronomy, mathematics, geography, optics, chemistry, botany and medicine has a tremendous impact in the history of science. The Arabs made remarkable contributions to the development of mathematics. They had the notion of zero(Sifr) which greatly facilitated manipulation of numbers. The terms Ciphra (in Latin); Cipher ( in English); and Chiffre ( in French) have been derived from the Arabic word, "Sifr" meaning empty or nil.

The Arabs also improved further the works of ancient mathematics particularly in the field of algebra. The knowledge so emerged provided the basis for the 9th century Arab mathematician Al-Khwarizmi's famous compendium of algebraic equations,"Kitab-Al-Jabr w'-al Muqabalah" which means,"book of restoring & balancing”. The word "algebra" comes from al-jabr in the treatise's title. Al-Khwarizmi is also called as "the father of mathematics" (Al-jebra). The word "algorithm" in programming languages has been derived from the name,"Al-Khwarizmi". A famous Arab scientist Ibn Al-Haitham (965- 1040 AD); latin name, Al-Hazen had great contribution in optics, physics and mathematics. He produced algebraic solutions of quadratic and cubic equation. He also wrote a book," Kitab-Al-Manazir" that is book of optics.

The Arabs also developed another branch of mathematics "trigonometry". The astronomical observations provided the necessary incentive to develop mathematics. The theory of conic sections was applied to solve problems in optics by Al-Farsi Kamal-Al-Din. Astronomer Muhammad ibn Ibrahim Al-Fazari translated an Indian astronomical book based on Siddhanta into Arabic. A great mathematician & astronomer of 9th century Abu Abd Allah Mohammad Ibn Isa Al-Mahani wrote commentaries of Euclid and Archimedes and improved the translation made by Ishaq Ibn Hunain of Mendous Spherical. He solved cubic equations,called as al-Mahani's equation.

Abul Wafa Mohammad Al-Buzjani's contribution to the development of trigonometry is remarkable. He was the first to show the generality of the 'Sine Theorem' relative to spherical triangles. Abu Raihan Al-Biruni made significant contribution in mathematics; that include theoretical and practical arithmetics method of solving algebraic equations, geometry and Archimede's theorem. A tenth century mathematician and engineer Abu Bakr Muhammad Ibn Al Hassan Al-Karaji completed the algebra of polynomials (mathematical expression that are the sum of a number of terms) of Al-Khwarizmi.

He worked on polynomials with an infinite number of terms. Ghiyath Al-Din Abu al-Fath Omar Ibn Ibrahim Al-Khayam ( Khayam means tent maker) was one  of the greatest mathematicians, astronomer and a Persian poet of the middle ages. His algebra marks a considerable advance on the work of the Greeks and that of Al-Khwarizmi, while Al-Khawarizimi dealt only with quadratics, Omar Al-Khayam mostly discussed the cubic equations geometrically by using conic sections.

Astronomer Nasir al-din Tusi created the mathematical basis of plane and spherical trigonometry in the 13th century and was the first to treat trigonometry seperately from astronomy. He also wrote some books on geometry, trigonometry and algebra. A good number of muslim mathematicians made significant contribution to the theory of numbers and the numerical methods for solving equations.

The widespread use of Indian and Arab system of numbers simplified calculations and had the great effect on the knowledge of mathematics. This way the Arab scholars rescued Greek science from the decadent state it had fallen into under the later Roman Empire. They created a live and growing science thereby creating a link for modern science. In doing so they drew on the observations and investigations of Persia, India and China and extended further the techniques and narrow basis of Greek mathematics, algebra and trigonometry, there by laid the foundation of optics, scientific chemistry and medical science.

The developments contained till 11th century AD, after which the best days of Arab science faded out due to several reasons.

(vi) Modern Mathematics: The development of modern science with the advances in mathematics have been a collective  endeavour spread over several millenniums. The influence of Islamic Science & Mathematics in this progression cannot be ignored, as it has linked the ancient mathematics and science of  Egyptian, Babylonian, Greek, Indian and Chinese origins and laid basis for modern science which began in Europe in the renaissance period.

Modern science was born in Europe during the period from the middle of the 15th century to the end of the 17th century. The first major break through from the whole system of ancient scientific though was due to the work of Copernicus, who in 15th century gave a clean and detailed explanation for the rotation of earth and other planets on their axes and their motion around a fixed sun, which was at the centre. This model simplified astronomical calculation and made more precise. This was the first phase of what we now call as the Scientific Revolution.

After Copernicean Model new form of  mathematics was advanced by many great mathematicians. The Scientific Revolution of the 16th and 17th centuries marked a pivotal period where mathematics became deeply interwined with scientific inquiry. Great mathematicians and philosophers Galileo Galilee, Johannes Kepler, Rene Descart and Issac Newton used mathematics to describe and predict natural phenomena with unprecedented precision.

Galileo's work on astronomical instruments and the study of moon & planets; Kepler's laws of planetary motion; and Newton's laws of motion & that of universal gravitation where the remarkable strides achieved in the history of science. Galileo and Kepler could formulate mathematical descriptions of the motion of bodies because they were masters of the new mathematics that had grown during the Renaissance. Algebra, geometry and the decimal system, taken from the ancients and the Arabs, as well as the introduction of logarithm by Napier (1550-1617) greatly simplified astronomical calculations. The developments in astronomy led to the new mathematical explanation of the universe, finally arrived at by Newton.

This was a great achievement in science. Newton's spectacular contributions to mathematics and natural philosophy-discovery of calculus, the unification of terrestrial and celestial mechanics under a single law, brought the era of the Scientific Revolution to a spectacular close. Newton's work epitomized the integration of mathematics into science.

His development of calculus, independently along side Gottfried Wilhelm Leibniz, provided the tools necessary to describe changing quantities. Newton's laws of motion and universal gravitation were formulated mathematically, that allowed for accurate predictions of planetary motion and other physical phenomena.

The 19^th  and 20^th  centuries saw rapid advancements in both mathematics and science, with each field propelling the other forward. James clark Maxwell best known for his formulation of electromagnetic theory of light and kinetic theory of gases had the greatest influence on 20th century physics.The development of non-Euclidean geometry by mathematicians such as Carl Friedrich Gauss, Nikolal Lobachevsky and Jains Bolyai opened new realms of understanding, impacting theories of space and time.

All these theories applicable to macroscope systems are now known as classical theories. Besides the mathematicians and theoretical physicists there were many experimentalists who carried out their studies using sophisticated high precision instruments and techniques that were developed or invented side by side

In the 20th century Albert Einstein's theory of relatively revolutionized physics with its profound implications for our understanding of space, time and gravity. These theories were grounded in complete mathematical framework that challenged and expanded the boundaries of traditional Euclidean geometry. Quantum mechanics, another major scientific breakthrough of the 20th century, relies heavily on advanced mathematics, including linear algebra differential equations, integration methods, matrices and determinants, polynomials & power series, spherical harmincs, wave mechanics and probability theory etc.

The mathematical formulations developed by Scientists such as Max Planck, Niel Bohr, Werner Heisenberg, Louis de Broglie and Erwin Schrodinger were essential for describing the behaviour of particles at atomic and subatomic scales. Similarly, scientists such as Georges Lamaitre (Father of Big Bang Theory), Subrahmanyam Chandrsekhar, S.N.Bose, C.V.Raman, Srinivasa Ramanujan, Abdul Salam, Pieter Higgs, Roger Penrose and Stephen Hawking formulated advanced theories for understanding the great universe, and cosmology. Mathematics remains fundamental in theoretical physics, where researchers seek to unify the force of nature through theories such as string theory and quantum theory. These theories require sophisticated mathematical structures and abstract reasoning.

In the contemporary era, mathematics continues to be indispensable across all scientific disciplines. Without the knowledge of mathematics we cannot understand the subjects and specializations like Physics, Chemistry, Geography, Astronomy, Cosmology, Hydraulics, Architecture, Engineering, and all the Modern Technologies. The advent of computer and digital technology has further amplified the importance of mathematics in science.

Computational mathematics and models, simulations, grounded in mathematical algorithms, are now standard tools in fields ranging from climate science to genomics. Computational methods- numerical methods and algorithms derived from mathematics are employed for solving complex scientific problems that are analytically intractable. Computational science combines mathematics, computer science, and domain specific knowledge to tackle problems in various fields from engineering to biology.

Mathematics is inherently interdisciplinary, linking various scientific fields. For instance, mathematical biology applies mathematical techniques to solve biological problem. In physics, mathematical methods are used to explore physical phenomena; origin and structure of matter in the universe. In chemistry, automic/molecular structure and reactions are described using mathematical principles.

Similarly in earth sciences mathematical approach is used to investigate and analyse the real situations. Geometry and calculus allow scientists to visualise and understand structures and changes in physical space. In the fields like astronomy mathematical models help visualize the structure of the universe, study of planets, stars and galaxies; and in medicine all the imaging technologies rely on mathematical principles.

Big data and machine learning represent new frontiers where mathematics is crucial. Algorithms rooted in statistical theory and linear algebra drive advancements in artificial intelligence, robotics and quantum computing, enabling the analysis of vast data sets and the creation of predictive models. These technologies have transformative implications for medicine, economics and social sciences.

Conclusion: From the ancient mathematicians who first explored the properties of numbers and shapes, to the modern scientists developing algorithms for artificial intelligence, mathematics has been a continuous thread in the tapestry of scientific discovery. Its relevance has not diminished over time; it has instead evolved and expanded, underpinning the ever-growing complexity and scope of scientific inquiry. Mathematics is not just a tool for science; it is the very foundation upon which our understanding of the universe is built.

We cannot neglect the relevance of mathematics as its applications are vast and integral to the advancement of scientific knowledge. It is said that the mathematics is king of all sciences. To quote John Perin, "I have taught mathematics to all kind of students and in my opinion there is not a single student who cannot be a discoverer of facts, inventor of things instead, provided you give him, the rights stuff and the early you give him, the better the chances are".

by: Prof (Dr) Mohammad Aslam Baba, Former Principal/Dean Engineering and Technology, Cluster University Srinagar