Piyush Goyal, the Union minister for railways, recently asserted – incorrectly – that mathematics did not help Albert Einstein with his “discovery” of gravity. The minister did later correct his supposedly minor mistake in ascribing the discovery of gravitation to Einstein. The other part of his statement, about the importance or otherwise of mathematics, is even more problematic.
In the received history of western science, Isaac Newton is credited with the discovery of the idea of the universal force of gravitation, which causes an attraction between all bodies that have mass, otherwise known as inertia, the resistance the body offers to any force that attempts to set it in motion from rest. Newton provided a precise description of this force. He asserted that it is proportional to the product of the masses of the two attracting bodies and that it falls off inversely as the square of the distance between them[footnote]If two bodies are separated by X, the force between them is Y. If they are separated by 2X, the force between them is now Y/4.[/footnote]. The constant of proportionality that determines the force’s strength is universal: it is the same for all pieces of matter.
Newton’s proposition and subsequent work solved many problems associated with the study of solar systems and those connected to terrestrial motion, but they also raised more questions. For example, Earth must attract the Sun with the same force with which the Sun attracts Earth. How do these forces cover the enormous physical distance between the bodies? Do these forces cover this distance in an instant or does it take some time? What prevents the Sun and Earth from falling into each other? What would be the geometry of Earth’s orbit around the Sun? For example, from his own observations as of those of Tycho Brahe, Johannes Kepler had already shown that this shape was that of an ellipse. Could Newton confirm Kepler or would Kepler be proven wrong by Newton’s description?
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Newton could fashion answers to many of these questions. This was in part because he could accurately calculate the observable consequences of his speculations, particularly with regard to the geometrical details of the orbits of planets, using the precise form of the law he had proposed. He did this with the help of a new mathematics that he constructed at the same time but independently of Gottfried Wilhelm Leibniz (and without knowledge of the earlier work of mathematicians in Kerala): called differential and integral calculus. It is now taught to schoolchildren. Newton also proved these results using purely geometrical arguments.
The use of mathematics to answer physical questions about nature and precisely so was novel. Such methods, together with the process of testing results through controlled experiments, were in the long run one of the profound consequences of Newton’s work. This is how what we call modern science emerged. Since the time of Newton, the use of mathematics in science has not been a luxury but an essential part of scientific enquiry. Mathematics gives science precision, power and depth.
This said, Newton had left aside the question of how the proposed action at a distance, and the apparently instantaneous transmission of the force, worked. It was as if time played no role in this process. It was to this question of transmission that Einstein provided an answer. He did this by changing the intrinsic nature of the theory of gravitation.
In Einstein’s theory, matter alters the space and time immediately around itself, changing the scales of measurement of both space and time as well as the flatness of the space-time continuum. These changes then spread outward from the source with the speed of light in vacuum – a large but finite number.
The mathematics that Einstein’s theory employs was not completely known even to Einstein when he started thinking about gravitation, around 1909. He laboured with his friend and fellow mathematician Marcel Grossman to master other mathematical concepts: tensor calculus, developed by the Italian mathematician Tullio Levi-Civita, and a branch of geometry developed by Georg Bernhard Riemann that described curved surfaces. He then successfully applied them in the theory of gravitation he developed in 1916.
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It has been more than 100 years since this theory first took shape, and we use it to this day to describe the external universe, as much as we continue to use Newton’s theory more than 350 years after it was discovered because of its relative simplicity. The study of Einstein’s theory includes the study of the existence and behaviour of gravitational waves, which physicists directly detected only in 2015. Their existence could hardly have been predicted without the use of mathematical tools.
Indeed, to speak of the absence of any need for mathematics in these developments is to invite ridicule. Paul Dirac, one of the great physicists of the last century, has said that nature in its deepest aspects has a structure that cannot be described or understood in spoken words alone. Physical reality is very strange, and we cannot comprehend it even in part without mathematical symbols, equations and ideas. Maybe the minister will correct himself a little more comprehensively now, and be more cautious in his pronouncements on such matters in the future.