Can policymakers plan to make India a $5 trillion economy in five years without worrying about the basic mathematics of economic growth? Some of our leaders seem to think that wishful thinking can take us there. Albert Einstein’s discovery of gravity has been invoked as a successful example for achieving seemingly unattainable goals through “out-of-the-box” thinking. Two obvious questions arise. First, did Einstein indeed explain relativity sans any inputs from mathematics? And, two, can economic growth models be devoid of mathematics?
Einstein’s general theory of relativity is the most accurate theory of gravity available to us at present. True, Einstein was certainly not the first one to make contributions to our understanding of gravity. More than two centuries prior to him, Isaac Newton had proposed a universal law of gravitation. However, Newtonian theory of gravity, though remarkably accurate most of the times, had its limitations. When gravity was extremely strong or when the motions involved were extremely fast, the calculations became imprecise. For example, the theoretical calculation of the orbit of Mercury — the planet closest to the Sun — turned out to have a small disagreement with the actual observations of Mercury’s motion. Einstein’s theory not only predicted the orbit of Mercury accurately, but also predicted a number of interesting phenomena not anticipated earlier.
A theory of ‘space-time’
Einstein’s description of gravity was radically different from that of Newton. Newton assumed the existence of an absolute space and universal time. According to Einstein, space and time are part of a single entity called ‘space-time’. What we identify as space or time heavily depend on the frame of reference of the observer. However, space-time is universal. And gravity is the manifestation of curved space-time. Any massive object would curve the space-time around it.
It is hard to imagine a curved space-time, an entity that spans four dimensions — three spatial dimensions and the time. Typically we can see the curvature of a surface when we have access to a higher dimension. For example, we see the curvature of the surface of a football because we have access to a third spatial dimension. It is impossible to directly observe the curvature of the space-time since we don’t have access to a fifth dimension. However, it is possible to infer the curvature of a space without accessing extra dimensions. All the familiar axioms of Euclidean geometry cease to be valid on curved spaces. For example, according to Euclidean geometry, two parallel lines always remain parallel. However, this is not true, for example, on the Earth’s surface. Consider lines of constant longitude: on the equator, meridians are parallel to each other; but on the poles all of them meet. Thus, one could do a measurement to check whether lines that are originally parallel remain parallel. If they don’t, this is an evidence that the space-time is curved. Several astronomical observations conducted in the last century confirm that space-time is indeed curved in the presence of massive objects.
Einstein himself was not well-versed in the geometry of curved spaces. Here, Einstein turned to his friend, mathematician Marcel Grossmann, to master the necessary techniques and tools. Armed with these tools, and driven by some unique physical insights which are marks of a genius, Einstein was able to construct an elegant mathematical theory of space-time.
However, mathematical elegance is not the primary touchstone of a theory of nature. The key yardstick of success is the theory’s ability to describe the natural phenomenon that it seeks to describe — in this case, gravity. General relativity remained inaccessible to most of the scientists during its initial years. However, Einstein and many others were able to extract specific observable consequences of the curved nature of space-time by mathematically solving the equations. Even though space-time itself is not directly observable, all of these observable predictions were verified by a variety of astronomical observations and laboratory tests.
Is math useful?
Not all areas of sciences are able to construct theories or models that have the level of mathematical rigour that theories of physics enjoy. This is due to the highly complex nature of the phenomena they seek to describe. Most of the social sciences are in this end of the spectrum, due to obvious reasons. However, economics is probably one notable exception, where models and techniques employing higher mathematics have proven to be highly fruitful.
However, “math” is also commonly used as shorthand for quantitative reasoning, which is the backbone of all scientific enquiry. Ideally, planning and policy should be largely informed by quantitative reasoning, including the purported goal of doubling the size of Indian economy in five years. Wishful thinking and ideological propaganda are poor substitutes to quantitative reasoning.
Parameswaran Ajith is a physicist at the International Centre for Theoretical Sciences, Bengaluru. Views are personal