Scientists make their lives (and the lives of young students) very difficult by swearing by mathematics. So why do they do that? To get an answer to that question, let us look at this complex subject of mathematics.
Mathematics came to humans originally as a counting system, presumably to keep track of the number of their livestock or children, and the numbers 1, 2, 3 and so on are natural needs. They were then coded, typically as strokes. Since we have 10 fingers, most early cultures had special symbols for units of 5 or 10. That is, they had individual signs for numbers up to 5 or 10 and then a special symbol for the number 5 or ten. I am sure, most of you remember the Roman numerals I (1), V (5), X (10), L (50), C (100) etc. So, in some sense the decimal system is an early invention. But if you ever did sums with these numbers, then you could barely add or subtract in these number systems. (There is an excellent set of 3 books called Universal History of Number by Georges Ifran, French author and historian, brought out by Penguin Books in 1994. The entire second volume of the book is dedicated to developments in India.)
Arithmetic took a major step forward when, in India, early mathematicians evolved and accepted a system where there would be a fixed set of signs for each number. However, its value would depend on where it was, in a sequence of numbers. That is, a number, say 3, at left extreme would indicate the value 3 but as second position from left it would indicate a quantity 10 times as large and so on. This implicitly, brought the concept of multiplication and division and the invention of the sign for a null value number 0 representing a number with no value.
Before long, mathematicians came up with the idea of multiplying and dividing any two numbers and undividable numbers. The way early Indians solved the problem was to make numbers very large so that the two numbers were dividable with integer outcome. But the stage was set for assuming that a quantity was represented by ‘x’ and one could divide the other numbers in the problem to arrive at the value of ‘x’ and the field of algebra came in along with the expansion of what could be counted – not just the livestock but even landmass, quantity of material, and the like. However, apart from this obvious trade related numbers and their operations, algebra, for a long time, remained a mental game where curious mathematicians defined rules about how two numbers or variables could interact and how their relations could be defined. It then went on to become so complex that it acquired a language of its own and added features that were very pretty in their own way, but now the subject was truly far from its original utilitarian route and became an independent subject of study. Many of these features were added by Indian mathematicians and the Kerala School remains the most prominent contributor It derived its scholarship from the works of Aryabhata and his contemporaries. The Arabs and the Europeans also added new ideas and features to this great enterprise of mathematics, pure, abstract and elegant. It was left for physicists to occasionally jump into this vast pool of ideas and discover concepts that they could use to understand real life situations and explore their science.
While mathematicians kept coming up with solutions to tricky trade related problem by mixing physics and mathematics – like weighing an elephant or dividing land between different claimants and so on, the field took a quantum jump when, around 16th century, scientists realised that the results of their experiment could be expressed not just as numbers, but that these numbers obeyed simple algebraic equations. Newton’s laws are an obvious example. Soon, as more and more complex aspects of nature were studied, more and more it relied on algebra and geometry to show us patterns in the working of nature.
Another field that got entangled in this is geometry. Initially a curiosity of shapes and their relations, it soon turned out that the objects of symmetries – they could be transformed in some ways and they looked the same, but when you transformed them in some other way, they did not look similar. This mix of shapes, symmetries, numbers and their interplay is what makes mathematics so fascinating.
As our faith in the ability to explain the working of nature through equations began to expand, we soon realised that nature is extremely consistent. If it obeys one equation once, it will obey the equation under all conditions, and if several equations were together valid for a situation, nature obeyed all the equations – there were no favourites for nature.
This realisation of nature simultaneously obeying all the previously noted rules was truly a sensational development. Soon complexities in mathematics and science began to march hand in hand and a whole bunch of physical processes – such as light beyond what is visible to us – from Radio to X-rays – were being predicted by mathematics.
The other interesting aspect of mathematics combined with physics was that it defined the most important entity in any experiment. For example, a pendulum with a wooden blob or metal blob behaved the same way, and hence the important thing for a pendulum was to have a thick blob at the end. There is one more thing that mathematics made possible and that is generalisation. If a physical system, say a pendulum obeyed an equation, and then all those other situations obeying the equation will behave the same way. For example the equation that is obeyed by a pendulum is also obeyed by a spring and by an oscillating crystal. So I can make clocks with pendulums, springs or crystals and they would all be good and depending on the nature of material be more or less accurate.
In fact, this faith in mathematics has taken people so far from conducting an experiment, that there are fields of science today where the word “experiment” has taken a different meaning and often it is not even possible to give a good description of what the equation implies. Theoretical physicists dwell in this realm where physical nature is described entirely on the basis of equations it obeys, rather than describe an observed experimental data in the form of an equation. This kind of abstraction is what gives mathematics the power it does and that is the reason why scientists swear by mathematics rather than any other description of how nature works.
Encyclopaedia Britannica defines mathematics as “The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.” (for example if I put a label “-” between two signs the result will be that the amount equivalent to the number on the right will be removed from the number on the left and so on). From these rules and with additional ideas of symmetry and logical consistency, they create a whole universe of intricately behaving mathematical entities and ideas that make a world of their own. Some of these ideas on operations (for example subtraction, use of the Pythagoras principle to create shapes of specific dimensions) are useful in real life, some are not. To demand utilitarianism from language of mathematics is like demanding use of a poem. That it provides pleasure to the initiated is an end in itself. If occasionally it becomes useful in real life, so be it. Yet, Godel established in 1931 that the language of mathematics, with its predefined set of rules can never be internally complete and there will always be ideas and operations that will lie outside which will be true but the axioms will not be able to prove it. These are called Godel’s incompleteness theorems
However, considering that humans were designed to live in a hunter gatherer environment where the primary purpose of life is to eat, not be eaten and reproduce, mathematics is not considered to be crucial. In fact, mathematics is often defined as an art that became science. So one would think that human mind is not adapt or optimised for mathematical studies, unlike say it having special talent to receive and analyse visual signals with great efficiency. But very recent studies seem to suggest that this is not entirely true. That children as young as 6 months old, begin to keep count of how many of any kind of objects are around and that primitive algebra seems to have been hard wired into our brain, just as sense of beauty, and geometrical symmetry are known to be hard wired in our brain and we are born equipped to appreciate this from the moment we can focus our eyes and see the world.
At the root, mathematics is a human construct. So is mathematics a language of nature that humans have learnt, and it is a coincidence that a system invented by humans is obeyed by nature? There are distinguished mathematicians who believe that mathematics was a human construct and it is just a coincidence that a large fraction of rules defined by mathematicians also reveal patterns in the manner in which nature works. These are generally called formalists. There are others who say that mathematics was in fact the language of nature that was discovered by humans who are called Platonists. One can of course argue indefinitely on the subtleties of these positions. But one thing is clear: nature works with mathematical precision. But an interesting question to ponder on is this, does mathematics exist because we do or do we exist because mathematics does?
If you want to know more on this, two beautiful books on “Best Writing in Mathematics 2012” and “Best Writing in Mathematics 2013” for interested general public have been published recently.
Dr Mayank Vahia is a scientist working at the Tata Institute of Fundamental Research since 1979. His main fields of interest are high-energy astrophysics, mainly Cosmic Rays, X-rays and Gamma Rays. He is currently looking at the area of archeo-astronomy and learning about the way the our ancestors saw the stars, and thereby developed intellectually. He has, in particular, been working on the Indus Valley Civilisation and taking a deeper look at their script.