The Fear, and the Use, of Mathematics and Physics
The two areas of human enquiry that inspire the greatest terror in the hearts of students are undoubtedly Mathematics and Physics. You may find history, or chemistry, or economics difficult, but your reaction to these subjects, and most others, is almost certainly not fear. On the other hand, when you encounter an equation, your first reaction is to escape to more amiable company. If you compare subjects to people, you will realize that your reaction to maths or physics is very similar to you reaction to a stern, quiet person who is famous for his wisdom but who makes you very uncomfortable indeed. When he speaks you listen dutifully, because you’ve been told his words contain a lot of meaning, but you understand almost nothing, and you end up feeling foolish and exposed; and what is worse, this person does not need to shout to make you feel this way – he just has to look at you. When you see an equation or a mathematical expression you react in the same way. Let us try to understand what mathematics is and why it is so difficult.
Pure mathematics is a kind of language: its symbols carry meaning and these symbols can be combined into expressions in well-defined ways to carry more complex meaning. The rules for constructing expressions in mathematics are very precise and well-defined. As a result it is possible in mathematics to start from simple ideas and rapidly build up mathematical structures that are unbelievably complex and far-reaching. Even to a professional mathematician, the heights that can be reached by this method are astonishing; in fact a true mathematician never loses his joyful amazement at the power and reach of mathematics.
In applied mathematics, of which theoretical physics is the most outstanding example, the meanings carried by the mathematical structures are closely reflected in the physical world. In a typical piece of reasoning in physics we start from a universal law, e.g. Newton’s second law, which relates the acceleration of an object to its physical interactions with other objects. When we throw a ball, its acceleration is due to its gravitational interaction with the Earth. We express this in mathematical form, and, once we do that, we have all the power and reach of pure mathematics at our disposal – i.e. we can now use the rules of mathematics to travel far away from the apparently simple physical facts expressed by our original equation. But because the physical situation has been captured in some fundamental sense by our original equation, the results we arrive at using mathematics remain representations of actual physical situations – including ones that we may never have guessed. For example, in the case of the ball thrown up, our mathematics tells us that if the velocity of the ball is more than 11.2 km/sec, it will escape the gravitational pull of the earth. So, starting from a mathematical representation of an apparently simple situation – a ball thrown upward – we arrive at the conclusion that rockets and interplanetary travel are possible!
This description of pure and applied mathematics also reveals the various reasons why they are reputed to be – and indeed are – so difficult. First, if you see just the final result of a long process of mathematical reasoning, it is virtually impossible to see how it could have been arrived at from its simple origins. Second, if you decide to go through the process of reasoning yourself, you have to make sure that you walk only along the paths allowed by the rules of mathematics – else you fall! When a mathematical structure is first being built there are often big gaps in it, and seasoned mathematicians must therefore become adept at leaping across these gaps, rather like a monkey jumping from tree to tree. In pure mathematics, as the structure is completed, mathematicians try to ensure that every gap is closed – the tree-tops are linked by a network of sky-bridges – and that it becomes possible for anyone with basic skills and courage to walk from one point to another. In physical mathematics, on the other hand, it is common practice to leave many of the gaps unclosed – we are expected to use our “physical intuition” to leap across the gaps. A mathematician is more sensitive to the beauty and intricacy of the structures that they have built, many of which seem to float in the air. A physicist is more interested in how he can get from one point in the forest to another, and to do that he moves sometimes along the ground – this is called reasoning physically – and sometimes high above the ground – this is called mathematical physics.
From my description above you can deduce that someone wishing to be a pure mathematician must possess two distinct abilities: first, he must be able to guess what kinds of mathematical structures can be built – is this conjecture provable? – and are worth building – is it mathematically important? –; and second, he must have the ability to build them, i.e. he must learn the necessary techniques. To be able to walk along mathematical structures that others have built is no mean ability, and certainly a very useful one, but a creator of mathematics must do much more. On the other hand a physicist must be able to walk boldy along the ground level of physical reality, climb up the elevated network called mathematical physics and walk along it, and always keep in mind the connection between the two.
There are of course many areas outside physics in which mathematics is indispensable. All branches of engineering use a combination of physical and mathematical reasoning, much as physicists do. Theoretical computer science is pure mathematics. Even in some areas of science that have traditionally been practised without mathematics, it is beginning to make inroads. For example, biology, once the most non-mathematical science, has recently begun to discover that mathematical reasoning can go a long way in explaining its phenomena. As a result, physicists are moving into biology in droves, leading to the a new area called biophysics. In chemistry too sophisticated mathematical models are used to determine the details of the chemical bond. Economics and its offshoots, e.g. finance, have of late become increasingly mathematical. The stock market and its various goods – options, derivatives, etc – are priced according to mathematical models. The increasing complexity of these models means that many people with PhDs in mathematics and physics and computer science have begun working in finance, especially in the US.
You may ask what role mathematics and physics have to play in the ordinary lives that most of us lead. Is it necessary for an IAS officer or journalist or a doctor to be familiar with the methods of mathematics and physics? One sometimes reads articles condemning the common man for being mathematically and scientifically illiterate in a world where our technology depends so much on advanced mathematics. I feel, however, that this line of argument misses the point in one way. When calculators didn’t exist, most of us knew our multiplication tables. Now that calculators are available to everyone, do we insist that everyone still remember their multiplication tables? Of course not. Music compression using mp3 depends on a beautiful and powerful mathematical idea called the Fourier transform. Does that mean that all of us must learn about Fourier transforms? Of course not.The experts must of course know their mathematics, or our computers and mobile phones will not work, but it is not just futile but pointless to insist that everyone should be familiar with the mathematics that underlies computers and mobile phones and other conveniences. Of course, everyone is welcome to learn about such mathematics, and feel enriched and inspired by it, but it is unlikely to help administer districts better or write more vivid reports or heal patients more effectively.
Does that mean I think mathematics and physics can be dispensed with altogether by the ordinary person? Not at all. If we believe that an educated person should be familiar, at some level, with the world in which he lives, then perhaps we should encourage him to develop an appreciation of some ideas in mathematics and physics – ideas that are reasonably simple yet convey the spirit and the methods of these disciplines. For example, the idea of the proof, so central to mathematics, is conveyed by fairly simple theorems in geometry, of the kind that used to be taught in all schools when I was a child. The subtlety of the concept of number can be appreciated by trying to undertand why mathematicians are so taken up with prime numbers. The idea of symmetry, which is fairly intuitive and physical, leads to the beautiful and abstract mathematics of group theory. Similarly, in physics the average person can try to learn from popular sources some of the central ideas of physics. A ball follows a path, but an electron does not – this is the strange world of quantum mechanics. One can never catch up with light – this is electromagnetic theory, optics, and relativity –, and so on. You can go beyond the basics too, and discover from popular sources what the major areas of research in mathematics and physics are. A large number of books explain these ideas for the non-scientific reader, and of course the internet is an inexhaustible resource for someone who wishes to learn.
That brings us to my final point. Suppose that you do not wish to spend your time appreciating mathematics and physics. What you want to know is whether they can help you in your day-to-day life. If I were asked to choose one area of mathematics – beyond addition and multiplication – that could help virtually everyone, it would be statistics. And one idea from physics that every minister and IAS officer and journalist should learn is how to make an intelligent estimate.
The basic ideas of statistics appear in virtually all areas of life, though we are often unaware of them. Consider for example the result of a blood test. You find that your cholesterol is 190. Unhappy that it is close to the danger mark, you get it tested again. This time you find that it is 180, and you feel reassured. Being an “optimist” you prefer to believe the second result! But suppose instead that you tried to understand this disparity in test results; then you would have to learn about the basic ideas of a statistical distribution and its mean and standard deviation. You would realize that if you got your blood tested a very large number of times, the results would yield a distribution, which would resemble the famous bell curve. Its mean would tend towards your true cholesterol level. More importantly – and this point is usually missed, even by doctors – the width of the distribution, or its standard deviation, would tell you how far you can trust the number produced by any one test, and help you to distinguish a significant difference between two test results from a insignificant difference. Another important statisical idea is that of a correlation: if you have data on two variables in a populations – e.g. height and weight –, how do you test whether they are related? Obviously, taller people are, on the whole, heavier than shorter people (in spite of the existence of short fat people and tall thin people). If you make a plot of height versus weight of a population, this correlation between height and weight will show itself as a band. If you understand this method, and it refinements, you can use correlations between variables not so obviously connected, to understand how much one factor influences another. And by looking at the way correlations change with location, and with time, you can draw important conclusions on how to invest resourses. For example, if you plot an indicator of health, say longevity, versus income, you are likely to find that in a sample of poor countries longevity increases with wealth, whereas in a sample of rich countries, longevity is not as strongly correlated with wealth. Whether you are talking about medical tests, the census, average rainfall, change in prices, variation in nutrition levels – the ideas of statistics will help you to draw useful conclusions from the information you have.
The idea from physics that I choose as the most useful for the common man is intelligent estimation. Before embarking on a calculation or an experiment that may consume a lot of time and resources, it is often important to have some idea where one is headed. For this, physicists must learn to do what they call “order-of-magnitude” calculations. The great physicist Enrico Fermi expressed this in his own way: he said that every physicist should know how to estimate the number of barbers in the city of Chicago! Let us try to estimate the number of barbers in the city of Delhi. – The average man has one haircut a month, i.e. one-thirtieth of a haircut per day. The male population of Delhi is between 60 lakhs and 90 lakhs, which means that there are 2 to 3 lakh haircuts per day. The average barber probably cuts 15 to 20 heads of hair in a day. … Put these together and you will find that the city of Delhi has between 10,000 and 20,000 barbers. To determine exactly how many barbers there are you need a survey, but notice how easily you can deduce that the number can’t be 1000 (or Delhi-ites would have to grow their hair like sadhus) or 100,000 (or the barbers would be mostly unemployed). A problem that when first posed sounds impossible to answer becomes simple when broken appropriately into parts. The same method can be used to estimate the power consumption or the petrol consumption in a city, the appropriate fare for a half-hour autorickshaw ride, the amount of grain needed to feed a population, the number of doctors needed in a district – the questions are endless. Normally one looks to experts for answers to these questions – and indeed experts are needed for precise answers –, but if you learn the physicist’s science of intelligent estimation, you can check whether the authorities know what they are talking about, and you don’t always have to wait for their answers.