The two areas of human equity 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 mathematics and physics is similar to your 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 stupid and exposed; and what is worse, this person doesn’t need to shout to make you feel this way- he just has to look at you.
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 of constructing expressions in mathematics are very precise and well- defined. As a result, it is possible in mathematics to start form simple ideas and rapid buildup mathematical structures that are unbelievably complex and far-reaching. Even for 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 with 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 the gravitational interaction with 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 upwards-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 mathematics reasoning, it is virtually impossible to see how it could have been arrived at from its simple origins. Second, if you 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 intuitions to leap across the gaps. A mathematician is more sensitive to the beauty and intimacy 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 the description above, you can deduce that someone wishes to become a pure mathematician must possess two different abilities: fiat, he must be able to guess what kind of mathematical structures can be built-is this conjecture provable?-and worth building-is it mathematically important?-and second, he must have to 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 boldly along the ground level of physical reality, climb up the elevated network called mathematical physics and walk along it, always keep in mind the connection between the two.