Why learn pure math?
Why learn pure math?
In a world consumed by vast chaos and abhorrent confusion,
one discipline will rise
to bring structure and meaning to the universe...
Meet Math, a troubled subject on a quest to unravel the secret patterns.
Feared by many and studied by few, Math will have to overcome stigma and curricular changes to achieve greatness.
But, as we delve deeper into this abstract knowledge, we may discover hidden theorems with the power to change EVERYTHING.
From the creators of the question "How long is the diagonal of a square?", comes a mind-bending millennia-long journey that will challenge all you thought you knew about reality.
MATH
Starring: numbers,+, -, shapes, chalk, sets, functions, formulas, lots and lots of scratch paper, abstract theory, weird symbols, and deductive reasoning.
I think most people can imagine ways in which applied maths is being used to make our lives better, but what about pure math? Why should we study abstract theory and not just the concrete methods used in applications?
The development of science is getting more specialized over time as people work on hard problems and come up with new tools and innovative ways to use them. This increased specialization of the different areas of science presents a serious difficulty in communication, as it makes it really difficult for scientists to communicate the value of their work to the general audience. Most of us rely on science for making many decisions about our daily lives. For example, many of us look for weather predictions to check whether or not it will rain today. But, as a general member of the audience who does not work in meteorology, ask yourself, if you had to make that prediction, what kind of information would you want to see to make that decision? You can probably guess that any kind of data you want to know will probably look chaotic and random, with no real discernible patterns. But still, a decision about whether or not to take an umbrella today needs to be made. Is there any way we can make that decision in a safe way?
This communication problem is not just between the scientific community and the rest of society, but even just among scientists working in different areas there can be a hard time understanding each other. Aside from the differences in methods and techniques, there is usually the obstacle of language, by which I mean that every area, every field, every discipline has its own local jargon, creating an obstacle for people from other areas to engage in interdisciplinary work.
Pure math is there to make things precise, it gives us a language that serves as a common ground on which we can say things without ambiguity. And as science continues to become more and more specialized, the need for an effective communication method will always be there. I should make the distinction here that mathematics is not there to tell you whether or not it will rain today, instead it gives you tools to analyze the data you have chosen to use to make a prediction, so that you can give a precise estimate of the error of making any possible prediction.
This kind of bridges that pure math has been building are not instantaneous things that someone can just immediately elucidate after looking at two areas of science, but instead they usually take decades, even generations of mathematicians doing hard work to eventually find the right route on which a solid bridge can be built. And since things are continuously getting more involved, we should keep studying mathematics to be able to understand what other scientists are saying.
Most people have had very little interaction with pure mathematics, and most likely their experience of doing mathematics at all amounts to just learning how to do calculations. Even if you had to learn trigonometry in school, most of the questions in tests were probably about computing angles, lengths or areas of geometric shapes. There are formulas for all those things and these days we have computers that can do all of that for us, so it is only fair to ask: why bother?
Some of the computations that need to be done involve a infinite amount of steps. This may seem weird to you if you have never seen an integral, but such kinds of computations do arise naturally and frequently. Now, if people can perform calculations that involve an infinite amount of steps, then computers probably can do that too. This is true for the most part, but computers do struggle at times.
Things start to get complicated when the calculations that you need to do require not just infinitely many steps, but in fact two series of infinitely many steps that depend on each other in sequence, so that the steps of one series require the computations done in the other series of steps. This is quite common. When doing finitely many operations, this kind of phenomenon is never an issue because we all know that addition and multiplication are both commutative operations, so 3+5 is the same thing as 5+3, and 3·5 equals 5·3. But this property does not hold in general when doing infinitely many computations, sometimes it really does matter in which order you do things. Then again, other times it does not matter, and it can actually make the calculations much easier to perform them in a specific convenient order. So how can we tell when we can swap the order? That is the kind of thing that abstract theory, specifically mathematical analysis, can tell you.
Many problems in maths can be phrased using equations, and so solving the problem amounts to finding the solutions of an equation. Indeed, for most people, their day to day finances amounts to the following situation.
I have a certain amount of money, and I also have a list of things I want to buy. I don't have enough money to buy everything, so I need to figure out what is more convenient for me to buy.
This kind of problem can be phrased in terms of solving (systems of) equations. Computers are very helpful in these cases, but things can get tricky very quickly.
Suppose that the problem you are trying to solve doesn't just want you to find the solutions of an equation, but it wants you to find those solutions which are whole numbers; these are called integer solutions. These kinds of equation are called a Diophantine equations, and the general study of integer solutions of equations is called Diophantine geometry. Many things in life appear only in whole number quantities, so if a solution of an equation happens to have decimal places, it may not be a useful solution.
Can we use a computer to find all the integer solutions? Unfortunately, we know that there is no general formula or method for finding the integer solutions of equations (see Hilbert's Tenth Problem). So a lot of research is devoted to figuring out whether certain kinds of equation have integer solutions, and if so, trying to find all of them. Although computers can help us find some integer solutions by performing checks of many possible values, even when they find some, they cannot tell us whether we have found all of them. We need some abstract theory to help us here.
Of course, it is not always going to be the case that we always want to find the integer solutions of an equation, we can also be interested in finding other kinds of solutions. It does happen though that the solutions we look for may be too complicated to write down as meaningful expressions, if we asked the computer to tell us the value of the number it must just seem like a complicated sequence of digits without much information we can use. You may find it weird that anyone would be interested in the solutions of an equation, but not in the actual numerical values of that equation. What else is there to find if not the numerical values? Well, sometimes we want to know whether the solutions follow a discernible pattern; even if each solution by itself is difficult to describe because it is some weird random number, they may still satisfy a pattern that is easy to describe. But what kind of pattern? Again this is something were asking a computer may be pointless, but abstract theory can give us a hand.