Why learn pure math?
Why learn pure math?
In a world consumed by chaos and confusion,
one discipline will rise
to bring order and structure 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, if we delve deeper into the theory, 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 and didn't know about reality.
MATH
Starring: numbers,+, -, shapes, sets, functions, formulas, weird symbols, and frustrating phrases like: "it is easy to see" and "we leave the proof as an exercise to the reader".
While it may be easy to imagine ways in which applied math is being used to make our lives better, there is a valid question regarding the value of developing pure math. Why should we study abstract theory and not just the concrete methods used in applications?
Let me start by stating the perhaps obvious observation that it can be very difficult to anticipate what methods and tools will be useful before you have them. When mathematicians prove a theorem, it is not usually clear what ingredients will go into the proof, and as such it can be
For example, if we had an oracle (or a very sophisticated AI) that we trusted and that told that the twin prime conjecture is true, that would not really be satisfactory because what we want to know why it is true.
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 makes it really difficult for scientists to communicate the value of their work to the general audience.
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 gives us a language that serves as a common ground on which we can make precise statements. And as science continues to become more and more specialized, the need for an effective communication method will always be there.
This kind of bridge that pure math has been building is not an instantaneous thing that someone can just immediately elucidate after looking at two areas of science, but instead it usually takes a long time 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 scientists are saying.
For most people their experience of doing mathematics 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 do we still bother?
Some of the computations that need to be done involve an 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.
Most of us rely on science for making many decisions about our daily lives. For example, people look for weather predictions to check whether or not it will rain today. 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? By "safe" I mean that this decision is not simply being made for yourself, but others will rely on your predictions, and as such, you bear some responsibility.
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.