The Map of Mathematics

The mathematics we learn in school doesn’t quite do the field of mathematics justice. We only get a glimpse at one corner of it, but the mathematics as a whole is a huge and wonderfully diverse subject. My aim with this video is to show you all that amazing stuff.

We’ll start back at the very beginning. The origin of mathematics lies in counting. In fact counting is not just a human trait, other animals are able to count as well and e vidence for human counting goes back to prehistoric times with check marks made in bones.

There were several innovations over the years with the Egyptians having the first equation, the ancient Greeks made strides in many areas like geometry and numerology, and negative numbers were invented in China. And zero as a number was first used in India. Then in the Golden Age of Islam Persian mathematicians made further strides and the first book on algebra was written.

Then mathematics boomed in the renaissance along with the sciences. Now there is a lot more to the history of mathematics then what I have just said, but I’m gonna jump to the modern age and mathematics as we know it now. Modern mathematics can be broadly be broken down into two areas, pure maths: the study of mathematics for its own sake, and applied maths: when you develop mathematics to help solve some real world problem.

But there is a lot of crossover. In fact, many times in history someone’s gone off into the mathematical wilderness motivated purely by curiosity and kind of guided by a sense of aesthetics. And then they have created a whole bunch of new mathematics which was nice and interesting but doesn’t really do anything useful.

But then, say a hundred hears later, someone will be working on some problem at the cutting edge of physics or computer science and they’ll discover that this old theory in pure maths is exactly what they need to solve their real world problems! Which is amazing, I think! And this kind of thing has happened so many times over the last few centuries.

It is interesting how often something so abstract ends up being really useful. But I should also mention, pure mathematics on its own is still a very valuable thing to do because it can be fascinating and on its own can have a real beauty and elegance that almost becomes like art. Okay enough of this highfalutin, lets get into it.

Pure maths is made of several sections. The study of numbers starts with the natural numbers and what you can do with them with arithmetic operations. And then it looks at other kinds of numbers like integers, which contain negative numbers, rational numbers like fractions, real numbers which include numbers like pi which go off to infinite decimal points, and then complex numbers and a whole bunch of others.

Some numbers have interesting properties like Prime Numbers, or pi or the exponential. There are also properties of these number systems, for example, even though there is an infinite amount of both integers and real numbers, there are more real numbers than integers. So some infinities are bigger than others.

The study of structures is where you start taking numbers and putting them into equations in the form of variables. Algebra contains the rules of how you then manipulate these equations. Here you will also find vectors and matrices which are multi-dimensional numbers, and the rules of how they relate to each other are captured in linear algebra.

Number theory studies the features of everything in the last section on numbers like the properties of prime numbers. Combinatorics looks at the properties of certain structures like trees, graphs, and other things that are made of discreet chunks that you can count. Group theory looks at objects that are related to each other in, well, groups.

A familiar example is a Rubik’s cube which is an example of a permutation group. And order theory investigates how to arrange objects following certain rules like, how something is a larger quantity than something else. The natural numbers are an example of an ordered set of objects, but anything with any two way relationship can be ordered.

Another part of pure mathematics looks at shapes and how they behave in spaces. The origin is in geometry which includes Pythagoras, and is close to trigonometry, which we are all familiar with form school. Also there are fun things like fractal geometry which are mathematical patterns which are scale invariant, which means you can zoom into them forever and the always look kind of the same.

Topology looks at different properties of spaces where you are allowed to continuously deform them but not tear or glue them. For example a Möbius strip has only one surface and one edge whatever you do to it. And coffee cups and donuts are the same thing - topologically speaking.

Measure theory is a way to assign values to spaces or sets tying together numbers and spaces. And finally, differential geometry looks the properties of shapes on curved surfaces, for example triangles have got different angles on a curved surface, and brings us to the next section, which is changes. The study of changes contains calculus which involves integrals and differentials which looks at area spanned out by functions or the behaviour of gradients of functions.

And vector calculus looks at the same things for vectors. Here we also find a bunch of other areas like dynamical systems which looks at systems that evolve in time from one state to another, like fluid flows or things with feedback loops like ecosystems. And chaos theory which studies dynamical systems that are very sensitive to initial conditions.

Finally complex analysis looks at the properties of functions with complex numbers. This brings us to applied mathematics. At this point it is worth mentioning that everything here is a lot more interrelated than I have drawn.

In reality this map should look like more of a web tying together all the different subjects but you can only do so much on a two dimensional plane so I have laid them out as best I can. Okay we’ll start with physics, which uses just about everything on the left hand side to some degree. Mathematical and theoretical physics has a very close relationship with pure maths.

Mathematics is also used in the other natural sciences with mathematical chemistry and biomathematics which look at loads of stuff from modelling molecules to evolutionary biology. Mathematics is also used extensively in engineering, building things has taken a lot of maths since Egyptian and Babylonian times. Very complex electrical systems like aeroplanes or the power grid use methods in dynamical systems called control theory.

Numerical analysis is a mathematical tool commonly used in places where the mathematics becomes too complex to solve completely. So instead you use lots of simple approximations and combine them all together to get good approximate answers. For example if you put a circle inside a square, throw darts at it, and then compare the number of darts in the circle and square portions, you can approximate the value of pi.

But in the real world numerical analysis is done on huge computers. Game theory looks at what the best choices are given a set of rules and rational players and it’s used in economics when the players can be intelligent, but not always, and other areas like psychology, and biology. Probability is the study of random events like coin tosses or dice or humans, and statistics is the study of large collections of random processes or the organisation and analysis of data.

This is obviously related to mathematical finance, where you want model financial systems and get an edge to win all those fat stacks. Related to this is optimisation, where you are trying to calculate the best choice amongst a set of many different options or constraints, which you can normally visualise as trying to find the highest or lowest point of a function. Optimisation problems are second nature to us humans, we do them all the time: trying to get the best value for money, or trying to maximise our happiness in some way.

Another area that is very deeply related to pure mathematics is computer science, and the rules of computer science were actually derived in pure maths and is another example of something that was worked out way before programmable computers were built. Machine learning: the creation of intelligent computer systems uses many areas in mathematics like linear algebra, optimisation, dynamical systems and probability. And finally the theory of cryptography is very important to computation and uses a lot of pure maths like combinatorics and number theory.

So that covers the main sections of pure and applied mathematics, but I can’t end without looking at the foundations of mathematics. This area tries to work out at the properties of mathematics itself, and asks what the basis of all the rules of mathematics is. Is there a complete set of fundamental rules, called axioms, which all of mathematics comes from?

And can we prove that it is all consistent with itself? Mathematical logic, set theory and category theory try to answer this and a famous result in mathematical logic are Gödel’s incompleteness theorems which, for most people, means that Mathematics does not have a complete and consistent set of axioms, which mean that it is all kinda made up by us humans. Which is weird seeing as mathematics explains so much stuff in the Universe so well.

Why would a thing made up by humans be able to do that? That is a deep mystery right there. Also we have the theory of computation which looks at different models of computing and how efficiently they can solve problems and contains complexity theory which looks at what is and isn’t computable and how much memory and time you would need, which, for most interesting problems, is an insane amount.

Ending So that is the map of mathematics. Now the thing I have loved most about learning maths is that feeling you get where something that seemed so confusing finally clicks in your brain and everything makes sense: like an epiphany moment, kind of like seeing through the matrix. In fact some of my most satisfying intellectual moments have been understanding some part of mathematics and then feeling like I had a glimpse at the fundamental nature of the Universe in all of its symmetrical wonder.

It’s great, I love it. Ending Making a map of mathematics was the most popular request I got, which I was really happy about because I love maths and its great to see so much interest in it. So I hope you enjoyed it.

Obviously there is only so much I can get into this timeframe, but hopefully I have done the subject justice and you found it useful. So there will be more videos coming from me soon, here’s all the regular things and it was my pleasure se you next time.