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.