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.docxThe mathematics we learn in school only scratches the surface of the vast and fascinating field of mathematics. We only get a glimpse of one aspect of this incredible subject, but the truth is that mathematics is a vast and wonderfully diverse field. In this video, I aim to show you all the amazing things you don't know about mathematics.
History of Mathematics
Let's start at the very beginning. The origin of mathematics lies in counting, a skill that is not unique to humans but is also found in other animals. Evidence for human counting dates back to prehistoric times, with check marks made in bones providing a glimpse of early human counting practices.
Throughout history, there have been numerous innovations in mathematics. The Egyptians created the first equation, the ancient Greeks made significant advancements in many areas such as geometry and numerology, negative numbers were invented in China, and zero as a number was first used in India.
In the Golden Age of Islam, Persian mathematicians made further progress, writing the first book on algebra. Mathematics experienced a boom during the Renaissance, along with the sciences, leading to new discoveries and advancements in the field.
Modern Mathematics
Modern mathematics can be divided into two main areas: pure mathematics and applied mathematics. Pure mathematics is the study of mathematics for its own sake, while applied mathematics is the development of mathematical tools to solve real-world problems. However, these two areas often overlap.
Many times in history, mathematicians have been driven by curiosity and a sense of aesthetics, creating new and interesting theories that may not have immediate practical applications. However, a hundred years later, when working on cutting-edge problems in physics or computer science, they may discover that these old theories are exactly what they need to solve their real-world problems, which is truly amazing. This kind of discovery has happened many times over the past few centuries, highlighting the fascinating interconnections between abstract theories and their practical applications.
But pure mathematics on its own is still a valuable field. It can be fascinating and have a beauty and elegance that is almost like art.
Pure Mathematics
Pure mathematics can be divided into several sections:
Numbers
The study of numbers begins with natural numbers and arithmetic operations. It then moves on to other types of numbers, such as integers, which contain negative numbers, rational numbers like fractions, real numbers that include numbers like pi, which have infinite decimal points, and complex numbers. Some numbers have interesting properties, such as prime numbers, pi, and 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. This means that some infinities are larger than others.
Structures
The study of structures involves taking numbers and putting them into equations in the form of variables. Algebra contains the rules for manipulating these equations. Here you will also find vectors and matrices, which are multi-dimensional numbers, and the rules for how they relate to each other are captured in linear algebra.
Group Theory
Number theory studies the features of all the numbers we have discussed, such as the properties of prime numbers. Combinatorics looks at the properties of certain structures like trees, graphs, and other things made of discrete chunks that you can count. Group theory looks at objects that are related to each other in groups, such as a Rubik's cube. Order theory investigates how to arrange objects following certain rules, such as 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.
Geometry
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 from school. There are also fun things like fractal geometry, which are mathematical patterns that are scale invariant, meaning you can zoom into them forever and they 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. 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.
Finally, differential geometry looks at the properties of shapes on curved surfaces. For example, triangles have different angles on a curved surface, which brings us to the next section, which is changes.
Changes
The study of changes contains calculus, which involves integrals and differentials. It looks at area spanned out by functions or the behaviour of gradients of functions. Vector calculus looks at the same things for vectors. Here we also find a bunch of other areas like dynamical systems, which look at systems that evolve in time from one state to another, like fluid flows or things with feedback loops like ecosystems. Chaos theory studies dynamical systems that are very sensitive to initial conditions. Finally, complex analysis looks at the properties of functions with complex numbers.
Applied Mathematics
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 more like 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.
Physics
We'll start with physics, which uses almost 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, such as mathematical chemistry and biomathematics, which look at loads of stuff from modelling molecules to evolutionary biology.
Mathematics is also extensively used 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. 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. 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. 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 to model financial systems and get an edge to win all those fat stacks.
Computer Science
Related to this is optimisation, where you are trying to calculate the best choice amongst a set of many different options or constraints. You can normally visualise this 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. 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.
Foundations of Mathematics
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 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. 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.
five questions on the text:
How did the early history of mathematics influence modern mathematical concepts?
What are the key differences between pure and applied mathematics, and how do they overlap?
How has the study of numbers evolved from natural numbers to complex numbers?
In what ways does mathematics play a crucial role in computer science and cryptography?
What are Gödel's incompleteness theorems, and how do they impact our understanding of mathematical foundations?