The Map of Mathematics

Summary

This video script offers an insightful journey through the vast and diverse field of mathematics, from its origins in counting to its modern branches of pure and applied mathematics. It highlights the interplay between pure curiosity-driven research and its unexpected usefulness in solving real-world problems. The script delves into various mathematical disciplines, including number theory, algebra, geometry, and calculus, and their applications in physics, engineering, and computer science. It also touches on the philosophical foundations of mathematics, pondering the human-made nature of a field that so elegantly explains the universe.

Takeaways

• 📚 Mathematics is a vast and diverse field that extends far beyond what is taught in schools.
• 🔍 The origins of mathematics are rooted in counting, with evidence dating back to prehistoric times.
• 🌏 Contributions to mathematics have been made by various civilizations, including the Egyptians, Greeks, Chinese, Indians, and Persians during the Golden Age of Islam.
• 🌟 The Renaissance period saw a significant boom in mathematics along with the sciences.
• 📈 Modern mathematics is divided into pure and applied mathematics, with pure mathematics focusing on the study of math for its own sake, and applied mathematics using math to solve real-world problems.
• 🔄 There is often a crossover between pure and applied mathematics, where abstract concepts developed out of curiosity later find practical applications.
• 🎨 Pure mathematics is not only valuable for its applications but also for its inherent beauty and elegance, akin to art.
• 🔢 The study of numbers in pure mathematics includes natural numbers, integers, rational numbers, real numbers, and complex numbers, each with unique properties.
• 📐 Geometry and its branches, such as fractal geometry and topology, explore the properties and behaviors of shapes in various spaces.
• 📉 Calculus and its related fields study changes, including the areas under curves and the behavior of gradients, which are essential in understanding dynamical systems and chaos theory.
• 🔧 Applied mathematics intersects with various fields like physics, engineering, economics, and computer science, providing tools and theories for practical applications.
• 🏗️ The foundations of mathematics, including mathematical logic, set theory, and category theory, explore the axiomatic basis and consistency of mathematical rules.
• 🤖 Fields like machine learning, cryptography, and the theory of computation are deeply connected to pure mathematics and have significant implications in the modern world.

Q & A

What is the origin of mathematics according to the script?

The origin of mathematics lies in counting, which is not just a human trait but is also observed in other animals. Evidence for human counting goes back to prehistoric times with check marks made on bones.

Which ancient civilization is credited with the invention of the first equation?

The Egyptians are credited with having the first equation in the history of mathematics.

What significant mathematical concept was first used in India?

Zero as a number was first used in India, marking a significant development in the field of mathematics.

What is the first book on algebra attributed to?

The first book on algebra was written during the Golden Age of Islam by Persian mathematicians.

How is modern mathematics divided in the script?

Modern mathematics is broadly divided into two areas: pure mathematics, which is the study of mathematics for its own sake, and applied mathematics, which involves developing mathematics to help solve real-world problems.

What is an example of a field in pure mathematics that studies numbers and their properties?

Number theory is an example of a field in pure mathematics that studies the features of numbers, such as the properties of prime numbers.

What is the significance of the interrelation between pure and applied mathematics mentioned in the script?

The interrelation between pure and applied mathematics is significant because often theories developed in pure mathematics, initially without practical applications, later prove to be essential for solving complex problems in fields like physics or computer science.

What does the script suggest about the nature of mathematics as an art form?

The script suggests that pure mathematics can be seen as an art form due to its beauty, elegance, and the aesthetic sense that guides its development.

What is the role of numerical analysis in applied mathematics?

Numerical analysis is a mathematical tool used in situations where the mathematics becomes too complex to solve analytically. It involves using simple approximations and combining them to get approximate answers, which is particularly useful in fields like engineering and physics.

How does the script describe the relationship between mathematics and computer science?

The script describes the relationship between mathematics and computer science as deeply intertwined, with the rules of computer science being derived from pure mathematics, and many areas of mathematics being essential for the development of intelligent computer systems and cryptography.

What are the foundational areas of mathematics that the script discusses?

The foundational areas of mathematics discussed in the script include mathematical logic, set theory, and category theory, which attempt to understand the properties of mathematics itself, such as the existence of a complete and consistent set of axioms.

What philosophical question does the script raise about the nature of mathematics?

The script raises the philosophical question of how a discipline made up by humans, such as mathematics, can explain so much about the universe, highlighting the deep mystery of the connection between human-created mathematical concepts and the fundamental nature of the universe.

Outlines

00:00

📚 Introduction to the Vastness of Mathematics

The video script begins by emphasizing the vastness and diversity of mathematics, which is often only partially introduced in schools. It traces the origins of mathematics to counting, a trait shared by humans and some animals, with evidence dating back to prehistoric times. The script highlights key historical developments, such as the first equations by the Egyptians, the Greeks' contributions to geometry and numerology, the invention of negative numbers in China, and the concept of zero in India. It also mentions the significant advancements made during the Golden Age of Islam and the Renaissance. The modern era of mathematics is divided into pure and applied mathematics, with pure mathematics being driven by curiosity and aesthetics, often leading to solutions for real-world problems discovered later. The script also appreciates the beauty and elegance of pure mathematics, likening it to an art form.

05:04

🔍 Deep Dive into Pure Mathematics and Its Subfields

This paragraph delves into the various branches of pure mathematics, starting with the study of numbers, including natural numbers, integers, rational numbers, real numbers, and complex numbers, each with unique properties. It discusses the concept of different infinities and the study of structures through algebra, linear algebra, number theory, combinatorics, group theory, and order theory. The paragraph then explores the study of shapes and spaces in geometry, fractal geometry, topology, measure theory, and differential geometry. The interrelation and crossover between these fields are acknowledged, with the script noting the complex web of connections that cannot be fully represented in two dimensions.

10:04

🌐 Applied Mathematics and Its Real-World Applications

The script transitions to applied mathematics, illustrating how it is deeply intertwined with various fields such as physics, chemistry, biology, engineering, and computer science. It discusses the application of mathematical principles in understanding and modeling natural phenomena, building complex systems, and solving optimization problems. The paragraph covers topics like dynamical systems, chaos theory, complex analysis, numerical analysis, game theory, probability, statistics, and cryptography. It also touches on the foundations of mathematics, exploring the philosophical questions about the nature of mathematical truth and the limits of mathematical knowledge as suggested by Gödel's incompleteness theorems. The theory of computation and complexity theory are briefly mentioned, highlighting the computational aspects of mathematics.

💡 Personal Reflections on the Joy of Mathematical Learning

In the final paragraph, the script shares a personal perspective on the joy of learning mathematics, particularly the moment of clarity when a complex concept becomes clear, leading to a sense of understanding the fundamental nature of the Universe. The author expresses a deep appreciation for the beauty and satisfaction that comes from mathematical epiphanies. The script concludes by acknowledging the popularity of the request to create a map of mathematics and expresses a hope that the video has met the viewers' expectations, promising more content in the future.

Keywords

💡Mathematics

Mathematics is a broad field that encompasses the study of numbers, quantities, shapes, and patterns. In the video, it is portrayed as a vast and wonderfully diverse subject, with roots in counting and evolving through various historical innovations. The script discusses how mathematics is divided into pure and applied branches, each with its own sub-disciplines and applications, illustrating the depth and breadth of the field.

💡Pure Mathematics

Pure mathematics refers to the study of mathematical concepts independently of any application to real-world problems. It is driven by curiosity and a sense of aesthetics. The script highlights pure mathematics as a valuable pursuit in itself, with its own beauty and elegance, and as a source of solutions for applied problems that were initially developed without practical intentions.

💡Applied Mathematics

Applied mathematics involves the development and application of mathematical concepts to solve real-world problems. The script mentions that there is a significant crossover between pure and applied mathematics, with theories initially developed in pure mathematics often finding practical uses in fields like physics, engineering, and computer science.

💡Natural Numbers

Natural numbers are the set of positive integers starting from 1, used for counting and ordering. In the video, natural numbers are the starting point of the study of numbers in pure mathematics, and they are foundational to arithmetic operations and further explorations in number theory.

💡Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols in equations. The script describes algebra as a part of the study of structures, where numbers are put into equations as variables, and it includes the study of vectors and matrices within linear algebra.

💡Geometry

Geometry is the study of shapes, sizes, and the properties of space. The script mentions geometry as having its origins with Pythagoras and being closely related to trigonometry. It also touches on more complex areas such as fractal geometry and topology, which are part of the broader study of spatial properties.

💡Calculus

Calculus is the mathematical study of continuous change, involving the concepts of limits, derivatives, integrals, and infinite series. The video describes calculus as a part of the study of changes, focusing on the behavior of functions and their rates of change, which is crucial for understanding phenomena in physics and engineering.

💡Probability

Probability is the measure of the likelihood that a given event will occur. In the script, probability is discussed as the study of random events and is integral to statistics, which in turn is essential for analyzing data and making informed decisions in various fields, including economics and finance.

💡Optimization

Optimization involves finding the best solution among a set of available options, often visualized as finding the maximum or minimum of a function. The video script mentions optimization as a common human endeavor, from getting the best value for money to maximizing happiness, and as a critical tool in fields like economics and computer science.

💡Computer Science

Computer science is the study of algorithms, data structures, and the theoretical foundations of information and computation. The script highlights the deep relationship between computer science and pure mathematics, noting that the rules of computer science were derived from mathematical principles and that areas like machine learning and cryptography rely heavily on mathematical concepts.

💡Foundations of Mathematics

The foundations of mathematics is a field that investigates the basic principles and concepts that underlie mathematical thought. The video discusses this area in the context of exploring the completeness and consistency of mathematical axioms, referencing Gödel’s incompleteness theorems, which suggest that mathematics may not have a fully consistent and complete set of foundational rules.

Highlights

The origin of mathematics lies in counting, with evidence dating back to prehistoric times.

Egyptians, ancient Greeks, and Chinese made significant early contributions to mathematics.

Zero as a number was first used in India, marking a pivotal development in mathematics.

Persian mathematicians in the Golden Age of Islam contributed to algebra, a foundational branch of mathematics.

Renaissance saw a boom in mathematics alongside the sciences, indicating its integral role in scientific advancement.

Modern mathematics is divided into pure and applied maths, reflecting its intrinsic and extrinsic value.

Pure mathematics is driven by curiosity and aesthetics, often leading to unexpectedly useful theories.

Applied mathematics solves real-world problems, showing the practicality of mathematical theories.

The interplay between pure and applied maths highlights the crossover and mutual enrichment of both fields.

Number theory, algebra, and combinatorics are key areas within pure mathematics, each with unique properties and applications.

Geometry and topology explore the properties of shapes and spaces, contributing to our understanding of physical laws.

Calculus and vector calculus are fundamental to the study of changes and dynamics in various systems.

Physics, chemistry, and engineering rely heavily on applied mathematics for modeling and problem-solving.

Numerical analysis and game theory are applied to complex scenarios where exact solutions are unattainable.

Probability and statistics are essential in understanding and predicting random events and large data sets.

Computer science and cryptography are deeply intertwined with pure mathematics, showcasing its versatility.

The foundations of mathematics question the axiomatic basis and consistency of mathematical systems.

Gödel’s incompleteness theorems reveal the inherent limitations of mathematical axioms, sparking philosophical debates.

The theory of computation and complexity theory explore the limits and capabilities of problem-solving in computing.

The video concludes with a personal reflection on the joy of understanding mathematics and its profound connection to the universe.

Transcripts

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.