Is Mathematics invented or discovered? – this is one of the classic ToK maths questions. Read this essay and decided for yourself!

## General ToK

1. A huge number of articles from Plus Maths exploring mathematics and the nature of reality. For example, read Plus Maths article, “Is God a Mathematician?”
2. Max Tegmark’s paper on physics, parallel universes and whether maths describes reality or is reality. Also a Youtube video explaining Tegmark’s views.
3. A detailed article on the question of mathematics and reality.
4. The Numperphile video channel. For example, “Do numbers exist?”

## 1.1 Basic Algebra

TOK: Mathematics and the knower. To what extent should mathematical knowledge be consistent with our intuition?
TOK: Mathematics and the world. Some mathematical constants (pi, e, , Fibonacci numbers) appear consistently in nature. What does this tell us about mathematical knowledge?
TOK: Mathematics and the knower. How is mathematical intuition used as a basis for formal proof? (Gauss’ method for adding up integers from 1 to 100.)
Aim 8: Short-term loans at high interest rates. How can knowledge of mathematics result in individuals being exploited or protected from extortion?
1. Are the Constants of Nature Really Constant?
2. A mathematical look at Fibonacci Numbers in Nature
3. Youtube video of Fibonacci numbers in nature.
4. Carl Sagan on the speed of light and time dilation.
5. Brain Greene TED talk about universal “fine tuning”.
6. Daniel Tammet – An autistic Savant who “sees” numbers (synathesia)..
7. Nrich article on Gauss
8. Nrich discussion on the compounding chess squares legend.
9. Plus Maths article on compound interest.
10. Plus Maths article on radioactive decay.
11. Cesaro Summation – does the series 1-1+1-1…= 1/2 ?

## 1.2 Logs

TOK: The nature of mathematics and science. Were logarithms an invention or discovery?
1. Open University discussion about the invention of logs by Napier.
2. Wikipedia article about the various log scales used in real life.

## 1.3 Counting Principles

TOK: The nature of mathematics. The unforeseen links between Pascal’s triangle, counting methods and the coefficients of polynomials. Is there an underlying truth that can be found linking these?
Aim 8: How many different tickets are possible in a lottery? What does this tell us about the ethics of selling lottery tickets to those who do not understand the implications of these large numbers?
1. Maths is Fun discussion on Pascal’s Triangle, the Chinese equivalent and links to binomial distribution.
2. Wikipedia on Lottery mathematics.
3. Game theory and expected value for Deal or No Deal.

## 1.5 Imaginary numbers

TOK: Mathematics and the knower. Do the words imaginary and complex make the concepts more difficult than if they had different names?
TOK: The nature of mathematics. Has “i” been invented or was it discovered?
TOK: Mathematics and the world. Why does “i” appear in so many fundamental laws of physics?
1. A brief history of complex numbers.
2. Interactive applications demonstrating the use of i in modelling real world problems.

## 1.6 Imaginary numbers

TOK: The nature of mathematics. Was the complex plane already there before it was used to represent complex numbers geometrically?
TOK: Mathematics and the knower. Why might it be said that e^ip + 1 = 0 is beautiful?
1. Plus Maths article on Euler’s Identity.
2. Introduction to Julia and Mandelbrot sets
4. Youtube video describing how to generate Julia sets.
5. A video explaining how to derive Euler’s equation. (Requires integration and complex numbers).
6. Wikipedia article on Euler’s Identity

## 1.4 and 1.7 Induction

TOK: Nature of mathematics and science. What are the different meanings of induction in mathematics and science?
TOK: Knowledge claims in mathematics. Do proofs provide us with completely certain knowledge?
TOK: Knowledge communities. Who judges the validity of a proof?
TOK: Reason and mathematics. What is mathematical reasoning and what role does proof play in this form of reasoning? Are there examples of proof that are not mathematical?

## 1.8 Solutions in 3D

TOK: Mathematics, sense, perception and reason. If we can find solutions in higher dimensions, can we reason that these spaces exist beyond our sense perception?
1. Maths Illuminated resources on the fourth dimension.
2. Carl Sagan video on flatland and tesseracts.
3. Youtube video on how to imagine the 10th dimension.
4. Michael Kaku youtube video on string theory and multi-dimensions.
5. Blog Posts on fractals, the Koch Snowflake and Dragon Curves (fractional dimensions)

## 2.1 Concepts of functions

TOK: The nature of mathematics. Is mathematics simply the manipulation of symbols under a set of formal rules?

## 2.2-2.5 Graphing functions

TOK: Mathematics and knowledge claims. Does studying the graph of a function contain the same level of mathematical rigour as studying the function algebraically (analytically)?

## 3.1-3.4 Triangles

Int: The origin of degrees in the mathematics of Mesopotamia and why we use minutes and seconds for time.
TOK: Mathematics and the knower. Why do we use radians? (The arbitrary nature of degree measure versus radians as real numbers and the implications of using these two measures on the shape of sinusoidal graphs.)
TOK: Mathematics and knowledge claims. If trigonometry is based on right triangles, how can we sensibly consider trigonometric ratios of angles greater than a right angle?
Appl: Triangulation used in the Global Positioning System (GPS).
Int: Why did Pythagoras link the study of music and mathematics?
1. Donlald Duck in Math Magicland video re Pythagoras and music.
2. Youtube video of Bach’s Crab Cannon looped on a Mobius strip.
3. Babylonian Mathematics.
4.An explanation of why we use radians.

## 3.5-3.6 Periodic graphs

TOK: Mathematics and the world. Music can be expressed using mathematics. Does this mean that music is mathematical, that mathematics is musical or that both are reflections of a common “truth”?
1. A number of Plus Maths articles on the link between maths and music.
2. Excellent Maths Illuminated unit on the mathematics of sound and waves

## 3.7 Cosine rule

TOK: Mathematics and knowledge claims. How can there be an infinite number of discrete solutions to an equation?
TOK: Nature of mathematics. If the angles of a triangle can add up to less than 180°, 180° or more than 180°, what does this tell us about the “fact” of the angle sum of a triangle and about the nature of mathematical knowledge?
Int: The use of triangulation to find the curvature of the Earth in order to settle a dispute between England and France over Newton’s gravity.
2. Plus Maths article on Non Euclidean geometry, art, fractals and Indra’s Pearls.
3. Plus Maths article on why triangles don’t always add up to 180 degrees.
4. Youtube video on triangulation and parallax
5. Excellent Maths Illuminated unit on non-Euclidean geometry

## Vectors 4.1

TOK: Mathematics and knowledge claims. You can perform some proofs using different mathematical concepts. What does this tell us about mathematical knowledge?
Aim 8: Vectors are used to solve many problems in position location. This can be used to save a lost sailor or destroy a building with a laser-guided bomb.
1. A large number of Plus Maths articles on vectors in the real world.

## 4.2 Scalar product

TOK: The nature of mathematics. Why this definition of scalar product?

## 4.3 Vector equations

TOK: The nature of mathematics. Why might it be argued that vector representation of lines is superior to Cartesian? Appl: Physics SL/HL 6.3 (magnetic force and field).
Youtube video – honey bees use vectors dancing to describe the position of flowers.

## 4.7 Vector planes in 3 dimensions

TOK: Mathematics and the knower. Why are symbolic representations of three-dimensional objects easier to deal with than visual representations? What does this tell us about our knowledge of mathematics in other dimensions?
1. Maths Illuminated resources on the fourth dimension.
2. Carl Sagan video on flatland and tesseracts.
3. Youtube video on how to imagine the 10th dimension.
4. Michael Kaku youtube video on string theory and multi-dimensions.
5. Excellent Maths Illuminated unit on extra dimensions
6. Blog Posts on fractals, the Koch Snowflake and Dragon Curves (fractional dimensions)

## 5.1 Statistics and Probability

TOK: The nature of mathematics. Why have mathematics and statistics sometimes been treated as separate subjects?
TOK: The nature of knowing. Is there a difference between information and data?
Aim 8: Does the use of statistics lead to an overemphasis on attributes that can easily be measured over those that cannot?
Appl: Misleading statistics in media reports.
1. Youtube double slit experiment and wave function collapse.
2. Heisenberg’s Uncertainty Principle, Schrodinger’s Cat and probability as the fabric of reality.
3. Kaku video discussing determinism, free will and probability.
4. Collection of resources demonstrating misleading data.
5. Youtube BBC video discussing determinism, probability and chaotic systems.
6. Collection of Plus Maths articles discussing a variety of chaotic systems in real life.
7. TED talk about statistics and misconceptions in real life.
8. Excellent Maths Illuminated unit on chaos.
9. 60 seconds in thought – Schodinger’s cat (number 6)

## 5.2 Basic probability

Aim 8: Why has it been argued that theories based on the calculable probabilities found in casinos are pernicious when applied to everyday life (eg economics)?
1. Article on casino probabilities.
2. Detailed Plus maths article on the importance of statistics in trials, and the “Prosecutor’s fallacy”

## 5.6 Binomial Distribution

TOK: Mathematics and the real world. Is the binomial distribution ever a useful model for an actual real-world situation?
1. Online ESP test – test ESP claims using binomial model.
2. A fake ESP trick.

## 5.7 Normal Distribution

Aim 8: Why might the misuse of the normal distribution lead to dangerous inferences and conclusions?
TOK: Mathematics and knowledge claims. To what extent can we trust mathematical models such as the normal distribution?
1. Plus Maths article on medical science, probability and normal distributions.

## Calculus

TOK: The nature of mathematics. Does the fact that Leibniz and Newton came across the calculus at similar times support the argument that mathematics exists prior to its discovery?
Int: How the Greeks’ distrust of zero meant that Archimedes’ work did not lead to calculus.
Int: Investigate attempts by Indian mathematicians (500–1000 CE) to explain division by zero.
TOK: Mathematics and the knower. What does the dispute between Newton and Leibniz tell us about human emotion and mathematical discovery?
1. Wikipedia article on the history of calculus.
2. A timeline and introduction to famous mathematicians involved in the development of calculus.
3. 60 Seconds in Thought – Achilles and the Tortoise.
4. Youtube video on Newton’s beliefs and eccentricities.
5. A video on the history of calculus as told through cartoons – Newton and Leibniz.
6. A BBC video – A history of infinity (3 parts). Part 2 discusses Achilles and the Planck unit.
7. Excellent Maths Illuminated unit on the size of infinity
8. Cantor’s diagonal argument which proves there are different sizes of infinity.
9. A detailed article on the history of infinity.
10. Short BBC documentary on fractals and Mandelbrot
11. How long is the coastline of Britain?
12.Hilbert’s Infinite Hotel – 60-Second Adventures in Thought (4-6)

## Calculus 6.2-6.7

TOK: Mathematics and knowledge claims. Euler was able to make important advances in mathematical analysis before calculus had been put on a solid theoretical foundation by Cauchy and others. However, some work was not possible until after Cauchy’s work. What does this tell us about the importance of proof and the nature of mathematics?
TOK: Mathematics and the real world. The seemingly abstract concept of calculus allows us to create mathematical models that permit human feats, such as getting a man on the Moon. What does this tell us about the links between mathematical models and physical reality?
1. A huge number of articles from Plus Maths exploring mathematics and the nature of reality.
2. A short article discussing some different careers which use calculus.
3. A fun youtube video – “Calculus Rhapsody.”
4. Wolfram explanation of Gabriel’s Horn – a volume of revolution with finite volume and infinite surface area.
5. A video explaining how to derive Euler’s equation. (Requires integration and complex numbers).

There are already quite a few opportunities for cross-curricular links above, but I’ve also included some extra ideas below that don’t fit in so well with the syllabus, but are interesting areas to explore.