1. Mathematics
Michael Lacewing

2. Some topics for tok maths
Maths as a formal system, including
Gödel’s incompleteness theorem
Infinity
Maths and reality, including
Non-Euclidean geometry
Is maths invented or discovered?
Weirdly special numbers and patterns, such as i, e, and the Fibonacci sequence
Maths and beauty …

3. I. Maths as a formal system
A formal system is a structure of reasoning, with three key elements:
Axioms
Deductive reasoning
Theorems
Axioms are starting points, basic assumptions. They aren’t derived from anything else. In a set of axioms, the axioms should be
Consistent: they don’t give rise to contradictions
Independent: you cannot deduce any axiom from the others
Simple
Fruitful: they give rise to many theorems

4. Maths as a formal system: euclid
Euclid is often called the founder of geometry
He is the one that listed the five axioms that we to this day still use

5. Euclid’s 5 axioms
1:It shall be possible to draw a straight line joining any two points
2: A finite straight line may be extended without limit in either direction
3: It shall be possible to draw a circle with a given centre and through a given point
4: All right angles are equal to one another
5: There is just one straight line through a given point which is parallel to a given line
All Euclidian geometry is based on, and provable using, only those five claims
E.g. Lines perpendicular to the same line are parallel
Two straight lines do not enclose an area

6. What can you prove with a formal system?
Here you’re going to take your own set of axioms, and see what you can deduce from it
We’re going to use the primitive terms, ‘Bees’ and ‘Hives’
P1: Every Hive is a collection of bees
P2: Any two distinct Hives have one and only one Bee in common
P3: Each Bee belongs to two and only two Hives
P4: There are exactly four Hives

7. What can you prove with a formal system?
Try to, using the axioms above, show the following:
A) This set of axioms is consistent (none contradict each other)
B) Postulates 2 and 3 are independent (one does not entail another)
C) What Theorems can we conclude from this?
Hint: try drawing it out with different coloured/labelled bees

8. The case of the barber
Here is a Barber He has two rules He shaves everyone in town that does not shave themselves He does not shave anyone that does shave themselves Does he succeed? (Is this system consistent?)

9. Gödel's incompleteness theorem
In the early 20th century, the foundations of mathematics came into doubt, as different attempts to provide a complete and consistent set of axioms for all mathematics produced paradoxes and inconsistencies.
In 1931, Kurt Godel proved two extraordinary claims
no consistent set of axioms can prove all truths about natural numbers
No set of axioms (about natural numbers) can prove its own consistency
In other words, for any formal system, there are statements that it can state but neither prove nor disprove – so the system is incomplete
No contradictions have been discovered so mathematicians aren’t too worried
Does this mean that even maths is not completely certain?

10. II. Infinity: Hilbert’s hotel
Just to clarify: what is infinity?

11. III. Maths and reality Recall Euclid’s 5 theorems:
1:It shall be possible to draw a straight line joining any two points
2: A finite straight line may be extended without limit in either direction
3: It shall be possible to draw a circle with a given centre and through a given point
4: All right angles are equal to one another
5: There is just one straight line through a given point which is parallel to a given line
Reimann suggested some alternatives:
1’: Two points may determine more than one straight line.
2’: All straight lines are finite in length but endless (i.e. circles)
5:’: There are no parallel lines.

13. Reimann geometry Some new theorems:
All perpendiculars to a straight line meet at one point
Two straight lines enclose an area
The sum of the angles of any triangle is greater than 180 degrees
(An aside: A hunter leaves his house one morning and walks one mile due south. He then walks one mile due west and shoots a bear, before walking a mile due north back to his house. What colour is the bear?)

14. The geometry of space
Euclid’s geometry describes flat surfaces – 3 straight dimensions
Reimann’s geometry describes hyperbolic and elliptical curvatures
So which system of geometry – Euclid’s or Reimann’s – describes space itself?
Einstein showed it was Reimann’s: space is curved!

15. III. MATHS AND REALITY
Discovered or invented? (5.11)
Stephen Law, The Philosophy Gym, ‘The strange realm of numbers’
Some special numbers:
i: (5.46) real applications, e.g. in electrical engineering and fluid dynamics
e: approx : (1.57)
Fibonacci sequence: (3.20)

16. IV. Maths and beauty
On mathematics and symmetry: (8.08)
Is beauty mathematical?
Is mathematics beautiful?
Elegance in mathematics:
There are 1,024 people in a knock-out tennis tournament (i.e. each game has 1 winner and 1 loser). What is the total number of games that must be played before a champion can be declared?
= 1023
1024 – 1 = 1023

17. Maths the tok way We’ve looked at
Maths as a formal system
Infinity
Maths and reality
Maths and beauty
Research a topic that interests you in a TOK way.
What are some KQs that it generates? Is there an overarching one, and then some subsidiary ones?
Write an outline for an essay that would discuss those KQs.

18. Some Maths KQs
Why is there sometimes an uneasy fit between mathematical descriptions and the world? (For example, if I had four cows and then took five away, how many would be left?)
Is mathematics invented or discovered?
If mathematics is created by man, why do we sometimes feel that mathematical truths are objective facts about the world rather than something constructed by human beings?
If mathematics is an abstract intellectual game (like chess) then why is it so good at describing the world?
Why should elegance or beauty be relevant to mathematical value?
Is certainty possible in mathematics?
How did the rise of non-Euclidean geometry change people’s conception of mathematics?