Mathematics Michael Lacewing

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 …

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

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

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

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

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

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?)

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?

II. Infinity: Hilbert’s hotel Just to clarify: what is infinity? https://youtu.be/Uj3_KqkI9Zo

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.

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?)

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!

III. MATHS AND REALITY Discovered or invented? https://www.youtube.com/watch?v=X_xR5Kes4Rs (5.11) Stephen Law, The Philosophy Gym, ‘The strange realm of numbers’ Some special numbers: i: https://www.youtube.com/watch?v=T647CGsuOVU; (5.46) real applications, e.g. in electrical engineering and fluid dynamics e: approx. 2.718: https://www.youtube.com/watch?v=R0oUeLQIbIk (1.57) Fibonacci sequence: https://www.youtube.com/watch?v=wTlw7fNcO-0 (3.20)

IV. Maths and beauty On mathematics and symmetry: www.youtube.com/watch?v=V5tUM5aLHPA (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? 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 1023 1024 – 1 = 1023

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.

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?