Mathematics Notes

1. Mathematics Notes ToK

2. Introduction Certainty Search for abstract patterns Useful, practical Mathematical literacy essential to a successful career in any branch of science Wrong answers are simply wrong – which perhaps serves to explain why those who are not very good at math tend to fear the subject so much Requires selective attention – context must be ignored to operate at the abstract level

3. The Mathematical Paradigm Mathematical imperialism Can everything that is important in life be “counted”? Definition: “the science of rigorous proof” Formal system, consisting of three key elements: axioms, deductive reasoning, theorems

4. Axioms Starting points/basic assumptions Self-evidently true Need to be consistent, independent, simple, and fruitful The five axioms of Euclidean geometry are an example

5. Deductive Reasoning Axioms are like premises Theorems are like conslusions

6. Theorems Deduced from axioms via proofs In a proof, a theorem is shown to follow logically from the relevant axioms A conjecture is a hypothesis that seems to work but has not been shown to be necessarily true Conjectures are based on inductive reasoning, and cannot provide certainty Goldbach’s conjecture: every even number is the sum of to primes – still unproven

7. Beauty, Elegance, and Intuition Proofs need to be clear, economical, and generalizable Creative imagination and intuition (educated, rather than natural) appear to play a key role Is mathematical insight inborn, or can it be taught?

8. Mathematics & Certainty • mathematical propositions are either true by definition (analytic propositions) or they aren’t (synthetic propositions) • Mathematical propositions are either knowable independent of experience (a priori) or they aren’t (a posteriori)

9. Three Options Mathematics as empirical (empiricism) – mathematical propositions are based on many, many real-world experiences; although we can use concrete examples to illustrate mathematical propositions, mathematics tends to generalize beyond experience at the abstract level Mathematics as analytic (formalism) – when solving a problem, perhaps we are only unpacking what is already contained within it; unfortunately, only trivial truths tend to be uncovered this way, which fails to explain why mathematics “works” so well Mathematics as synthetic a priori (Platonism) – the idea that human beings can uncover truths about the nature of reality using reason alone; but then who or what put these truths there? Those who believe in God have a ready response to this question; those who don’t, don’t. For example, it is unclear how mathematical ability provides us with an evolutionary advantage, if evolution is the answer.

10. Discovered or Invented? Platonists would argue that mathematics is discovered whereas formalists would argue that it is invented Mathematical objects do not exist in the real world – they are idealisations. How, then, do they enable us to make real-world discoveries on the basis of them? Plato believed that mathematics is more certain than perception and that its claims are timelessly true (e.g. Pythagorean theorem), making mathematical claims even “more real” than perceived reality Unfortunately, this does not explain how we physical entities are able to know about them Perhaps mathematical claims merely have a “social existence,” as the formalists would have use believe

11. Non-Euclidean Geometry & the Problem of Consistency Mathematicians became suspicious that one of Euclid’s axioms (that of parallels) was less self-evident than the others Riemannian geometry – replaced some of Euclid’s axioms with their contraries – found no contradictions, though he was unable to prove it Only makes sense on a sphere Austrian mathematician Kurt Gödel (1906-78) came up with his incompleteness theorem, in which he proved that it is impossible to prove that a formal mathematical system is free from contradiction What this means on an abstract level is that even mathematics cannot provide us with absolute certainty Which system – Euclidean or Riemannian – provides the best description of the world? Both…sort of. Euclidean geometry works at the local level, but the universe appears to obey the rules of Riemannian geometry

12. Applied Mathematics The fact that many mathematical ideas that begin as purely intellectual exercises eventually end up having practical utility is a highly mysterious, even unreasonable, aspect of this area of knowledge Raises the question of whether governments and institutions should provide funding for apparently “useless” mathematical research Einstein believed that mathematical systems are invented, but that their applicability to the real world is discovered Although some mathematical rules probably were originally suggested to us by reality, many connections are more difficult to explain (e.g. Buffon’s needle problem)

13. Reference van de Lagemaat, R. (2011). Theory of Knowledge for the IB Diploma. Cambridge, UK: Cambridge University Press.