Mathematics in ToK

1. Mathematicsin ToK Area of Knowledge 1: How do we apply language, emotion, sensory perception, and reason/logic to gain knowledge through Mathematics?

2. THIS IS NOT MATH CLASS!! …but rather, a unit designed to look at how we arrive at knowledge, truth, and wisdom through the use of mathematics. How we use math to make sense of the world, and how far we can trust its certainty.

3. What words come to mind when you think about the term “mathematics”?

4. Where do you see mathematics in everyday life?

5. May 3, 2013 • Blog!! • EE to do—you should spend some time researching this weekend. Also, update your research question on Managebac! • Need help with research direction, citation, etc? Come to room 328 today at lunch! • Do we need to update our calendar?

6. Galileo proposed that we find mathematics everywhere in nature. Think of some examples.

7. Fibonacci series The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it.

8. Perception: The Golden ratio…is it perfection? This mask of the human face is based on the Golden Ratio. The proportions of the length of the nose, the position of the eyes and the length of the chin, all conform to some aspect of the Golden Ratio. Remember WHY we are attracted to people…procreation! Does symmetry correlate to a healthy mate? http://www.intmath.com/numbers/math-of-beauty.php

9. Where do you benefit from mathematics?

10. Math Riddle Time! How can you add eight 8's to get the number 1,000? (only use addition)

11. Emotion Think of the emotion experiments you did. Where did math come into play in those?

12. Math and magic… The Magic Gopher

13. Some Definitions of Math • The science of rigorous proof. • The study of patterns and relationships between numbers and shapes. • Patterns amidst the chaos. • Right and wrong answers. • The most accurate reflection of reality. • The use of numbers an symbols as metaphors for understanding even the incomprehensibly abstract. • What is a metaphor?

14. Why are some people drawn to mathematics as an area of knowing? Conversely, why are some people averse to mathematics? Similarly, do you believe that being ‘good’ at mathematics is inborn, intuitive, or learned?

15. Why do we study mathematics? What’s the point? Is, as it says in the gold-packet, mathematics a retroactive science? (meaning, the math hasn’t yet been discovered for some of the problems we attempt to solve) (p. 148)

16. Some possible answers…from real people like you • In math, there is a right and a wrong. • Other subjects are too relative. • Math requires you to ignore context and operate on a purely abstract level. • Math is useful. • You can use numbers and statistics in an argument or as evidence and it’s more convincing. • Failure and success are equally important teachers. • Numbers are merely symbols that can stand for anything; they are not things or people. They are not concrete.

17. Imperialism in mathematics Imperialism: “my way is better than your way.” If you can’t express something in mathematical symbols then it has no intellectual value. Agree or disagree?

18. Do you believe that applied mathematics came first or that pure mathematics came first? Why?

19. Math Riddle Time! How many two cent stamps are there in a dozen?

20. Euclid • Father of geometry (on a plane). • “The pursuit of knowledge is an end in itself.” Euclidian geometry, until the 19th century, was looked at as a model for ALL knowledge. Euclid used in Lincoln’s argument that all men are created equal. “Things which are equal to the same thing are equal to each other.”

21. Euclid’s formal reasoning was in syllogism (Begin with) AXIOMS  Premise (Use) DEDUCTIVE REASONING Premise (Arrive at) THEOREMS  Conclusion Why not use inductive reasoning?

22. What’s an axiom? • Basic assumption, self-evident truths, used to create firm foundations of understanding on which to build new ideas. (19th century) • Required to be: consistent, independent, simple, and fruitful. (Review pg. 190 in text for explanations) • Current: axioms are not ‘self-evident’ truths, but assumptions premises, definitions, or givens at the base of a mathematical system.

23. Euclid’s Axioms • It shall be possible to draw a straight line joining any two points. • A finite straight line may be extended without limit in either direction. • It shall be possible to draw a circle with a given center and through a given point. • All right angles are equal to one another. • There is just one straight line through a given point which is parallel to a given line.

24. And theorems… • Lines perpendicular to the same line re parallel. • Two straight lines do not enclose an area. • The sum of the angles of a triangle is 180 degrees. • The angles on a straight line sum to 180 degrees.

25. To what extent do you think the geometric paradigm can be applied to other areas of knowledge? What are the strengths and limitations of applying this type of knowledge to other areas?

26. Non-Euclidian Geometry (19th century onward) • Riemannian geometry: what if the surface on which we work is a curve, not a plane? • The reverse of Euclid’s axioms cannot be disproved based on the curvature of space. • Einstein used Riemannian geometry. • Math game break.

27. Correspondence and Coherence in Mathematics Correspondence: accurately explains what exists. Coherence: axioms used as foundations are logically consistent. Then, one can manipulate ideas in a process of ‘pure’ thinking, creating new knowledge. (?) How do correspondence and coherence lead to consistency in mathematics?

28. 5/8/10 Homework… • Blog is due Saturday • Bring a statistic to class that you read/saw; take note of the source and purpose. Write it down. • N. Science—not due until next Monday. • Book: 221-255—Notes • Gold packet • Vocab • 1 KI • 1 Quote

29. I will be collecting comp books again the week before finals. You will have them with you over the summer. • Start looking at the “Recommended Further Reading” list on the blog. You will need to pick one book and read it over the summer. Keep an informal reading log in your comp book. At the beginning of next semester you will be responsible for sharing with the class what you read and learned. • Final Presentation assignment coming soon!

30. Invented or Discovered? Platonists: believe math is discovered; it exists in a realm we cannot fully comprehend. Plato’s arguments: 1. Mathematics is more certain than perception 2. mathematics is timelessly true (you CAN step in the same river twice) *What does this remind you of from our unit on perception? Scientific realism? Phenomenalism?

31. Criticism • Too much mysticism when dealing with an infinite number or mathematical possibilities. If mathematical objects have an idealized existence, how can we (as physical beings) comprehend that they even exist, let alone allow them to make sense?

32. Formalist • Math is invented by man to help us make sense of reality. • Math consists of man-made definitions, axioms, and theorems. • The “perfect” circle and “perfect” line (by definition) do not exist. It is the idea of these things that we use in mathematics.

33. Euclid: formal systems are suggested to us by reality in response to practical problems, then turn out to be a useful way of describing reality. Einstein: Mathematical systems are invented, but it is a matter of discovery which of the various systems apply to reality. You can invent any formal system and prove theorems from axioms with complete certainty; however, once you ask which system applies to the world, you are faced with an empirical question which can only be answered on the basis of observation. Thus, the Riemannian geometry is a better descriptor of physical space.

34. Proofs and conjectures • In a proof, a theorem is shown to follow logically from axioms. • A conjecture is a hypothesis that may not necessarily be true.

35. Goldbach’s Great Unproven Conjecture Every even number is the sum of two primes.

36. You can test something 1,000,000 times but it is still a relatively small ratio of tested to non-tested when taking infinity into account. • How far do you have to go before you can say something is proven? • When does a conjecture become an axiom?

37. Is Descartes’ statement, “I think therefore I am,” a theorem, an axiom, or neither?

38. Creativity, Intuition, Beauty, Elegance How are these words associated with mathematics? When is intuition helpful and harmful in mathematics?

40. Math or Art words? Symmetry Proportion Sequence Frequency Medium

41. The universal language Are mathematical concepts something that extend beyond the way human begins make sense of the world? Film: Contact (1997) Hollywood and Mathematics…emotion/passion Film/Play: Proof Film: Good Will Hunting

42. But remember… Mathematics is somewhat reliant on being explained in a non-mathematical language and classification systems. To test how different it can be, take a moment and briefly jot down the definition of “to add”.

43. Math Riddle Time! As I was going to St. Ives I met a man with seven wives. Each wife had seven sacks, Each sack had seven cats, Each cat had seven kits; Kits, cats, sacks and wives, How many were going to St. Ives?

44. Intuition • Should math that we trust intuitively be put to the test? • Is any formal system free from contradiction? • What are the local, global, and universal implications of math we intuitively trust?

45. Mathematics and Certainty Analytic propositions: true by definition Synthetic Propositions: every propositions that is not analytic. SO: All propositions are either analytic or synthetic.

46. A priori: a proposition that is true independent of experience. A posteriori: cannot be known independent of experience. SO: all true propositions can be known a priori or pa posteriori.

47. Review: pg. 197--201 in text • Mathematics as empirical? • Mathematics as analytic? • Mathematics as synthetic a priori? • Re-read these pages and discuss.

48. 3 Philosophies on Math and certainty • Math truths are empirical generalizations based on a number of experiences. • Math is analytic: true by definition. • Math gives us knowledge independent of experience. Which do you agree with the most?

49. Then there was Godel Who believed that any system of logic is, by its nature, incomplete. Godel’sincompleteness theorem: (1931) It is impossible to prove that a formal mathematical system is free from contradiction. Godel did not prove that maths contains contradictions, but that we cannot be certain that it doesn’t. It is always possible that one day we will find a contradiction; and one small contradiction in a formal system would destroy the system. What does he mean by this? How does this apply to ideas in reason/logic?

50. Statistics Revisited Rhetorical device Right and wrong Upon reading a statistic…keep in mind…