1. TOK Mathematics Lesson 2: How do we justify mathematics? ? © Roy White, International College Hong Kong 2013

2. Sum of the interior angles of a triangle is 180° How can we justify this claim? Two Choices: we test a ‘number’ of the triangles and assume the rule follows we try to give a reason why it must be true without testing Two Choices: we test a ‘number’ of the triangles and assume the rule follows we try to give a reason why it must be true without testing

3. These methods are: Induction: We see something occur repeatedly and generalize. Eg. Last 3 times I had TOK it rained. Today I have TOK so it will rain. Eg. Last 3 times I had TOK it rained. Today I have TOK so it will rain. Deduction: From general principles we conclude that something will work in a specific case or we create another general principle. E.g. All teachers have a degree. Ms…. Is a teacher, so she has a degree. E.g. All teachers have a degree. Ms…. Is a teacher, so she has a degree.

4. Mathematical Proof A mathematical proof: a deductive argument in which the validity of each step and the connections between the steps is explicitly made clear in such a way that the result follows with certainty. validity DO you see any difference between induction and deduction? How do we create the general principles in the first place?

5. Even + Even = Even Induction: 2 + 4 = 6, 10 + 24 = 34, 12 + 6 = 18 …. 2 + 4 = 6, 10 + 24 = 34, 12 + 6 = 18 …. Therefore, even + even = even Therefore, even + even = evenDeduction: A even number can be written as 2 x another integer, i.e. even = 2k for some integer k. A even number can be written as 2 x another integer, i.e. even = 2k for some integer k. Proof : Let e 1, e 2 be any two even numbers. Proof : Let e 1, e 2 be any two even numbers. Now, e 1 + e 2 = 2k 1 + 2k 2 = 2(k 1 + k 2 ) = 2k, which is even. Now, e 1 + e 2 = 2k 1 + 2k 2 = 2(k 1 + k 2 ) = 2k, which is even.

6. Problem with Induction TOK Mathematics Lesson 2: Reasoning 6 P = 0, R =1 P = 2, R =2 P = 3, R =4 P = 4, R =8 P = 5, R =16

7. Problem with Induction TOK Mathematics Lesson 2: Reasoning 7 P = n 2 + n + 11 This formula generates a prime number Problems with testing: a political example. Preceding the latest Iraq war, the UN required the Iraq Government to allow weapons inspectors into the country. The UN demanded that the Iraq government prove for certain, via the inspectors, that weapons of mass destruction do not exist. Why would it be impossible for the weapons inspectors to satisfy the requirements of the UN?

8. Deduction: A Closer Look Deduction: A even number can be written as 2 x another integer, i.e. even = 2k for some integer k. A even number can be written as 2 x another integer, i.e. even = 2k for some integer k. Proof : Let e 1, e 2 be any two even numbers. Proof : Let e 1, e 2 be any two even numbers. Now, e 1 + e 2 = 2k 1 + 2k 2 = 2(k 1 + k 2 ) = 2k, which is even. Now, e 1 + e 2 = 2k 1 + 2k 2 = 2(k 1 + k 2 ) = 2k, which is even. How do we know this is true? Hey! We used a definition but we discussed before problems with these!

9. The Building Blocks Why can a perfect dictionary not exist?Why can a perfect dictionary not exist? Why is it impossible to deduce all mathematical claims using rigourous proof?Why is it impossible to deduce all mathematical claims using rigourous proof?

10. How do we justify mathematics? Axioms, Definitions, Theorems Theorems

11. How do we justify mathematics ? Induction: Deduction: Pros: Cons: Pros: Cons :

12. Our starting point..Axioms Can we have a different set of axioms? How can we be sure the mathematics is true? How did the Greeks come up with their axioms? Yes. We then simply generate a different body of knowledge. Mathematicians aim to create mathematic, rather than be sure it reflects real life. The Greeks used axioms which they believed represent reality…. But you could also start with axioms which do not represent reality. Who knows! But probably began with using induction as we generally do in mathematics.

13. POINTS to PONDER What subjects (Aok) use induction ? Deduction? Is there any difference between the method of justification used in the Human sciences and the Natural science? What about methods in Natural Science & History? Are there other types of reasoning? What methods of justification are used in the arts to support a knowledge claim?

14. POINTS to PONDER What other means are used to sway or justify a claim? What method is used in a court room? What method is being used by a drug company to convince you that a drug is effective?

15. POINTS to PONDER Examine the summary of this radio program in which the Head of The Cheshire home for the elderly tries to convince listeners of the need to increase public funding for his home.summary TOK Mathematics Lesson 2: Reasoning 15 What methods does he use to convince listeners Which methods have an effect on you? Induction? Deduction? Which parts of the argument are NOT reasonable?

16. Student Reflections TOK Mathematics Lesson 2: Reasoning 16 What is the difference between induction and deduction? What are the pros and cons of both? If inductive conclusions are not certain, why is experimental science founded on induction rather than deduction? Besides the method of reasoning in an argument, what other factors may sway your view? Should these factors sway you? Is there any difference in the inductive methods used in experimental science vs history or human sciences?

17. IB Questions TOK Mathematics Lesson 2: Reasoning 17 How can knowing a friend be the same as knowing a scientific theory? How is knowing a friend different from knowing a scientific theory? How is knowing a historical period similar to and different from knowing a friend? How is knowing a friend similar and different from knowing how to swim?