Method[s] & Tool[s] in Math[s]

Starter: Write down the names of any theories or theorems you can think of from any class.

Reveal some possible answers...

Theories: Special relativity, general relativity, Gravity, Big Bang

Theorems: Pythagorean, Binomial Theorem, Baye’s Theorem, Triangle Sum Theorem, Fundamental

Theorem of Calculus, De Moivre’s Theorem, Factor and Remainder Theorems, Hairy Ball Theorem

How can we prove the angles of a triangle add up to 180 degrees? Try to work it out and explain your method. Have you achieved proof?

How do we prove this is true for all cases? Formal proof of triangle sum theorem

3. Stand up for Euclid's postulates! Sit down when one is revealed you don't believe to be true.....

That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Using just these 5 (and some definitions), all of the geometry that we study in IB is able to be proven (Euclid himself did a huge part of this).

4. What are some problems with proof & an axiomatic system?

A Errors aren’t always obvious:

The problem with this is that when going from (a+b)(a-b)=b(a-b) we have to divide both sides by (a-b) which is zero since a=b

B Intuition/inductive reasoning can lead us astray

C Some proofs require so much computation that only computers can really verify it.

D Even if we are perfect, there is still Gödel’s Incompleteness Theorem

6. Why does it matter that some true statements are unproveable?

What does this show about our own inherent knowledge of maths?
What implications are there for the human brain not being able to effectively comprehend math and the nuance of numbers?
What does this show us about the importance of clear methods and tools in mathematics?

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