Chapter 5 Mathematics

Lesson 1 Math and Deduction What is this saying?

Mathematical knowledge claims Sound and hard to argue with because they are based on logical deduction Different than knowledge in some other areas of knowledge because there is a possibility of proving claims completely Are objective claims which anyone who understands math can agree on

Logical deduction The foundation on which mathematics is built Deduction can be defined as: making conclusions based on premises known to be true Mathematics proves itself through deduction

Logic Logic is the science of correct reasoning Reasoning is any argument in which certain assumptions of premises are stated, and then some other conclusion or fact necessarily follows. Logic sometimes called the science of necessary inference Aristotle 384-322 B.C.

Math and metaphysics The logic behind mathematics is also viewed as being a member of the branch of philosophy known as metaphysics Metaphysics is the study and the description of the nature of reality Math in the west originated as a branch of philosophy. It attempted to describe and understand the nature of reality The assumption that reasoning is the best way at getting to the nature of reality is a philosophical assumption which may not necessarily be true in all or any circumstances

Subjects and Predicates A subject is considered an individual phenomenon or entity, such as a tree or a bird. A predicate is an attribute of the subject, such as the tree being big or the bird being grey. There are agreed upon rules in math regarding the use of subjects and predicates

Fundamental principles regarding subjects and predicates Identity: Everything is what it is and acts accordingly Non-contradiction: It is impossible for something to both be and not be. A given predicate cannot both belong and not belong to a given subject in a given respect at a given time. Contradictions do not exist. Either-or: Everything must either be or not be. A given predicate either belongs or does not belong to a given subject in a given respect at a given time.

Syllogisms A syllogism is the basic Aristotelian unit of reasoning. A an indisputable conclusion reached based on premises known to be true

Examples of Syllogisms Some A is B All B is C Therefore, some A is C The conclusion is indisputable

Examples of Syllogisms All cats that live in Sweden have six toes Cindy the cat lives in Sweden Therefore, Cindy the cat has six toes The conclusion is indisputable

Examples of Syllogisms Premise 1: a2 + b2 = c2 in right triangles Premise 2: a = 6, b = 8 in the right triangle ABC Therefore, C = 10 The conclusion is indisputable

Paradoxical Logic Math is based on logic But logic can prove the impossible What does this say about math? Is it reflective of reality?

When will Jessie catch the Zombie? If Jesse is to catch up to the zombie, first, Jesse must pass the spot where the zombie started This spot will be called point A. But when Jesse gets to point A, the Zombie will have moved on to point B Jesse Owens. Winner of four Olympic gold medals in the 1936 Berlin Olympics

When will Jessie catch the Zombie? Then to catch up, Jesse must get to point B before he can actually get to the zombie. So, he gets to point B But when he gets there, the zombie has moved on to point C. Now, to catch up to the zombie Jesse must first pass through point C When he gets to point C though, the Zombie has already moved on to point D, and so on and so on for all eternity. A zombie. Winner of nothing. Loves brains

So when will Jessie catch the Zombie? NEVER!!!! The distance between them is getting smaller and smaller, but the fact remains that the zombie will always be moving and as soon as Jessie gets to any point (point n) the zombie will have moved and the zombie will be at a new point.

Absurdity? The paradox may be an absurdity but the logic is sound. What does this say about math since math is based on logic and logic can prove the impossible?

Lesson 2 Does Mathematics Even Exist? Mathematics is a game played according to certain simple rules with meaningless marks on paper. --David Hilbert

Does math have anything to do with reality? Does math even exist? Is it only a construct of the human mind? Just because it is used does it prove that it is real? Would math exist without human culture? Is math logical nonsense?

Mathematic realism or Platonism According to Plato the ideal world was a place that was made up of numbers and mathematical relationships Mathematicians, such as Euclid and Pythagoras had discovered how the world was made According to the mathematic realist, math is something discovered. It is something that exists and awaits discovery Is it here that math resides?

However… If there is a mathematical reality somewhere, where exactly is this reality? How and where do all of these mathematical entities exist? Is it a separate world? Is it an internal world? Perhaps nothing is mathematically related to anything else unless we, human beings, say they are related and explain that relationship with the language of numbers.

Is Math an Arbitrary Game? As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality --Albert Einstein If mathematics is an arbitrary creation, the mathematical concepts and rules are not something that correspond to reality in any way, but instead are simply created and given meaning by the people that create them. Perhaps math is arbitrary exactly like language.

It is possible that math was created to accommodate the needs of people? Positive numbers were needed to facilitate the organization of societal interaction Much later other complex elements like zero were added Axiomatic understanding changed at will. Logic itself can be changed Why is there no “zero” bead on this Chinese abacus?

The arbitrariness continues Other abstract components continue to be added to mathematics as needed. These are things such as: Negative numbers Irrational numbers Imaginary numbers

Mathematics changes and evolves With the needs of society As society becomes more complex, so to does the demands put on mathematics New mathematics are developed to deal with the needs Does this mean math exists? That’s a good question.

Lesson 3 How Do We Know What We Know in Math?

Faith Not necessarily the first basis for knowledge claims in mathematics, but it still plays a role An axiom in math is a “fact” that is assumed to be true Axioms lead to assumptions that are based on faith in the truth of the axiom

Proof Proof means there has to be 100 percent certainty that what is being claimed is really the case The certainty comes through deductive reasoning Proof in mathematics is also based upon consensus

Authority The complexity of much mathematical thought means the “consumers” of mathematics can only trust in the authority of the mathematical experts

Authority contd. Formulas are like recipes. Most people can use them but do not fully understand why they work On the consumer of math level, knowledge comes to a great degree from following the authority of the math book.

Pure mathematics Pure mathematics is mathematics which is done purely for the sake of doing math (like abstract algebra) The aim is not to apply the knowledge to a real world setting “Knowledge” produced in pure mathematics often has little or no importance to everyday life. it may be argued that no real knowledge is actually produced at all as long as there is no practical application to real life settings.

Pure mathematics contd. As far as pure mathematicians are concerned, pure mathematics does produce knowledge because it explores the boundaries of mathematics and pure reason It produces knowledge about reasoning and how mathematical structures function Perhaps Rodin’s thinker is pondering the boundaries of mathematics and pure reason.

Applied mathematics Applied mathematics is all about application to real world settings. Used in such professions as engineering, economics, statistical analysis, and computer science Applied mathematics is using mathematical knowledge to do something In applied mathematics, knowledge is gained at a pragmatic level This astronaut is sure glad someone did the math right at Houston.

Lesson 4 Paradoxes Impossible images, like this one from M.C. Escher are types of visual paradoxes.

What is a paradox? A paradox is a statement or a group of statements that do (or seem to) lead to a contradiction They lead to a situation that defies logic and intuition They “prove” the impossible

Why of interest for math and TOK? Paradoxes attack the power of reasoning Mathematics is seen by many people as a pure science and an expression of logic perfected At the same time though, that same logic can prove the most absurd claims to be true What does this say about reason?

Paradox one: 2=1 Let x and y be equal, non-zero quantities x = y Add x to both sides 2x = x + y Take 2y from both sides 2x − 2y = x − y Factorize 2(x − y) = x − y Divide out (x − y) 2 = 1 That is an odd elephant!

Burdian’s ass The point to be made by this paradox is that reason is not necessarily the best tool when considering the choices made during life Can you think of areas where reason will not provide the answers? Houston, we have a problem!

Zeno’s dichotomy To get the plate of brains bill must get from point A to point B But, before he gets to point B from point A, the zombie must first reach the halfway point between the two points The zombie formerly known as Bill.

Zeno’s dichotomy contd. However, when reaching the halfway point, another halfway point comes into existence which he must reach However, to his despair, upon reaching the new halfway point, yet another halfway point is created, and again and again and again…. The zombie is doomed to failure, because he always has to reach a halfway point before he reaches the final destination. The zombie formerly known as Bill.

The Monty Hall Paradox Here is how it works: Behind one of the three doors below is a fancy red sports car, behind the other two are goats. You get whatever it is behind the door. If you pick the door with the sports car, you drive away in luxury. If you pick the door with the goat, well, you will probably need some hay. So make your pick.

The Monty Hall Paradox Behind one of these doors is a fancy red sports car. Behind the other two is a goat Pick the door you think the car is behind.

The Monty Hall Paradox For the sake of the example let’s say you pick door number two The game show host is just about to open door number two, but then he presents you with an offer. He says, I’ll open one of the doors (which he does and a goat is revealed) He then makes you the offer that you may switch doors from the door you picked to the other door If you switch doors, are your chances of winning increased?

The Monty Hall Paradox Obviously the car is not behind door number three Before door number two is opened you may switch from door number two to door number one What do you choose? Will your chance of winning increase?

Believe it or not your chances of winning are increased by switching doors. Here is why: In the first pick you choose goat number 1. The game host picks the other goat. Switching will win the car In the first pick you choose goat number 2. The game host picks the other goat. Switching will win the car In the first pick you choose the car. The game host picks either of the two goats. Switching will lose So, by switching, your chances of winning actually increase from 1/3 to 2/3

Omnipotence paradox If a god is truly all powerful, then it should be able to do anything. Well then, can this god create a stone so big that he/she/it can not move it? Is this the big stone?

The reasoning runs this way: Either this omnipotent god can create a stone it cannot lift or it cannot create a stone which it cannot lift If this god can create a stone which it cannot lift, then there is one thing it cannot do; namely lift the stone it just created If this god cannot create a stone which it cannot lift, then there is one thing it cannot do; namely create such a large stone Therefore there is at least one task this god cannot perform Omnipotent means that this being can do anything Subsequently, this god is not omnipotent

Concluding thoughts about logic and reason Logic and reason is not always consistent and is not always the answer to the nature of reality Reasoning and rationale can play tricks on us and make us believe in the impossible since the impossible sometimes seems so rationally likely Mathematics is bound by the rules of logic. As has been shown through several examples, the rules of logic are not always the best way to get at the nature of reality Therefore math can not always be the best way to understand the nature of reality There are many sides to every issue and knowledge is not as straightforward as it seems at first glance