Chapter 5 Mathematics. Lesson 1 Math and Deduction What is this saying?

Chapter 5Chapter 5MathematicsMathematics

Lesson 1 Math and Lesson 1 Math and DeductionDeduction

What is this saying?

Mathematical knowledge Mathematical knowledge claims claims

• Sound and hard to argue with because Sound and hard to argue with because they are based on logical deductionthey are based on logical deduction

• Different than knowledge in some other Different than knowledge in some other areas of knowledge because there is a areas of knowledge because there is a possibility of proving claims completelypossibility of proving claims completely

• Are objective claims which anyone who Are objective claims which anyone who understands math can agree onunderstands math can agree on

Logical deductionLogical deduction

• The foundation on which The foundation on which mathematics is builtmathematics is built

• Deduction can be defined as: Deduction can be defined as: making making conclusions based on premises conclusions based on premises known to be trueknown to be true

• Mathematics proves itself through Mathematics proves itself through deduction deduction

LogicLogic

• LogicLogic is is the science of the science of correct reasoningcorrect reasoning

• ReasoningReasoning is is any any argument in which argument in which certain assumptions of certain assumptions of premises are stated, premises are stated, and then some other and then some other conclusion or fact conclusion or fact necessarily followsnecessarily follows..

• Logic sometimes Logic sometimes calledcalled the science of the science of necessary inferencenecessary inference Aristotle 384-322 B.C.

Math and metaphysicsMath and metaphysics

• The logic behind mathematics is also viewed as The logic behind mathematics is also viewed as being a member of the branch of philosophy being a member of the branch of philosophy known as known as metaphysicsmetaphysics

• Metaphysics Metaphysics is is the study and the description of the study and the description of the nature of realitythe nature of reality

• Math in the west originated as a branch of Math in the west originated as a branch of philosophy. It attempted to describe and philosophy. It attempted to describe and understand the nature of realityunderstand the nature of reality

• The assumption that reasoning is the best way at The assumption that reasoning is the best way at getting to the nature of reality is a philosophical getting to the nature of reality is a philosophical assumption which may not necessarily be true in assumption which may not necessarily be true in all or any circumstances all or any circumstances

Subjects and PredicatesSubjects and Predicates

• A A subjectsubject is considered an individual is considered an individual phenomenon or entity, such as a phenomenon or entity, such as a treetree or a or a birdbird..

• A A predicatepredicate is an attribute of the subject, is an attribute of the subject, such as the tree being such as the tree being bigbig or the bird or the bird being being greygrey..

• There are agreed upon rules in math There are agreed upon rules in math regarding the use of subjects and regarding the use of subjects and predicatespredicates

Fundamental principles Fundamental principles regarding subjects and regarding subjects and

predicatespredicates • IdentityIdentity:: Everything is what it is and acts Everything is what it is and acts

accordinglyaccordingly • Non-contradictionNon-contradiction: : It is impossible for It is impossible for

something to both be and not be. A given something to both be and not be. A given predicate cannot both belong and not predicate cannot both belong and not belong to a given subject in a given belong to a given subject in a given respect at a given time. Contradictions do respect at a given time. Contradictions do not exist.not exist.

• Either-orEither-or: : Everything must either be or Everything must either be or not be. A given predicate either belongs or not be. A given predicate either belongs or does not belong to a given subject in a does not belong to a given subject in a given respect at a given time. given respect at a given time.

SyllogismsSyllogisms

• A A syllogismsyllogism is the is the basic Aristotelian basic Aristotelian unit of reasoning.unit of reasoning.

• A an indisputable A an indisputable conclusion conclusion reached based reached based on premises on premises known to be trueknown to be true

Examples of SyllogismsExamples of Syllogisms

• Some Some AA is is BB

• All B is CAll B is C

• Therefore, some A is CTherefore, some A is C

• The conclusion is indisputableThe conclusion is indisputable


• All cats that live in Sweden have six toesAll cats that live in Sweden have six toes

• Cindy the cat lives in SwedenCindy the cat lives in Sweden

• Therefore, Cindy the cat has six toesTherefore, Cindy the cat has six toes



• Premise 1: a2 + b2 = c2 in right Premise 1: a2 + b2 = c2 in right triangles triangles

• Premise 2: a = 6, b = 8 in the right Premise 2: a = 6, b = 8 in the right triangle triangle ABCABC

• Therefore, C = 10Therefore, C = 10


Paradoxical LogicParadoxical Logic

• Math is based on Math is based on logiclogic

• But logic can But logic can prove the prove the impossibleimpossible

• What does this What does this say about math?say about math?

• Is it reflective of Is it reflective of reality?reality?

When will Jessie catch the When will Jessie catch the Zombie?Zombie?

• If Jesse is to catch up to If Jesse is to catch up to the zombie, first, Jesse the zombie, first, Jesse must pass the spot must pass the spot where the zombie where the zombie startedstarted

• This spot will be called This spot will be called point point AA. But when Jesse . But when Jesse gets to point gets to point AA, the , the Zombie will have Zombie will have moved on to point moved on to point BB

Jesse Owens. Winner of four Olympic gold medals in the 1936 Berlin Olympics

When will Jessie catch the When will Jessie catch the Zombie?Zombie?• Then to catch up, Jesse must Then to catch up, Jesse must

get to point get to point BB before he can before he can actually get to the zombie. actually get to the zombie. So, he gets to point So, he gets to point BB

• But when he gets there, the But when he gets there, the zombie has moved on to zombie has moved on to point C. Now, to catch up to point C. Now, to catch up to the zombie Jesse must first the zombie Jesse must first pass through point Cpass through point C

• When he gets to point When he gets to point CC though, the Zombie has though, the Zombie has already moved on to point already moved on to point DD, and so on and so on for , and so on and so on for all eternity.all eternity. A zombie. Winner of nothing. Loves brains

So when will Jessie catch the So when will Jessie catch the Zombie?Zombie?

•NEVER!!!!NEVER!!!!• The distance between them is getting The distance between them is getting

smaller and smaller, but the fact remains smaller and smaller, but the fact remains that the zombie will always be moving and that the zombie will always be moving and as soon as Jessie gets to any point (as soon as Jessie gets to any point (point point nn) the zombie will have moved and the ) the zombie will have moved and the zombie will be at a new point.zombie will be at a new point.

Absurdity?Absurdity?

• The paradox may be an absurdity but The paradox may be an absurdity but the logic is sound.the logic is sound.

• What does this say about math since What does this say about math since math is based on logic and logic can math is based on logic and logic can prove the impossible?prove the impossible?

Lesson 2 Lesson 2 Does Mathematics Does Mathematics Even Exist?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 Does math have anything to do with reality?with reality?

• Does math even exist?Does math even exist?

• Is it only a construct of the human Is it only a construct of the human mind?mind?

• Just because it is used does it prove Just because it is used does it prove that it is real?that it is real?

• Would math exist without human Would math exist without human culture?culture?

• Is math logical nonsense?Is math logical nonsense?

Mathematic realism or Mathematic realism or PlatonismPlatonism

• According to Plato According to Plato the ideal the ideal world was a place that was world was a place that was made up of numbers and made up of numbers and mathematical relationshipsmathematical relationships

• Mathematicians, such as Mathematicians, such as Euclid and PythagorasEuclid and Pythagoras had had discovereddiscovered how the world how the world was madewas made

• According to the According to the mathematic realistmathematic realist, math is , math is something discovered. It is something discovered. It is something that exists and something that exists and awaits discoveryawaits discovery

Is it here that math resides?

However…However…

• If there is a mathematical reality somewhere, If there is a mathematical reality somewhere, where exactly is this reality?where exactly is this reality?

• How and where do all of these mathematical How and where do all of these mathematical entities exist?entities exist?

• Is it a separate world? Is it an internal world?Is it a separate world? Is it an internal world? • Perhaps nothing is mathematically related to Perhaps nothing is mathematically related to

anything else unless we, human beings, say anything else unless we, human beings, say they are related and explain that relationship they are related and explain that relationship with the language of numbers.with the language of numbers.

Is Math an Arbitrary Game?Is Math an Arbitrary Game?

• As far as the laws of mathematics refer to reality, As far as the laws of mathematics refer to reality, they are not certain; and as far as they are they are not certain; and as far as they are certain, they do not refer to realitycertain, they do not refer to reality -- --Albert EinsteinAlbert Einstein

• If mathematics is an arbitrary creation, the If mathematics is an arbitrary creation, the mathematical concepts and rules are not mathematical concepts and rules are not something that correspond to reality in any way, something that correspond to reality in any way, but instead are simply created and given but instead are simply created and given meaning by the people that create them.meaning by the people that create them.

• Perhaps math is arbitrary exactly like language.Perhaps math is arbitrary exactly like language.

It is possible that math was It is possible that math was created to accommodate the created to accommodate the

needs of people?needs of people? • Positive numbers were needed to Positive numbers were needed to

facilitate the organization of societal facilitate the organization of societal interactioninteraction

• Much later other complex elements Much later other complex elements like zero were addedlike zero were added

• Axiomatic understandingAxiomatic understanding

changed at will. Logicchanged at will. Logic

itself can be changeditself can be changedWhy is there no “zero” bead on this Chinese abacus?

The arbitrariness continuesThe arbitrariness continues

• Other abstract components continue Other abstract components continue to be added to mathematics as to be added to mathematics as needed. These are things such as:needed. These are things such as:• Negative numbersNegative numbers• Irrational numbersIrrational numbers• Imaginary numbersImaginary numbers

Mathematics changes and Mathematics changes and evolvesevolves

• With the needs of societyWith the needs of society

• As society becomes more complex, so As society becomes more complex, so to does the demands put on to does the demands put on mathematicsmathematics

• New mathematics are developed to New mathematics are developed to deal with the needsdeal with the needs

• Does this mean math exists?Does this mean math exists?

• That’s a good question.That’s a good question.

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

FaithFaith

• Not necessarily the first basis for Not necessarily the first basis for knowledge claims in mathematics, knowledge claims in mathematics, but it still plays a rolebut it still plays a role

• An An axiomaxiom in math is a “fact” that is in math is a “fact” that is assumed to be trueassumed to be true

• Axioms lead to assumptions that are Axioms lead to assumptions that are based on faith in the truth of the based on faith in the truth of the axiomaxiom

ProofProof

• Proof means Proof means there has to be 100 there has to be 100 percent certainty that what is being percent certainty that what is being claimed is really the caseclaimed is really the case

• The certainty comes through The certainty comes through deductive reasoningdeductive reasoning

• Proof in mathematics is also based Proof in mathematics is also based upon upon consensusconsensus

AuthorityAuthority

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

Authority contd.Authority contd.

• Formulas are like Formulas are like recipes. Most people recipes. Most people can use them but do can use them but do not fully understand not fully understand why they workwhy they work

• On the consumer of On the consumer of math level, knowledge math level, knowledge comes to a great comes to a great degree from following degree from following the authority of the the authority of the math book.math book.

Pure mathematicsPure mathematics

• Pure mathematics is mathematics which is Pure mathematics is mathematics which is done purely for the sake of doing math (like done purely for the sake of doing math (like abstract algebra)abstract algebra)

• The aim is not to apply the knowledge to a The aim is not to apply the knowledge to a real world settingreal world setting

• ““Knowledge” produced in pure mathematics Knowledge” produced in pure mathematics often has little or no importance to everyday often has little or no importance to everyday life. it may be argued that no real knowledge life. it may be argued that no real knowledge is actually produced at all as long as there is is actually produced at all as long as there is no practical application to real life settings.no practical application to real life settings.

Pure mathematics contd.Pure mathematics contd.

• As far as pure As far as pure mathematicians are mathematicians are concerned, pure concerned, pure mathematics does produce mathematics does produce knowledge because it knowledge because it explores the boundaries of explores the boundaries of mathematics and pure mathematics and pure reasonreason

• It produces knowledge It produces knowledge about reasoning and how about reasoning and how mathematical structures mathematical structures functionfunction

Perhaps Rodin’s thinker is pondering the boundaries of

mathematics and pure reason.

Applied mathematicsApplied mathematics

• Applied mathematics is all about Applied mathematics is all about application to real world settings.application to real world settings.

• Used in such professions as Used in such professions as engineering, economics, engineering, economics, statistical analysis, and statistical analysis, and computer sciencecomputer science

• Applied mathematics is using Applied mathematics is using mathematical knowledge to do mathematical knowledge to do somethingsomething

• In applied mathematics, In applied mathematics, knowledge is gained at a knowledge is gained at a pragmatic levelpragmatic level

This astronaut is sure glad someone did

the math right at Houston.

Lesson 4 ParadoxesLesson 4 Paradoxes

Impossible images, like this one from M.C. Escher

are types of visual paradoxes.

What is a paradox?What is a paradox?

• A paradox is a statement or a group A paradox is a statement or a group of statements that do (or seem to) of statements that do (or seem to) lead to a contradictionlead to a contradiction

• They lead to a situation that defies They lead to a situation that defies logic and intuitionlogic and intuition

• They “prove” the impossibleThey “prove” the impossible

Why of interest for math and Why of interest for math and TOK?TOK?

• Paradoxes attack the power of Paradoxes attack the power of reasoningreasoning

• Mathematics is seen by many people Mathematics is seen by many people as a pure science and an expression of as a pure science and an expression of logic perfectedlogic perfected

• At the same time though, that same At the same time though, that same logic can prove the most absurd logic can prove the most absurd claims to be trueclaims to be true

• What does this say about reason?What does this say about reason?

Paradox one: 2=1Paradox one: 2=1

• Let x and y be equal, non-Let x and y be equal, non-zero quantitieszero quantities

• xx = = yy• Add x to both sidesAdd x to both sides • 22xx = = xx + + yy• Take 2y from both sidesTake 2y from both sides• 2x − 2y = x − y2x − 2y = x − y• FactorizeFactorize• 2(x − y) = x − y2(x − y) = x − y• Divide out (x − y)Divide out (x − y)• 2 = 12 = 1

That is an odd elephant!

Burdian’s assBurdian’s ass

• The point to be made The point to be made by this paradox is that by this paradox is that reason is not reason is not necessarily the best necessarily the best tool when considering tool when considering the choices made the choices made during lifeduring life

• Can you think of areas Can you think of areas where reason will not where reason will not provide the answers?provide the answers?

Houston, we have a problem!

Zeno’s dichotomyZeno’s dichotomy

• To get the plate of To get the plate of brains bill must get brains bill must get from from point Apoint A to to point Bpoint B

• But, before he gets to But, before he gets to point Bpoint B from from point Apoint A, , the zombie must first the zombie must first reach the halfway reach the halfway point between the two point between the two pointspoints

The zombie formerly known as Bill.

Zeno’s dichotomy contd.Zeno’s dichotomy contd.

• However, when reaching the However, when reaching the halfway point, another halfway point, another halfway point comes into halfway point comes into existence which he must existence which he must reachreach

• However, to his despair, upon However, to his despair, upon reaching the new halfway reaching the new halfway point, yet another halfway point, yet another halfway point is created, and again point is created, and again and again and again….and again and again….

• The zombie is doomed to The zombie is doomed to failure, because he always failure, because he always has to reach a halfway point has to reach a halfway point before he reaches the final before he reaches the final destination.destination.

The zombie formerly known as Bill.

The Monty Hall ParadoxThe Monty Hall Paradox

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


• Behind one of these doors is a fancy red Behind one of these doors is a fancy red sports car.sports car.

• Behind the other two is a goatBehind the other two is a goat • Pick the door you think the car is behind.Pick the door you think the car is behind.


• For the sake of the example let’s say you pick For the sake of the example let’s say you pick door number twodoor number two

• The game show host is just about to open door The game show host is just about to open door number two, but then he presents you with an number two, but then he presents you with an offer. offer.

• He says, I’ll open one of the doors (which he does He says, I’ll open one of the doors (which he does and a goat is revealed)and a goat is revealed)

• He then makes you the offer that you may switch He then makes you the offer that you may switch doors from the door you picked to the other doordoors from the door you picked to the other door

• If you switch doors, are your chances of If you switch doors, are your chances of winning increased?winning increased?


• Obviously the car is not behind door number threeObviously the car is not behind door number three• Before door number two is opened you may switch Before door number two is opened you may switch

from door number two to door number onefrom door number two to door number one• What do you choose? Will your chance of winning What do you choose? Will your chance of winning

increase?increase?

Believe it or not your chances of Believe it or not your chances of winning are increased by winning are increased by

switching doors. Here is why:switching doors. Here is why:• In the first pick you choose goat number 1. In the first pick you choose goat number 1.

The game host picks the other goat. The game host picks the other goat. Switching will Switching will winwin the car the car

• In the first pick you choose goat number 2. In the first pick you choose goat number 2. The game host picks the other goat. The game host picks the other goat. Switching will Switching will winwin the car the car

• In the first pick you choose the car. The In the first pick you choose the car. The game host picks either of the two goats. game host picks either of the two goats. Switching will Switching will loselose

• So, by switching, your chances of winning So, by switching, your chances of winning actually increase from 1/3 to 2/3actually increase from 1/3 to 2/3

Omnipotence paradoxOmnipotence paradox

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

Is this the big stone?

The reasoning runs this The reasoning runs this way:way:

• Either this omnipotent god can create a stone it Either this omnipotent god can create a stone it cannot lift or it cannot create a stone which it cannot lift or it cannot create a stone which it cannot liftcannot lift

• If this god can 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 then there is one thing it cannot do; namely lift the stone it just createdstone it just created

• If this god cannot create a stone which it cannot lift, If this god cannot create a stone which it cannot lift, then there is one thing it cannot do; namely create then there is one thing it cannot do; namely create such a large stonesuch a large stone

• Therefore there is at least one task this god cannot Therefore there is at least one task this god cannot performperform

• Omnipotent means that this being can do anythingOmnipotent means that this being can do anything • Subsequently, this god is not omnipotentSubsequently, this god is not omnipotent

Concluding thoughts about Concluding thoughts about logic and reasonlogic and reason

• Logic and reason is not always consistent and is not Logic and reason is not always consistent and is not always the answer to the nature of realityalways the answer to the nature of reality

• Reasoning and rationale can play tricks on us and Reasoning and rationale can play tricks on us and make us believe in the impossible since the make us believe in the impossible since the impossible sometimes seems so rationally likelyimpossible sometimes seems so rationally likely

• Mathematics is bound by the rules of logic. As has Mathematics is bound by the rules of logic. As has been shown through several examples, the rules of been shown through several examples, the rules of logic are not always the best way to get at the logic are not always the best way to get at the nature of realitynature of reality

• Therefore math can not always be the best way to Therefore math can not always be the best way to understand the nature of realityunderstand the nature of reality

• There are many sides to every issueThere are many sides to every issue and and knowledge knowledge is not as straightforward as it seems at first glanceis not as straightforward as it seems at first glance

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Chapter 5 Mathematics. Lesson 1 Math and Deduction What is this saying?