CHAPTER 1: INTRODUCTION MISCONCEPTIONS IN MATHEMATICS 1. Math requires special or inherent intellectual abilities. 2. Math is gender dependent, ethnic based, hereditary. 3. Math in modern issues is too complex for the average person to understand. 4. It is difficult and dull. 5. Math makes you less sensitive to the romantic and aesthetic aspects of life. 6. Math makes no allowance for creativity. 7. Math provides exact answers. 8. Math is irrelevant to my life. MATHEMATICS (definition) The word mathematics is derived from the Greek word mathematikos which means “inclined to learn” (thus literally, to be mathematical is to be curious, open-minded, and interested in always learning more) 3 DIFFERENT WAYS TO LOOK AT MATHEMATICS 1. Mathematics as the sum of its branches 2. Mathematics as a way to model the world 3. Mathematics as a language MATHEMATICS AS THE SUM OF ITS BRANCHES 1. logic- study of the principles of reasoning 2. arithmetic- methods for operating on numbers 3. algebra- methods for working with unknown quantities 4. geometry-study of size and shape 5. trigonometry-study of triangles and their uses 6. probability- study of chance 7. statistics - methods for analyzing data 8. calculus - study of quantities that change MATHEMATICS AS A WAY TO MODEL THE WORLD ● Mathematics can be viewed as a tool for creating models, representations that allow us to study real phenomena. ● Models allow us to gain insight into otherwise intractable problems; and point to areas where further research is needed. ● Mathematical models take the form of tables, graphs, equations, etc. There are numerous techniques for building mathematical models and use them to study meaningful problems. ● However, models are not the “real thing”, they are only as good as the equations and observations from which they are made. MATHEMATICS AS A LANGUAGE 1 ● ● ● It is the language of nature because it is very useful for modeling the natural world. It has its own grammar and vocabulary. You must be “fluent” in this language. Quantitative literacy is the level of mathematical fluency required for success in today’s world. POPULAR QUOTES John Kerneny The man ignorant of Mathematics will be increasingly limited in his grasp of the main forces of civilization. Roger Bacon Mathematics is the door and key to the sciences. Edward Gibbon, Decline and Fall of the Roman Empire Mathematics is distinguished to have a particular privilege, that is, in the course of ages, they may always advance and can never recede. *Quantitative literacy is fundamental to nearly every discipline of study and to different issues in society that an individual faces. INTERDISCIPLINARY THINKING ● Important issues, whether personal or societal, are interdisciplinary in nature. They can best be understood when examined from various perspectives. ● Issues are better studied in an approach that recognizes how the various branches of human knowledge are interconnected. ● The danger of compartmentalized education is the lack of perspective and the inability to see the ‘big picture’. QUANTITATIVE LITERACY ● Quantitative literacy is literacy in terms of information involving mathematical ideas or numbers. It is the ability to interpret and reason with quantitative information. ● Quantitative reasoning is the process of interpreting and reasoning with quantitative information. INNUMERACY ● It is the lack of quantitative literacy. ● Since quantitative literacy is a survival skill, the lack of it can lead to financial trouble and personal problems. ● Innumeracy leads to a misunderstanding of logic, probability and statistics, thus to an inability to distinguish between legitimate science and fraudulent science ADVANTAGES OF QUANTITATIVE LITERACY 1. It enriches the appreciation of both ancient and modern culture. 2 2. It helps you appreciate literature. 3. It helps you appreciate the substantial contribution to mathematics, science and technology of achievers in art, literature or politics. 4. It provides greater chances for employment. FOUR LEVELS OF QUANTITATIVE LITERACY 1. Theoretical Mathematics ● discovery of entirely new mathematical principles [mathematicians, theoretical scientists] 2. Applied Mathematics ● known mathematical tools are applied to problems of immediate interest such as analyzing risk in insurance policies, developing mathematical models to assess human impact on the environment, or teaching mathematics [engineers, scientists, teachers, statisticians, business analysts] ● Applied mathematics can be viewed as a central resource for addressing and solving problems in a wide and growing variety of disciplines such as business management, economics, engineering, biology/ecology, computer science/artificial intelligence, physics, chemistry, medicine and physiology, and even in psychology and sociology 3. Vocational Mathematics ● the use of mathematical tools routinely such as computer programming, and accounting and banking wherein mathematical methods are used to analyze financial records and investment strategies ● does not involve discovery of new principles or application of principles in new ways [computer programmers, accountants, statisticians] 4. Quantitative Literacy ● necessary for everyone ● survival skill in today’s technological society HOW TO ATTAIN QUANTITATIVE LITERACY 1. Identify your personal goals and strategies Goal - an end toward which effort is directed Strategy - a plan or method for achieving a goal 2. Break down your psychological barriers 3. Set your course CHAPTER 2: NATURE OF MATHEMATICS 3 CHAPTER 2.1: INTRODUCTION TO LOGIC AND REASONING LOGIC - the study of reasoning QUANTITATIVE REASONING - the ability to reason with quantitative or mathematical information. WHAT IS LOGIC ● Logic is the study of the methods and principles used to distinguish correct reasoning from wrong reasoning. Copi ● Logic deals with arguments and inferences. It provides methods for distinguishing those which are logically correct from those which are not. Salmon ● Logic is concerned with the question of adequacy or probative value of different kinds of evidence. Cohen ● Logic is both science and art; it is concerned with the quest of knowledge and truth; it is also the study of the validity or correctness of our reasoning. Mourant ● The main business of logic is the systematic evaluation of arguments for internal cogency. Smith WHAT LOGIC IS NOT ● It is not a science of thought because psychology also is and they are two different fields of study. ● Thought refers to any process that occurs in the mind and not all thought is an object for a logician. ● All reasoning is thinking but not all thinking is reasoning. ● Reasoning is a special kind of thinking in which inference takes place, where conclusions are drawn from premises. INFORMAL LOGIC ● Study of natural language arguments. ● Its branches include the study of fallacies, critical thinking and argumentation theory. ● There is no method of establishing the invalidity of an argument since there is no underlying theory. FORMAL LOGIC ● Study of the inference with purely formal content so that the inference can be expressed as a particular application of a given abstract rule. ● Lacks reference to meaning or content and simply evaluates the correctness of the form (or structure) of the argument. *insert logical arguments slides here CHAPTER 2.2: PROPOSITIONAL AND PREDICATE LOGIC PROPOSITIONS 4 ● ● ● ● ● ● A proposition (or statement) is a declarative sentence that makes a distinct claim, such as an assertion or denial; it proposes something to be true or false. A proposition is that which is expressed by certain sentences in certain context and of which it is proper to say that it is true or false. Examples: The Earth revolves around the sun. I am hungry. In an argument, both premise and conclusion are propositions. Each makes a distinct claim that, depending on your viewpoint, is either true or false. A proposition must have a subject and predicate. A proposition must be capable of being true or false, but not both at the same time, though we may not know which it is. PROPOSITIONAL AND PREDICATE LOGIC PREDICATE LOGIC is the part of logic that deals with the inner structure of the propositions, that is, it analyzes the subject-predicate structures of a proposition. PROPOSITIONAL LOGIC deals with connections between propositions only. It does not break propositions into smaller constituents. PREDICATE LOGIC *TYPES OF PROPOSITIONS 1. Categorical Proposition - a proposition that expresses a relationship between two categories or sets, the subject set S and predicate set P. The Four Standard Categorical Propositions 1. All S are P 2. No S are P 3. Some S are P 4. Some S are not P *A proposition always makes a claim of truth. This claim however is not necessarily true. Although a proposition is capable of being either true or false, determining which it is may not be possible. *CLAIMS OF TRUTH A proposition maybe 1. unambiguous - no one can reasonably disagree with its truth or falsity 2. unverifiable - would require impossible or impractical procedures to determine its truth or falsity 5 3. matter of opinion - truth can be argued endlessly PROPOSITIONAL LOGIC cont *TYPES OF PROPOSITIONS 2. Compound Proposition - a proposition that consists of two or more simple (or prime) propositions joined together by logical connectors. not (negation) and (conjunction) or (disjunction) if – then (conditional) if and only if (biconditional) Conjunction - Let p and q be propositions. If both p and q are true, then the compound proposition p and q (denoted as p q) is true. Otherwise p q is false. Disjunction - Let p and q be propositions. If at least one of p or q is true, then the compound proposition p or q (denoted as p q) is true. Otherwise, p q is false. Conditional Proposition - a proposition joined by the words if and then. - p is called the antecedent, while q is called the consequent - The conditional proposition if p then q (denoted as p q) is true unless p is true and q is false. - Non-sequitur - logic is the art of going wrong with confidence Biconditional Proposition - The proposition p if and only if q (denoted p q) is true if p and q have the same truth values. It is false if p and q have opposite truth values. LOGICAL EQUIVALENCE Two compound propositions are logically equivalent (denoted ≡) if they have the same truth tables. RULES OF REPLACEMENT 6 1. De Morgan’s Theorems i.) ~(p˄q) ≡ ~p˅~q ii.) ~(p˅q) ≡ ~p˄~q 2. Commutation i.) p˄q ≡ q˄p ii.) p˅q ≡ q˅p 3. Association i.) (p˄q)˄r ≡ p˄(q˄r) ii.) (p˅q)˅r ≡ p˅(q˅r) 4. Distribution i.) p˄(q˅r) ≡ (p˄q)˅(p˄r) ii.) p˅(q˄r) ≡ (p˅q)˄(p˅r) 5. Double Negation p ≡ ~~p 6. Transposition or Contraposition p → q ≡ ~q → ~p 7. Material Implication p → q ≡ ~p ˅q 8. Exportation p → (q → r) ≡ (p˄q)→ r 9. Idempotence i.) p ≡ p˅p ii.) p ≡ p˄p 7 DEDUCTIVE ARGUMENTS Goal: to investigate how arguments actually proceed from premises to conclusions. This process is called inference (we infer the conclusion from the premises). 2 BASIC TYPES OF INFERENTIAL PROCESSES 1. Deductive Inference - a specific conclusion is deduced (or logically derived) from general premises 2. Inductive Inference - a conclusion is formed by generalizing from specific premises IS THE ARGUMENT VALID? ● An argument is valid if its conclusion necessarily follows from its premises - even though we may not agree that its premises are true or that its conclusion is true. ● In logic, there is a distinction between validity and truth. Validity is concerned only with the logical structure (or form) of an argument, not the truth of its premises or conclusions. Summary of Possibilities for an Argument: 1. Valid and sound (logical and with true premises and conclusion) 2. Valid but not sound (logical but the premises and conclusion are false) 3. Invalid with a false conclusion 4. Invalid with a true conclusion Deductive Arguments with One Conditional Premise (The Four Basic Conditional Arguments) 1. Affirming the antecedent (valid argument: modus ponens) Form: Premise: If p, then q. Premise: p. Conclusion: q. 2. Affirming the consequent (invalid argument) (Converse error) Form: Premise: If p then q. Premise: q. Conclusion: p The conclusion does not follow from the premises. 3. Denying the consequent (valid argument: modus tollens) If p then q. Not q ∴Not p 4. Denying the antecedent (invalid argument) (Inverse error) If p then q. Not p Not q 8 HYPOTHETICAL SYLLOGISM deductive arguments with a chain of conditionals If p then q. // If q then r. // = If p then r. RULES OF INFERENCE 1. Disjunctive Syllogism p v q // ~p // ∴ q 2. Simplification: p˄q // ∴ p 3. Conjunction: p // q // ∴p˄q 4. Addition: p // ∴p˅q 5. Absorption: p→q // ∴ p →(p˄q) 6. Constructive Dilemma: (p→q) ˄(r→s) // p˅r // ∴ q˅s *self-review proof of validity INDUCTION AND DEDUCTION IN MATHEMATICS THEOREMS are statements of mathematical truth which requires proof which is possible only through deductive logic. AXIOMS are the starting points for mathematical proof, the “givens”, assumed to be true without proof. Goldbach Conjecture (1742) “Every even number (except 2) can be expressed as a sum of two prime numbers.” Fermat’s Last Theorem (Pierre Fermat, 1601-1665) “For any natural number n except 1 or 2, it is impossible to find natural numbers a, b, and c that satisfy the relationship an + bn = cn.” 9 CHAPTER 2.3: THE SEARCH FOR TRUTH AND KNOWLEDGE PART 1: THE SEARCH FOR TRUTH SCIENTIFIC METHOD We describe the inter-relationships among logic, mathematics and science, which open the way to understanding the scientific method, the principal means by which knowledge is acquired today CLASSICAL LOGIC AND MATHEMATICS Mathematics establishes the truth of a theorem by constructing a proof, which essentially, is an argument wherein established mathematical facts serve as premises, and the theorem is the conclusion (the last line of the proof). Pythagorean Theorem. For any right triangle whose legs measure a and b units and whose diagonal measures c units, a2 + b2 = c2. Proof: Proof of Bhaskara (12th century Hindu mathematician) ARISTOTLE (384-322 B.C.) ● the similarity between logic and mathematics explains why many philosophers were also mathematicians. ● well-known Greek philosopher, tutor of Alexander the Great ● probably the first person who attempted to give logic a rigorous foundation. ● believed that truth could be established from three basic laws Three Basic Aristotelian Laws 1. The law of identity (A thing is itself.) 2. The law of excluded middle (A statement is either true or false.) 3. The law of non-contradiction (No statement is both true and false.) EUCLID ● ● Aristotle’s laws were the basis of the logic used by the Greek mathematician Euclid to establish the foundations of geometry (in his famous treatise The Elements (300 BC). Euclid began with only 5 postulates or premises from which he derived all of classical geometry, also known as Euclidean geometry. 10 Gottfried Wilhelm von Leibniz (1646-1716) “If we could find characters or signs appropriate for expressing all our thoughts as definitely and as exactly as arithmetic expresses numbers or geometric analysis expresses lines, we could in all subjects in so far as they are amenable to reasoning accomplish what is done in Arithmetic and Geometry. For all inquiries which depend on reasoning would be performed by the transposition of characters and by a kind of calculus... And if someone would doubt my results, I should say to him: “let us calculate, Sir” and thus by taking a pen and ink, we should soon settle the question. “ Leibniz’s Dream ● Recognizing that logic could be used to establish mathematical truths, could logic also be used to establish other truths? Could it be used to determine “universal truths”? ● Leibniz (1646-1716) attempted to establish a calculus of reasoning which can be used to decide all arguments; suggested that an international symbolic language for logic be developed with which equations of logic could be written and used to calculate a “solution” to any argument. So what really became of Leibniz’s dream? ● Kurt Godel in 1931 proved that the dream could never be achieved. ● Leibniz’s dream was shattered! ● But this ushered in a new period in the relationship between logic and mathematics, often termed the period of modern logic. THE HISTORY OF LOGIC Classical Logic (300 BC to mid 1800’s) Aristotelian Logic // Euclidean Geometry Symbolic Logic (mid 1800’s to 1931) Algebra of Sets Modern Logic (since 1931) Godel’s Theorem 11 PART 2: THE LIMITATION OF LOGIC GÖDEL’S THEOREM Mathematicians believed that for an ultimate system of logic to be realized, a first step is to show that mathematics could be wholly understood as a system of logic. Only then could mathematical logic be developed into Leibniz’s dream of a calculus of reasoning. DAVID HILBERT sought to formalize mathematics as a system in which all mathematical truths, or theorems, could be derived from a few basic assumptions called axioms, by applying rules of logic. Required Properties of an Axiom System 1. It must be finitely describable, that is, the number of basic axioms should be limited. 2. It must be consistent, that is, it should have no internal contradictions (statements that are both true and false). 3. It must be complete, that is, the basic axioms should allow analysis of every possible situation. *In 1931, Kurt Gödel, an Austrian mathematician, proved that no formal system of logic can possess all three required properties. He proved that no system can be simultaneously complete, consistent and finitely describable. The Value of Logic ● ● ● ● ● ● Logic allows the discovery of new knowledge and the development of new technology. Logic provides ways to address disputes, even if it cannot always ensure their resolution. Through logic, you can study your personal beliefs and societal issues. Logic can help you study the nature of truth, though logic cannot ultimately answer all questions. Though logic alone may fail under some circumstances, logical reasoning is an excellent tool for understanding and acquiring knowledge. Finding the proper balance between logic and other processes of decision making is one of the greatest challenges of being human. PART 3: LOGIC AND SCIENCE 12 SCIENCE Lat. scientia which means “having knowledge” or “to know”. Science is knowledge acquired through careful observation and study; knowledge as opposed to ignorance or misunderstanding. SCIENTIFIC METHOD - It is a set of principles and procedures, based on logic, for the systematic pursuit of knowledge. - an idealization of the process used to discover or construct new knowledge Some Terminology: ● Fact - a simple statement that is indisputably or objectively true, or close as possible to being so. ● Law - a statement of a particular pattern or order in nature ● Hypothesis - a tentative explanation for some set of natural phenomena, sometimes called “an educated guess” ● Scientific Theory - an accepted (that is, extensively tested and verified) model that explains a broad range of phenomena THE SCIENTIFIC METHOD 1. Recognition and formulation of a problem 2. Construction of a hypothesis 3. New predictions 4. Unbiased and reproducible tests of new predictions 5. Modification of hypothesis 6. Hypothesis passes many tests and becomes a theory 7. Theory continually challenged and re-tested for refinement, expansion, and/or replacement SCIENCE, NONSCIENCE AND PSEUDOSCIENCE Nonscience - any attempt to search for knowledge that knowingly does not allow the scientific method Pseudoscience - that which purports to be science but, under careful examination, fails tests conducted by the scientific method PART 4: PARADOXES 13 PARADOX - is a situation or statement that seems to violate common sense or to contradict itself. - allow for the recognition of problems which may lead to new principles, to new facts, or to a new scientific theory. - may or may not be resolved. EXAMPLES OF PARADOXES ● The “up-and-down” problem ● “I never tell the truth.” ● Zeno’s Paradox (baffled mathematicians for 2,000 years but resolved already today) ● The Paradox of Light (now scientifically understood, but still baffles the common sense) ● The Paradox of Creation (not yet resolved and may never be) ● Liar’s Paradox “This sentence is false.” ● Barber’s Paradox “There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved themselves or were shaved by the barber. And the barber only shaved the men who did not shave themselves.” CHAPTER 2.4: FALLACIES FALLACY 14 Latin “fallacia” meaning deceit or trick *Previously we considered formal fallacies in which logical errors occur through a flaw in the form or structure of the argument. Here, we consider informal fallacies, in which an argument is deficient because of its content. PART 1: FALLACIES OF RELEVANCE FALLACIES OF RELEVANCE committed when the premise is irrelevant to the conclusion of an argument. 1. SUBJECTIVISM The fallacy of subjectivism has the form “I believe/want p to be true, therefore p is true.” 2. APPEAL TO IGNORANCE The fallacy of appeal to ignorance has the form “p has not been proven false, therefore p is true.” Its other form is “p has not been proven true, therefore p is false.” 3. LIMITED CHOICE (OR FALSE CHOICE) The limited choice fallacy has the form “p is false, therefore q is true.” 4. APPEAL TO EMOTION The appeal to emotion fallacy has the form “p evokes a strong emotional response, therefore p is true.” 5. APPEAL TO FORCE Appeal to force has the form “I say p is true and if you don’t agree with me, you will be hurt or ridiculed; therefore p is true.” 6. INAPPROPRIATE APPEAL TO AUTHORITY The fallacy of inappropriate appeal to authority is committed when the support for a proposition relies on the testimony of an inappropriate or unqualified authority. The appeal to emotion fallacy has the form “An authority says p is true, therefore p is true.” 7. PERSONAL ATTACK (AD HOMINEM) Ad Hominem - (Latin) “to the person” This fallacy involves attacking the character, circumstances, or motives of a person making an argument. The ad hominem fallacy has the form “Person X says that p is true and person X is a bad person; therefore p is NOT true.” 8. BEGGING THE QUESTION(CIRCULAR REASONING) p is true, p is true (often expressed using different words). 9. NON SEQUITUR Non Sequitur - (Latin) “does not follow” Two types of non sequitur: Diversion or Red Herring attention is diverted from the real issue to another issue 15 Straw Man an argument is made against a distortion of someone’s idea or position PART 2: FALLACIES OF NUMBERS AND STATISTICS FALLACIES OF NUMBERS AND STATISTICS few common fallacies that involve numbers or the collection and analysis of statistical data. These fallacies are particularly important to our work in this course, and you will see them arise again and again. 1. APPEAL TO POPULARITY The fallacy of appeal to popularity has the form “many people believe p is true, therefore p is true.” The fact that large numbers of people believe a proposition is used as evidence of its truth 2. APPEAL TO NUMBERS The appeal to numbers fallacy has the form “p has been observed many times, therefore p is true.” A conclusion is drawn solely on the basis of quantity 3. HASTY GENERALIZATION The fallacy of hasty generalization has the form “p is true one or a few times, therefore p is always true.” Supports a proposition with an inadequate number of instances or instances that are atypical 4. BIAS AND THE AVAILABILITY ERROR Availability error - the human tendency to make judgments based on what is available in the mind. The availability error takes the form “the first thing that comes to mind is p, therefore p is true.” 5. FALSE CAUSE The false cause fallacy has the form “A came before B, therefore A caused B.” Correlation Versus Cause Correlation - exists between two different events when the incidence of one event is related in some way to the incidence of another. 16 Positive Correlation - when the incidence of both events rises and falls together. Negative Correlation - When the incidence of one event increases while the other decreases. How de we know if one event is a causal condition for another? An event is a: •Necessary condition - if the effect cannot happen in its absence. •Sufficient condition - if the effect always happens when the event occurs. Confidence in Causality •Possible Cause: An apparent linkage exists between two events, such as a correlation, but no other evidence suggest causality. •Probable Cause: A good reason to suspect causality exists •Cause Beyond Reasonable Doubt: A model is so successful in explaining the linkage between events that it seems unreasonable to doubt the causal connection. PART 3: FALLACIES INVOLVING PERCENTAGES FALLACIES INVOLVING PERCENTAGES fallacies that involve a misunderstanding of how to work with percentages. ABSOLUTE AND RELATIVE CHANGE Absolute change = new value - previous value Relative change = absolute change/previous value CHAPTER 2.5: AXIOMATIC SET THEORY CANTOR’S (INTUITIVE) THEORY OF SETS Definition (Cantor, Georg): A set is any collection into a whole of definite distinguishable objects of our intuition or thought. The objects are called the elements or 17 members of the set, denoted by ∈. 1. The Intuitive Principle (Axiom) of Extension Two sets are equal if they have the same members. 2. The Intuitive Principle (Axiom) of Abstraction A formula P(x) defines a set A by the convention that the members of A are exactly those objects a such that P(a) is a true statement. ● This provides a set for each condition or property. ● It allows the creation of ”too large” sets. ● Leads to contradiction like the Russel Paradox ZERMELO-FRAENKEL SET THEORY This eventually led to the restructuring of set theory along axiomatic lines. By the Zermelo-Fraenkel Set Theory, Cantor’s Set Theory was axiomatized in such a way that: i. All undesirable features (like paradoxes) are avoided; and ii. All desirable features are retained. ZF1 (Axiom of Existence) There exists a set which has no element. Remark: Intuitively, there should only be one set that has no element. ZF2 (Axiom of Extensionality) Suppose A and B are sets. If for all x, x ∈ A if and only if x ∈ B then A=B. Lemma 1: There exists only one set with no element. The Empty Set - unique set with no element is called the empty set and is denoted ∅. Theorem 1: For any sets A, B, C. The equality relation satisfies the following: i. A = A (reflexivity); ii. A = B implies B = A (symmetry); and iii. A = B and B = C imply A = C (transitivity) ZF3 (Axiom Schema of Specification/Restricted Comprehension) Let P(x) be a property of x. For any set A, there exists a set B such that for any x, x ∈ B if and only if x ∈ A and P(x) holds. Theorem 2: B is unique Proof: 18 Assume that there exists another set B’ such that for all x, x ∈ B’ if and only if x ∈ A and P(x). Note that x ∈ B implies x ∈ A and P(x) which gives x ∈ B’ . Also x ∈ B’ implies x ∈ A and P(x), hence x ∈ B. Thus, x ∈ B if and only if x ∈ B’ . Therefore, B = B’ ZF4 (Axiom of Pairing) If A and B are sets, then there exists a set C such that x ∈ C if and only if x = A or x = B. ZF5 (Axiom of Union) For any set S, there exists U such that x ∈ U if and only if x ∈ A for some A ∈ S Properties of the Inclusion Relation i. A ⊆ A. (reflexivity) ii. A ⊆ B and B ⊆ A imply A = B. (antisymmetry) iii. A ⊆ B and B ⊆ A imply A ⊆ C. (transitivity) ZF1: Axiom of Existence ZF2: Axiom of Extensionality ZF3: Axiom Schema of Specification/Restricted Comprehension ZF4: Axiom of Pairing ZF5: Axiom of Union ZF6: Axiom of Power Set ZF7: Axiom of Infinity (to be discussed later in the course) ZF8: Axiom Schema of Replacement ZF9: Axiom of Choice (Well-Ordering Theorem) (to be discussed later in the course) CHAPTER 2.6: NUMBERS 19 Mathematics is said to be the language of nature. PART 1: A BRIEF HISTORY OF NUMBERS The concept of numbers has evolved over time. It developed in parallel with methods for writing numerals which are symbols that represent numbers. THE ORIGIN OF MODERN NUMERALS ● ● ● Numeral systems relied on tallies with fingers or toes, piles of stones, or notches cut on a bone or a piece of wood. But these systems are inadequate for large numbers. To simplify the process of counting, counts are grouped by 2’s, 3’s, then eventually, by 5’s, 10’s, and 20’s. In 3000 B.C., the Egyptians and Babylonians independently introduced the first numeral system to go beyond tallying. 20