Are the truths of mathematics invented or discovered?
|
|
- Felix McCarthy
- 7 years ago
- Views:
Transcription
1 Are the truths of mathematics invented or discovered? By Vincent Wen Imagine you are a prehistoric caveman living a nomadic life. One bright sunny morning, a fellow caveman comes to you, borrows a few apples, and promises to give them back. To make sure your fellow caveman will return them in the same quantity so the days of gathering will not turn out to be fruitless, you raise your hands, bend your fingers, and see if you can use them to represent the number of apples you lent. Does this sound familiar to you? Yes, it is finger counting, the predecessor of our numerical system. Ever since the dawn of humanity, people have been using mathematics to make our lives easier. Whether you think of math as a tool used to resolve our daily trivia; a study aimed at explaining natural phenomena, or a system aided in the advancement of our galloping technology, it has been and will always be an integral part of our life. After twelve years of systematic study in this discipline, the importance of math in my life has surpassed any other subject. Nonetheless, I still find it challenging to answer this question: are the truths of mathematics invented or discovered? After much thought, in this essay, I will argue that the truths of math are invented rather than discovered. Mathematics, a field of knowledge which has been developed by mankind for thousands of years, is generally defined as an axiomatic study of quantity, space and relations. It involves the formulation of new theorems by a rigorous deductive process from several chosen axioms. These axioms are the primitive assumptions of
2 mathematics in which all other theorems and formulas must depend upon. These axioms cannot be deducted nor proven using any logical means. They are considered to be self-evident and are taken to be true (Penrose, 2004). For example, in our arithmetic system, for every natural number x, x+0=x. This is one of Peano axioms and it is an axiom because it cannot be proven using any other axioms; it is self-evident; theorems such as for all natural numbers a and b, a+b=b+a can be proven using this axiom (Hawking, 2005). Having the definition of the truths of math established, it will not take a great deal of mental exercise to arrive at the conclusion that these truths are invented. Invention is the creation of new things such as computers, TV, automobiles, which have never existed before. On the contrary, discovery is not creation. The object discovered must exist before it is discovered. It could be something you have never seen before or it could be a different way of thinking about something. A quote by Albert Szent-Gyorgyi, a Hungarian Nobel Prize laureate, elaborates on the meaning of discovery in a clever and concise manner, Discovery consists of seeing what everybody has seen and thinking what nobody has thought (Good, 1963, p. 222). Unlike discoveries, axioms did not exist before they were artificially defined (notice the word artificially is used instead of explicitly, because the axioms can also be implicitly defined when our ancestors did mathematical operations before they wrote the axioms down). Therefore, using disjunctive syllogism, we can conclude the axioms (the truths of mathematics) are invented. At this point, many questions may arise which ponder the validity of the
3 arguments presented in the previous paragraph. What if someone proves an axiom to be false? Why are electricity and gravity discovered but not axioms? Why can we not say that axioms are taken to be true due to observation? For example, a Neanderthal might have the experience that when one apple is put next to another apple, there are two apples altogether. When the same principle is applied to other objects, it would still work. Thus, the Neanderthal would come to the conclusion that 1+1=2 and use it as an axiom for further reasoning. To answer these questions, we must revisit our definition of the truths of mathematics and re-examine its subtlety closely. The axioms, being the absolute starting point of any deduction, cannot even be proven, let alone proven false. This is somewhat similar to the uncaused cause of the universe 1 except the axioms are defined by humans instead of created by a supreme being. This also implies that if 1+1=3 was presupposed to be true, it could be used as an axiom, though all deductions thereof must be consistent with this axiom. Although it may sound absurd at first, there are actual mathematical systems in which 1+1=2 does not hold true. For example, in the binary number system, 1+1=10; in the Boolean logic system, 1 AND 1 will still yield 1. Albert Einstein (1923, p. 28) once said: 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. In the course of this essay, the latter part of that statement holds true. Unlike!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "! #$%!&'()*%+,!-.!,$%!!"#$!%&'(#$!%&()*(+,&(!"-.&/%&!/-)01!2%!3,&,%1!&3! !561%7!899:;<! =>%'?,$@+(!$&3!&!/&)3%A!! 6!/&)3&0!/$&@+!/&++-,!2%!@+.@+@,%0?!0-+(A! #$%'%.-'%7!&+!)+/&)3%1!/&)3%!*)3,!%B@3,A!
4 science, math is neither observation based nor experiment based. Unlike chemistry, math is not concerned with matter. It could be based on axioms which are entirely abstract. Unlike biology, math does not set hypotheses and accumulate knowledge inductively. Unlike physics, math does not attempt to explain natural phenomena and understand the world. Although all three sciences (biology, chemistry, physics) use math as a tool to model relationships and draw conclusions, the math used by each science actually depends on different axioms. Although some axioms seem to be observed in nature and then taken to be true, nature actually inspired human to define these axioms in the same way as the boiling kettle inspired James Watt to invent steam engines. Although physics has always been trying to perfect our understanding of the world, as impeccable as it may seem, it depends on the inductive notion of causality 2 which makes it unable to stand the scrutiny of mathematics and its skepticism. After all the struggles between these arguments and counter-arguments, I could not help but think that this is not mathematics. It is supposed to be a complete, elegant and dazzling system where the most rational thoughts in the world are encapsulated within. However, each time I tried to defend my thesis, clarify the definitions, an invisible net placed upon the realm of mathematics seemed to contract. Fortunately, all we are looking at here is only a snippet of mathematics, namely the axioms. The applications of mathematics are far beyond its confined domain of its axioms. As indicated by Allegory of the Cave 3, the search for truth always requires courage,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 8! C%%!D&E)%,,%7!DA!FA!%,!&0A7!5899G7!HA!IJ;7!,$%!1%+@&0!-.!/&)3&0@,?!2?!K&>@1!L)*%A! G! C%%!D0&,-!5GM9!NAOA=;7!,$%!1@&0-()%!2%,4%%+!C-/'&,%3!&+1!F0&)/-+7!N--P!QRR7!#$%!S%H)20@/A!
5 discomfort, and sometimes even sacrifice. What is sacrificed is the flexibility of the axioms, what is bestowed is the certainty of mathematics. To sum it up, the truths of mathematics, or the axioms, are invented because they are defined by humans and served as the foundation of mathematics due to our desire for certainty. Work Cited: Ade, H. (2009). Course Lecture. AY Jackson S. S., Toronto, ON, Canada. Einstein, A. (1923). Sidelights on Relativity (Geometry and Experience). P. Dutton., Co. Good, I. J. (1963). The Scientist Speculates. Basic Books (New York). Hawking, S. (2005). God Created the Integers. Running Press. Paquette, P. G., Gini-Newman, L., Flaherty, P., Horton, M., Jopling, D., Miller, H., et al. (2003). Philosophy: Questions and Theories. McGraw-Hill Ryerson. Penrose, R. (2004). The Road to Reality. Jonathan Cape. Plato, The Republic. (B. Jowett, Trans.). The Internet Classics Archive. Available from < (Original work done in 360 B.C.E).
1/9. Locke 1: Critique of Innate Ideas
1/9 Locke 1: Critique of Innate Ideas This week we are going to begin looking at a new area by turning our attention to the work of John Locke, who is probably the most famous English philosopher of all
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationPlato gives another argument for this claiming, relating to the nature of knowledge, which we will return to in the next section.
Michael Lacewing Plato s theor y of Forms FROM SENSE EXPERIENCE TO THE FORMS In Book V (476f.) of The Republic, Plato argues that all objects we experience through our senses are particular things. We
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationA Few Basics of Probability
A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationCosmological Arguments for the Existence of God S. Clarke
Cosmological Arguments for the Existence of God S. Clarke [Modified Fall 2009] 1. Large class of arguments. Sometimes they get very complex, as in Clarke s argument, but the basic idea is simple. Lets
More informationOne natural response would be to cite evidence of past mornings, and give something like the following argument:
Hume on induction Suppose you were asked to give your reasons for believing that the sun will come up tomorrow, in the form of an argument for the claim that the sun will come up tomorrow. One natural
More informationBackground Biology and Biochemistry Notes A
Background Biology and Biochemistry Notes A Vocabulary dependent variable evidence experiment hypothesis independent variable model observation prediction science scientific investigation scientific law
More informationMathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
More informationAn Excerpt from THE REPUBLIC, BOOK VI. The Simile of the Divided Line. by Plato
An Excerpt from THE REPUBLIC, BOOK VI The Simile of the Divided Line by Plato (Written 360 B.C.E) Translated by Benjamin Jowett Summary of excerpt, and a graphical depiction of the divided line: STATES
More informationSet theory as a foundation for mathematics
V I I I : Set theory as a foundation for mathematics This material is basically supplementary, and it was not covered in the course. In the first section we discuss the basic axioms of set theory and the
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationdef: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.
Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true
More informationQuine on truth by convention
Quine on truth by convention March 8, 2005 1 Linguistic explanations of necessity and the a priori.............. 1 2 Relative and absolute truth by definition.................... 2 3 Is logic true by convention?...........................
More informationThe Slate Is Not Empty: Descartes and Locke on Innate Ideas
The Slate Is Not Empty: Descartes and Locke on Innate Ideas René Descartes and John Locke, two of the principal philosophers who shaped modern philosophy, disagree on several topics; one of them concerns
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationSECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationDEVELOPING HYPOTHESIS AND
Shalini Prasad Ajith Rao Eeshoo Rehani DEVELOPING 500 METHODS SEPTEMBER 18 TH 2001 DEVELOPING HYPOTHESIS AND Introduction Processes involved before formulating the hypotheses. Definition Nature of Hypothesis
More informationPREPARATION MATERIAL FOR THE GRADUATE RECORD EXAMINATION (GRE)
PREPARATION MATERIAL FOR THE GRADUATE RECORD EXAMINATION (GRE) Table of Contents 1) General Test-Taking Tips -General Test-Taking Tips -Differences Between Paper and Pencil and Computer-Adaptive Test 2)
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
More information3. Logical Reasoning in Mathematics
3. Logical Reasoning in Mathematics Many state standards emphasize the importance of reasoning. We agree disciplined mathematical reasoning is crucial to understanding and to properly using mathematics.
More informationAn Innocent Investigation
An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number
More informationInductive Reasoning Page 1 of 7. Inductive Reasoning
Inductive Reasoning Page 1 of 7 Inductive Reasoning We learned that valid deductive thinking begins with at least one universal premise and leads to a conclusion that is believed to be contained in the
More information4.2 Euclid s Classification of Pythagorean Triples
178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple
More information6.080/6.089 GITCS Feb 12, 2008. Lecture 3
6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
More information6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationFive High Order Thinking Skills
Five High Order Introduction The high technology like computers and calculators has profoundly changed the world of mathematics education. It is not only what aspects of mathematics are essential for learning,
More informationExhibit memory of previously-learned materials by recalling facts, terms, basic concepts, and answers. Key Words
The Six Levels of Questioning Level 1 Knowledge Exhibit memory of previously-learned materials by recalling facts, terms, basic concepts, and answers. who what why when where which omit choose find how
More informationWRITING A RESEARCH PAPER FOR A GRADUATE SEMINAR IN POLITICAL SCIENCE Ashley Leeds Rice University
WRITING A RESEARCH PAPER FOR A GRADUATE SEMINAR IN POLITICAL SCIENCE Ashley Leeds Rice University Here are some basic tips to help you in writing your research paper. The guide is divided into six sections
More informationMath 223 Abstract Algebra Lecture Notes
Math 223 Abstract Algebra Lecture Notes Steven Tschantz Spring 2001 (Apr. 23 version) Preamble These notes are intended to supplement the lectures and make up for the lack of a textbook for the course
More informationDigitalCommons@University of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University
More informationReality in the Eyes of Descartes and Berkeley. By: Nada Shokry 5/21/2013 AUC - Philosophy
Reality in the Eyes of Descartes and Berkeley By: Nada Shokry 5/21/2013 AUC - Philosophy Shokry, 2 One person's craziness is another person's reality. Tim Burton This quote best describes what one finds
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationWriting a Project Report: Style Matters
Writing a Project Report: Style Matters Prof. Alan F. Smeaton Centre for Digital Video Processing and School of Computing Writing for Computing Why ask me to do this? I write a lot papers, chapters, project
More informationRead this syllabus very carefully. If there are any reasons why you cannot comply with what I am requiring, then talk with me about this at once.
LOGIC AND CRITICAL THINKING PHIL 2020 Maymester Term, 2010 Daily, 9:30-12:15 Peabody Hall, room 105 Text: LOGIC AND RATIONAL THOUGHT by Frank R. Harrison, III Professor: Frank R. Harrison, III Office:
More informationGuide to Leaving Certificate Mathematics Ordinary Level
Guide to Leaving Certificate Mathematics Ordinary Level Dr. Aoife Jones Paper 1 For the Leaving Cert 013, Paper 1 is divided into three sections. Section A is entitled Concepts and Skills and contains
More informationWhat is Undergraduate Education?
Education as Degrees and Certificates What is Undergraduate Education? K. P. Mohanan For many people, being educated means attending educational institutions and receiving certificates or degrees. This
More informationComputation Beyond Turing Machines
Computation Beyond Turing Machines Peter Wegner, Brown University Dina Goldin, U. of Connecticut 1. Turing s legacy Alan Turing was a brilliant mathematician who showed that computers could not completely
More informationPrime Factorization 0.1. Overcoming Math Anxiety
0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
More informationPhilosophical argument
Michael Lacewing Philosophical argument At the heart of philosophy is philosophical argument. Arguments are different from assertions. Assertions are simply stated; arguments always involve giving reasons.
More informationTheorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive
Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationThomas Kuhn and The Structure of Scientific Revolutions
Thomas Kuhn and The Structure of Scientific Revolutions The implications of the story so far is that science makes steady progress That the process of science cycles round and round from Induction to Deduction...
More informationSYMBOL AND MEANING IN MATHEMATICS
,,. SYMBOL AND MEANING IN MATHEMATICS ALICE M. DEAN Mathematics and Computer Science Department Skidmore College May 26,1995 There is perhaps no other field of study that uses symbols as plentifully and
More informationIntroduction. Appendix D Mathematical Induction D1
Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to
More informationINCIDENCE-BETWEENNESS GEOMETRY
INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationProblem of the Month: Perfect Pair
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
More informationPositive Philosophy by August Comte
Positive Philosophy by August Comte August Comte, Thoemmes About the author.... August Comte (1798-1857), a founder of sociology, believes aspects of our world can be known solely through observation and
More informationINTERSECTION MATH And more! James Tanton
INTERSECTION MATH And more! James Tanton www.jamestanton.com The following represents a sample activity based on the December 2006 newsletter of the St. Mark s Institute of Mathematics (www.stmarksschool.org/math).
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationAutomated Theorem Proving - summary of lecture 1
Automated Theorem Proving - summary of lecture 1 1 Introduction Automated Theorem Proving (ATP) deals with the development of computer programs that show that some statement is a logical consequence of
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationOrganizing an essay the basics 2. Cause and effect essay (shorter version) 3. Compare/contrast essay (shorter version) 4
Organizing an essay the basics 2 Cause and effect essay (shorter version) 3 Compare/contrast essay (shorter version) 4 Exemplification (one version) 5 Argumentation (shorter version) 6-7 Support Go from
More informationGeometry, Technology, and the Reasoning and Proof Standard inthemiddlegradeswiththegeometer ssketchpad R
Geometry, Technology, and the Reasoning and Proof Standard inthemiddlegradeswiththegeometer ssketchpad R Óscar Chávez University of Missouri oc918@mizzou.edu Geometry Standard Instructional programs from
More informationDIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents
DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition
More informationHow does the problem of relativity relate to Thomas Kuhn s concept of paradigm?
How does the problem of relativity relate to Thomas Kuhn s concept of paradigm? Eli Bjørhusdal After having published The Structure of Scientific Revolutions in 1962, Kuhn was much criticised for the use
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationPreparing for the GRE Writing test Erin Jensen Assistant Director University of Utah Writing Center erin.jensen@utah.edu www.writingcenter.utah.
Preparing for the GRE Writing test Erin Jensen Assistant Director University of Utah Writing Center erin.jensen@utah.edu www.writingcenter.utah.edu What is the GRE? GRE stands for Graduate Record Examination
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationWhy I Wrote this Packet
Things All Political Science Majors Should Know About Writing and Research Chris Cooper Department of Political Science and Public Affairs Western Carolina University Why I Wrote this Packet Many of our
More informationAppendix A: Science Practices for AP Physics 1 and 2
Appendix A: Science Practices for AP Physics 1 and 2 Science Practice 1: The student can use representations and models to communicate scientific phenomena and solve scientific problems. The real world
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationChapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M.
31 Geometric Series Motivation (I hope) Geometric series are a basic artifact of algebra that everyone should know. 1 I am teaching them here because they come up remarkably often with Markov chains. The
More informationGCDs and Relatively Prime Numbers! CSCI 2824, Fall 2014!
GCDs and Relatively Prime Numbers! CSCI 2824, Fall 2014!!! Challenge Problem 2 (Mastermind) due Fri. 9/26 Find a fourth guess whose scoring will allow you to determine the secret code (repetitions are
More informationThe History of Logic. Aristotle (384 322 BC) invented logic.
The History of Logic Aristotle (384 322 BC) invented logic. Predecessors: Fred Flintstone, geometry, sophists, pre-socratic philosophers, Socrates & Plato. Syllogistic logic, laws of non-contradiction
More informationReview. Bayesianism and Reliability. Today s Class
Review Bayesianism and Reliability Models and Simulations in Philosophy April 14th, 2014 Last Class: Difference between individual and social epistemology Why simulations are particularly useful for social
More informationScience and Scientific Reasoning. Critical Thinking
Science and Scientific Reasoning Critical Thinking Some Common Myths About Science Science: What it is and what it is not Science and Technology Science is not the same as technology The goal of science
More informationWhy STEM Topics are Interrelated: The Importance of Interdisciplinary Studies in K-12 Education
Why STEM Topics are Interrelated: The Importance of Interdisciplinary Studies in K-12 Education David D. Thornburg, PhD Executive Director, Thornburg Center for Space Exploration dthornburg@aol.com www.tcse-k12.org
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationWriting a Literature Review in Higher Degree Research. Gillian Colclough & Lindy Kimmins Learning & Teaching Support
Writing a Literature Review in Higher Degree Research Gillian Colclough & Lindy Kimmins Learning & Teaching Support This presentation: Aims of a literature review Thoughts about a good literature review
More informationWhat Is Induction and Why Study It?
1 What Is Induction and Why Study It? Evan Heit Why study induction, and indeed, why should there be a whole book devoted to the study of induction? The first reason is that inductive reasoning corresponds
More informationWriting an essay. This seems obvious - but it is surprising how many people don't really do this.
Writing an essay Look back If this is not your first essay, take a look at your previous one. Did your tutor make any suggestions that you need to bear in mind for this essay? Did you learn anything else
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationWriting = A Dialogue. Part I. They Say
Writing = A Dialogue You come late. When you arrive, others have long preceded you, and they are engaged in a heated discussion, a discussion too heated for them to pause and tell you exactly what it is
More informationLecture 2. What is the Normative Role of Logic?
Lecture 2. What is the Normative Role of Logic? What is the connection between (deductive) logic and rationality? One extreme: Frege. A law of logic is a law of rational thought. Seems problematic, if
More informationThis puzzle is based on the following anecdote concerning a Hungarian sociologist and his observations of circles of friends among children.
0.1 Friend Trends This puzzle is based on the following anecdote concerning a Hungarian sociologist and his observations of circles of friends among children. In the 1950s, a Hungarian sociologist S. Szalai
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationFinancial Mathematics
Financial Mathematics For the next few weeks we will study the mathematics of finance. Apart from basic arithmetic, financial mathematics is probably the most practical math you will learn. practical in
More informationProblem of the Month: Fair Games
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
More informationRigorous Software Development CSCI-GA 3033-009
Rigorous Software Development CSCI-GA 3033-009 Instructor: Thomas Wies Spring 2013 Lecture 11 Semantics of Programming Languages Denotational Semantics Meaning of a program is defined as the mathematical
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationRules of Inference Friday, January 18, 2013 Chittu Tripathy Lecture 05
Rules of Inference Today s Menu Rules of Inference Quantifiers: Universal and Existential Nesting of Quantifiers Applications Old Example Re-Revisited Our Old Example: Suppose we have: All human beings
More information3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.
SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid
More informationIndices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková
Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead
More informationThey Say, I Say: The Moves That Matter in Academic Writing
They Say, I Say: The Moves That Matter in Academic Writing Gerald Graff and Cathy Birkenstein ENTERING THE CONVERSATION Many Americans assume that Others more complicated: On the one hand,. On the other
More informationGeneral Philosophy. Dr Peter Millican, Hertford College. Lecture 3: Induction
General Philosophy Dr Peter Millican, Hertford College Lecture 3: Induction Hume s s Fork 2 Enquiry IV starts with a vital distinction between types of proposition: Relations of ideas can be known a priori
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More information