To Infinity and Beyond
|
|
- Mervyn Doyle
- 5 years ago
- Views:
Transcription
1 To Infinity and Beyond 25 January 2012 To Infinity and Beyond 25 January /24
2 The concept of infinity has both fascinated and frustrated people for millenia. We will discuss some historical problems about infinity, some modern (around 100 years old) ideas about infinity, and some interesting puzzles about the concept. To Infinity and Beyond 25 January /24
3 Zeno s Paradoxes There are a series of paradoxes possibly coming from the Greek philosopher Zeno ( BC). One we will discuss is the paradox of Achilles and the Tortoise. Aristotle is quoted referring to this paradox: In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. To Infinity and Beyond 25 January /24
4 Here is the statement of the paradox (borrowed in large part from Wikipedia): In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Suppose Achilles allows the tortoise a head start of 100 yards. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 yards, bringing him to the tortoise s starting point. During this time, the tortoise has run a much shorter distance, say, 10 yards. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. To Infinity and Beyond 25 January /24
5 Clicker Question Do you think Achilles will never catch up to the Tortoise? A Yes B No To Infinity and Beyond 25 January /24
6 A Football Paradox The 49ers and Cowboys are playing. The 49ers have the ball on the 1 yard line. As they start the play the Cowboys are called off sides. The penalty moves the ball half the distance to the goal line, so now it is on the 1/2 yard line. Again, the Cowboys are off sides; the penalty moves the ball half the distance to the goal line. They keep being called for off sides. No matter how many penalties, the ball is not quite to the goal line. If they get called for infinitely many penalties, shouldn t the 49ers end up in the end zone? To Infinity and Beyond 25 January /24
7 Galileo s Paradox To Infinity and Beyond 25 January /24
8 The following comes from Galileo s book Dialogue Concerning the Two New Sciences (from books.google.com) In readable form... To Infinity and Beyond 25 January /24
9 Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension. To Infinity and Beyond 25 January /24
10 Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension. Salviati: This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty. I take it for granted that you know which of the numbers are squares and which are not. To Infinity and Beyond 25 January /24
11 Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves. To Infinity and Beyond 25 January /24
12 Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves. Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not? To Infinity and Beyond 25 January /24
13 Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves. Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not? Simplicio: Most certainly. To Infinity and Beyond 25 January /24
14 Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves. Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not? Simplicio: Most certainly. Salviati: If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. To Infinity and Beyond 25 January /24
15 Simplicio: Precisely so. To Infinity and Beyond 25 January /24
16 Simplicio: Precisely so. Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as the numbers because every number is the root of some square. This being granted, we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said that there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers taken all together. To Infinity and Beyond 25 January /24
17 Sagredo: What then must one conclude under these circumstances? To Infinity and Beyond 25 January /24
18 Sagredo: What then must one conclude under these circumstances? Salviati: So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes equal, greater, and less, are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. To Infinity and Beyond 25 January /24
19 To put this more simply, the paradox is that it seems that the set of all square numbers {1, 4, 9, 16, 25,...} is both smaller than and the same size as the set of all whole numbers {1, 2, 3, 4, 5,...}. Both cannot be true. To Infinity and Beyond 25 January /24
20 To put this more simply, the paradox is that it seems that the set of all square numbers {1, 4, 9, 16, 25,...} is both smaller than and the same size as the set of all whole numbers {1, 2, 3, 4, 5,...}. Both cannot be true. Galileo gets the idea that these two sets may be the same size by seeing that elements can be paired off, without leaving out anything. To Infinity and Beyond 25 January /24
21 Giving a simpler example of this idea, Clearly if I have 10 M&Ms and you take 8 of them, then you have fewer M&Ms than I started with. Galileo s paradox makes this basic fact not clear for infinite sets. To Infinity and Beyond 25 January /24
22 Giving a simpler example of this idea, Clearly if I have 10 M&Ms and you take 8 of them, then you have fewer M&Ms than I started with. Galileo s paradox makes this basic fact not clear for infinite sets. So, is the set of all square numbers smaller than the set of all whole numbers? Is it the same size? Does the question not make sense? To Infinity and Beyond 25 January /24
23 Georg Cantor To Infinity and Beyond 25 January /24
24 Cantor, a German mathematician, whose career was mostly in the later part of the 19th century, did important work in set theory. He formalized the idea of the size of a set, and defined what it means for one set to be larger, smaller, or the same size as another set. He used the idea Galileo discussed. To Infinity and Beyond 25 January /24
25 Cantor, a German mathematician, whose career was mostly in the later part of the 19th century, did important work in set theory. He formalized the idea of the size of a set, and defined what it means for one set to be larger, smaller, or the same size as another set. He used the idea Galileo discussed. His work, applied to infinite sets, was harshly criticized by many, including some of the most famous mathematicians of the time. One, David Hilbert, strongly supported his work. We will revisit Hilbert in the next class. To Infinity and Beyond 25 January /24
26 How to Compare Sizes of Sets Kids can compare two sets before knowing their numbers by pairing off elements. For example, a very young child can understand that there are just as many M&Ms as cars in the following picture. To Infinity and Beyond 25 January /24
27 Roughly, two sets have the same size if one can pair off elements of one set with elements of the other, leaving no elements left out of the pairing. To Infinity and Beyond 25 January /24
28 Roughly, two sets have the same size if one can pair off elements of one set with elements of the other, leaving no elements left out of the pairing. One set is larger than another if the second is the same size as a subset of the first, but not vice-versa. To Infinity and Beyond 25 January /24
29 Roughly, two sets have the same size if one can pair off elements of one set with elements of the other, leaving no elements left out of the pairing. One set is larger than another if the second is the same size as a subset of the first, but not vice-versa. To Infinity and Beyond 25 January /24
30 Infinite Sets Cantor extended this idea to arbitrary sets by saying two sets have the same size, whether or not they are finite or infinite, if elements of one can be paired with elements of the other, leaving no elements out. To Infinity and Beyond 25 January /24
31 Infinite Sets Cantor extended this idea to arbitrary sets by saying two sets have the same size, whether or not they are finite or infinite, if elements of one can be paired with elements of the other, leaving no elements out. With his definition, it is true that the set of whole numbers {1, 2, 3, 4, 5,...} is the same size as the set of squares {1, 4, 9, 16, 25,...}, by what Salviati says in Galileo s dialogue. To Infinity and Beyond 25 January /24
32 Infinite Sets Cantor extended this idea to arbitrary sets by saying two sets have the same size, whether or not they are finite or infinite, if elements of one can be paired with elements of the other, leaving no elements out. With his definition, it is true that the set of whole numbers {1, 2, 3, 4, 5,...} is the same size as the set of squares {1, 4, 9, 16, 25,...}, by what Salviati says in Galileo s dialogue. Cantor also defined a set to be infinite if it is the same size as a proper subset. It is not at all obvious that this corresponds to our intuition. To Infinity and Beyond 25 January /24
33 Infinite Sets Cantor extended this idea to arbitrary sets by saying two sets have the same size, whether or not they are finite or infinite, if elements of one can be paired with elements of the other, leaving no elements out. With his definition, it is true that the set of whole numbers {1, 2, 3, 4, 5,...} is the same size as the set of squares {1, 4, 9, 16, 25,...}, by what Salviati says in Galileo s dialogue. Cantor also defined a set to be infinite if it is the same size as a proper subset. It is not at all obvious that this corresponds to our intuition. For example, the set of whole numbers is infinite, because it is the same size as the set squares. To Infinity and Beyond 25 January /24
34 Infinite Sets Cantor extended this idea to arbitrary sets by saying two sets have the same size, whether or not they are finite or infinite, if elements of one can be paired with elements of the other, leaving no elements out. With his definition, it is true that the set of whole numbers {1, 2, 3, 4, 5,...} is the same size as the set of squares {1, 4, 9, 16, 25,...}, by what Salviati says in Galileo s dialogue. Cantor also defined a set to be infinite if it is the same size as a proper subset. It is not at all obvious that this corresponds to our intuition. For example, the set of whole numbers is infinite, because it is the same size as the set squares. To Infinity and Beyond 25 January /24
35 Galileo s paradox arises from the thought that one should be able to say two sets are the same size if you can pair up elements, which makes perfect sense for finite sets. Galileo thought that the set of squares shouldn t be as large as the set of whole numbers, even though they can be paired off. To Infinity and Beyond 25 January /24
36 Galileo s paradox arises from the thought that one should be able to say two sets are the same size if you can pair up elements, which makes perfect sense for finite sets. Galileo thought that the set of squares shouldn t be as large as the set of whole numbers, even though they can be paired off. To Infinity and Beyond 25 January /24
37 Galileo s paradox arises from the thought that one should be able to say two sets are the same size if you can pair up elements, which makes perfect sense for finite sets. Galileo thought that the set of squares shouldn t be as large as the set of whole numbers, even though they can be paired off. With Cantor s definition of infinity, any infinite set would be just as mysterious to Galileo. To Infinity and Beyond 25 January /24
38 Quiz Question Cantor s work on infinite sets was well received by the mathematicians of the time. A True B False To Infinity and Beyond 25 January /24
39 Clicker Question Are all infinite sets the same size? To Infinity and Beyond 25 January /24
40 Clicker Question Are all infinite sets the same size? A Yes To Infinity and Beyond 25 January /24
41 Clicker Question Are all infinite sets the same size? A Yes B No To Infinity and Beyond 25 January /24
42 Clicker Question Are all infinite sets the same size? A Yes B No We ll discuss this next time! To Infinity and Beyond 25 January /24
43 Back to Zeno Zeno s paradoxes can be resolved with the use of calculus, which makes sense of the notion of adding infinitely many numbers together. Zeno s paradoxes require one to think that adding infinitely many numbers together would result in an infinite sum. However, this is not always true. To Infinity and Beyond 25 January /24
44 Back to Zeno Zeno s paradoxes can be resolved with the use of calculus, which makes sense of the notion of adding infinitely many numbers together. Zeno s paradoxes require one to think that adding infinitely many numbers together would result in an infinite sum. However, this is not always true. The football example may make this easier to see. No matter how many penalties the Cowboys incur, the total distance the ball gets moved is never more than 1 yard. If they got infinitely many penalties, then the distance would be 1 yard. To Infinity and Beyond 25 January /24
45 Next Time We will look at some puzzles about infinity, coming from an idea of Hilbert, which is called the Hilbert Hotel. To Infinity and Beyond 25 January /24
To Infinity and Beyond
To Infinity and Beyond 22 January 2014 To Infinity and Beyond 22 January 2014 1/34 In case you weren t here on Friday, Course website: http://sierra.nmsu.edu/morandi/math210 Get a copy of the syllabus
More informationTo Infinity and Beyond. To Infinity and Beyond 1/43
To Infinity and Beyond To Infinity and Beyond 1/43 Infinity The concept of infinity has both fascinated and frustrated people for millennia. We will discuss some historical problems about infinity, some
More informationInfinite Sequences and Series Section
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Infinite Sequences and Series Section 8.1-8.2 Dr. John Ehrke Department of Mathematics Fall 2012 Zeno s Paradox Achilles and
More informationDale Oliver Professor of Mathematics Humboldt State University PONDERING THE INFINITE
Dale Oliver Professor of Mathematics Humboldt State University PONDERING THE INFINITE PLAN FOR TODAY Thought Experiments Numbered Ping Pong Balls Un-numbered Ping Pong Balls Zeno s Paradox of Motion Comparing
More informationMA 105 D3 Lecture 3. Ravi Raghunathan. July 27, Department of Mathematics
MA 105 D3 Lecture 3 Ravi Raghunathan Department of Mathematics July 27, 2017 Tutorial problems for July 31 The numbers refer to the tutorial sheet. E.g. 1.1 (iii) means Problem no. 1 part (iii) of the
More informationIntroduction to Paradoxes
Introduction to Paradoxes LA Math Circle Beginner Group Designed by Sian Wen Warm Up!. Assume that three sailors are caught by a group of pirates and kept blindfolded on the pirate ship. Lucky for them,
More informationZeno s Paradox #1. The Achilles
Zeno s Paradox #1. The Achilles Achilles, who is the fastest runner of antiquity, is racing to catch the tortoise that is slowly crawling away from him. Both are moving along a linear path at constant
More informationCantor and Infinite Sets
Cantor and Infinite Sets Galileo and the Infinite There are many whole numbers that are not perfect squares: 2, 3, 5, 6, 7, 8, 10, 11, and so it would seem that all numbers, including both squares and
More informationLecture 5. Zeno s Four Paradoxes of Motion
Lecture 5. Zeno s Four Paradoxes of Motion Science of infinity In Lecture 4, we mentioned that a conflict arose from the discovery of irrationals. The Greeks rejection of irrational numbers was essentially
More information8. Reductio ad absurdum
8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, 10 Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual
More informationThe paradox of knowability, the knower, and the believer
The paradox of knowability, the knower, and the believer Last time, when discussing the surprise exam paradox, we discussed the possibility that some claims could be true, but not knowable by certain individuals
More informationTo Infinity and Beyond
University of Waterloo How do we count things? Suppose we have two bags filled with candy. In one bag we have blue candy and in the other bag we have red candy. How can we determine which bag has more
More informationPHY1020 BASIC CONCEPTS IN PHYSICS I
PHY1020 BASIC CONCEPTS IN PHYSICS I The Problem of Motion 1 How can we predict the motion of everyday objects? ZENO (CA. 490 430 BC) AND ONENESS Motion is impossible! If all is one as Parmeinides said
More information8. Reductio ad absurdum
8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual infinities
More informationFACTORIZATION AND THE PRIMES
I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the natural numbers 1, 2, 3,... of ordinary
More information35 Chapter CHAPTER 4: Mathematical Proof
35 Chapter 4 35 CHAPTER 4: Mathematical Proof Faith is different from proof; the one is human, the other is a gift of God. Justus ex fide vivit. It is this faith that God Himself puts into the heart. 21
More informationHow does certainty enter into the mind?
1 How does certainty enter into the mind? Ching-an Hsiao h_siao@hotmail.com Any problem is concerned with the mind, but what do minds make a decision on? Here we show that there are three conditions for
More informationMath 144 Summer 2012 (UCR) Pro-Notes June 24, / 15
Before we start, I want to point out that these notes are not checked for typos. There are prbally many typeos in them and if you find any, please let me know as it s extremely difficult to find them all
More informationMathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur. Lecture 1 Real Numbers
Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Lecture 1 Real Numbers In these lectures, we are going to study a branch of mathematics called
More informationSequences and infinite series
Sequences and infinite series D. DeTurck University of Pennsylvania March 29, 208 D. DeTurck Math 04 002 208A: Sequence and series / 54 Sequences The lists of numbers you generate using a numerical method
More informationThe Two Faces of Infinity Dr. Bob Gardner Great Ideas in Science (BIOL 3018)
The Two Faces of Infinity Dr. Bob Gardner Great Ideas in Science (BIOL 3018) From the webpage of Timithy Kohl, Boston University INTRODUCTION Note. We will consider infinity from two different perspectives:
More information1 Continued Fractions
Continued Fractions To start off the course, we consider a generalization of the Euclidean Algorithm which has ancient historical roots and yet still has relevance and applications today.. Continued Fraction
More informationPROOF-THEORETIC REDUCTION AS A PHILOSOPHER S TOOL
THOMAS HOFWEBER PROOF-THEORETIC REDUCTION AS A PHILOSOPHER S TOOL 1. PROOF-THEORETIC REDUCTION AND HILBERT S PROGRAM Hilbert s program in the philosophy of mathematics comes in two parts. One part is a
More informationAxiomatic set theory. Chapter Why axiomatic set theory?
Chapter 1 Axiomatic set theory 1.1 Why axiomatic set theory? Essentially all mathematical theories deal with sets in one way or another. In most cases, however, the use of set theory is limited to its
More informationAristotle on continuity of time in Physics VI 2. Piotr Bªaszczyk
Aristotle on continuity of time in Physics VI 2 Piotr Bªaszczyk Abstract In Physics, 233 Aristotle proves that all time is continuous and denes a sequence of points that divide the time ZH into innitely
More informationAlternative Technologies
Alternative Technologies Zeno's Tortoise by David McGoveran, Alternative Technologies A Greek philosopher by the name of Zeno of Elea (ca. 490-430 BC) is alleged to have contrived a set of paradoxes regarding,
More informationEE 144/244: Fundamental Algorithms for System Modeling, Analysis, and Optimization Fall 2016
EE 144/244: Fundamental Algorithms for System Modeling, Analysis, and Optimization Fall 2016 Discrete Event Simulation Stavros Tripakis University of California, Berkeley Stavros Tripakis (UC Berkeley)
More information4 Infinity & Infinities
4 Infinity & Infinities As we shall see, time within certain closed environments in a host spacetime can lie beyond the infinite future of time in the host spacetime, or equivalently time in the host lies
More informationIntegration Made Easy
Integration Made Easy Sean Carney Department of Mathematics University of Texas at Austin Sean Carney (University of Texas at Austin) Integration Made Easy October 25, 2015 1 / 47 Outline 1 - Length, Geometric
More informationMITOCW MITRES18_006F10_26_0000_300k-mp4
MITOCW MITRES18_006F10_26_0000_300k-mp4 NARRATOR: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational
More informationChapter 10. Definition of the Derivative Velocity and Tangents
Chapter 10 Definition of the Derivative 10.1 Velocity and Tangents 10.1 Notation. If E 1 (x, y) and E 2 (x, y) denote equations or inequalities in x and y, we will use the notation {E 1 (x, y)} = {(x,
More informationMath 3361-Modern Algebra Lecture 08 9/26/ Cardinality
Math 336-Modern Algebra Lecture 08 9/26/4. Cardinality I started talking about cardinality last time, and you did some stuff with it in the Homework, so let s continue. I said that two sets have the same
More informationInfinity. The infinite! No other question has ever moved so profoundly the spirit of man. David Hilbert
ℵ ℵ The infinite! No other question has ever moved so profoundly the spirit of man. David Hilbert ℵℵ The Mathematics of the Birds and the Bee Two birds are racing towards each other in the heat of passion.
More information3 The language of proof
3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
More informationCSCI3390-Lecture 6: An Undecidable Problem
CSCI3390-Lecture 6: An Undecidable Problem September 21, 2018 1 Summary The language L T M recognized by the universal Turing machine is not decidable. Thus there is no algorithm that determines, yes or
More informationIntroduction This is a puzzle station lesson with three puzzles: Skydivers Problem, Cheryl s Birthday Problem and Fun Problems & Paradoxes
Introduction This is a puzzle station lesson with three puzzles: Skydivers Problem, Cheryl s Birthday Problem and Fun Problems & Paradoxes Resources Calculators, pens and paper is all that is needed as
More informationStudy skills for mathematicians
PART I Study skills for mathematicians CHAPTER 1 Sets and functions Everything starts somewhere, although many physicists disagree. Terry Pratchett, Hogfather, 1996 To think like a mathematician requires
More informationAffine Planes: An Introduction to Axiomatic Geometry
Affine Planes: An Introduction to Axiomatic Geometry Here we use Euclidean plane geometry as an opportunity to introduce axiomatic systems. Keep in mind that the axiomatic approach is not the only approach
More informationUNIT 1 MECHANICS PHYS:1200 LECTURE 2 MECHANICS (1)
1 UNIT 1 MECHANICS PHYS:1200 LECTURE 2 MECHANICS (1) The topic of lecture 2 is the subject of mechanics the science of how and why objects move. The subject of mechanics encompasses two topics: kinematics:
More informationSolution to Proof Questions from September 1st
Solution to Proof Questions from September 1st Olena Bormashenko September 4, 2011 What is a proof? A proof is an airtight logical argument that proves a certain statement in general. In a sense, it s
More informationAppendix 2. Leibniz's Predecessors on the Continuum. (a) Aristotle
Appendix 2 Leibniz's Predecessors on the Continuum (a) Aristotle Aristotle discusses the continuum in a number of different places in his works, most notably in the Physics (Books 5 and 6), before this
More informationComplex Numbers: A Brief Introduction. By: Neal Dempsey. History of Mathematics. Prof. Jennifer McCarthy. July 16, 2010
1 Complex Numbers: A Brief Introduction. By: Neal Dempsey History of Mathematics Prof. Jennifer McCarthy July 16, 2010 2 Abstract Complex numbers, although confusing at times, are one of the most elegant
More informationPhilosophy of QM First lecture.
Philosophy of QM 24.111 First lecture. WHAT IS PHILOSOPHY? One (passable) answer: The discipline that studies RIGOROUSLY! questions too fundamental to be of interest to anyone else. Examples: Is there
More informationKRIPKE S THEORY OF TRUTH 1. INTRODUCTION
KRIPKE S THEORY OF TRUTH RICHARD G HECK, JR 1. INTRODUCTION The purpose of this note is to give a simple, easily accessible proof of the existence of the minimal fixed point, and of various maximal fixed
More informationThe following are generally referred to as the laws or rules of exponents. x a x b = x a+b (5.1) 1 x b a (5.2) (x a ) b = x ab (5.
Chapter 5 Exponents 5. Exponent Concepts An exponent means repeated multiplication. For instance, 0 6 means 0 0 0 0 0 0, or,000,000. You ve probably noticed that there is a logical progression of operations.
More informationMODULE 1. Infinity. Eternity? said Frankie Lee, with a voice as cold as ice.
MODULE Infinity Eternity? said Frankie Lee, with a voice as cold as ice. That s right, said Judas, eternity, though some call it paradise. Bob Dylan The essence of mathematics lies in its freedom. Georg
More informationAristotle on Space. Physics, Book IV
Aristotle on Space Physics, Book IV The existence of place is held to be obvious from the fact of mutual replacement. Where water now is, there in turn, when the water has gone out as from a vessel, air
More informationBasic Ideas in Greek Mathematics
previous index next Basic Ideas in Greek Mathematics Michael Fowler UVa Physics Department Closing in on the Square Root of 2 In our earlier discussion of the irrationality of the square root of 2, we
More information(1) If Bush had not won the last election, then Nader would have won it.
24.221 Metaphysics Counterfactuals When the truth functional material conditional (or ) is introduced, it is normally glossed with the English expression If..., then.... However, if this is the correct
More information= 5 2 and = 13 2 and = (1) = 10 2 and = 15 2 and = 25 2
BEGINNING ALGEBRAIC NUMBER THEORY Fermat s Last Theorem is one of the most famous problems in mathematics. Its origin can be traced back to the work of the Greek mathematician Diophantus (third century
More informationLecture 2 - Length Contraction
Lecture 2 - Length Contraction A Puzzle We are all aware that if you jump to the right, your reflection in the mirror will jump left. But if you raise your hand up, your reflection will also raise its
More informationLecture 4: Constructing the Integers, Rationals and Reals
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 4: Constructing the Integers, Rationals and Reals Week 5 UCSB 204 The Integers Normally, using the natural numbers, you can easily define
More informationEE 144/244: Fundamental Algorithms for System Modeling, Analysis, and Optimization Fall 2014
EE 144/244: Fundamental Algorithms for System Modeling, Analysis, and Optimization Fall 2014 Discrete Event Simulation Stavros Tripakis University of California, Berkeley Stavros Tripakis (UC Berkeley)
More informationFinal Exam Extra Credit Opportunity
Final Exam Extra Credit Opportunity For extra credit, counted toward your final exam grade, you can write a 3-5 page paper on (i) Chapter II, Conceptions in Antiquity, (ii) Chapter V, Newton and Leibniz,
More informationfrom Euclid to Einstein
WorkBook 2. Space from Euclid to Einstein Roy McWeeny Professore Emerito di Chimica Teorica, Università di Pisa, Pisa (Italy) A Pari New Learning Publication Book 2 in the Series WorkBooks in Science (Last
More informationIncompatibility Paradoxes
Chapter 22 Incompatibility Paradoxes 22.1 Simultaneous Values There is never any difficulty in supposing that a classical mechanical system possesses, at a particular instant of time, precise values of
More informationMATH10040: Chapter 0 Mathematics, Logic and Reasoning
MATH10040: Chapter 0 Mathematics, Logic and Reasoning 1. What is Mathematics? There is no definitive answer to this question. 1 Indeed, the answer given by a 21st-century mathematician would differ greatly
More informationConnectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).
Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.
More informationCountability. 1 Motivation. 2 Counting
Countability 1 Motivation In topology as well as other areas of mathematics, we deal with a lot of infinite sets. However, as we will gradually discover, some infinite sets are bigger than others. Countably
More informationSection 7.1: Functions Defined on General Sets
Section 7.1: Functions Defined on General Sets In this chapter, we return to one of the most primitive and important concepts in mathematics - the idea of a function. Functions are the primary object of
More informationDominoes and Counting
Giuseppe Peano (Public Domain) Dominoes and Counting All of us have an intuitive feeling or innate sense for the counting or natural numbers, including a sense for infinity: ={1,, 3, }. The ability to
More informationRussell s logicism. Jeff Speaks. September 26, 2007
Russell s logicism Jeff Speaks September 26, 2007 1 Russell s definition of number............................ 2 2 The idea of reducing one theory to another.................... 4 2.1 Axioms and theories.............................
More informationCHAPTER 11. Introduction to Intuitionistic Logic
CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated
More informationChurch s undecidability result
Church s undecidability result Alan Turing Birth Centennial Talk at IIT Bombay, Mumbai Joachim Breitner April 21, 2011 Welcome, and thank you for the invitation to speak about Church s lambda calculus
More informationParadoxes in Mathematics and the Meaning of Truth. West Virginia University Benedum Distinguished Scholar Award Lecture
Paradoxes in Mathematics and the Meaning of Truth West Virginia University Benedum Distinguished Scholar Award Lecture Krzysztof Ciesielski Department of Mathematics, West Virginia University, Morgantown,
More informationBasic methods to solve equations
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 Basic methods to solve equations What you need to know already: How to factor an algebraic epression. What you can learn here:
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let
More informationCardinality of Sets. P. Danziger
MTH 34-76 Cardinality of Sets P Danziger Cardinal vs Ordinal Numbers If we look closely at our notions of number we will see that in fact we have two different ways of conceiving of numbers The first is
More informationWhat is proof? Lesson 1
What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might
More informationIntroduction to Logic
Introduction to Logic L. Marizza A. Bailey June 21, 2014 The beginning of Modern Mathematics Before Euclid, there were many mathematicians that made great progress in the knowledge of numbers, algebra
More informationModern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur
Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture 02 Groups: Subgroups and homomorphism (Refer Slide Time: 00:13) We looked
More informationIntroduction to Logic and Axiomatic Set Theory
Introduction to Logic and Axiomatic Set Theory 1 Introduction In mathematics, we seek absolute rigor in our arguments, and a solid foundation for all of the structures we consider. Here, we will see some
More information2.5. INFINITE SETS. The Tricky Nature of Infinity
2.5. INFINITE SETS Now that we have covered the basics of elementary set theory in the previous sections, we are ready to turn to infinite sets and some more advanced concepts in this area. Shortly after
More informationGödel s Argument for Cantor s Cardinals Matthew W. Parker Centre for Philosophy of Natural and Social Science
www.logicnest.com Gödel s Argument for Cantor s Cardinals Matthew W. Parker Centre for Philosophy of Natural and Social Science The Hume Cantor Principle: If there is a 1-1 correspondence between two collections,
More informationOne-to-one functions and onto functions
MA 3362 Lecture 7 - One-to-one and Onto Wednesday, October 22, 2008. Objectives: Formalize definitions of one-to-one and onto One-to-one functions and onto functions At the level of set theory, there are
More informationSequences and Series
Sequences and Series What do you think of when you read the title of our next unit? In case your answers are leading us off track, let's review the following IB problems. 1 November 2013 HL 2 3 November
More informationCHAPTER 5. The Topology of R. 1. Open and Closed Sets
CHAPTER 5 The Topology of R 1. Open and Closed Sets DEFINITION 5.1. A set G Ω R is open if for every x 2 G there is an " > 0 such that (x ", x + ") Ω G. A set F Ω R is closed if F c is open. The idea is
More informationElementary Linear Algebra, Second Edition, by Spence, Insel, and Friedberg. ISBN Pearson Education, Inc., Upper Saddle River, NJ.
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. APPENDIX: Mathematical Proof There are many mathematical statements whose truth is not obvious. For example, the French mathematician
More informationNotes for Lecture 21
U.C. Berkeley CS170: Intro to CS Theory Handout N21 Professor Luca Trevisan November 20, 2001 Notes for Lecture 21 1 Tractable and Intractable Problems So far, almost all of the problems that we have studied
More informationSeminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)
http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product
More informationNotice how these numbers thin out pretty quickly. Yet we can find plenty of triples of numbers (a, b, c) such that a+b = c.
THE ABC Conjecture Mark Saul, Ph.D. Center for Mathematical Talent Courant Institute of Mathematical Sciences New York University I The abc conjecture was formulated independently by Joseph Oesterle and
More informationAlfred North Whitehead (English Mathematician and Philosopher; )
Artwork (cropped) from Shots of Awe YouTube series by Jason Silva Jason Silva (Venezuelan Media Artist, Futurist, Philosopher, 1982 - ) http://thisisjasonsilva.com/ Infinite Algebra Our minds are finite,
More informationLecture 1: Axioms and Models
Lecture 1: Axioms and Models 1.1 Geometry Although the study of geometry dates back at least to the early Babylonian and Egyptian societies, our modern systematic approach to the subject originates in
More informationcis32-ai lecture # 18 mon-3-apr-2006
cis32-ai lecture # 18 mon-3-apr-2006 today s topics: propositional logic cis32-spring2006-sklar-lec18 1 Introduction Weak (search-based) problem-solving does not scale to real problems. To succeed, problem
More informationLeibniz and Cantor on the Actual Infinite
1 A well known feature of Leibniz s philosophy is his espousal of the actual infinite, in defiance of the Aristotelian stricture that the infinite can exist only potentially. As he wrote to Foucher in
More informationThe SI unit for Energy is the joule, usually abbreviated J. One joule is equal to one kilogram meter squared per second squared:
Chapter 2 Energy Energy is an extremely loaded term. It is used in everyday parlance to mean a number of different things, many of which bear at most a passing resemblance to the term as used in physical
More information2. Limits at Infinity
2 Limits at Infinity To understand sequences and series fully, we will need to have a better understanding of its at infinity We begin with a few examples to motivate our discussion EXAMPLE 1 Find SOLUTION
More informationConsequences of special relativity.
PHYS419 Lecture 12 Consequences of special relativity 1 Consequences of special relativity. The length of moving objects. Recall that in special relativity, simultaneity depends on the frame of reference
More informationMath 300 Introduction to Mathematical Reasoning Autumn 2017 Inverse Functions
Math 300 Introduction to Mathematical Reasoning Autumn 2017 Inverse Functions Please read this pdf in place of Section 6.5 in the text. The text uses the term inverse of a function and the notation f 1
More informationA Little History Incompleteness The First Theorem The Second Theorem Implications. Gödel s Theorem. Anders O.F. Hendrickson
Gödel s Theorem Anders O.F. Hendrickson Department of Mathematics and Computer Science Concordia College, Moorhead, MN Math/CS Colloquium, November 15, 2011 Outline 1 A Little History 2 Incompleteness
More informationCHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC
CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called,
More informationWe introduce one more operation on sets, perhaps the most important
11. The power set Please accept my resignation. I don t want to belong to any club that will accept me as a member. Groucho Marx We introduce one more operation on sets, perhaps the most important one:
More informationA proof has to be rigorously checked before it is published, after which other mathematicians can use it to further develop the subject.
Proof in mathematics is crucial to its development. When an idea is formulated or an observation is noticed it becomes necessary to prove what has been discovered. Then again, the observation may prove
More informationConsequences of special relativity.
PHYS419 Lecture 12 Consequences of special relativity 1 Consequences of special relativity. The length of moving objects. Recall that in special relativity, simultaneity depends on the frame of reference
More informationInternational Journal of Pure and Applied Mathematics Volume 52 No , CONSCIOUSNESS AND CYCLICITY OF THE UNIVERSE
International Journal of Pure and Applied Mathematics Volume 52 No. 5 2009, 687-692 CONSCIOUSNESS AND CYCLICITY OF THE UNIVERSE Todor Zh. Mollov Department of Algebra University of Plovdiv Plovdiv, 4000,
More informationPhysics 141 Dynamics 1 Page 1. Dynamics 1
Physics 141 Dynamics 1 Page 1 Dynamics 1... from the same principles, I now demonstrate the frame of the System of the World.! Isaac Newton, Principia Reference frames When we say that a particle moves
More informationNotes on Chapter 2 of Dedekind s Theory of Algebraic Integers
Notes on Chapter 2 of Dedekind s Theory of Algebraic Integers Jeremy Avigad October 23, 2002 These notes are not a comprehensive summary of Chapter 2, but, rather, an overview of the main ideas. 1 Background
More informationThe Limit of Humanly Knowable Mathematical Truth
The Limit of Humanly Knowable Mathematical Truth Gödel s Incompleteness Theorems, and Artificial Intelligence Santa Rosa Junior College December 12, 2015 Another title for this talk could be... An Argument
More informationMA103 STATEMENTS, PROOF, LOGIC
MA103 STATEMENTS, PROOF, LOGIC Abstract Mathematics is about making precise mathematical statements and establishing, by proof or disproof, whether these statements are true or false. We start by looking
More informationMechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras
Mechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras Lecture - 21 Central Potential and Central Force Ready now to take up the idea
More information