Lecture One
The Ever-Evolving Notion of Number
Scope: In this lecture, we will introduce the concept of number and foreshadow an interesting paradox: Although numbers are precision personified. a precise definition of number still eludes us. In fact, one of the central themes of this course is that the concept of number is not a fixed, rigid idea but an ever-evolving notion. As our understanding of the world expands and our capacity for abstract thinking grows throughout history, so too does our view of what number means. We will see numbers move from useful tools for measuring quantities to abstract objects of independent interest. This lecture previews the main themes of the course, from an exploration of the life of numbers to the endless world of the transfinite.
Outline
I. Welcome to a world of number.
A. What is your definition of number?
B. The distinction between number and numbers is subtle.
C. Numbers are at once practical notions in our everyday world and abstract objects from our imagination.
D. Before our ancestors could write, they contemplated quantities.
E. Historically, the study of numbers was a central component of one's education—one of the original liberal arts.
II. Many people incorrectly believe that mathematics is completely understood; most of mathematics, in fact, remains mysterious.
A. Forward progress is extremely slow moving.
B. New discoveries in mathematics are made by building on the work of others who came before.
III. Our knowledge of the early origins of number is vague; we must depend
on relics that archaeologists uncover.
A. Some ancient civilizations recorded their work on materials that stood the test of time.
B. Others employed materials that, over time. disintegrated; thus, our knowledge is as fragmented as the ancient. broken tablets we try to understand.
C. In this course, we will study moments in time to produce a mosaic of small pieces that. when viewed from afar, will allow us to see how numbers grew in our understanding and sparked our imagination.
IV. This course is a blend of mathematics in a historical framework.
A. Although these lectures offer a fluid conceptual development of the notion of number. at times we will gently glide back and forth through history so that we can appreciate and better understand the allure of number as our story unfolds.
B. The course covers three main themes.
C. We will discover that numbers are truly difficult to define precisely, despite what most people believe.
1. We will come to appreciate the notion of number as one that is always evolving.
2. We will also see the recurring theme that what at first appears familiar and commonplace is, in fact, rare and exotic; conversely, what first appeared exotic will later be viewed as the norm.
D. The first series of lectures focuses on early attempts to quantify
1. We will journey back to 30.000 B.C.E. and see some of the earliest attempts to count.
2. Numbers slowly evolved into adjectives (e.g., "three" apples).
3. Counting numbers (also known as natural numbers) became the most familiar numbers (e.g., 1, 2, 3, 4, 5 ...).
E. We will investigate the challenges of communicating and manipulating numbers.
1. Through these investigations, we will see the notion of number expand further, as our ancient ancestors struggled with zero and negative numbers.
2. We will explore how individuals were moved to associate personalities, magic, and even cosmic significance to numerical notions.
F. We will explore numbers in nature and discover how Fibonacci strove to make them more natural. We will then focus on the nature of numbers themselves.
1. By the 6th century B.C.E., the Pythagoreans were studying numbers as objects in their own right, rather than using them , solely as tools for calculation and recordkeeping.
2. Pythagoras may have been inspired by the religious sect in India known as the Jains, whose members may have been the first number theorists. Today, this exploration into the study of numbers is known as number theory.
3. Cultures share their passion for number theory; the more we explore, the more our field of vision of number widens.
G. We will celebrate two of the most important numbers in our universe, π and e, using these famous quantities as the inspiration to see subtle distinctions between different types of numbers.
H. We will consider two mathematical views of number evolution that allow us to expand our notion of number in new directions. We will also encounter "numbers" that challenge our very notion of what number means.
I. We will journey beyond the universe of number and delve into the more abstract world of infinity. Using the very first method for counting. we will discover that, just as with numbers. infinity can be understood and can hold many surprising features.
1. Although our discussions will become a bit technical at some points, those details are not the central focus of this course.
2. Our main goal is the realization that the study of number is a beautiful endeavor that has captured humankind's imagination throughout the ages and continues to inspire us to explore its endless frontier.
Questions to Consider:
1. What is your definition of number? You are encouraged to write down your definition after this lecture to see how it changes throughout the course.
2. For what purpose are numbers used? What types of numbers have you encountered in your life?
Taught by Edward B. Burger
Williams College | Ph.D., The University of Texas at Austin
---------------------------
memo :
・数というのは外界に存在するものではないということ。これは、汎心論*1というものが我々の認識についてより根源的であるということを示唆しているかもしれない。
(Numbers are at once practical notions in our everyday world and abstract objects from our imagination. )
*1 from wiki
http://en.wikipedia.org/wiki/Panpsychism
Panpsychism, in philosophy, is either the view that all parts of matter involve mind, or the more holistic view that the whole Universe is an organism that possesses a mind (see pandeism, pantheism, panentheism and cosmic consciousness). It is thus a stronger and more ambitious view than animism or hylozoism, which holds only that all things are alive. This is not to say that panpsychism believes that all matter is alive or even conscious but rather that the constituent parts of matter are composed of some form of mind and are sentient.
Panpsychism claims that everything is sentient and that there are either many separate minds, or one single mind that unites everything that is. The concept of the unconscious, made popular by the psychoanalysts, made possible a variant of panpsychism that denies consciousness from some entities while still asserting the ubiquity of mind.
これの無意識については少し注意。
The Ever-Evolving Notion of Number
Scope: In this lecture, we will introduce the concept of number and foreshadow an interesting paradox: Although numbers are precision personified. a precise definition of number still eludes us. In fact, one of the central themes of this course is that the concept of number is not a fixed, rigid idea but an ever-evolving notion. As our understanding of the world expands and our capacity for abstract thinking grows throughout history, so too does our view of what number means. We will see numbers move from useful tools for measuring quantities to abstract objects of independent interest. This lecture previews the main themes of the course, from an exploration of the life of numbers to the endless world of the transfinite.
Outline
I. Welcome to a world of number.
A. What is your definition of number?
B. The distinction between number and numbers is subtle.
C. Numbers are at once practical notions in our everyday world and abstract objects from our imagination.
D. Before our ancestors could write, they contemplated quantities.
E. Historically, the study of numbers was a central component of one's education—one of the original liberal arts.
II. Many people incorrectly believe that mathematics is completely understood; most of mathematics, in fact, remains mysterious.
A. Forward progress is extremely slow moving.
B. New discoveries in mathematics are made by building on the work of others who came before.
III. Our knowledge of the early origins of number is vague; we must depend
on relics that archaeologists uncover.
A. Some ancient civilizations recorded their work on materials that stood the test of time.
B. Others employed materials that, over time. disintegrated; thus, our knowledge is as fragmented as the ancient. broken tablets we try to understand.
C. In this course, we will study moments in time to produce a mosaic of small pieces that. when viewed from afar, will allow us to see how numbers grew in our understanding and sparked our imagination.
IV. This course is a blend of mathematics in a historical framework.
A. Although these lectures offer a fluid conceptual development of the notion of number. at times we will gently glide back and forth through history so that we can appreciate and better understand the allure of number as our story unfolds.
B. The course covers three main themes.
C. We will discover that numbers are truly difficult to define precisely, despite what most people believe.
1. We will come to appreciate the notion of number as one that is always evolving.
2. We will also see the recurring theme that what at first appears familiar and commonplace is, in fact, rare and exotic; conversely, what first appeared exotic will later be viewed as the norm.
D. The first series of lectures focuses on early attempts to quantify
1. We will journey back to 30.000 B.C.E. and see some of the earliest attempts to count.
2. Numbers slowly evolved into adjectives (e.g., "three" apples).
3. Counting numbers (also known as natural numbers) became the most familiar numbers (e.g., 1, 2, 3, 4, 5 ...).
E. We will investigate the challenges of communicating and manipulating numbers.
1. Through these investigations, we will see the notion of number expand further, as our ancient ancestors struggled with zero and negative numbers.
2. We will explore how individuals were moved to associate personalities, magic, and even cosmic significance to numerical notions.
F. We will explore numbers in nature and discover how Fibonacci strove to make them more natural. We will then focus on the nature of numbers themselves.
1. By the 6th century B.C.E., the Pythagoreans were studying numbers as objects in their own right, rather than using them , solely as tools for calculation and recordkeeping.
2. Pythagoras may have been inspired by the religious sect in India known as the Jains, whose members may have been the first number theorists. Today, this exploration into the study of numbers is known as number theory.
3. Cultures share their passion for number theory; the more we explore, the more our field of vision of number widens.
G. We will celebrate two of the most important numbers in our universe, π and e, using these famous quantities as the inspiration to see subtle distinctions between different types of numbers.
H. We will consider two mathematical views of number evolution that allow us to expand our notion of number in new directions. We will also encounter "numbers" that challenge our very notion of what number means.
I. We will journey beyond the universe of number and delve into the more abstract world of infinity. Using the very first method for counting. we will discover that, just as with numbers. infinity can be understood and can hold many surprising features.
1. Although our discussions will become a bit technical at some points, those details are not the central focus of this course.
2. Our main goal is the realization that the study of number is a beautiful endeavor that has captured humankind's imagination throughout the ages and continues to inspire us to explore its endless frontier.
Questions to Consider:
1. What is your definition of number? You are encouraged to write down your definition after this lecture to see how it changes throughout the course.
2. For what purpose are numbers used? What types of numbers have you encountered in your life?
Taught by Edward B. Burger
Williams College | Ph.D., The University of Texas at Austin
---------------------------
memo :
・数というのは外界に存在するものではないということ。これは、汎心論*1というものが我々の認識についてより根源的であるということを示唆しているかもしれない。
(Numbers are at once practical notions in our everyday world and abstract objects from our imagination. )
*1 from wiki
http://en.wikipedia.org/wiki/Panpsychism
Panpsychism, in philosophy, is either the view that all parts of matter involve mind, or the more holistic view that the whole Universe is an organism that possesses a mind (see pandeism, pantheism, panentheism and cosmic consciousness). It is thus a stronger and more ambitious view than animism or hylozoism, which holds only that all things are alive. This is not to say that panpsychism believes that all matter is alive or even conscious but rather that the constituent parts of matter are composed of some form of mind and are sentient.
Panpsychism claims that everything is sentient and that there are either many separate minds, or one single mind that unites everything that is. The concept of the unconscious, made popular by the psychoanalysts, made possible a variant of panpsychism that denies consciousness from some entities while still asserting the ubiquity of mind.
これの無意識については少し注意。