INFINITY and the IMAGINATIOIN

 

POESIS

 

“Infinity is a fathomless gulf into which all things vanish.”  Marcus Aurelius. 121-180.  Roman Emperor and Philosopher

 

“Infinity is where things happen which can’t.”  Young student quoted by Arthur Koestler in the Act of Creation.

 

“Infinity is a dark illimitable ocean without bound.”  Milton.  1608-1674.  English Poet

 

An infinite set is one that can be put into one to one correspondence with a subset of itself.  George Cantor  1845-1918.  German Mathematician.

 

“Infinity converts the possible into the inevitable.”  Norman Cousins.  April 15.  1978

 

The universe is not bounded in any direction.  If it were, it would necessarily have a limit somewhere.  But clearly a thing cannot have a limit unless there is something outside to limit it.   Lucretius.   99-55 BC

 

Who can number the sand of the sea, and the drops of rain, and the days of eternity?  Ecclesiastes 1:2

 

“Oh moment, one and infinite!”  Robert Browning.  Poet  1812-1889

 

To see a world in a grain of sand

And a heaven in a wild flower,

Hold infinity in the palm of your hand

And eternity in an hour.   William Blake.  Poet, artist.  1757-1827

 

It was a dark and stormy night Images.… 

 

We must not take the One as infinite in extent, but rather as infinite in fathomless depths of power.  Plotinus .  3rd century mystical philosopher.  Ennead 6.9.6

 

The One cannot be measured: it is the measure, not the measured.  Ennead 5.5.11

 

Nothing bothers me more than the possible infinity of space and time.  Yet, nothing bothers me less, as I never think about them.   Anonymous joke.

 

 

LONGER REFERENCES:

Archimedes:  The Sand Reconer

Dedekind: on the irrational

Hwa Yen: Fa Tsang’s hall of mirrors

 

 

A.     ANCIENT  QUESTIONS

 

COSMOLOGY SPACE AND INFINITY

 

From ancient times people asked: how did the universe begin?  Where did it come from?  They looked up into the night sky, wondered at the multitude of stars, and about what was beyond the stars.  Did the universe go on forever, or did it end? They pondered the composition of our Earth and physical world: were there some minute particles which made up the world?  And as they thought about the universe, and imagined about its source and endlessness, they wondered also about themselves-- the unlimited nature of their own imaginations, and their thoughts.

 

If you imagine traveling through space toward the supposed boundary of the universe, what would you expect to find?  It is mind boggling to imagine that the universe is endless, actually infinite.  And it is inconceivable to suddenly meet with a boundary to the universe.  If there is a boundary to the universe, what is beyond the boundary?  Since the universe is “everything”, there can’t be anything beyond it.  So why is there a limit or boundary?   Is beyond the boundary the emptiness of the Void, absolute nothing, or the pure freedom of God?  These themselves are good candidates for infinites.  And if there is no boundary, and the universe goes on forever, infinitely, this is beyond imagination. ?   Either conclusion: that the universe is somehow limited, or that it is actually infinite, are frightening, or impossible, or paradoxical.

 

Questions about our own world, planet Earth, gave people some ideas about the universe.  If you went very far, you came to the oceans.  What is beyond the ocean?  Although somewhere along the line European culture forgot that the earth is round, and Columbus had to recall this fact to us in 1492, the Greek Eratosthenes measured the circumference of the earth before 300 BC to a great degree of accuracy.  This is a remarkable feat.  First of all, he had to know it was round, without going around it, and secondly he used some clever geometry to estimate its circumference. 

 

If you look at a ship leaving at sea, and the water is very calm, and the Earth is flat, you should see the whole ship receding into the distance.  But as the earth is round, you see the bottom of the ship disappear and the mast sticking up from the horizon longer. 

 


Eratosthenes started at a point A on earth where the sun was directly overhead at noon.  The next year, he observed the sun on the same day from a town B some 400 miles north.  Using basic geometry, and supposing about the sun’s rays are parallel, we find that the angle the sun deviates from direct overhead is equal to the central angle which the arc AB subtends (see picture).  A  simple ratio gives us that 6/360 = 400/Circumference.  With these numbers, we get Circumference = 24000.

 

 


When one looks at the sky during the day, the sun seems to be moving, and the stars seem to move across the sky during the night.  It looks as if there is a huge sphere carrying the heavens along.   If the universe is rotating, and there is no “empty space”, then what is the shape of the universe?   If we consider a bug which lives on just on a circle, then the bug can go only forward or backward.  As it moves forward, it encounters no boundary, but it is in a finite universe. 

 

Similarly, if we are in a spherical universe, then we can go on forever without hitting a boundary.  At some point we sees ahead a very familiar sight: the earth from which we originated.  We have come back to the origin, although we traveled in a “straight” path, a “geodesic.”  It will be important to say more about this idea of “straight” since it is intrinsically bound up with the question about the shape of space. 

 

We may also use dreams to see how the universe can be contained and yet unlimited.  If you wake up in a dream , or find yourself in a dream, you do not know how or when it began.  You were not there.  And the you in the dream experiences trees, a world, of the imagination.  If you begin traveling, the dream world is as vast as your imagination.  Although the universe of the dream is not “there” in an objective sense before you explore it, as you do explore you find it has no boundary.  Moreover, when you wake you find that the entire dream was “in” your imagination, even the dream starts, trees and so on.  Whereas the dream contents were, at any moment, finite, the imagination or mind of the dreamer was not.  And from the perspective of the person really having the dream, it took place within a finite time.  An entire sequence of years of events in the dream leading up to the alarm ringing in the dream was precipitated by the beginning of the alarm ringing, perhaps 30 seconds of wakeful time.

 

Several possible ways to think about the universal space/time can be imagined:

 

Unbounded and infinite, unending, eternal. 

Bounded and finite. 

Bounded but infinite, in the sense of an infinite number of infinitesimal parts or particles. Unbounded but finite, like traveling in a circle.

 

 

 

Exploration 1

 

1.   a.   What does infinity mean to you?  Write or draw pictures about two meanings of infinity.

  1. Is there anything in your experience which is infinite?  Explain or give a description.

 

2. In what sense are each of these finite/infinite:

1/3   

The border of Cayuga lake.

A circle.   

The graph of y = x3 .

The universe

Your mind.

 

3.  a.  An ancient debate about the divisiblity of matter led to speculation about the nature of atoms.  If you imagine dividing a line segment in half, then in half again and so on, what happens "eventually"? Think about the question in terms of your imagination and also in terms of a piece of paper that you keep cutting in half.  What kind of a geometric object will you get if you imagine continuing to cut a line segment in half indefinitely? 

 

b. A related problem in the number realm concerns .9.  Many people think (believe) that .9 = .999...   is not equal to 1 and some forms of ("non-standard” analysis allow .9 ≠ 1.  However most mathematics books say that .9 = 1.   Which seems most reasonable to you?  .9 = 1 or .9 ≠ 1?  What are the most convincing arguments for your answer?  Make as explicit as you can what assumptions you are making and why you accept these assumptions as true.

 

4.  How would YOU answer the argument in Zeno’s Achilles and the Tortoise dilemma?  What happens when a thought process or thought model like Zeno gave seems to not fit the physical world?  What are the assumptions being made to say Achilles does not reach the tortoise? 

 

5.  How many pieces of paper would you need to pile up to make a pile as high as the Moon?

If you fold a piece of paper in half, and then in half again you get 2,4,8, … pieces of paper thick.  How many times would you need to FOLD a piece of paper to get a pile as high as the Moon?

(You can consider that the moon is 400,000 km from Earth)

 

6.  How many earths would fit into a sphere the size of the sun? 

How many earths would fit into a sphere with diameter equal to the diameter of the Earth’s orbit around the Sun?

If you make a scale model of the solar system and you make the sun 1 meter in diameter, how far is the Sun from the Earth?

(You can consider that the Sun diameter is 100 times the Earth, and the Sun distance to the Earth is about  100 times the sun diameter.  The moon diameter is about ¼ of the earth, and the moon distance to Earth is about 100 times its diameter.)

 

If an atom was as big as the earth, how big would the earth be?   (You may consider that an atom is on the scale of 10–10 meters in diameter.

 

INFINITY in a cultural context

The Greek word aiperon , which we usually translate as “infinite” means disordered, chaos, indefinite, indeterminate, and also was slang for a crumpled piece of stuff.  It was not a positive meaning. 

 

For Aristotle in 5th century BC, “being infinite is a privation, not a perfection but the absence of limit.”  Aristotle also had a “fear” of the Void, of a vacuum, of the conception that there would be anywhere which was really and totally empty.  Questions about space and time being infinite or unending, can be asked in terms of the number system.  Are the whole numbers infinite?  In what sense? Aristotle skirted the issue by saying that the integers were potentially infinite, but not actually so, as at any point you could only count finite.  He was quite a pragmatist.

 

Plato, however, says that “God exhibited the limit and the unlimited.”  So he takes it that unlimitedness is a divine quality.

 

In the Vedas, much more than 3000 years old, the sages state that Brahman is the full (purna), and whatever the Absolute Brahman manifests does not affect Brahman, for taking the full from the full leaves the full.  You can see how this can be represented mathematically: taking all the odd numbers from the numbers leaves an infinite number of numbers. And in the Rig veda:  “Whence is this universe and who created it?  The Gods are later than this world’s production, and so cannot say its origin.  The one who lives in highest heaven, who alone precedes the universe, alone knows about its origin.  Or perhaps, that one also knows not.”

 

Another key notion here is Measure.  For Hindu philosophy, for example, the universe is “measured out” from the Infinite.  The name for the principle of manifesting, Maya, means measure.  In the Timeaus of Plato, the Infinite divinity--the Demiurge-- measures out the World Soul and the manifest universe.  This is all accomplished through the word for ratio: logos.  The world soul inserts ratios into the living stuff of its own being in order to manifest space and time.  And “in the beginning was the word, and the word was with God…” the word for Word is also Logos.  So is some view the universe is finited out of the infinite.

 

Although the universe may be finite, that which gives rise to the universe need not be.  Attributes of The One of Plotinus, the Brahman of the Upanisads, the Tao, are infinity, timelessness, omnipresence. And for other traditions, the absolute is empty, sunya, Void.  We will see how the Infinite and the Zero are intimately connected.

 

Chuang Tsu presents the idea of the “absolute necessity of what has no use” in a short story with that title.  At any given time you stand on one square meter of earth.  But take away the rest of the Earth you are not standing on right now, and you will face a terrifying void, rendering the square meter you are on right now useless.  Thus, Chuang points out, is the absolute necessity of what has no use.

 

In one sense, all of our finite math depends on the infinite, on there being no end to the system of numbers.  So we are actually safe using as large number as we like.  It is like saying that the images in the dream in any given dream are finite, but the mind which dreams is not limited or bounded by the particular dreams. 

 

If our own mind and imagination can think about these questions, does that mean that we are infinite?  By the principle of reflection, does that mean that the infinite actually exists? 

 

 

UNCOUNTABLY LARGE: ARCHIMEDES SAND RECONER

Questions about the universe being endless can be made more precise in terms of the counting numbers.  There seems to be no last counting number (if there were, just add 1 to it), yet we cannot easily grasp the entirety of the counting numbers.  If you start with 2 and double, double, (no trouble) what will happen?    Is there any numerical end to the doubling? You should also think about this geometrically: repeatedly scaling a picture by 2.   Even imagining very large numbers is difficult for the imagination.  Can you really picture a million dots?  A million anything?  Many early cultures had names for  1,2,3 and then just said “many.” Any number beyond their counting system was therefore large, or uncountably finite.  Many people, up to 5th century BC, thought that the number of grains of sand were infinite.  

 

The great 3rd century BC mathematician Archimedes, in his little work “the sand reconer” gives a remarkable sequence of  stages of scale: grains of sand in a grain of rice, grains of rice in a foot, feet in a mile, miles in the earth (known through Erotosthenes) and even the distance to the Sun.  Archimedes fills the whole known universe with Sand and still calculates a finite, but very large, number.   Maybe it is on the order of 10100.  But it is finite.  In the Vedas there is made use of a number 10140 .  This is larger than the number of atoms in the known universe.  So in what sense is it actual? 

 

TIME

Similar questions apply to time as well.  See four views of time in ancient philosophyFor Plato, time was a moving image of eternity.  Later on, especially in the 1st-10th centuries, mystical philosophy also thought more deeply about the nature of Reality and the infinite.  Fa Tsang hall of mirrors.  Tibetan Buddhism.  Plotinus and Proclus.  Eckhart.  For Plotinus, time was the motion of the (divine) soul. For Augustine (?), the God was  sphere whose circumference was nowhere, and center everywhere.   See my four views of time:  as psychological, cosmological, geometric, and cyclic.

 

PROBLEMS:

7.       How long ago is 1 million, 1 billion, 1 trillion years?

8.      A king gives as a prize a grain of rice on the first day, and double that for every day in a month.  How many grains of rice is that after 30 days? Would that much rice fill the classroom?  The Earth? 

9.       Are the counting numbers actually infinite?  They don’t end.  But each number you can name is finite.  Are the even numbers infinite?  They also don’t end. But the even numbers is a subset of the counting numbers.  Aren’t there fewer even numbers?  This leads to a paradox about infinite sets

.

CAUSALITY AND INFINITE REGRESS

Causality also presented a problem.  If this moment is caused by a previous one M2, then that must have its cause in the previous M3 and so on.  This is called an infinite regress.  [There is a famous story about a person at a lecture:  turtles all the way down.]

 

Heraclitus, in 5th century BC, said you cannot enter the same stream twice, as each moment the entire universe has changed.   In fact, you cannot even enter the stream once.

 

VERY SMALL: THE INFINITESIMAL Matter

If one looks at the very small, one is also led to the infinite. Is there a smallest particle?  What is the constitution of the universe? The search for a smallest particle, or for the ultimate constituent of universe, for the nature of motion, leads either to the exploration of infinitesimals, or to divisibility to 0.

 

Democritus argued that there are smallest particles, atoms.  But can not these atoms be divided further? 

Whenever you have a small interval, such as 0 to 1, you can divide it in half.  You  can keep doing this.  Does this imply that there are an infinite number of decreasing intervals?  What happens eventually? 

 

Archimedes helped make this physical question quite precise mathematically, using the concept of measure.   He felt that there was no particle which has measure 0.  Acceptance of  his “Axiom” about measuring segments is at the heart of  2000 years of exploration which followed Archimedes. 

 

Axiom of Archimedes:

No matter how small a particle P, and no matter how large a measuring unit U, we can always find a counting number N so that multiplying the length of P by N will be larger than U.

 

For example, if a particle is only 1/1014 meters in length (about the size of a nucleus of an atom), then simply multiply by 1015 (or line up that many atoms in a row) to get longer than a meter.

 

Does anything have 0 measure?  Is there some infinitesimally small particle, which results from dividing a line in half, in half again, and so on forever?  If there is, it cannot satisfy the Axiom of Archimedes.  Because if an infinitesimal blip has length 0, then multiplying it by any integer, no matter how large, will give 0: by the axiom that 0 x N is 0, for all whole numbers N.  So you must give up that Archimedes axiom, or you must give up that 0 x N must be 0.  

 

ZENO’S PARADOXES

Zeno (5th century BC) gave famous examples to show the paradox of motion, also based on the problem of the infinitely small.  Achilles goes faster than the Tortoise, but cannot catch her, because first he must get half way to her, then 3/4 of the way, then 7/8 and so on.  An arrow cannot really move in getting from  A to B.  Because at any moment we see it the arrow cannot move to the next instant.  If it did, it would have to move halfway, and half of that etc.  And you cannot leave this room.  For the same reason.

[Let’s stop here to appreciate what the problems are.  Draw pictures. Etc.]

 

Do these processes actually yield infinity or infinitesimals, or only ‘potentially” infinite numbers or sets, as Aristotle thought, or is this only a semantic distinction, as Cantor, the 19th century mathematician who went far to solve some of the problems, thought? 

 

 

 

B. INFINITY and PROCESS in GREEK MATHEMATICS  6th – 4th c. BC

 

Although the Greeks did not seem to like the infinite, they encountered several situations where the infinite was either absolutely necessary, or at least was very useful. Before Pythagoras, the Greeks thought, or perhaps they thought, that all of mathematics could be encompassed by the finite, or at least by whole number ratios.  Everything was ratio.  But Pythagoras, Eudoxus, Euclid, Archimedes, Applonius--the greatest mathematicians of the golden age of Greece in 6th to 4th century BC—actually considered the infinite, grappled with the problems through mathematical explorations.   They circumvented the inability to grasp the infinite in the large or small directly, in one gulp, by attempts to grasp it through potentially infinite processes or repeated activities of getting larger and larger or smaller and smaller.  Through their work in the realm of mathematics, they helped to make sense of practical problems, as well as helping us to think more clearly about philosophic ideas of infinity. 

 

We will explore some of the following major developments about infinity from the 6th-4th c. BC:

 

 

A prime number is a number which has only itself and 1 as divisors.  Are the number of primes finite or unending?  How could one prove a set is infinite, without counting all examples?  Euclid gave a proof about the infinity of the number of primes in the 5th century BC. And Eratosthenes gave a method called a “sieve” to find all prime numbers.

 

A straight line seems like a straightforward finite object.  But: can a line can be extended indefinitely?  Euclid’s definition of a straight line assumed it could.

 

What is the relation of the diagonal of a square to its side?  Can you find a measuring unit small enough so that it will exactly measure the diagonal and side a whole number of times?  Can you express the relation of side to diagonal as a whole number fraction?  Pythagoras explored this question in the 6th century BC and found a dismaying answer which invoked the infinite.

 

How to you find the area of a circle, or other curved regions, in terms of square units?  What is the value of pi, the ratio of circumference and diameter of a circle?   Archimedes answered these questions in the 4th century BC, through using a potentially infinite process.

 

 

Theorem of Euclid: The area of any triangle with a given base and height is the same. 

So: What if we allow the base to stay fixed at one unit, and move the third vertex way far away along a line parallel to the base.  How can the area stay fixed? 

 

 

THE IRRATIONAL/INCOMMENSURABILITY

We can try to fill in or name all the points on the line with ratios of whole numbers, which can be constructed individually or all at once by a remarkable method using the number lattice.  It can be shown that between every two rational numbers there is another.  This is called the property of being dense.  But this means that if r is a rational number, there is no “next” rational number. 

 

PROBLEM 11:  Show how to find a rational number between any two given rational numbers r and s. 

 

Pythagoras, in the 6th century BC, understood that there are numbers, or relations, which are very finitely and easily constructible, such as the diagonal and side of a square, but which cannot be represented by whole number ratios. Another way to say this is that if there is a square with side length 1 unit, the diagonal cannot  be represented as any finite ratio of two whole numbers.  Its length does not fall on any of the rational number points on the number line.  So there are numbers which are NOT the ratio of any two whole numbers.  Pythagoras and his group kept this idea secret for some time. The existence of so- called “irrational” numbers was as a tremendous blow to finitist views held by some of the mathematicians of the time.  .

 

Let us explore this problems a little further.  It can be shown that the diagonal of a square d with side 1 is such that d2 is 2.  In other words, d is the side of a square with area 2.  You can then try to express d as a ratio of two numbers m and n.  By a proof by contradiction, or by exhaustion, or by self-similarity, you will see that it is impossible.  Thus the length d does not correspond to any point on the number line with rational name, even though these points are dense.  And even though, in fact, one can write down a sequence of rational number which get as close as one likes to d.  In fact, one can write as many elements as one likes of a set, all of whose members is less than d, but for which d is the smallest number not in the set.  Thus, although d names a point on the line not hit by a ratio of whole numbers, we can approximate d as closely as we like by a sequence of rational numbers.  This fact lead Dedekind, 2000 years later, to formulate a careful and precise definition of irrational numbers as limits of sets of rational numbers.

 

Embedded in proofs of the incommensurability of the side and diagonal of a square is the notion of self-similarity: if something is self similar (that is, identical to a magnified piece of itself) then that thing must be infinite.  (see the pictures here, land of lakes, mirrors and so on.)  Self-similarity is crucial for an understanding of the infinite in a mathematical sense

 

PROBLEMS

12.  How do you know that every whole number division either ends or repeats?  For example, 1/5 = .2  But .3 = .333…

13. What does it mean that there is no rational number corresponding to √2?  Do the divide and average algorithm for √2.  Show that the sequence of numbers gets closer to √2.

14  What happens when you take √ over and over?  Try starting with at least three different numbers, using your calculator.  Note how long it takes.

15    Large and small perimeter problems.  An attempt to measure a two dimensional region using length results in infinite measure.  An attempt to measure one dimensional length as area, results in 0.  Explain this giving your own examples and pictures. 

16    To get a potentially infinite perimeter with finite area, imagine iterating the process of dividing a paper in half and adding it on to itself to get larger and larger perimeter.  Show that the perimeter can be made greater than any given whole number of units.

17    How do you find the area of a curvy figure such as a circle, using square units?  Come up with at least two ways of your own, before reading the method of Archimedes.

18    Explain what Pi is.  Explain Archimedes’ itertive  process by which he could (potentially) estimate pi to any desired accuracy.  


WHOLE NUMBER LATTICE AND COUNTING.

We can get a picture of this relation of rationals and irrationals in another way.  Consider the whole number lattice: points on the plane with coordinates whole numbers.  We can correlate each fraction m/n to a point in the number lattice (n,m) and the slope of the line joining the origin and the point (n,m) is m/n as well. 

 

PROBLEM

19: Draw a line through the origin and a point (n,m) in the lattice. 

a.         Prove that the line must go through an infinite number of points in the lattice.  

b.         Prove that, any other point (p,q) in the lattice that is on the line has the same fraction value n/m. 

c.          Prove that if you look from the origin down a line which goes through some point in the lattice, the point seen first represents the ratio of slope of the line in lowest terms, say 2/3, and all the other points have coordinates which reduce to the ratio 2/3. 

 

20.   Draw line L vertically at x=1.  Prove that a line OM through the origin and any point M(n,m) will intersect L at the point  (1,m/n).     Thus we can construct all fraction lengths along the line L. 

 

21.  Can there be any lines through the origin which do not pass through any of the other points on the whole number lattice?  If so, how would you describe one?  If not, why not? 

 

COUNTING THINGS

If you want to count things, you can list them.  By doing this you are essentially putting down one thing for each successive counting number.  So you are making a 1-1 correspondence between counting numbers and your set under investigation.  We can use the 1-1 principle to tell if two sets have the same number of elements, even without counting them!  For example, if I have a large number of shoes in the closet, I can know if the number of left and right are the same even without knowing how many of each there are.  So we can compare the magnitude or number of sets by putting their elements into this 1=1 correspondence.  What happens when we do this with infinite sets?   For example, Gallileo showed that every square number can be corresponded to the whole number it is the square of, and so the whole numbers and squares are just as numerous.

 

1          2          3          4          5          6          7          8          9          …

1          4          9          16         25         36         49         64         81         …

 

 

So we have put a 1-1 correspondence between a set and one of its subsets.  George Cantor used this property as his definition of infinite: a set is infinite if it can be put into 1-1 correspondence with a subset of itself.  He then went on to show that there are an infinite number of different orders of infinity, as we shall see But not every whole number is a square. 

 

Hilbert, 19th- 20th century mathematician, ran a famous hotel.  It had an infinite number of rooms numbered 1,2,3,…   When a busload of 40 people came to visit, even though the rooms were all full, he had no problem  He moved each guest into a room with number 40 higher than they started, leaving the first 40 rooms empty for the new guests.  Then a busload with an infinite number of guests wearing numbers 1,2,3,… came to town.  He moved each of the already present guests to a room which had an odd number: double plus one the number they started with.  He had each of the new guests move into an even number room  which was double the number on their t shirt.  This left room 1 for himself. 

 

THE WHOLE NUMBERS ARE JUST AS NUMEROUS AS THE RATIONAL NUMBERS

There is a Systematic way to list or name all of the rational numbers.  It “proves” that the number of rational numbers or fractions is the same as the number of whole numbers. 

 

List all fractions whose num. and den. sum to, successively, 1,2,3,4,5,…N...  Within each group order the fractions by numerator from 0 to N-1.

So the Rational numbers are, in order, 0/1, 0/2, 1/1, 0/3, 1/2, 2/1, 0/4, 1/3, 2/2, 3/1,…

 

Can we do this with the irrational numbers?  How do we describe all the numbers which fill in the “gaps” in the rationals?  Can we fill in the gaps?  Can points ever fill in a line, which has dimension? It was George Cantor in the 19th century who formulated clearly the view that the set of all the irrational numbers could NOT be put in 1-1 correspondence with the whole numbers: there was no way to make a list of all the irrationals as you could do with the rationals.  And there was not only the infinity of the whole numbers, but there were other orders of infinity. And if there was more than one level of infinite, there were infinite infinities.  And perhaps an Absolute infinite.

 

The idea of 1-1 correspondence created problems in geometry.  Consider two concentric circles with radii 1 and 2 inches repectively.  One circumference is double the other.  But on the other hand, draw radii from the center through any point on the smaller out to the larger, or from any point on the larger through the smaller to the center.  You see easily a 1-1 correspondence between points on the two circles.  You can also show a 1-1 correspondence between points on the entire line and the semicircle.  However, you have to leave out the endpoints of the semicircle, because otherwise it is too big to be put in correspondence with the (infinite) line. 

 


 


 

 

 

 

 

 


PROBLEMS:

22. Show how to put the points of a one inch line in 1-1 correspondence with points on a two inch line.

23.  Show that:  Between every two rational numbers is an irrational.  Between every two irrationals is a rational.  But there are many more irrationals: in a sense we will show later.

 

C. ZERO AND THE INFINITE.  SELF-SIMILARITY INFINITE

 

The next mathematical jump really involves the “invention” and use of the 0, sunya or cipher.  The mathematics of the infinite in this time period is parallel to the  attempt to grapple with the infinite in the domain of the mystical and religious philosophy, the great speculative debates between the Hindu and Buddhist worlds in north India, and creative minds in China, Tibet and south Asia.  And with the infinitely complex visions of the Sufi masters in the middle east.  And the continued explorations of the Infinite, Ain Soph, of the Kabbalists.  And the Scholastics grappling with the infinite mind of God. 

 

 

READ:

Fa Tsang:  “The Hall of Mirrors” and the excerpt of “On the Golden Lion”

 

Krishnamurthi:  “Advaita and Mathematics”

 

St. Augustine.  Excerpts from “The Vision of God”

 

D. INFINITE SEQUENCES AND SERIES

From there we move to the 17th century:  three great and connected ideas.  The calculus of infinitesimal tangent slope, the calculus of summing more and more smaller and smaller parts, and the notion of the series. 

 

In the “infinite” sequences and series of Wallis, Euler etc. a developing notion of a limit.  Infinite series  which converge to pi.  We have to explain these terms: converge, infinite series, and limit. 

 

1671:  Gregory:   1 – 1/3 + 1/5 – 1/7 + - …    = π/4

1736   Euler         1/12 + 1/22 + 1/32 +1/42 + …   = π2/6

 

In the calculus of Leibnitz we have a revival of the infinitesimal idea: a tiny segment with no length, but able to form a ratio to other infinitesimal segments such that dy/dt makes sense.

 

Newton and Leibnitz both formalize the insights of Archimedes, 2000 years earlier, that you can take more and more of smaller pieces of a region to find its area.  And with this we can approach the questions of motion left by Zeno.  If a function v(t) represents the velocity of a particle in terms of time t, how far has the particle traveled from time a to b?  The answer can be represented as the area of a region in the plane bounded by the graph of the function and the axis.

 

[Here we need to visit some remarkable properties of sequences and series.  We also went to look at them as generated by recursion and multiple copy machines. 

And the idea of a limit.

And: does a line straighten or not.

 

PROBLEMS:

Explore the convergence of these canonical series:

 

Geometric series with ratio 1/2:   1/2 + 1/4 + 1/8 + …

 

Harmonic series:  1/2 + 1/3 + 1/4 + …