math 1700: the nature and growth of ideas in mathematics · math 1700 review of fall 2008 1 math...

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Math 1700 Review of Fall 2008 11

Math 1700: The Nature and Growth of Ideas in Mathematics

Where we are now, with a capsule review of the main points in the course in September and October – before the strike


Our new timetableWe have three remaining classes in the “Fall” term: February 3, 10, & 17.Then there follows a 12 day fall exam period. There is no exam in this period in this course, but you may have exams in other courses at this time.We resume on March 10 and continue without interruption every Tuesday night until our last class on May 19.Then follows the spring exam period from May 22 to June 2. Our final exam will be during this period.


Significant dates:

March 31 – the 2nd in-class test, worth 20% of the mark.April 3 – last day to drop the course without receiving a grade. Note that results from the March 31 test will notbe available by then.

You will have only the first test mark and the fall participation mark (which will be posted during the “fall” exam period).

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New schedule of lectures

Take note of the new lecture and assignment schedule, handed out in class, or available on the course website, www.yorku.ca/bwall/math1700

This is quite different from the original schedule.


New optional marking schemeThe original marking scheme for this course was:

Test 1, 20%Test 2, 20%Class participation, 20%Final exam, 40%

The new optional scheme will be:

Test 1, 25%Test 2, 25%Final exam, 50%

Your course mark will be calculated both ways and you will automatically get the higher of the two.


A review of September and October

Recall that my thesis was that mathematical thinking is the backbone of the organization of our society:

That mathematical ideas pervade every aspect of our culture: commerce, science, government, even the arts.That it is mathematics that makes human civilization strong and resilient.

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Mathematical thinking is apparent from the earliest human cultures.

The basic mathematical abstraction, counting, is evident in the earliest human remains.E.g., the Tally stick where some count is represented by notches on a stick.


Early written number systems and rules for calculations: Egypt

Ancient Egypt had a decimal (10-base) system, but numbers were written without place value. A separate, unique symbol was used for each different order of magnitude, units, tens, 100s, 1000s, etc., as above.


Early written number systems and rules for calculations: Egypt, 2

Written numbers were unambiguous, but cumbersome to write in formal hieroglyphics.The same symbol had to be repeated again and again for multiples of the same order of magnitude. E.g., 7 different horseshoe symbols in the number 276.

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In Hieratic, the everyday shorthand writing system used by the scribes, it was even more complex as different symbols were introduced for every value in each order of magnitude. E.g. different symbols for 4, 5, 40, 50, 400, 500, etc.



Though the writing of particular numbers was cumbersome, it was exact. Egyptians developed an efficient method of performing multiplications and divisions using a system of successive doubling and then adding the doubled quantities together.



Multiplication: 13x24

Division: 300 ÷ 14

312

192192

8

96964

482

24241

294

224

56

14

21

22416161128

5644

282

1411

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Early written number systems and rules for calculations: Babylonia

In Mesopotamia and the later Babylonia, writing developed as a system of scratches in wet clay that was left to dry.Written numbers were marks made with a stylus in combinations of two shapes, a vertical stroke, indicating one, and a horizontal stroke for ten.


Early written number systems and rules for calculations: Babylonia, 2

Curiously, the number system chosen was 60-based, or sexagesimal.Likely the reason for chosing 60 as their base was that 60 can be factored evenly by so many different numbers. Fractions, and division, were then relatively easier to express.



Using the symbols for one and ten, combinations of these made up the basic numbers from 1 to 59

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The Babylonian system used the same numerals for different orders of magnitude, unlike the Egyptians.They had a place value system (as we do), but without a symbol for zero or a decimal point, their numbers were ambiguous.


The route to mathematics proper

Egypt, Babylonia, and other ancient cultures became proficient in arithmetical calculation and in using, for example, geometric ideas in construction and surveying, but showed little interest in developing an abstract body of knowledge.That appeared to begin in ancient Greece.


The roots of mathematics

The earliest recorded ideas of mathematics as a unified body of knowledge appear among the writings of those credited with the origin of western philosophy: the pre-Socratic Greek philosophers.These were remembered for asking questions such as, “What is the world made of?”

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The Pre-Socratics

Among the noteworthy pre-Socratic philosophers, who attempted to answer these basic questions are:

Thales Anaximander Anaximenes Heraclitos


The predominance of logic

The pre-Socratics were noted for attempting to find the truth about the world through reason. They developed the art of critical thinking to a high level.The abstract and precise version of reasoning is logic. The use of logic brought difficult questions that defied easy answers.


Parmenides and ZenoAmong the most influential were Parmenides and his (supposed) student Zeno from the Greek colony of Elea.

Parmenides Zeno

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The law of contradiction

Parmenides argued that you can use the existence or non-existence of anything in an analysis, because something either is or it is not, it cannot be anything else.

Example: There can be no such thing as empty space, because space is something and empty is nothing.


Zeno’s paradoxes

Zeno continued from where Parmenides left off, making his master’s points much easier to grasp by expressing them as a number of paradoxes:

The stadium

Achilles and the tortoise

The flying arrow


The mathematical view of Nature

One significant viewpoint that emerged from Greek philosophy had special importance for the role it gave to mathematics in understanding Nature.This view gave paramount importance to the mathematical structure that was seen to underly Nature.There are two chief versions of this:

The extreme view of Pythagoras.The Platonic view of forms.

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The Pythagorean viewpoint

Pythagoras of Samos lived from about 580-500 BCE.He established a cult of followers who took vows to follow the beliefs and customs of Pythagoras.


Everything is Number

The Pythagoreans viewed number as the underlying structure of everything in the universe.Pythagorean numbers take up space.

Like little hard spheres.Everything in Nature was number.This view was supported by some surprising discoveries of mathematical structure where it was not previously suspected.


Numbers and Music

Pythagoras discovered that the means to produce sounds that would be considered harmonious bore a simple numerical ratio to each other, such as the length of vibrating strings or of vibrating lengths of metal (or columns of air).

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Number magicPythagoras began to look for numbers with special meaning, that described some inexplicable feature of the universe.Where he did not find them readily, he began to imagine them, because they must be there, in his view.

Of special significance was the number 10, called the tetractysbecause it could be formed into a triangle with four on each side.


The unusual Pythagorean Cosmos and the Tetractys

So convinced was Pythagoras that the tetractys was the key number of the universe, that he postulated an unseen tenth heavenly body, the counter earth, or antichthon, always on the other side of his central fire, and invisible to human eyes.


The Pythagorean TheoremOne of the strongest supports for the Pythagorean idea of a hidden mathematical structure that explains all was the Pythagorean theorem, that asserts a simple mathematical relationship among the sides of any right triangle.

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The awful problem of incommensurability

Unfortunately, it was the Pythagorean theorem that proved to be the undoing of the Pythagorean simplistic idea of a numerical structure for the universe.In the simple case of the diagonal of a square, which makes two isoceles right triangles, Pythagoreans realized that there were no compatible numbers (in their sense) that could be used to measure both the lengths of the sides of the square and the diagonal.

They were incommensurable magnitudes. But for Pythagoreans, there could be no such thing.


The Decline of the Pythagoreans

The incommensurability of the diagonal and side of a square sowed a seed of doubt in the minds of Pythagoreans.They became more defensive, more secretive, and less influential.But they never quite died out.


The Platonic view

Plato (427-348 BCE) lived about 200 years after Pythagoras.Plato founded the most influential and long-lasting school in ancient times, the Academy.

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The Academy

“Let no one who does not know geometry enter here.”The Academy was intended to be a school for future statesmen, but Plato believed that statesmen needed clear thinking and the ability to reason exactly. For that they required a thorough grounding in mathematics.Hence the (supposed) inscription over the entrance gate, quoted above.


The Divided Line: Plato’s world view summed up

The line represents everything that is.

All below the major division are the world as we perceive it with our senses.

But the “real” world, from which the sensible world derives, is that above the major division, the intelligible world, which can be apprehended only by the mind.

The structural part of this is represented by mathematics and logic.

Hence the “true structure” and the “true reality” of the world is its mathematical structure. All else is illusion.


Saving the Phenomena

One of the duties that Plato prescribed for would-be philosophers was to explain the world of the senses in terms of the intelligible world.

To show how a lower part of the divided line is accounted for by a higher part.

A prime example of saving the phenomena was to show how the mysterious movements of the heavenly bodies—the planets in particular—are accounted for by mathematically exact rules.

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Retrograde motion

The planets not only move relative to the fixed stars, they change direction.


Retrograde motion


The Spheres of EudoxusEudoxus of Cnidus, 408-355 BCE, was a prominent mathematician and astronomer of ancient Greece and a former student at Plato’s Academy.Eudoxus came up with a mathematical scheme to “save” the planets by accounting for their weird motions with simple geometric manipulations.

He imagined a series of concentric spherical shells for each planet, turning on different axes nested inside each other.On the innermost spherical shell would be the only part visible: the planet.

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The Spheres of Eudoxus, 2The outer sphere is aligned north and south and turns simultaneously with the celestial sphere.This swings the planet around daily.


The Spheres of Eudoxus, 3

Next is the Ecliptic Sphere, which is aligned with the motion of the sun, i.e. a 23.5° tilt to the axis of the celestial sphere.This causes the slow west to east migration of the planet


The Spheres of Eudoxus, 4

The third and fourth spheres are aligned differently for each planet and produce the looping retrograde motions.The planet is on the innermost sphere.

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A complex (invisible) system in the sky

Eudoxus required 27 different concentric spheres.

3 for each of the sun and moon, 4 for each of the other 5 planets, and the celestial sphere for the fixed stars.


Yes, but…

The main problem with Eudoxus’brilliant solution is that it did not work.Despite all the possibilities, Eudoxuscould never figure out the relative sizes, angles, and rates of revolution to put the planet in the right place in the sky.


Aristotle

Plato’s most important student at his Academy was Aristotle, who was by far the most influential thinker of ancient times.

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Two contrasting views

Plato and Aristotle differed fundamentally on their views on Nature.In consequence Aristotle gave little importance to mathematics.

Plato (left) and Aristotle, from Raphael’s School of Athens in the Vatican


But he affected mathematics

Aristotle, like the pre-Socratic philosopher Parmenides, gave fundamental importance to reason, and formalized it as logic.Aristotle’s logic became the backbone of western philosophy, and also began to structure mathematics.


Mathematical Reasoning

Plato’s Academy excelled in training mathematicians.Aristotle’s Lyceum excelled in working out logical systems.They came together in a great mathematical system.

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Euclid’s ElementsAfter the death of Alexander the Great, the centre of Greek culture, at least of Greek learning, moved from Athens to Alexandria in Egypt. Around 300 BCE, a man named Euclid headed up mathematical studies at the Museum in Alexandria, which was the greatest academic institution in ancient times.

Little is known about Euclid’s personal life.


Euclid’s Elements, 2Euclid is now remembered for only one work, called The Elements.13 “books” or volumes.Contains almost every known mathematical theorem, with logical proofs.Euclid’s Elements is the second most widely published book in the world, after the Bible.It was the definitive and basic textbook of mathematics used in schools up to the early 20th century.


The power of Euclid’s ElementsWhat is distinctive about the Elements is the logical structure.The work begins with its stated, explicit assumptions, the axioms.Every theorem that follows from then on depends logically only on the axioms themselves, or on earlier theorems that were logically derived from the axioms.The style of argument is Aristotelian logic.The subject matter is Platonic forms.

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The scope of the Elements

There are 13 books in the Elements.Included are almost all known mathematical truths from ancient times.Notably, the Elements contains a remarkable general proof of the Pythagorean theorem, many proofs about the properties of numbers, and, the final theorem, a proof that there are precisely five regular solids, the so-called Platonic solids (which we will be looking at later).


Proposition I.47 (the Pythagorean Theorem)

The famous proof of I.47 proceeds by showing that each of the smaller squares is equal in area to part of the larger square on the hypotenuse.


Proposition I.47 (the Pythagorean Theorem)

It does this by showing that the smaller square, e.g., ABFG, is twice the area of the constructed triangle FBC, while the corresponding rectangle BDL is twice the area of triangle ABD, and that triangles FBC and ABD are equal to each other.

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The other example shown in class: Proposition IX.20

There is no limit to the number of prime numbersProved by

1. Constructing a new number.2. Considering the consequences whether it is prime or not (method of exhaustion).3. Showing that there is a contraction if there is not another prime number. (reduction ad absurdum).


The main point about Euclid:Building Knowledge with an Axiomatic System Start from generally agreed upon premises ("obviously" true)Tight logical implicationProofs by:

1. Construction2. Exhaustion3. Reductio ad absurdum (reduction to absurdity)

-- assume a premise to be true-- deduce an absurd result

math 1700: the nature and growth of ideas in mathematics · math 1700 review of fall 2008 1 math...

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