Survey Results

Question 7 of Extra Credit II asked students to fill out the survey in section 9.2 of the book. 25 people filled out part of the survey, and 23 completed it entirely. Here is the statistical abstract!




1 Gender
Among respondents: 16 females, 9 males, or 64.0% female vs. 36.0% male.

In the whole class: 45 females and 25, or 64.3% female vs 35.7% male.




2 Weight
Overall: Average: 137.4 lbs,   Standard Deviation: 29.9 lbs
Females: Ave 123.8,   Stdev: 21.6

Males: Ave 161.7,   Stdev: 27.8

3 Height   Note: 60in = 5ft.
Overall: Ave 64.8in,   Stdev 4.15
Females: 62.6,   Stdev 2.4
Males: Ave 68.8,   Stdev 3.6

Among respondents, males on average are 6.2in taller than the females. That's huge.




4 Favorite Body Part

Overall:

It is interesting to see the breakdown by gender on this one.

Females:

Males:

There appears to be more variability among what males consider their favorite body parts.




5 Number of times per month people call home
Overall: Ave: 4.9, Stdev: 2.55
Female: Ave 4.3, Stdev 2.5
Male: Ave: 6, Stdev 2.5

The data reports that males call home more often than females.




6 Number of times per month people do Laundry
Overall: Ave: 6.08, Stdev 2.17
Female: Ave: 5.9, Stdev 2.1
Male: Ave 6.4, Stdev 2.4

The data shows that males do somewhat more washes than females, but the difference may not be statistically significant.




7 How many times per month do you drink alcohol?
Overall: Ave 6.08, Stdev 2.17
Female: Ave 5.87, Stdev 5.07
Male: Ave 6.67, Stdev 4.80

Males drink more frequently on average, but there is a wider range of variability in female drinking.




8 How many dates do you go on per month?
Overall: Ave 3.32, Stdev 2.56
Female: Ave 3.25, Stdev 2.41
Male: Ave 3.44, Stdev 2.96

Males and Females go on approximately the same number of dates, but the variability in the number of male dates is wider.




9 Number of months since you last saw a movie in a theater
Overall: Ave 1.21 months since last theater movie
Females: Ave 1.02 months since last theater movie
Males: Ave 1.56 months since last theater movie

Among respondents, females attend movie theaters more often than males.




10 Number of hours per day spent browsing the internet.
Overall: Ave 3.24, Stdev 0.97
Females: Ave 3.38, Stdev 0.89
Males: Ave 3.00, Stdev 1.12

Among respondents, females use the Internet more often, but among males the usage rate has a wider range of variability. In this case, the data are clearly skewed because the highest allowable number of hours was 4, but some people use the internet more than 4 hours per day.




11 Number of emails received per day
Overall: Ave 15.20, Stdev 4.76
Females: Ave 16.06, Stdev 4.91
Males: Ave 13.67, Stdev 4.33

Females receive more emails on average.




12 Hours per week spent studying
Overall: Ave 21.80, Stdev 11.80  (Wow! That's a HUGE range of variability!)
Females: 21.25, Stdev 10.25
Males: 22.78, Stdev 14.81

Females and males spend about the same number of hours studying, but the variability among males is greater.




13 Number of college courses taken so far
Overall: Ave 16.60, Stdev 6.60
Females: Ave 17.31, Stdev 6.70
Males: Ave 15.33, Stdev 6.61

On average, females have taken more classes.




14 Number of years left before graduation
Overall: Ave 1.92, Stdev 1.10
Females: Ave 1.833, Stdev 1.18
Males: Ave 2.06, Stdev 1.01

On average, females are closer to graduation.




15 Post-Graduation Plans

Overall:

16 How many people in the class do you find attractive?
Overall: Ave 4.22, Stdev 2.75 (Very wide range of variability!)
Females: Ave 4.00, Stdev 3.01
Males: Ave 4.56, Stdev 2.41

On average, the females found fewer people attractive, but the range of variability was significantly higher.

For this one, I was highly curious about the data. Here is a closer look.

Overall:

So the Mode is "at most 2." That's cold, people, cold! What's wrong with kids today?

The averages between females and males are not that different, but just look at how differently the data breaks down:

Females:

Males:

So there we have it. According to this data set, females cluster around not being attracted to many people, but the distribution is wider. Males cluster around the average, but have a tighter distribution.




17 Favorite topic in class

Overall

18 Number of hours of sleep per night
Overall: Ave 6.69, Stdev 1.25  (6.69 seems pretty low to me!)
Females: Ave 6.50, Stdev 1.42
Males: Ave 7, Stdev 0.87

Geeze, ladies, you need to get more sleep at night! Guys, too! You should be getting 8-9 hours!




19 Number of Siblings
Overall: Ave 1.42, Stdev 1.14
Females: Ave 1.27, Stdev 0.96
Males: Ave 1.67, Stdev 1.41

The variability is very large: the standard deviation is larger than the mean.




20 Hours spent doing athletic activities
Overall: Ave 5.65, Stdev 8.46
Females: Ave 2.97,  Stdev 4.06
Males: Ave 10.11, Stdev 11.88

Among respondents, males do far more athletic activities per week. The variability for both males and females is huge.

Teaching Blog Test Time!

Preparing for a test requires figuring out what you need to know. That involves two things: figuring out what's important, and figuring out what the teacher thinks is important. And what is that? In the broadest perspective, this class has two main themes: learn the skills of mathematical reasoning, and learning about the subject matter of mathematics.

I'll say just a word about the first part, mathematical reasoning. Math isn't numbers. It is the study of formal structures, which are systems of specific, clearly-defined objects along with rules by which the objects relate. Perfect examples include arithmetic in the integers, and the game of Chess. Imperfect examples include the system of traffic laws and the American legal system. Much of what we experience in daily life has nothing to do with formal systems, but in whatever you do in life, having the ability to formulate problems in formal ways, and to use formal reasoning to analyze problems, will be a benefit.

As for the second part, the subject matter itself, I present the following outline.

Proofs you need to know:

• There are infinitely many primes.
• square roots of 2, 3, etc are irrational
• The cardinality of the reals is larger than the cardinality of the natural numbers
• The Pythagorean Theorem
• There are only 5 regular polyhedra (aka Platonic solids)

Chapter 1
No real content. Just some puzzles meant to make a point, for instance that formal reasoning can be used to solve real-world problems.

Chapter 2 - Numbers (Sections: 2.1- 2.4, 2.6)

1. The pigeon hole principle
2. The Fibonacci sequence. Recursive formulas and direct formulae. Ratios of successive numbers in a recursive sequence. Limits. Infinite fractions.
3. Prime numbers. Divisibility (for instance, the exact  meaning of m|n). Unique factorization. The division algorithm. Quotient and remainder. Proof that there are infinitely many primes.
4. Modular arithmetic. Reduction mod n. UPC codes and check-digits.
5. Rational (numbers that are ratios) and irrational numbers (numbers that are not ratios). Proof that square roots of 2, 3, etc are not rational

Chapter 3 - Infinity (Sections: 3.1-3.3)

1. One-to-one correspondence. Something more primitive than counting: Cardinality.
2. The even numbers, the prime numbers, the integers, and the rationals all have the same cardinality as the natural numbers.
3. Countable and uncountable sets. Examples of countable and uncountable sets. The diagonal argument. Proof that the reals are uncountable.

Chapter 4 - Geometry (Sections 4.1-4.3, 4.5, 4.7)

1. The Pythagorean theorem.
2. Triangulation. The art gallery theorem.
3. Definition of the Golden Rectangle. Computation of the value of the Golden Ratio. The Golden Spiral
4. Polygons. Polyhedra. Duals of polyhedra. Stellation and truncation. Regular polygons. Regular polyhedra. Proof that there are only 5 regular polyhedra.
5. Definition of dimension (a degree of freedom, or a mode of measurement). Real-life examples of multi-dimensional problems and data sets. Imagining the 4th dimension. Analogy with 1, 2 and 3 dimensions.

Chapter 5 - Geometry and Topology (Sections 5.1-5.3)

1. Geometry is the study of shape. Topology is more primitive: it is the study of connectedness. Topological changes and geometric changes. How to prove two objects are topological equivalent, and how to prove two objects are topologically different.
2. Gluing patterns. The Torus. The Mobius strip. The Klein bottle.
3. Knots and links. Equivalent and inequivalent knots. The unknot.

Chapter 6 - Graphs (Sections 6.1-6.3)

1. Graphs. Euler Circuits. The Konigsberg bridge problem.
2. The Euler characteristic. Second proof that there are only 5 regular polyhedra.
3. Planar and non-planar graphs. E<3V-6 for planar graphs without multiple edges. The graphs K5 and K3,3. Application of graphs: maps and the 4 color problem.

Chapter 7 - Fractals (Sections 7.1-7.3)

1. Self-similarity. Infinite detail. Fractal-like objects and phenomena in nature.
2. Examples of fractals: simple instructions repeated indefinitely. Sierpinski triangle. Sierpinki carpet. Menger sponge. Koch curve and Koch snowflake.
3. Fractal dimension. Scale and content.

Chapter 8 - Chance and Probability (Sections 8.1-8.3, 8.5)

1. Definition of the probability of an event. Definition of relative frequency. P(not E) = 1-P(E). Yatzhee question. Birthday question. Law of independent events: P(E and F) = P(E)P(F) provided E and F are independent.
2. Streaky-ness and probability.
3. Nash equilibrium. Payoff matrix. Probability strategies (eg. pgs 630-631)

Chapter 9 - Statistics (Sections 9.1-9.3, 9.5)

1. Statistics is the organization of data. Ways statistics can be misused. Coin-flip strategies for poll-taking. Correlation.
2. Histograms. "Center" of a distribution: mean, median, and mode. Skewed and normal distributions. The desert island example. Distributions that are bimodal, trimodal, etc. 
3. The bell curve. Standard deviation. Coin flips.
4. Deeper analysis of data: the hospital example, and the university gender discrimination example.

Chapter 10 - Applications (Portions of Sections 10.1-10.3)

1. Expected value. Weighted standard deviation.
2. Testing for a disease.
3. Interest and compounding.

Fiction

Math is not the real world, nor even in the real world; the real world exists on its own. So is math just made up? Is it fiction?

I'm not going to answer this question for you---but do think about it. Eugene Calabi, one of Penn's most distinguished emeriti (who just celebrated his 90th birthday) and someone well worth listening to, is fond of saying "math is like science fiction." The reason he says so is that, in science fiction, just like in mathematics, the bounds of what is real is no obstruction. In math, you can take limits to infinity, or perform some operation ad infinitum, and likewise find no obstruction. I'd like to point out some other similarities between math and fictitious universes.

The Seirpinski Triangle

The Koch Curve

The Koch Snowflake

Lightsabers don't exist in the real world, but there is nothing stopping them from existing in fiction. In fiction Peter Parker is Spider Man (Spider Man exists!) and not just that but he winds up with Mary Jane; Elizabeth has her happy ending with Mr. Darcy (yes I had to look that one up---it's been a while since 10th grade English); Silas Marner finds his gold, though not the same gold he lost. All of this is a bit Walter Mitty---the point is that in fiction, or shall I say in good fiction, there is nothing stopping the imagination from traveling where ever it wills. The only constraints are the story's internal logic, and the natural constraints of character and common sense.

And inside of mathematics, what stops us from going where we want to go? What wild extremes can we explore, what great structures structures can we build? What finesse of detail can we obtain?

A piece of the Mandelbrot Set, a fractal of the early computer age.

In math the utmost extremes are open to us. And by utmost, I mean beyond the ridiculous, beyond anything our imaginations can readily conjure up---I mean nothing less than infinitude, or the infinitesimal, itself.

Fractal artists exists today; this is an example of such work.

A segment of a Julia set.

Take fractals as an example. In building our fractals, we took a simple set of instructions, and applied them again and again and again. In reality, you could only do this a certain number of times before the lines became too small for your pencil, or your computer ran up against its limits of floating point precision. But these are physical constraints, not logic constraints. in the realm of pure thought---of formal reasoning---there is nothing blocking you from carrying out a set of instructions infinitely often. The result of a simple process carried out to the absurd (that is, carried out to infinity) is in many cases the creation of patterns of unrivaled intricacy.

A Fractal Ball

Today fractals are used extensively in computer graphics; for instance landscapes, cloudscapes, etc are constructed with fractal techniques.

A Fractal Landscape

The first "fractal" artists was probably M.C. Escher himself, who mostly worked before the widespread availability of computers---he crafted his works by hand. Here is an example of a fractal-like piece of his:

Small and Smaller (1956)

Today the world of fractal art is immense, in part thanks to the video game industry. Works range from the abstract

A selection from http://www.fractal.art.pl

to the representational

A fractal tiger.

So yes, in a way mathematics is like fiction. But math is not even limited by what can be imagined---it has just one law: the law of the nuos, of formal reasoning itself.

Shapes

We've been moving forward with the geometry section of our course; the main topics we've seen to date are

• The Pythagorean theorem and its proof
• Triangulation
• The "Art Museum Theorem" and its proof
• The Golden Rectangle and the re-appearence of the Golden Ratio (!)
• The Golden Spiral
• Polygons and regular polygons
• Polyhedra and regular polyhedra

Of course another name for a "regular polyhedron" is a "Platonic solid." As a general reference on the Platonic solids, you might look at this fun website:

http://www.learner.org/interactives/geometry/platonic.html

Some of the exercises there make for good practice.



A point I have  emphasized in class and on this blog is the difference between the use of math, and the knowledge (or theory) of math. It is, for instance, the difference between knowing that a 3-4-5 triangle is a right triangle, and knowing why right triangles satisfy a²+b²=c².

What's the difference? Why is that important? Consider the following. The ancient Babylonians knew about many Pythagorean triples: they had 3-4-5, 5-12-13, and many others. But after the Pythagoreans developed the theory of right triangles, it became possible to produce infinitely many pythagorean triples! Plato himself developed such a method, as did Euclid (see this for a "fun" description of Euclid's method).

This represents a massive advance on the ancient science. What made the advance possible went beyond the heuristic techniques practiced by prior societies: with the concept of proof came theoretical knowledge, and with theory, the techniques of mathematics exploded in fecund usefulness.



Still, the roots of the Pythagorean theorem trace back back to profoundly ancient times: times as ancient to Pythagoras as the Roman legions are to us. The first set of Pythagorean triples are seen in Mesopotamian cuneiform tablets from the 20th century BC! Yes, people were studying right triangles 4000 years ago. They appear in formal Chinese writing somewhat later: around 200 BC, just after the Warring States period, but some centuries before their contact with the Greco-Roman world. But knowledge of geometry was likely developed much earlier. Sadly, much was lost to history when the Qing dynasty, in a bid to limit intellectual diversity after the destructive fracturing and perpetual warfare of the prior epoch, oversaw the burning of nearly all books in China around 210BC---a great deal of science and literature was lost permanently.

It is important to note that none of these earlier sources provided any actual proof of the Pythagorean theorem, at least in total. The first recorded proof appears in the writing of Euclid in 300 BC (although the Zho Bi Suan Jing gives a formal proof that 3-4-5 triangles are right).

With math so ancient, there is one question: why!!!???

I am sure many undergraduates in math classes have wondered this. Was math developed for torture? As a religious expression of perfection? As a high intellectual exercise? As a tool of the elite to oppress the working classes?

The Babylonians used Pythagorean triples, and an evidently well-developed computational system for understanding them, for the purpose of parcelling off land. In order to allot land equitably and to make efficient use of space, it was important to be able to mark off right angles. Without modern surveying equipment, this is a difficult task! Actually creating a right angle with real accuracy is hard to do! Sure, a person can eye-ball it fairly well, but when dealing with potentially large parcels of land, small errors in the angle could create very large deviations of actual area. That could cost people a great deal in lost produce, and could cost the state in lost taxes.

The Pythagorean triple is a fantastic way to precisely mark off right angles, and occurs simply by measuring lengths, not angles. One takes three cords of, say, lengths 30 meters, 40 meters, and 50 meters---it was well within the capability of ancient societies to manufacture such cord lengths with a great deal of precision. Then your people extend the cords to maximum length without stretching, until they just meet to produce a big triangle. The angle between the 30m length and the 50m length will be a perfect right angle. Land apportionment problem solved!

Now, is this why right triangles and Pythagorean triples were developed? Or did the bean-counters and actuaries of Babylon simply take advantage of math that was developed for other reasons?  A certain answer is impossible, as there is no historical record that far back but only archaeological remains---various tablets and stones with precious scant detail. But what is clear is this: societies of the time had use for this math, reasons to develop it, and to nurture its progress.

Figures such as triangles were studied in pre-history times: one reason, as I have discussed, is their usefulness in dividing up land resources. But what about solid shapes? When and where did they come into play? What was their usefulness? We'll discuss more about this next time.

Cantor and his sets

Lately in class we have been studying the concept of "infinity," one of the most frequently used and frequently mis-understood of all mathematical terms. In this post we will talk about a notion more basic even than number (known also as ordinality). That notion is cardinality.

As we all know, "infinity" is not a number: most people will say it is concept, not a number. But what sort of concept? What are its parameters? Most people cannot give a definition, but will say something imprecise, like "its the thing bigger than all the numbers" or "it what you get when you put all the numbers together," or, worse, "it's what you get when you divide by zero."

The person who showed us how to put away all this imprecise, silly nonsense was Georg Cantor. Born to a German merchant family in St. Petersburg in 1845, sixteen years before the emancipation of the serfs in Russia, and about the time Tolstoy was flunking out of college. But he wouldn't spend his life in Russia; at 11 years of age, his family returned to pre-unification Germany.

His family, like many from the non-noble classes with higher aspirations, took advantage of the advancement opportunities that the expanding economies and liberalizing mentalities of 19th century Europe offered. Because his family was modestly wealthy, they had the opportunity to enroll Georg in schools teaching broad curricula in the humanities and science, as opposed to the trade/training schools most would attend in his day. These were the Realschule of aristocratic, pre-Bismarckian Germany.

He showed considerable skill in mathematics, and found himself, with his family's backing, able to attend the top schools (rubbing elbows with the likes of Kronecker and the great Karl Weierstrass). He finally landed a spot at Halle, a university that dates to 1502.

Cantor single-handedly developed the notion of cardinality, today regarded as a cornerstone mathematics---this notion is as valuable to modern mathematics as division is.

Definition: Two sets have the same cardinality if there is a one-to-one correspondence between them.

Now there are one-to-one correspondences between sets of finite size, provided the number of elements in the sets are the same: any two sets of fifty elements, say, have the same cardinality. For instance, there is a (very natural) one-to-one correspondence between the States in the US and the junior senators in the US senate.

In math, the notion of cardinality correlates to some, but not all, of our intuitive ideas about "amount." Take for instance the natural numbers and the primes. It would seen that there are far fewer primes than there are numbers, right? Well there is a one-to-one correspondence:

N. Number  |   Prime
---------------------------------

        1          |      2

        2          |      3

        3          |      5

        4          |      7

        5          |      11

        6          |      13

        7          |      17

        etc.      |      etc. 

Each natural number appears once and only one on the left side of the list, and each prime appears once and only once on the right side. The list itself demonstrates a one-to-one correspondence. Thus the cardinality of the primes is the same as the cardinality of the natural numbers.

Other one-to-one correspondences are possible, such as between the natural numbers and the integers:

N. Number  |   Integer

---------------------------------

        1          |      0

        2          |      1

        3          |      -1

        4          |      2

        5          |      -2

        6          |      3

        7          |      -3

        8          |      4

        9          |      -4

        etc.      |      etc. 

In this correspondence, we can actually write down a formula: the nth natural number corresponds to the integer n/2 if n is even, and to -(n-1)/2 if n is odd.

There are also many correspondences between natural numbers and fractions. The correspondence below is NOT one-to-one:

N. Number  |   Fraction

---------------------------------

        1          |      1

        2          |      1/2

        3          |      2/2

        4          |     1/3

        5          |      2/3

        6          |      3/3

        7          |      1/4

        8          |      2/4

        9          |      3/4

        10        |      4/4

        11        |      1/5

        etc.      |      etc. 

Why? There are repeats. We can make a one-to-one correspondence between the natural numbers and all fractions (strictly) between 0 and 1 by removing all the repeats. We obtain

N. Number  |   Fraction

---------------------------------

        1          |      1/2

        2          |      1/3

        3          |      2/3

        4          |      1/4

        5          |      3/4

        6          |      1/5

        7          |      2/5

        8          |      3/5

        9          |      4/5

        10        |      1/6

        11        |      5/6

        12        |      1/7

        13        |      2/7

        etc.      |      etc. 

In class we demonstrated a one-to-one correspondence between the natural numbers and ALL positive fractions. It would seem, so far, that there is only one infinity.

But Cantor showed us otherwise. His proof was by contradiction: that is, assume there IS a one-ton-one correspondence between, say, all natural numbers and all real numbers between 0 and 1. This would mean there is some list of the sort

N. Number  |   Fraction

---------------------------------

        1          |      .123456789123...

        2          |      .101001000100001.....

        3          |      .1112131415161......

        4          |      .141518191819.....

        5          |      .1111111111111.....

        6          |      .8989898989898989....

        7          |      .0000001000000001000000.....

        etc.      |      etc. 

Given such a list, one first forms the diagonal decimal number: the number whose first digit is the first digit of the first number, whose second digit is the second digit of the second number, and so forth. For the list above, the diagonal is .10155191....

Then one create the modified diagonal, in which each digit of the diagonal is increase by 1, except the digit "9" which becomes the digit "1." In our case, the modified diagonal is .21266202....

It is a simple matter to show that the modified diagonal is not on the list: it differs from the first number on the list in the first digit, from the second number in the second digit, and so on.

Therefore, given any list of decimals, you can find a number that is not on the list. This means there is NO complete list of decimals, and therefore NO one-to-one correspondence between the decimals and the natural numbers.

Twelve hundred years of Pythagoras' cultural history

I post normally on Wednesday afternoons, but I've fielded some questions on last week's post that I wanted to address without taking space away from new topics. It is tempting to trace western cultural roots to the Greeks, but this is arbitrary: the reason they are given special emphasis is that they are the earliest culture to leave behind writing. Prior to, say, 550BC there is no real historical record. Prior to about 700BC there is little-to-no literary record. The reason, again, is the development of alphabets: the people's writing. For at least a millennium prior to that, writing was mainly used by centralized governments and religious authorities. Their purposes seemed mainly to have been religious observation, mundane record keeping and other functionary necessities, and state propaganda (inscriptions on monuments, etc). Exceptions certainly exist, such as the epic of Gilgamesh, the code of Hammurabi, and some scientific/technical works such as astronomical observations, mathematical texts, harvesting and planting techniques, etc.

Pythagoras' cultural inheritance would have seemed rather thin to us, which may help explain why he travelled so far to gather knowledge---he was seeking out the remnant heritage of the destroyed high cultures of a past age. With no writing, very little was handed down through the Greek dark age aside from the assortment of myths and observances, along with a more-or-less confining field of religious strictures (exemplified by Hesiod's Works and Days) of the isolated cultures that settled down after the fall of the old realms.

For us in the modern world, knowledge is sought in the present, or in the future. Pythagoras, too, wanted to break out of the outmoded traditions that were perhaps useful to a simpler age, but no more. But in his time, knowledge lay in a deep, vanquished past.

For the visual among us, I am attaching a rough historical timeline of the eras surrounding Pythagoras: from the fall of the great empires, through the long dark ages, and on to the next ages of empires.

Descent and dissent: math and the revolution (of 500 BC)

The world of Pythagoras' youth was a rapidly changing place, and, in his time, change was generally for the better. Yet he was unhappy.

Society was progressing out of a broad dark ages, and it found its material conditions improving. But spiritually and intellectually people found a growing alienation. Political and social ties were changing, and so too were previously stable religious forms and cultural modes.

A little background might be in order. The high civilizations of the eastern mediterranean, the "Bronze Age" Mycenaean, Egyptian, Anatolian, and Levant cultures, had developed from far more ancient Mesopotamian and Egyptian roots---and had followed the same arc into destruction. They had crested perhaps a millennium prior, and by Pythagoras' time (the 500's BC) these civilizations and those before were long, long gone. They were as much a memory to the bustling sea-side port town of Pythagoras' boyhood as Rome was to the artists of renaissance Florence.

In both cases, the long centuries of intervening time were troubled. The older cultures (extinguished by 1200 BC) had crumbled from over-extended militaries, collapsing agricultural productivity, declining trade, persistent warfare, and finally political revolutions and a destructive migratory period: Germans and the Huns in the Roman case, the Sea Peoples, Dorians, Indo-Europeans, and others in the Mediterranean case. What followed was recognizably similar: centuries of dark ages where economies were localized, cities virtually non-existent, populations in decline, social organization built around honor and kinship hierarchies, art reverting to simpler forms, written literature all but gone, and religious observance, though fragmented, absolute.

By about 800 BC, the alphabet (from obscure origins) was introduced into the eastern mediterranean cultural basin. This lead first to more effective trade, then to revitalized and more democratized governments (the common people could read and write), and then to new literary forms. Persistent warfare was brought under control, possibly through sea-faring innovations and new technologies in iron-based weaponry, which allowed strengthened municipalities to exert control and offer protection in larger geographical areas and in sea lanes.

These trends were a few centuries old when Pythagoras was born---roughly as old as colonial times are to us. By then the motley, atomized world of the past had coalesced into more uniform associations of varying sizes and strengths (the city-states and their dependents) that were connected by a variety of cultural and economic ties, and were sometimes in conflict. Broadening trade brought wide-reaching cultural exchange: the Ionians (Athens and most of the greek colonies) wrestled with incorporating the new ideas; the Dorians (Sparta and laconia) generally retrenched and built internal responses to re-inforce the old mores.

To some, this was confusion. To others, enlightenment. Pythagoras was born in Samos, a port city of the Anatolian littoral and one of the wealthiest independent cities in the Ionian cultural sphere----and at the crossroads of the growing trade networks of the mediterranean. Exactly what influenced Pythagoras' upbringing is difficult to determine, but ancient writers agree he had traveled widely in youth, learning everything he could about mind, spirituality, science, politics, and ethics. He was certainly in Egypt for a time, likely in the levant, possibly in mesopotamia, and could have been as far away as India.

As a result he was among the very most learned of his day. And what did he do with his knowledge? What anyone would do! He started a revolutionary humanistic movement.

I say humanistic, because he was among the first of his age to believe in the supremacy of the human mind: he did not deny the supernatural, but he completed the break, begun two generations earlier with Thales, from his era's confining religious heritage. To him, knowledge and society's norms were not of the gods, but could be questioned, and, maybe, re-formed. They were subjects of human understanding; humans were not subject to them.

Hubris? In the eyes of his contemporaries, yes, very much so. In fact he was considered dangerous: Pythagoras' political movement was violently suppressed, and at age 75 he was forced to flee his adopted hometown of Croton (which is still a town in southern Italy), and likely died stateless. Apparently nobody else wanted to admit an incendiary like this into their protection.

Of course Pythagoras' ethical, political, and spiritual ideas have little resonance today. But his mathematical achievement outlasted not just his political organization, not just the rise and fall of Greek civilization, but survives intact to the present and has become part of our common world culture. Specifically, the achievement begun by him and his contemporaries, and completed by the time of Euclid 200 years later, was this: formal proof.

What could Pythagoras do with proof? Well for one he could show that his rulers were idiots. They believed they had Knowledge (big knowledge with a big K) and so should govern the cities. But Pythagoras knew they could not answer the simplest of queries.

For instance, what is the ratio of the hypotenuse of an isosceles right triangle to its side length?

Pythagoras could show that there was no such ratio! This kind of pure thought, of knowledge-as-such, made up the cornerstone of Pythagoras' thinking. The direct human contact with truths, whether simple or profound, mathematical or political, was Pythagoras' revolution.

Returning to math, in class we gave two proofs that such a ratio does not exist: Pythagoras' original proof using the method of infinite descent, and a simpler proof using Euclid's theory of factorization applied to fractions. Here we will give just the simpler proof.

Proof that the square root of 2 cannot be expressed as a fraction.

For a proof by contradiction, assume that there is a ratio that squares to 2. Assume m/n is that ratio, meaning m and n are natural numbers and m²/n²=2. By using Euclid's theorem of unique factorization and then canceling common terms, we know we can write such an m and n with no common factors.

Rearranging, m²=2n². Therefore m² is even, so m is even, so m=2q for some natural number q.

The equation is then 4q²=2n², so 2q²=n². This means n² is even, so therefore n is even.

And we have derived a contradiction: although m and n had no common factors, we proved that m and n have a common factor of 2. This contradiction implies that the assumption, that the ratio m/n squares to 2, must be false.