1. History and Abstraction

Mathematics begins with the earliest civilizations 3000-1000 B.C.: India, China, Egypt, and Babylonia

Egyptians: Year 365 days, good approximation of Pi (for assessing property area for tax assessment), incredible engineering (certain tolerances in the Great Pyramids are good to 1 part in 27,000. Records exist from before 3,500 of tallies of over 1 million goats.

Bablyonians developed a base 60 number system, they were excellent astronmers, and began the field of algebra. They could calculate interest on loans and had lots of recreational mathematical puzzles.

All of the skill in these cultures was based on practical needs, however, and rules of calculating and geometrical reckoning, even though almost always correct, were never ìprovenî to be right generally. Mathematics, as we know it today, which involves systematic and rigorous definitions, foundational axioms, and deductive proofs, did not exist until the 6th Century B.C. in Greece.

Here is a very rough time line of some of the most important figures in Greek mathematics and philosophy:

Thales (~624 B.C. - 550 B.C.) is considered the father of deductive mathematics. He was a geometer, and is said to have proved that all isosceles triangles have equal base angles, that any angle inscribed in a semi-circle is a right angle. He was the first of a long line of semi-mystical Greek mathematician-philosophers that travelled to Egypt, Babylonia, and perhaps India, came home and taught disciples and ran schools.

Although Pythagoras is most famous for the theorem about the relation between the hypotenuse and sides of right triangles, he was the center of a cult that was obsessed with questions about number theory. His groups discovered prime numbers, odd and even, perfect numbers (numbers that equal the sum of their prime factors, e.g., 6 = 3 + 2 + 1). Little is actually known about the Pythagorean Brotherhood on the Greek colony of Croton, but it is conjectured to have engaged in all sorts of ritualistic practices, encouraged its initiates to become ascetics (denied sleep, rich food, and other worldly pleasures), and hazed its younger listeners unmercifully. What is clear is that Pythagoreanism took the phrase ìEverything is Numberî quite seriously, and in fact believed that the metaphysical foundation of our world was built on whole numbers. Because everything is made of eternal and perfect whole numbers, the ratio any two lengths must be a ìrationalî number: the fraction of two whole numbers.

The discovery of irrational numbers was a blow to the Pythagorean dream of a metaphysics of whole numbers, but it was accepted nonetheless, and in this way was perhaps a true beginning of mathematical reasoning.

Applying the Pythagorean theorem to a triangle made up of two sides of a unit square and its diagonal, (e.g., triangle B-C-D above) it can be proved that ratio of the length of a side to the diagonal cannot be the ratio of two whole numbers. Here is the "demonstration."

Let the length of each side of the square = 1. Call the the hypotenuse H (the line segment B-D).

1) By the Pythagorean theorem,

H² = 1² + 1²

H² = 2

H =

2) To do a reductio, assume the opposite of what we want to prove, i.e., that is rational. If we can derive a contradiction, then we will conclude that is irrational. If is rational, then there exist whole numbers p and q such that. Further at least 1 of these numbers must be odd (if they were both even we could divide them both by 2, and keep doing this, until at least 1 was odd).

3) So , and p² = 2q².

4) So p² is even, and by lemma 1, p is also even.

5) By lemma 2, p² = 4n, and since p² = 2q², 4n = 2q².

6) Thus 2n = q², and q² is even. Again by lemma 1, then q is also even.

7) So we have that p is even and that q is even, contrary to step 2. Q.E.D.

Legend has it that when the Pythagoreans discovered that not everything was rational, they swore oaths not to reveal this blasphemy to the world. Hippasus violated this oath, and was allegedly drowned for his sin. Not only did this shatter the Pythagorean dream of a "rational" universe, but Euclid later proved that the essential building blocks of numbers, the prime numbers, were in fact infinite. Here is his proof, which again is a reductio.

1) Suppose that the primes are finite.

2) Then the set of all primes P = {P₁, P₂, ... , P_n}, where P_n is the biggest prime.

3) Create number X that is the product of all the primes + 1,

i.e., X = (P₁ * P₂ * ... * P_n) + 1

4) Then X > P_n, so X cannot be prime.

5) But if X is not prime, there must be a prime P_i such that X = P_i *Y.

6) But for every prime P_i, when P_i is divided into X, there will be a remainder of 1, so X has no prime factors, and is therefore prime, contrary to what we proved in 4. Q.E.D.

2. Assignments

Reading:

Peterson, "The Burden of Proof," pp. 217 - 220, Reading Packet.

Glymour, Chapter 1 (pp. 3 - 15), Reading Packet.

Problems:

Type up and turn in Study question 1 on page 15 of chapter 1 of Glymour.