Ionian Mathematics

Many works on the history of math begin with the Ionian philosophers. Thales of Miletus is considered by many to be the first scientist as he was one of the first thinkers in written history to be concerned primarily with speculation on nature. Thales is reported to have predicted an eclipse and argued that water was the primary element. Anaximander argued sought a primary element that indeterminate. Anaximenes later argued that air was the primary element.

The Miletus school built their natural philosophy on ideas like the unity of opposites, frequently made use of metaphor, and clearly had a different concept of air and water than we do today; so it is hard to build on their work.

I would like to begin this work with the Cult of Pythagoras.

Pythagoras Squared

Pythagoras (circa 580BC-490BC) was from the island of Samos in the Aegean Sea off the coast of Ionia. Among his achievements was the discovery of the musical scale. Pythagoras realized that one can produce harmonic sounds when the size of the instruments formed certain ratios. Pythagoras is also attributed with the discovery of the Pythagorean Theorem which states that the square of the hypotenuse of a right triangle is equal to the sum of the square of its sides.

It is easy to demonstrate the Pythagorean Theorem for the unit squared (a 1x1 box). In the demonstration below I start with a single square. I cut the square along the diagonal. This new line is called the hypotenuse. I make four copies of my cut square. I arrange these copies in a new square so that hypotenuse faces inward as seen on the right side of the figure below.

The figure shows an outer box that has an area of four units. The lighter colored inner box is half of four (that is two). This inner box is the square of the hypotenuse; Hence, the square of the hypotenuse is two.

The general case of the Pythagorean Theorem is a bit more complex. In the second figure I position four copies of a triangle with leading edges A and B such that the hypoteuse C makes an inscribed square. Unfortunately, the inscribed square is skiwampus. (See the left side of the figure below).

Being immensely clever, I place labels from 1 to 4 on the outside triangles. I match up the triangles 1 & 3 in the upper left corner of the box, and the triangles 2 & 4 in the lower right corner. The upper right and lower left side of the reshuffled box now contain two squares that have size AxA and BxB. These two squares add up to the same size as the square of the hypotenuse CxC. Hence, the square of hypotenuse of a right angle is equal to the sum of the square of its sides.

In the simple case where A = 4, B = 3, we find that 4*4 + 3*3 = 5*5 or 16 + 9 = 25. In the case where a right triangle has sides 3 and 4, the length of the hypotenuse is 5.

A group of three positive integers (a,b,c) that solve the equation a² + b² = c² is called a pythagorean triplet. We will investigate these in detail later in the work.

The Metaphysics of Pythagoras

To the Pythagoreans, mathematics was more than just an intellectual curiosity. The Pythagoreans had built mathematics into the center of their world view. Apparently, the Pythagoreans saw number as the elemental force of nature.

By number, I mean natural numbers {1, 2, 3, 4 ...}. The Pythagoreans had found ratios of numbers the key to harmonious music and in other areas of physics. They assigned ratios to aspects of human relations.

The Society of Pythagoras is often likened to a cult. The Society had created a mystical view of the pure number. Legend holds that they Pytagoreans believed that all could be expressed as ratios of pure whole numbers.

Legend tells that a great crisis followed the realization that the hypotenuse of the unit square could not be expressed as a ratio of whole numbers. The myth around this discovery says that Pythagoras had the person who discovered this inconvenient truth, Hippasos, executed.

The proof starts with the assumption that the hypotenuse c can be expressed as a ratio of two integers p and r. That is c = p/r. As demonstrated in the previous section we know that the square of c is 2, or p²/r² = 2.

Divide both sides of the equation by 2, and we find that p²/2*r²=1. This implies that p is even. If p is even, then there is some number s such p = 2*s. If p = 2*s, then p² = 4*s².

Replacing p² gives the equation 4*s²/2*r² = 1. We can remove a 2 from both the numerator and denominator to get: 2*s²/r² = 1. This implies that r is even.

If the hypotenuse of the unit square could be expressed as the ratio (p : r); then both the numerator and denominator of the ratio would be even. This is a logical absurdity. This method where you take a premise, then show that it leads to an absurd conclusion is often called reductio ad absurdum.

An Irrational World

The Pythagorean tied their notion of reason to the ratio. Being able to express an idea as a ratio meant that it was subject to the known laws of mathematics. The discovery that a fundamental mathematical entity could not be expressed as a ratio was a tremendous blow to the Pythagorean world view.

In modern mathematics, we call numbers that can be expressed as a ratio "rational." We call those those that cannot be expressed rationally "irrational."

Our modern English terminology has created a problem: In common usage, people use the term "rational" to refer to the process of reason. In common usage, we use "irrational" to refer to those who act without reason.

As the mathematics of Irrational numbers is well known, one can say that mathematicians speak rationally of Irrational numbers. Sadly, some people are scared of fractions. In these case we would say they have an irrational fear of the Rational numbers.

When speaking of different numbering systems, I prefer the term "commensurable." In the case of the hypotenuse, I like to say the hypotenuse of the unit square is not commensurable with its sides.

Imposing the Grid

The fact that the hypotenuse is not commensurable with the edges of the unit square is more than just an idle curiousity. Finding that the hypotenuse is not commensurable with the edges of the unit shows that there are entire systems which are not commensurable.

If you were to take two pieces of graph paper and turn the second to a 45 degree angle of the first, you would find that it is impossible to get the vertices of the two pieces of paper to match up. See figure below:

[Click the picture to see a bigger graph]

The points on the grids won't match up because the rotation ends up multiplying all points (a,b) by 1/√2. Multiplying an integer by a radical yields a radical. (a,b) → (a/√2,b/√2).

Imagine that a person lays out a grid pattern on a section of land and makes a survey based on the grid pattern. Now, imagine a second surveyor who lays out a grid at a 45 degree angle to the first grid takes a survey.

The two surveyors would find that the numbers taken during their surveys don't mesh up. Although the two surveyors measured the same land using the same unit of measure, they would find their surveys were not commensurable.

In the case of perspective, we find that it is simple to create a grid with ourselves at the center looking toward the world. The measurements we take from the vantage point of our gril will not be commensurable with the measurements from the vantage point of other whol look at the same world through different angles.

Introducing the Infinite

If you are seeking a world view where everything can be assigned a clean definite natural number, then you will end up gravely disappointed.

The most promising path to resolving the conundrum of the hypotenuse is to accept that there are things that don't have a finite representation. So, here I introduce the concept of the "infinite".

People tend to associate the term "infinite" with the extremely large or extremely small. While introducing the "infinite," I wish to emphasize that "infinite" simply means "not finite." For example, the square root of two does not have a finite representation. In this case, we can use the finite symbol "√2" to represent the value. This symbol allows us to do a full range of calculations on this value.