Boxing Pythagoras

On Wildberger’s “Inconvenient Truths”

Dr. Norman Wildberger of the University of New South Wales has a wonderful and prolific YouTube channel in which he discusses a great deal of very interesting mathematics. I have discussed Dr. Wildberger before, regarding a very similar subject, but I wanted to take a moment to discuss a video from his Math Foundations series entitled, “Inconvenient truths about sqrt(2).”

In the video, Dr. Wildberger claims that there are three different ways in which $\sqrt{2}$ is commonly discussed: the Applied, the Algebraic, and the Analytical. He does a fairly good job of discussing the manner in which the ancient Greeks discovered that there exists no ratio of two whole numbers which can be equal to $\sqrt{2}$, which is a topic I have covered here, as well. He then explains what he means by each of the above three categories.

Some Unfortunate Choices in Mathematics Terminology

Words can be tricky things. The same word can often carry wholly different meanings depending upon the context in which it is used. Take, for instance, the semantic range of the word “light.” This word can carry very different meanings when used in different contexts, as the following sentences illustrate.

1. That feather is light.
2. That shade of pink is light.
3. That laserbeam is light.

Each one of these sentences is of the form “That <noun phrase> is light,” but the word “light” intends an entirely different thing, in each. In (1), “light” is a description of the weight of the feather. In (2), “light” is a description of the intensity of the shade of pink. In (3), “light” is a description of the physical nature of the laserbeam. There is a well known fallacy of logic called equivocation which involves conflating such definitions in order to arrive at a false conclusion. For example, if I said…

1. Light things weigh less than heavy things
2. This shade of pink is light
3. Therefore, this shade of pink weighs less than heavy things

…my logic would be invalid, because the definitions of “light” used in (1) and (2) are completely different.

Mathematics, unfortunately, contains some terminology which tends to lead to these same sorts of equivocation fallacies, because the common usage of a word very often differs from the mathematical usage of that word. While there are numerous examples from which I could likely choose, today I’m going to focus on a case which I believe to be particularly egregious. Today, I’m going to discuss Real and Imaginary numbers.

On the Irrationality of the Square Root of 2

Consider a triangle with two legs of equal length which meet at a right angle. What is the proportion of the length of the Hypotenuse to the length of one Leg?

The Legend of Hippasus

There was once an ancient Greek geometer named Hippasus who belonged to the Pythagorean Brotherhood. The Pythagoreans were a school of philosophers who held a special reverence for numbers and proportion. To these men, mathematics was more than just a method for quantifying and describing the world around them. The Pythagoreans held that numbers, themselves, were divine things, worthy of awe and worship. Relationships between these numbers– what we would now think of as a “proportion” or “ratio” of numbers– were intensely studied, as these proportions were thought to hold the secrets of the cosmos. If one were to divide a string according to some specific ratios, he could produce beautiful music. If one compared the proportions of two legs of a triangle, he could come to understand the remaining leg. Nothing in existence was more beautiful to the Pythagoreans than the discovery of these proportions and the properties they endowed.