# Boxing Pythagoras

## 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.

Occasionally in their calculations, the Pythagoreans would happen upon a proportion that was mysterious, to them. Discovering the exact ratio of numbers between these mysterious proportions was a delight and a treasure for members of the Brotherhood. Such finds were celebrated and lauded, and therefore these mysteries were pursued with diligent hearts. Hippasus was working on just such a problem. The particular case that Hippasus was attempting to resolve was actually a very common proportion, but its exact calculation had stumped many great minds for quite a long time. If he could discover the ratio behind this proportion, he would be guaranteed fame and glory amongst his brethren. So, Hippasus dedicated himself entirely to this pursuit, not even resting from his work while he travelled. And it just so happened that, while sailing across the Mediterranean, Hippasus discovered his answer.

Hippasus had been trying to find a ratio between two numbers which would describe his problematic proportion; but the answer that he discovered was that no such ratio could possibly exist. The number was, quite literally, irrational. This discovery was revolutionary. One of the principle tenets of the Pythagorean philosophy was the fundamental universality of these ratios. According to their beliefs, everything in the universe should be able to be described as a ratio between two numbers. And here, Hippasus had proven that there existed certain proportions which could not possibly be described rationally. Excited and thrilled at his discovery, Hippasus sprinted across the deck of the ship, yelling to his brothers about his breakthrough. But when they heard his claim, the Pythagoreans were appalled. In their eyes, Hippasus was not only preaching heresy, he had definitively PROVEN his heretical claim. Where Hippasus had expected to be met with praise and celebration, he instead found fear and denial and anger. The other Pythagoreans soon built into a murderous rage, and their curious brother became the target of their ire.

For the transgression of discovering the truth, Hippasus was thrown overboard and drowned in the waters of the Mediterranean Sea.

Now, this legend likely never actually happened, historically. The tale I’ve told expands quite a bit above and beyond what can be pieced together from ancient historians, and even those bits which I’ve accurately retold remain dubious, at best. But, as in any parable, the truth to be found in this story has nothing at all to do with whether or not it actually occurred, as written. There is a moral– a meaning to be discovered– in the text of the tale. And in the case of poor Hippasus, I have attempted to illustrate the dangers of constructing dogmatic views of reality based upon unjustified beliefs.

Throughout history, such dogmatism has often found itself in harsh opposition to evident reality. When Galileo Galilei showed the validity of the heliocentric model, he was arrested and threatened by men who believed, without ample justification, that the Earth must be the center of the universe. When Charles Darwin formulated his theory of Evolution by Natural Selection, he was harried and bullied by men who believed, without ample justification, that all life had been specially created. When Monseigneur Georges Lemaître showed that our universe likely began in an event later dubbed “the Big Bang,” he was belittled and ostracized by men who believed, without ample justification, that the universe was static and eternal.

When we attempt to assert how the universe must be structured, rather than attempting to learn how it truly is, we might as well be drowning Hippasus ourselves.

## 6 thoughts on “The Legend of Hippasus”

1. Reblogged this on Fairminded Notions and commented:
“We might as well be drowning Hippasus ourselves.” (the guy who discovered irrational proportions in the universe.

2. When Galileo Galilei showed the validity of the heliocentric model, he was arrested and threatened by men who believed, without ample justification, that the Earth must be the center of the universe.

According to Louis Dupré in Passage to Modernity, this is not quite correct:

As historians have repeatedly pointed out, the controversy turned not, or at least not primarily, around the change in the world picture. Pope Paul III himself had welcomed Copernicus’s theory and the Jesuit astronomers of the Roman College—the famous Father Clavius foremost among them—were quite willing to consider a heliocentric alternative. The main issue was causality. One revolutionary conclusion of Galileo’s new system was that power need not continuously flow from God once nature became endowed with a uniform intrinsic necessity of its own. The communication of motion, which had played such an important role in the ancient worldview and on which major arguments for the existence of God had rested,[7] lost its significance in a mechanistic order where bodies, once they moved, would continue to do so until stopped by an external cause. It needed no further assistance after it had received its initial impulse. The new science of mechanics did not dispense with a Creator who would initiate motion, but it appeared to withdraw God from nature after his creative act. Only the availability of an alternative interpretation of Scripture would justify the acceptance of Galileo’s “hypothesis,” Cardinal Bellarmine concluded. But none was available. (68)

It’s not clear that this concern was “without ample justification”; the mechanistic universe is not clearly a good model of reality and the metaphysics of causality are terribly muddled. A neat criticism of the very specific mathematical pattern into which many laws of nature fall is provided by Robert Rosen in Life Itself. He issues precisely the critique you have:

When we attempt to assert how the universe must be structured, rather than attempting to learn how it truly is, we might as well be drowning Hippasus ourselves.

In particular, carving reality up into ‘states’ which move from one to the next via ‘laws’ is an insistence that the universe has to be structured according to that formalism. And yet, Rosen shows that there are other ways that reality can operate, through the use of category theory. (Rosen is not a fuzzy thinker.)

• Thank you for your reply, Labreuer! I’ll have to add Rosen’s book to my ever-growing queue. It sounds wonderfully interesting.

That said, there seem to be a few oversimplifications in the excerpt from Dupré which you provided. While Copernican heliocentrism was widely lauded for its ability to simplify calculations which were otherwise exceedingly difficult, the Pope and Jesuits stopped noticeably short of believing that it represented an actual picture of the world. It was considered to be a useful tool, but not a factual description of cosmology. This is exceedingly clear by the edits made to Revolutions of the Heavenly Spheres by the Congregation of the Index in 1620. For example, where Copernicus had written that he had no shame in “admitting” that the Earth revolved around the Sun, the Church changed his book to instead read that he was “assuming” this behavior.

This is not to say that there were no religious advocates of Copernicus’ work. Certainly, this is not the case– Paolo Foscarini’s support immediately jumps to mind. However, the official position of the Church was clearly and definitively geocentric.

Now, I’ll grant that “ample justification” is a fairly vague and subjective term, and that another person’s understanding of this phrase might easily differ from mine; but it is certainly true that Galileo was arrested and threatened by men who believed that the Earth must be the center of the universe.

• It was considered to be a useful tool, but not a factual description of cosmology.

Oh, I didn’t mean to contest that. It’s just the reason they considered it important to argue that heliocentrism merely “saved the appearances” wasn’t necessarily because theology required the earth to be at the center of the universe. Indeed, according to Mano Singham’s 2007 Physics Today article The Copernican myths, “ancient and medieval Arabic, Jewish, and Christian scholars believed that the center was the worst part of the universe, a kind of squalid basement where all the muck collected.” Singham cites Dennis Danielson’s 2001 American Journal of Physics The great Copernican cliché.

Multiple times, I have seen it claimed that Ptolemaic astronomy was intended to “save the appearances”, and not necessarily be ‘literally true’. And so, the demands of Cardinal Bellarmine for Galileo to continue this tradition until he had really good reason otherwise seems to be much less outlandish than is often presented.

Furthermore, I think it’s important to note whether the the Church was worried more about geocentrism or the aspect of divine causation Dupré teases out. Both of these would cause the Church to advocate a “save the appearances” approach.