What Euclid Did : a section of The Unreasonable Influence ...
What Euclid Did : a section of The Unreasonable Influence of Geometry by Doug Muder
If Euclid did not invent his geometry, but just recorded knowledge that had been around for centuries before him, what did he do that was so important that his text was preserved and used for thousands of years?
Euclidean geometry was history's first example of a system of knowledge. Egyptian and Babylonian geometry texts are little more than lists of facts and formulas. But Euclid had not just put together a set of facts, he had assembled those facts into a structure. All the terms were defined, all the assumptions listed, and all the other statements were derived rigorously from those assumptions. By comparison all other fields of human thought were little more than grab-bags of ideas and tricks.
The assumptions – which are called postulates – are examples of what became known as self-evident truth. In other words, once these propositions were understood, they could not be denied. For a truth to be self-evident meant that even imagining its contradiction involved the mind in absurdities. Euclid's postulates include statements like “Any two points can be connected by a line” and “Any circle divides the plane into an inside and an outside.” These statements seemed to be fundamentally different from a truth of experience, like “Snow is white." You can easily imagine green snow or purple snow, even if you have never seen it. But how could a circle not divide a plane into inside and outside?
Euclid also did something more subtle, something that changed the course of human thought for all time. Prior to Euclid, arguments were private affairs. If I wanted to use reason to convince you of something, I would begin with premises that you would accept, and progress logically from there. The particular premises that you would grant might be very different from those of another person, and the argument that would result from our conversation would be unique to us. (Plato's dialogues are examples of such arguments. Changing any one of the characters would change the argument.) But Euclid's assumptions are not intended to be granted by you or me or any other particular person; they are intended to be universally acceptable. Likewise, the steps of Euclid's proofs are not intended to convince any particular person; they are intended to be beyond anyone's reproach. Euclid's structure establishes geometry as a public truth, not a private conviction.
Another way to say the same thing is that with Euclid, reason becomes coercive. If your assumptions are unquestionable, and your logic is rigorous, then all reasonable people are forced to agree with your conclusions, even if they would rather not.
What Kind of Knowledge is Geometry?
Pythagoras said, "Geometry is knowledge of the eternally existent." Plato said, "Geometry existed before the creation." This is not far from the kind of mysticism we find at the start of the Gospel of John: “In the beginning was the Word.” (The Greek word that is translated as word is actually logos, which is the root of our word logic. It meant something similar to what Star Trek's Mr. Spock meant by logic: the underlying order of the universe.)
Even thinkers who did not have such a mystical bent saw geometry as a very special kind of knowledge. It was useful in practical matters like architecture or surveying, and yet it did not appear to depend on experience. (What could you possibly see that would call geometry into doubt?) Its truth seemed unassailable, not open to contradiction by each new experiment. Its success was a monument to the power of pure reason, and it emboldened philosophers to see what else they could learn from reason alone. If reason unaided by experience could establish that the angles of a triangle add up to 180 degrees, perhaps it could also establish the existence of God, or the immortality of the soul. If one could only get the definitions right, and find assumptions whose truth was self-evident, then morality also could become a matter of public truth rather than private conviction. Perhaps people could be kept in line by the coercive power of reason, rather than physical force. Then, perhaps, power would belong to Plato's philosopher kings, to those best able to reason rather than those most able to wield force.
With so much at stake, it is no wonder that philosophers thought long and hard about geometry, about the kind of truth it represented, and the faculties that made it possible for humans to know such truth. In particular, where did Euclid’s assumptions come from? Why did they seem to be so self-evident, so immune to the doubts that plagued all other areas of human thought? How was it possible to have such knowledge – or any knowledge at all – without experience? And given that such knowledge was possible, how could we find more of it?
Some philosophers, most notably Hume, argued that the basic assumptions of geometry did come from experience. But the vast majority disagreed. How, they argued, could experience give us such certain convictions about perfectly thin, perfectly straight, infinitely long lines – objects that we have never seen even a single example of? How could we be more confident about the properties of these ideal objects than we are about the everyday objects that we allegedly abstracted them from?
Many different answers were proposed over the centuries. Plato claimed that our inborn geometric intuition was evidence of the soul's previous existence in another form. Prior to this life, he claimed, our souls lived in a realm of abstraction, where we beheld the forms directly. Our knowledge of right angles and parallel lines is a remembrance of that previous life, and so is our knowledge of Love, Justice, and all other abstractions.
Descartes believed that each human is born with a faculty of intuition, through which we can perceive simple truths immediately, without evidence or reasoning. “Intuition,” he writes in Rules for the Direction of the Mind, “is the undoubting conception of an unclouded and attentive mind, and springs from the light of reason alone; it is more certain than deduction itself.” This faculty was not limited to mathematics, but could perceive truth in any area, if the questions could be made simple enough. The goal of his Rules was to provide techniques for simplifying questions until they reached the point at which intuition could perceive the truth directly. He claimed to be developing a science which “should contain the primary rudiments of human reason, and its province ought to extend to the eliciting of true results in every subject.” He asserts: “All knowledge consists solely in combining what is self-evident.”
Spinoza saw the hand of divinity in such intuition. “Our mind, in so far as it truly perceives things, is a part of the infinite intellect of God, and therefore it must be that the clear and distinct ideas of the mind are as true as those of God.” The knowledge that such divine intuition makes available to us might have been lost forever behind a veil of superstition that “would have been sufficient to keep the human race in darkness to all eternity, if mathematics ... had not placed before us another rule of truth.”
The Encyclopedia Britannica tells us that “Leibniz distinguished necessary truths, those of which the opposite is impossible (as in mathematics), from contingent truths, the opposite of which is possible, such as ‘snow is white.’ But was this an ultimate distinction? At times Leibniz said boldly that if only man knew enough, he would see that every true proposition was necessarily true – that there are no contingent truths, that snow must be white.”
Kant theorized that we do not perceive the outside world directly at all, but that our senses present us with a preprocessed version of reality. Time and space, as we perceive them, do not really exist, but are merely properties of the way that our senses structure reality for us. Thus we are certain that the world will be Euclidean in the same way that a man wearing red glasses can be certain that the world he sees will be red. According to Kant, we see the world as Euclidean because we are unable to see it any other way.
Our ability to examine non-Euclidean geometry gives us an advantage over all these thinkers. And yet, none of them is entirely wrong. Our present-day understanding of geometry (as we will see in the Epilogue) borrows a little from all of these philosophers.
The Most Sincere Form of Flattery
It is no wonder then, that the Elements of Geometry is the most imitated text of all time. In one field of study after another, thinkers have wanted their systems of thought to become public truths in the same way as Euclidean geometry. Descartes' Meditations is one such effort. He attempts to assume nothing that is not self-evident, and to proceed rigorously to conclusions which he hopes will be universally acceptable. Thus, he is not even willing to assume that he himself exists, but rather concludes it from the observation that he thinks. This was self-evident to Descartes, because the opposite would be absurd: How could he think about the possibility that he was not thinking?
The high point of Catholic theological thought, the Summa Theologica of St. Thomas Aquinas, also imitates the style of Euclid, as does the Ethics of Spinoza, where we read “I shall consider human actions and appetites just as if I were considering lines, planes, or solids.”
The influence of the Euclidean form was amplified by the astounding success of Newton. His laws of motion may have been based on experiment and observation rather than pure reason or intuition, but once these laws were granted his reasoning proceeded to conclusions seemingly far removed from his assumptions. The same rules applied to falling rocks and planets moving across the sky. It is hard for us to recapture the shock of this discovery to Newton’s contemporaries. For more than a thousand years the Lord’s Prayer’s affirmation “Thy will be done on Earth as it is in Heaven” had been more than just a metaphor. The regularities of the planets were seen as a direct expression of the will of God. The learned man who read and believed the works of Newton must have felt an awe of the invisible order every bit as strong as the ancient Egyptians felt when they saw the miracles wrought by their priestly surveyors and architects.
With Spinoza’s claim that “the clear and distinct ideas of the mind are as true as those of God,” it became possible for human reason to oppose the claims of both tradition and divine revelation. Jefferson is claiming this authority when he writes "We hold these truths to be self-evident …" The rights of each individual, he was claiming, were not just his personal beliefs; they were universal truths that each person could validate by using his own Cartesian intuition.
By the middle of the 18th-century, the Euclidean form of reason with its definition/assumption/theorem structure reigned supreme in intellectual circles. It was the only rigorous way to think; it gave access to the highest form of truth; it forced all reasonable people to fall into line; and there was no limit to how far it could go.
Like the mythic Tower of Babel, the rational worldview of the Enlightenment was a structure that would reach to Heaven and make men as gods. In the myth, God deals with this possibility by splitting the language of the builders into many languages. Early in the 19th century something similar happened: geometry split into many geometries.
1. According to the author… in what ways has Euclid influenced philosophical thought?
2. What examples do you see of Euclid’s influence in your schooling and life? Include several examples.
3. What would a critic of the essay argue? Choose a excerpt to respond to and write a vigorous rebuttal.
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