What Kind of Knowledge is Geometry?

Part of the article “The Unreasonable Influence of Geometry” by Doug Muder

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.

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