When mathematics was in crisis: The birth of non-euclidean geometry

            The Enlightenment of the 17th and 18th century held up reason as the hallmark of human progress, while the romantic movement of the 19th century was a backlash against what it saw as the arrogance and naiveté of this reduction of the human to its brain. As such, there was a divide in the intellectual world between the sciences and the arts. The sciences largely bought into the Enlightenment presuppositions that the world was well-behaved, regulated by absolute laws that were accessible to human reason through rigorous processes of observation and logic. Acquisition of an understanding of these laws was paramount in our striving to move forward as a species.

The practitioners of the arts and letters, on the other hand, saw themselves as the loyal opposition, needing to correct what they saw as the overreach of the sciences, which seemed to miss the beauty, the joy, the experience of being human. The heart was as important as the brain and the fetish that scientists held for knowledge limited their true understanding.

But this divide was not impermeable. As we’ll see, there are important experienced by the sciences changed the way we saw the universe, the world, and human nature and this affected what was painted, built, written, directly. As the late 18th century philosopher Immanuel Kant pointed out, “Our understanding of reality is always mediated through concepts we use to create the ideas in our mind that only seem to come fully formed from our senses.”

But contrary to Kant, who held that these basic categories were necessary over time with advances in the sciences, and force us to radically revise how we make sense of ourselves and our environment. This provides fertile ground for scientists, who are often in need of new and exciting ways to try to organize the seemingly strange results they receive from the universe.

Sometimes art and science influence each other, but often not. There was a tension between the Enlightenment-influenced rationalists and thinkers who were romantically inclined. Those who latched onto reason would put forward the progress that science and technology made as evidence to that could come from nowhere else. First and foremost amongst these were the propositions of mathematics. They and they alone, provided humanity with truths that were absolutely certain, undeniably true.

Rene Descartes in the 17th century—one of the founders of this rationalistic movement and a major contributor to physics, mathematics, and philosophy—thought that the methods of the mathematician were so impressive that they ought to form the backbone of all further investigations. In all other areas of conversation, smart people disagreed about everything. But in mathematics you have universal assent. You had facts that could not be challenged by anyone who understood them and complex results that were derived from them with absolute rigor.

Mathematics was a thing of beauty, an absolute bedrock on which we could mathematized physics a generation later with his invention of calculus, it seemed like the rational worldview based on mathematics was well on its way to giving us an unassailable sense of reality itself. Mathematical propositions were self-evident, true beyond question. Those who doubted them showed themselves either to be lacking in understanding, mentally worrisome it was when in the 19th century, the very foundations of all of mathematics came into serious doubt.

Traditionally, we’ve thought the mathematical realm as having two parts: Geometry deals with shapes in space, and arithmetic deals with matters of number. Both had been rigorously grounded, we thought, and while there were interesting works showing some interconnections, we thought we had apart in both.

Since the 3rd century B.C., geometry was synonymous with the name Euclid. There had been centuries of work on geometry before Euclid, but what he did was take these results and order them. He created a structure based on a few simple and obvious propositions, and used a strict means of reasoning to derive hundreds of complex and intricate theorems. These theorems, because most basic truths.

These basic truths come in three groups. First are the definitions that simply describe what we mean by our basic geometric terms. A circle, for example, is the set of points in a plane some distance from a center point. In other words, pick a point, move a number, and now move that number of units away and in any direction you get the center point that makes a circle.

Definitions are true. They’re true because they simply tell us what we mean by words. We’re free to define any word in any way we want. As such Euclid’s definitions are true by definition. But there are two other categories of basic truths he used. One are Euclid’s axioms. The axioms were basic obvious truths that were not explicitly geometric. For example, equals added to equals yields equals. If John and Suzy have the same number of apples and I give them each some additional number of apples—giving them the same—then each still has the same number of apples as the other. How can you argue with that?

The postulates are similar except they are about purely geometric matters. Give us any two points and we can draw a line between them. Give us any line segment and we can continue that line as far as anyone wants in either direction. Give us a point and we can draw a circle around it any size we want. All right angles are equal to each other. No one could possibly doubt these.

These are just the first four of Euclid’s postulates. But the fifth, is different. Euclid put it like this: “That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, are less than two right angles.”

In other words, if two lines are approaching each other, they’ll eventually intersect. We usually think of it in terms of an equivalent formulation—take a line and a point not on that line. How many lines can we draw through the point that will be parallel with the line? One and only one. This postulate seemed less like the others and more like Euclid’s theorems, that is, the statements he proved from the other postulates. Maybe it’s a theorem. That is, maybe it would be possible to derive it from the other four. This would be a big deal because one thing mathematicians prize is elegance. A system is elegant if it makes the fewest possible assumptions.

To show how we could derive the fifth postulate from the other four, making it unnecessary as an assumption, it would shrink the set of presuppositions and this would improve the system—Euclid’s system, the greatest and seemingly most elegant and powerful system of all time. Improving Euclid would be like improving Shakespeare. It would assure one’s place in the annals of mathematical history, and thus much time was spent by brilliant so-called parallel postulate. But it was never found. It turns out there is a reason it was never found—it doesn’t exist. You can’t improve on Euclid as mathematicians had hoped.

The entire postulate is entirely independent from the other four, but we found this out the hard way. After mathematicians had failed in all their attempts to create a direct proof from the first four to the parallel postulate, the idea occurred to several different mathematicians to try what’s called an indirect proof. Now, we can show something is true by demonstrating that it can’t be false. If you know that I have a sibling and you want to prove that I have a brother, it suffices to prove that I can’t have a sister. If I have a sibling and it’s false that I have a sister, then it must be true that I have a brother.

What the mathematicians wanted to prove is that Euclid’s fifth postulate can be derived from the other four; that means that the truth of the other four postulates guarantees the truth of the fifth. So we start by assuming the opposite, that the other four postulates are true and that the fifth is false. Then we derive a contradiction, that is, any sentence of the form “A and not A.” Since either A or not A has to be true, but both can’t be, the contradiction A and not A has to be false.

The existence of this contradiction shows that if the postulates one, two, three, and four are held to be true then the denial of the fifth can’t be true. But if the denial of the fifth is false, then the fifth has to be true. This would show that Euclid could be simplified. But when mathematicians assumed one, two, three, and four, and the negation of five, they worked, and worked and worked but never found a contradiction. They found weird results— things like: You can’t have triangles with the same angles but of different size, the internal angles of triangles add to less than 180 degrees. Bizarre stuff—statements that seemed like they were false, but never a contradiction that had to be false.

In the first half of the 19th century, the Russian mathematician Nikolay Lobachevsky, among others, realized that they had found something incredibly deep and troubling. They had in their hands a new geometry, a different geometry—a non-Euclidean geometry.

This strange world was a new mathematical realm—if you’ll pardon the pun, a parallel mathematical universe. But if we have two geometries, which one’s true? When we had only Euclid’s, we assumed that it gave us the absolute truth about the nature of shape and space. But if there is a possible alternative, we can’t hold its truth to be absolute. We need a new sort of evidence to justify our belief in what seemed indubitable. But, what kind of evidence could this be? We can’t simply say that the alternatives are too weird—being weird doesn’t make it false.

(To be continued …)