Mathematical Proof and the Principles of Mathematics/History/The problem of parallels

A major step in reforming the foundations of mathematics was the development of what is now called non-Euclidean geometry. This developed from what was perceived to a major flaw in The Elements. Actually, as we've mentioned earlier, The Elements has quite a few flaws both major and minor, discovered in hindsight.

The fifth postulate

Euclid, following Aristotle, made a distinction between axioms, which were meant to include those truths which applied universally, and postulates, which included truths relevant to the topic at hand. So the five postulates of Euclid did not include statements like

When an equal amount is taken from equals, an equal amount results.

The postulates did include assumptions about points, lines, circles and angle. For example

It is possible to describe a circle with any center and any radius.

Most of these postulates seemed simple and intuitive, but the fifth postulate was both wordy and not immediately obvious. The actual statement is

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.

This seems like it could be a theorem which could be proved from the remaining postulates and axioms, yet it remained an open problem to find such a proof. Failing that, perhaps a new, simpler and more obvious postulate could be found as a replacement. False proofs were occasionally published, but these were found to have logical errors, and this only made the problem more tantalizing. There was progress on the problem before, but the first major advance came from Giovanni Saccheri in the early 18th century, coincidentally at the same time Berekley wrote The Analyst.

But before going into details on this problem, it's worthwhile to discuss some of the other problems found in The Elements. Starting right at the beginning in Book 1, Proposition 1, the construction of an equilateral triangle, Euclid assumes without proof that the two circles he created have a point of intersection. From the diagram it may seem clear that the circles intersect, but this is not a substitute for proof; there are many examples where what seems obvious from a diagram simply isn't true. A bit later in Book 1, Proposition 4, Euclid attempts to prove that if two triangle have two sides and their included angle equal then the triangles are congruent. The proof uses rigid motion, the idea that a triangle can be moved around on the plane without changing its properties. At the least it would require a postulate to make this a valid argument, but modern authors generally just take Proposition 4 itself as an axiom. There are other examples as well, but this should be enough to show that today's standards of proof are very different than those of Euclid's time.

Saccheri

Saccheri quadrilaterals

Saccheri attempted to find a proof and published the results in the rather ambitiously titled Euclides ab omni naevo vindicatus or "Euclid Freed of Every Flaw". His method was to consider a so-called Saccheri quadrilateral. (Actually, Omar Khayyam studied these earlier but the names used in mathematics are often inaccurate. On the other hand, since Saccheri's contribution was nearly lost to history and was only rediscovered by Beltrami in 1889, perhaps it's only fair after all.) A Saccheri quadrilateral ABCD has two opposite sides AD and BC equal and has right angles at A and B. In can be shown that angles C and D are then equal. Using the parallel postulate it can be shown that C and D are, in fact, right angles and the Saccheri quadrilateral is simply a rectangle. It can be shown as well that if C and D are right angles then the parallel postulate must hold, so to prove the parallel postulate it is enough to eliminate the cases where C and D are obtuse angles and where C and D are acute angles. This is one of many examples where the problem is reduced to three case and the case where the fifth postulate holds is somehow at the boundary between the remaining two.

The obtuse case can be eliminated using the remaining four axioms, but the proof makes critical use of the second postulate, which states that a line can be extended indefinitely. It turns out geometry on a sphere, studied for nearly as long as geometry on the plane because of its application to astronomy, is a perfectly consistent system of geometry where the second and fifth postulates fail to hold. One has to reinterpret the word "line" to mean "great circle" and make other changes in terminology, but from a purely logical point of view this "spherical geometry" is similar to plane geometry except new postulates replace the second and fifth postulates. The significance of this fact was not realized until much later.

That left the acute case. Saccheri developed an extensive theory for this hypothetical geometry in the hope of deriving a contradiction. In the end, he proved the theorem that two similar triangles must be congruent in this geometry, a fact which he held to be inconsistent with the nature of space. Saccheri left the proof there, despite the fact that there was no logical contradiction, only a statement that defied intuition and experience.

Gauss, Bolyai and Lobachevsky

Although Saccheri's attempt was technically a failure, it was influential in that helped later authors to consider the possibility that the fifth postulate was not, in fact, provable from the other four and a new type of geometry, a perfectly consistent geometry, might arise from the acute case. One of these was Johann Lambert who published Theorie der Parallellinien or "Theory of Parallel Lines" in 1766. This "non-Euclidean" geometry might still turn out to be contradictory, but in the mean time it produced interesting theorems and held a strange attraction.

By the start of the 19th century the idea that there might be some benefit in studying this new hypothetical geometry attracted the attention of Gauss. Although he never published his work, he did communicate his ideas to others through letters. (It has been said that Gauss feared damage to his reputation, the "uproar of the Boeotians", if he published on this controversial topic, but if anyone's reputation could stand the hit it would be that of Gauss.) Eventually, János Bolyai and Nikolai Lobachevsky, probably influenced by Gauss, both independently published treatises on the new geometry in the early 1830's.

Meanwhile, efforts to find a simpler alternative to the fifth axiom yielded a number of possibilities, for example:

If a line is perpendicular to one of two parallel lines then it is perpendicular to the other.
For two points on a given line, the distances to a parallel line are equal. (Simson)
Through a point not on a given line, exactly one parallel line can be drawn. (Playfair)

Other postulates, such as (following Saccheri) that a rectangle exists, have the advantage of simplicity but lack power. So it would take much more work to proceed to the rest of geometry using them as a starting point. Another such axiom is that there exist similar triangles which are not congruent.

It should be noted that Louis Bertrand published what has been considered the most convincing "proof" of the postulate in 1778. The proof relies on certain intuitive ideas about infinite areas, but these ideas turn out to be false in the new geometry. Another convincing argument was given by Bernhard Thibaut in 1809, relying on the idea that direction is preserved by motion in the plane. Examination of these so-called proofs leads to the conclusion that while this hypothetical plane might exist in some sense, it would have to be a very odd place.

Beltrami

The issue was finally settled in 1866 by Eugenio Beltrami. He constructed what is called a model of the new geometry within Euclidean geometry. In this model, every geometric term and property is given a different interpretation within the Euclidean plane. With these new interpretations, all the postulates of Euclidean Geometry hold except for the fifth postulate which is false.

In this way, any contradiction derived with the assumption that the fifth postulate is false would, in turn, lead to a contradiction with the assumption that the postulate is true. In effect, this proves that there is no proof of the fifth postulate, assuming of course that Euclidean geometry is free of contradictions.

Beltrami's model is somewhat complicated compared with more recent models, so we'll briefly describe what is known as the Poincaré disk model. A circle C on the plane is fixed.

Poincaré disk with hyperbolic parallel lines

A "point" is taken to mean a point in the interior of C.
A "line" is taken to that portion circle which intersects C at right angles.
Formulas for distances, angles and areas can be given in terms of properties of the point and lines involved. One early result was that the area of a triangle is proportional to angle deficit, defined as the difference between the sum of the angles and π.

Einstein

The new geometry may be logically consistent, but surely it can't be the geometry of the actual universe, or can it? Actually, at small scales new geometry is almost identical to Euclidean geometry. So if one were to try to determine by experiment whether the fifth postulate holds, say by measuring the sum of the angles in a triangle, the experiment would be inconclusive if the triangle were too small. But the universe is vast compared to our corner of it, so there is no triangle we could measure which would be big enough to find a difference between a Euclidean universe and a non-Euclidean one.

In fact, according to Einstein's general theory of relativity, space and time are curved in a way similar to the way lines "bend" toward or away from each other in non-Euclidean geometry. The degree to which space-time is curved depends on the presence of astronomical bodies, and the effects of gravity are seen as really being the effects of this curvature. So the theory of relativity states that space is actually non-Euclidean, but that turns out to be useful for explaining the physical universe.

Actually, allowing for local variation due to gravity, modern cosmology is closing in on an answer as to whether the universe is "flat", in other words Euclidean. Of course, we can only describe what the observable universe is like; the unobservable universe is outside the domain of science. Also, since Euclidean case sits at the boundary between a positively curved case (obtuse angles) and the negatively curved case (acute angles), a result of Euclidean can never be definitive; an increase in the accuracy of the test might still tilt the balance toward one of the other cases. This turns out to be exactly where we are, though the precision of the measurement says Euclidean to within a few percent.

Philosophical implications

The effect of these developments on the philosophy of mathematics was profound, perhaps matching the effect of the theory of evolution on biology. No longer could axioms and postulates be taken as self-evident truths. One kind of geometry might be better at describing what you see in the real world better than another one, but there is no such thing a geometry that's absolutely true. If there is no such thing as an absolute geometry of the real world, then perhaps the same holds for other mathematical ideas such as numbers.

In a way, this put mathematics at the same level as a game like chess or Monopoly. The axioms of a mathematical theory correspond to the rules of the game. Just as one set of rules might lead to a more entertaining game than another, so one set of axioms may lead to a more useful or interesting mathematical theory than another. But there is no such thing as a perfect set of rules to create a perfect game.

But this new found uncertainty was not an entirely bad thing. Freed from the restriction that geometry had to match intuition and the observed world, new types of geometries began to appear giving rise to new areas of mathematics.

The Fano plane, a "geometry" with seven lines and seven points.

One such is the Fano plane, which has points and lines, but no concepts of distance or angles. The number of points and lines is finite, so the entirety of the geometry can be represented by a simple diagram.

Cards from the game of Set. If the cards are taken to be "points" and the sets are taken to be "lines", then the game can be interpreted as a geometry with 81 points and 1080 lines.

Another benefit is that new interpretations may be found for existing sets of axioms. This means that new applications can be found for a theory that was developed for something entirely different.