Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)
CHAPTER IV. MATHEMATICS AND THE MODERN VIEW OF THE WORLD
27. THE GEOMETRY OF THE WORLD
To describe Einstein's theory of relativity with the correct perspective we must go back more than two millennia. We will repeat what axioms meant to the Greeks. They were basic working assumptions from which, by using the power of logic, they could develop mathematics further. The axioms themselves were obvious facts, physical truths, or ideal mathematical truths that did not need to be explained or substantiated. The geometry of the space in which we live could of course be examined mathematically, and Euclid summarized the mathematics of geometry in his book Elements, where he presented the axioms of geometric space that had been formulated in classical Greece. Euclid formulated ten axioms and postulates (modern formulations show fourteen), including “self-evident” axioms such as, given two points, there is a straight line between them, and that a point in a plane and a given length from it define a circle with the point at its center and the given distance as its radius. Euclid formulated his fifth axiom, which in time became the parallel-lines postulate, as follows (see the upper part of the diagram below).
If a line transversing two lines forms on one side of it two angles whose sum is less than that of two right angles (i.e., less than 180 degrees), the extensions of the two lines will meet on that side.
The formulation of the parallel-lines postulate accepted today states (see the lower part of the diagram):
Given a line and a point not on it, there is only one line that passes through that point that is parallel with the first line; parallel means that the extensions of the lines will never meet.
Euclid himself made no reference to parallel lines, that is, straight lines that never meet. The reason was, apparently, the desire to avoid the assumption that there are lines that are infinite, having no end. As we wrote in section 7, Aristotle distinguished between infinity and potential infinity. He claimed that potential infinity existed so that, for example, we can draw lines as long as we want, but he denied the existence of infinity, that is, he denied that an infinite line exists. Euclid adopted that view in his books and avoided presenting and dealing with infinite quantities.
His fifth axiom caused unease even in Euclid's lifetime. The critics claimed that it was not obvious nor apparent from observing nature, and therefore it was not suitable to be included as an axiom. Attempts were made, therefore, to prove that property using other, self-evident axioms, but those attempts failed. Interest in this question continued for hundreds of years. Particularly noteworthy were the contribution and discussions of Omar Khayyám (1048–1131) and his school in Persia. It was there that Euclid's original fifth axiom was replaced by the parallel-lines postulate, which is accepted still today. There too it was proven that if one assumes the existence of infinite lines, the two versions are the same.
An interesting development that illustrates the complexity of the connection between mathematics and nature was made by the Italian mathematician Geraolamo Saccheri (1667–1733) of the University of Pavia. We will give the details of the case to show the difficulties that logical arguments are likely to cause. Saccheri used proof by contradiction, that is, he chose to assume that the parallel-lines axiom was not correct and tried to obtain a contradiction between that assumption and the other statements that could be proven using the other axioms. In that way he would show that there was a contradiction deriving from the assumption that the axiom was incorrect so that the parallel-lines axiom is derived from the other axioms. We have shown previously that proof via reduction ad absurdum, or by contradiction, is not natural, and indeed it is difficult even to follow the steps that guided Saccheri, as we now show.
The parallel-lines axiom can be divided into two claims. The first is that there is a line parallel with the first line, and the second is that there is not more than one parallel line that passes through a given point not on that line. Saccheri first assumed that there is not even one parallel line that goes through the point, and he succeeded in finding a contradiction between that assumption (of no parallel line) and results that can be obtained from the other axioms. He concluded that there was at least one line parallel with the original line that went through the point not on the original line. He then assumed that there were two or more parallel lines that passed through the point. By means of a construction based on those two parallel lines he found such planes, however, with very strange properties, so much so that it was clear that they did not exist in the physical space we see around us. That was enough to convince Saccheri, and he declared that he had found the contradiction he was looking for and that the parallel-lines axiom is derived from the other axioms (because if we assume it is incorrect and we find a contradiction, then it is correct).
Saccheri, however, did not take the trouble to examine whether the fact that the strange properties do not exist in our space can be derived from Euclid's other axioms. Only if that is done can it be concluded that there is a contradiction. Otherwise, the only conclusion that can be drawn is that Euclid's axioms also allow the existence of strange spaces. That flaw derived, of course, from the belief that the axioms described the space around us, and, to establish a contradiction, it was sufficient to find a property that our physical space does not have in order to conclude that the property does not exist in the mathematical space. A short while later it was found that there was no mathematical contradiction between the properties discovered by Saccheri and the other axioms, and the question of whether the parallel-lines axiom derives from the other axioms was declared still open.
The German mathematician Georg Klügel (1739–1812) made a conceptual contribution to solving the problem. His doctoral thesis at the University of Göttingen, Germany, was devoted to a detailed review of the parallel-lines axiom and the hitherto failed attempts to reconcile it with the other axioms. Klügel completed his thesis with the hypothesis that Euclid's fifth axiom was based on the experience of our senses and hence may not be correct. In other words, there may be geometries that satisfy the other axioms but not the parallel-lines axiom. That declaration itself made researchers try to construct geometries in which that axiom did not hold, and this they did quickly. One was by Abraham Kästner (1719–1800), who constructed a geometry with properties similar to those discovered by Saccheri about a hundred years earlier, but this time the conclusion was the opposite, that is, the parallel-lines axiom does not depend on the other geometrical axioms, meaning there are mathematical spaces in which that axiom does not hold while the other axioms do. The conclusion drawn was that Euclid's axioms do not describe the physical space completely, a possibility that, if Saccheri had thought of it, might have led to the solution of the problem some hundred years sooner.
The geometries developed by Kästner and others at that time had such strange characteristics that it was clear they were not relevant in the context of a description of nature. The obvious conclusion was that new axioms had to be added to Euclid's original ones that would characterize the space we experience every day. Such an attempt was made by a student of Kästner's, Carl Friedrich Gauss (1777–1855), one of the greatest mathematicians of all time. Gauss, who worked for most of his life in Göttingen in the then Kingdom of Hanover, Germany, was born to a poor family, but the exceptional mathematical abilities he exhibited even at an early age brought him to the attention of the Duke of Braunschweig, who used his influence with the University of Göttingen to get them to accept Gauss as a student. Gauss made an enormous contribution to mathematics, which we cannot describe here. Most of his work related to the theory of numbers, but he contributed greatly in other areas too and was also one of the great philosophers of the natural sciences. Gauss learned from his tutor, Kästner, that the parallel-lines axiom did not derive from the other axioms, but at least at the outset he still thought that the axiom described the world around us. For years Gauss tried to suggest “correct” axioms, meaning obvious ones, from which the parallel-lines axiom could be proven. After years of unsuccessful attempts, his faith was shaken and he started to look for other axioms to replace the parallel-lines one. He saw that the parallel-lines axiom, that is, the property that parallel lines do not meet, applies in our daily experiences, experiences based on the measurement of small distances. Thus geometry that describes the world around us complies with a property similar to the parallel-lines axiom when dealing with small distances. For example, the parallel-lines axiom leads to the result that the sum of the angles of a triangle is 180 degrees. In other geometries that were found to fulfill the other axioms, not the parallel-lines one, the sum of the angles of a triangle was greater than 180 degrees. Gauss added the requirement that as the triangles get smaller and smaller, the sum of the angles has to come closer and closer to 180 degrees, or, put differently, for small distances, the geometry must be similar to the geometry we experience day to day. None of the geometries discovered previously that fulfilled the axioms of the plane apart from the parallel-lines axiom satisfied that requirement.
The question still remains unanswered whether the parallel-lines axiom can be proved from Euclid's axioms together with Gauss's new requirement, or whether perhaps even with that new requirement the parallel-lines axiom does not hold. If the latter is the case, the question of what is the true geometry of the physical space arises even more forcefully. In this connection, the following story is told about Gauss. Gauss, as a mathematician, developed a method for measuring the state's land. He thus served as a guide to the state's official surveyors and actually carried out measurements himself. According to the story, which has no historical authentication, Gauss tried to measure the angles of a triangle formed by three mountains in Germany that were situated a long distance from each other. If he would have found that the distances between them were large enough for the sum of the angles of the triangle to be more than 180 degrees, he would have proved that Euclid's mathematical geometry does not correctly describe the physical reality. The measurements did not reveal such a triangle. Be that as it may, the story is consistent with the fact that Gauss himself solved the open question, showing that the parallel-lines axiom cannot be proven from the plane axioms even if his new condition was applied, but that revelation he kept to himself. Gauss did not divulge whether he thought the parallel-lines axiom applies in the physical world or not.
An example that shows that a geometry can exist that fulfills all the plane axioms, including the new condition that Gauss proposed but excluding the parallel-lines axiom, was discovered by two young mathematicians independently of each other. One was a Russian from Kazan University, Nikolai Lobachevsky (1793–1856), and the other was a Hungarian, Johann Bolyai (1802–1860), an officer in the Hungarian army. Bolyai was the son of a well-known mathematician who had corresponded with Gauss and who had received letters in which Gauss wrote of his doubts about the geometry of the world. Lobachevsky and Bolyai both constructed geometries with the desired properties, namely, small triangles with the sum of their angles close to 180 degrees, but this did not apply to large triangles. When the young Bolyai revealed his discovery to Gauss, the latter showed him that he, Gauss, had already arrived at such a geometry, but he was generous enough not to claim the right to be acknowledged as its discoverer.
The mathematical problem was solved: the parallel-lines axiom is not derived from Euclid's axioms even if the requirement is added that in small distances space must behave as we experience it in our daily lives. The physical problem remains, however: Which of the various possible geometries according to the axioms is the one applicable to our world? This is not a trivial question. We must remember that Newton's theory, including his original equations and all the equations and other developments since Newton, was based on space as defined by Euclid's axioms, including the axiom about parallel lines. Is it possible that everything derived from this mathematics is not relevant in physical space?
At this point Bernhard Riemann comes into the picture. Despite his short life (he was born in 1826 and died at the young age of forty), he made crucial contributions to mathematics and physics. Georg Friedrich Bernhard Riemann was a student of Gauss, but even as a student he worked independently. Born to a poor family, he was a very sick child and youth. He started studying theology with the intention of becoming a priest. At the same time he showed great mathematical ability and tried to integrate the study of the Bible with mathematics, even attempting to examine the book of Genesis from a mathematical standpoint. His father, who recognized the young Bernhard's talent for mathematics, urged him to apply to the University of Göttingen, where he chose to work for his doctorate under the guidance of Gauss. The method of study and research required the candidate to submit three research topics, which the tutor and the thesis committee were supposed to approve. The tutor and committee were then to set for the student a predetermined time to write a thesis on them. The third topic proposed by Riemann brought about, many years after his demise, a change in the perception of the geometry of the world.
Riemann's approach was also to formulate axioms, but instead of looking for axioms that would describe what we feel and see, he developed a system of axioms that a physical space “ought to” satisfy, with “ought to” meaning that the concepts of “closest” and “shortest” make sense in the system. The technique is related to the angular structure of lines and planes and the curvature of different planes. This mathematical subject is called differential geometry.
It is not necessary to specialize in the subject to understand the concept underlying it, and that is the concept of a geodesic, the shortest line between two points. In a Euclidean space, a straight line is the shortest path joining two points. In general geometries, that is not necessarily the case. It may be that in the some geometries there may not be straight lines in the Euclidean sense, but there will be a shortest route that joins the two points. To illustrate, consider the following: in the geometry on the face of the Earth there are no straight lines, but there are geodesics. Airplanes flying between two towns along a similar latitude, for example between Madrid and New York, choose a route that takes them far north because that is the shortest. Geodesics in general space constitute the building blocks of Riemannian geometry. It is unclear where Riemann got his inspiration to define such a structure, but what is clear is that he was aware of the difficulties in describing and determining the geometry of nature and that he was familiar with the work of his teacher, Gauss. At the same time, he was aware of the least action principle and of Fermat's principle that preceded it. He therefore apparently wanted to construct general, not necessarily Euclidean, spaces in which the principles of shortest distance and least action could be applied. Riemann died before he could clarify his intentions. The mathematical tools he bequeathed, in particular the geometries based on shortest distances, served Albert Einstein in the construction of the new geometry of nature.