Mathematics and the Real World (24 page)

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.

28. AND THEN CAME EINSTEIN

Let us recall where matters stood toward the end of the nineteenth century with regard to the mathematical description of nature. On the one hand, Newton's mechanics based on arithmetic and Euclidean geometry had attained enormous success in both celestial mechanics and earthly engineering problems. The success of part of Newton's mechanics may be attributed to the existence of the mysterious matter, the ether. On the other hand, Maxwell had presented equations that predicted the existence of electromagnetic waves, which had indeed been found. The ether could not constitute the medium in which those waves moved, and no other medium was known. In addition, if Maxwell's equations were applied in Newtonian geometry, they lacked one very important element of Newton's theory—that the measurements could be taken in every inertial system. At the same time, mathematicians and physicists of the time started questioning whether Euclidean geometry was the right one for describing the world we live in. The feeling was, and Gauss stated it explicitly, that arithmetic could be relied upon as a tool for describing nature, but as it was not clear what the geometry of nature was, geometry could not be relied upon as such a tool.

This was the situation with which Einstein was familiar. Several additional discoveries and hypotheses were made that could have influenced him, but it is not clear to what extent he was exposed to and aware of
them. One such discovery was the famous experiment performed by Albert Michelson and Edward Morley, two American physicists. At that time the scientific community still believed in the existence of the ether as the medium through which forces are exerted (Maxwell's theory had not as yet been accepted; the Michelson-Morley experiment took place before electromagnetic radiation was discovered in a laboratory). One of the questions that arose was: In what direction does the ether move relative to the Earth? The idea of the experiments was to utilize the fact that the ether was the medium through which light waves propagate.

The principle was simple. Assume that a beam of light travels from a source,
A
, on Earth toward a mirror at point
B
and back, with the mirror arranged such that the direction from
A
to
B
is the direction of the Earth's orbit. At the same time, another beam of light was sent from the same source
A
to a mirror at point
C
and back, with the direction from
A
to
C
at right angles to the direction of the Earth's orbit. A simple calculation shows that the first beam of light would get back to
A
after the second beam. A calculation is needed because the claim is not at all intuitive. In order to understand that it is correct, imagine that the speed of light is just twice the speed at which the Earth is moving. In that case, the distance that the first beam has to travel to get from
A
to
B
is twice the actual distance between them. In the time that takes, the second beam will have already gotten back to
A
; that is, it will get back to the source sooner. In reality, the speed of light is much faster than twice the speed of the Earth, and the time difference between each of the beam's returns to the source would be minimal.