by Mario Livio
Merchant: But if I take the bet I shall be betting you that sum that I shall die within the year.
Captain: Why not if you lose the bet, as you certainly will?
Merchant: But if I am drowned you will be drowned too; and then what becomes of our bet?
Captain: True. But I will find you a landsman who will make the bet with your wife and family.
Merchant: That alters the case of course; but what about my cargo?
Captain: Pooh! The bet can be on the cargo as well. Or two bets: one on your life, the other on the cargo. Both will be safe, I assure you. Nothing will happen; and you will see all the wonders that are to be seen abroad.
Merchant: But if I and my goods get through safely I shall have to pay you the value of my life and of the goods into the bargain. If I am not drowned I shall be ruined.
Captain: That also is very true. But there is not so much for me in it as you think. If you are drowned I shall be drowned first; for I must be the last man to leave the sinking ship. Still, let me persuade you to venture. I will make the bet ten to one. Will that tempt you?
Merchant: Oh, in that case—
The captain has discovered insurance just as the goldsmiths discovered banking.
For someone such as Shaw, who complained that during his education “not a word was said to us about the meaning or utility of mathematics,” this humorous account of the “history” of the mathematics of insurance is quite remarkable.
With the exception of Shaw’s text, we have so far followed the development of some branches of mathematics more or less through the eyes of practicing mathematicians. To these individuals, and indeed to many rationalist philosophers such as Spinoza, Platonism was obvious. There was no question that mathematical truths existed in their own world and that the human mind could access these verities without any observation, solely through the faculty of reason. The first signs of a potential gap between the perception of Euclidean geometry as a collection of universal truths and other branches of mathematics were uncovered by the Irish philosopher George Berkeley, Bishop of Cloyne (1685–1753). In a pamphlet entitled The Analyst; Or a Discourse Addressed to An Infidel Mathematician (the latter presumed to be Edmond Halley), Berkeley criticized the very foundations of the fields of calculus and analysis, as introduced by Newton (in Principia) and Leibniz. In particular, Berkeley demonstrated that Newton’s concept of “fluxions,” or instantaneous rates of change, was far from being rigorously defined, which in Berkeley’s mind was sufficient to cast doubt on the entire discipline:
The method of fluxions is the general key, by help whereof the modern Mathematicians unlock the secrets of Geometry, and consequently of Nature…But whether this Method be clear or obscure, consistent or repugnant, demonstrative or precarious, as I shall inquire with the utmost impartiality, so I submit my inquiry to your own Judgement, and that of every candid Reader.
Berkeley certainly had a point, and the fact is that a fully consistent theory of analysis was only formulated in the 1960s. But mathematics was about to experience a more dramatic crisis in the nineteenth century.
CHAPTER 6
GEOMETERS: FUTURE SHOCK
In his famous book Future Shock, author Alvin Toffler defined the term in the title as “the shattering stress and disorientation that we induce in individuals by subjecting them to too much change in too short a time.” In the nineteenth century, mathematicians, scientists, and philosophers experienced precisely such a shock. In fact, the millennia-old belief that mathematics offers eternal and immutable truths was crushed. This unexpected intellectual upheaval was caused by the emergence of new types of geometries, now known as non-Euclidean geometries. Even though most nonspecialists may have never even heard of non-Euclidean geometries, the magnitude of the revolution in thought introduced by these new branches of mathematics has been likened by some to that inaugurated by the Darwinian theory of evolution.
To fully appreciate the nature of this sweeping change in worldview, we have first to briefly probe the historical-mathematical backdrop.
Euclidean “Truth”
Until the beginning of the nineteenth century, if there was one branch of knowledge that had been regarded as the apotheosis of truth and certainty, it was Euclidean geometry, the traditional geometry we learn in school. Not surprisingly, therefore, the great Dutch Jewish philosopher Baruch Spinoza (1632–77) entitled his bold attempt to unify science, religion, ethics, and reason Ethics, Demonstrated in Geometrical Order. Moreover, in spite of the clear distinction between the ideal, Platonic world of mathematical forms and physical reality, most scientists regarded the objects of Euclidean geometry simply as the distilled abstractions of their real, physical counterparts. Even staunch empiricists such as David Hume (1711–76), who insisted that the very foundations of science were far less certain than anyone had ever suspected, concluded that Euclidean geometry was as solid as the Rock of Gibraltar. In An Enquiry Concerning Human Understanding, Hume identified “truths” of two types:
All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are…every affirmation which is either intuitively or demonstratively certain…Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would forever retain their certainty and evidence. Matters of fact…are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction…That the sun will not rise tomorrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood.
In other words, while Hume, like all empiricists, maintained that all knowledge stems from observation, geometry and its “truths” continued to enjoy a privileged status.
The preeminent German philosopher Immanuel Kant (1724–1804) did not always agree with Hume, but he also exalted Euclidean geometry to a status of absolute certainty and unquestionable validity. In his memorable Critique of Pure Reason, Kant attempted to reverse in some sense the relationship between the mind and the physical world. Instead of impressions of physical reality being imprinted on an otherwise entirely passive mind, Kant gave the mind the active function of “constructing” or “processing” the perceived universe. Turning his attention inward, Kant asked not what we can know, but how we can know what we can know. He explained that while our eyes detect particles of light, these do not form an image in our awareness until the information is processed and organized by our brains. A key role in this construction process was assigned to the human intuitive or synthetic a priori grasp of space, which in turn was taken to be based on Euclidean geometry. Kant believed that Euclidean geometry provided the only true path for processing and conceptualizing space, and that this intuitive, universal acquaintance with space was at the heart of our experience of the natural world. In Kant’s words:
Space is not an empirical concept which has been derived from external experience…Space is a necessary representation a priori, forming the very foundation of all external intuitions…On this necessity of an a priori representation of space rests the apodictic certainty of all geometrical principles, and the possibility of their construction a priori. For if the intuition of space were a concept gained a posteriori, borrowed from general external experience, the first principles of mathematical definition would be nothing but perceptions. They would be exposed to all the accidents of perception, and there being but one straight line between two points would not be a necessity, but only something taught in each case by experience.
To put it simply, according to Kant, if we perceive an object, then necessarily this object is spatial and Euclidean.
Hume’s and Kant’s ideas bring to the
forefront the two rather different, but equally important aspects that had been historically associated with Euclidean geometry. The first was the statement that Euclidean geometry represents the only accurate description of physical space. The second was the identification of Euclidean geometry with a firm, decisive, and infallible deductive structure. Taken together, these two presumed properties provided mathematicians, scientists, and philosophers with what they regarded as the strongest evidence that informative, inescapable truths about the universe do exist. Until the nineteenth century these statements were taken for granted. But were they actually true?
The foundations of Euclidean geometry were laid around 300 BC by the Greek mathematician Euclid of Alexandria. In a monumental thirteen-volume opus entitled The Elements, Euclid attempted to erect geometry on a well-defined logical base. He started with ten axioms assumed to be indisputably true and sought to prove a large number of propositions on the basis of those postulates by nothing other than logical deductions.
The first four Euclidean axioms were extremely simple and exquisitely concise. For instance, the first axiom read: “Between any two points a straight line may be drawn.” The fourth one stated: “All right angles are equal.” By contrast, the fifth axiom, known as the “parallel postulate,” was more complicated in its formulation and considerably less self-evident: “If two lines lying in a plane intersect a third line in such a way that the sum of the internal angles on one side is less than the two right angles, then the two lines inevitably will intersect each other if extended sufficiently on that side.” Figure 39 demonstrates graphically the contents of this axiom. While no one doubted the truth of this statement, it lacked the compelling simplicity of the other axioms. All indications are that even Euclid himself was not entirely happy with his fifth postulate—the proofs of the first twenty-eight propositions in The Elements do not make use of it. The equivalent version of the “fifth” most cited today appeared first in commentaries by the Greek mathematician Proclus in the fifth century, but it is generally known as the “Playfair axiom,” after the Scottish mathematician John Playfair (1748–1819). It states: “Given a line and a point not on the line, it is possible to draw exactly one line parallel to the given line through that point” (see figure 40). The two versions of the axiom are equivalent in the sense that Playfair’s axiom (together with the other axioms) necessarily implies Euclid’s original fifth axiom and vice versa.
Figure 39
Over the centuries, the increasing discontent with the fifth axiom resulted in a number of unsuccessful attempts to actually prove it from the other nine axioms or to replace it by a more obvious postulate. When those efforts failed, other geometers began trying to answer an intriguing “what if” question—what if the fifth axiom did, in fact, not prove true? Some of those endeavors started to raise nagging doubts on whether Euclid’s axioms were truly self-evident, rather than being based on experience. The final, surprising verdict eventually came in the nineteenth century: One could create new kinds of geometry by choosing an axiom different from Euclid’s fifth. Furthermore, these “non-Euclidean” geometries could in principle describe physical space just as accurately as Euclidean geometry did!
Let me pause here for a moment to allow for the meaning of the word “choosing” to sink in. For millennia, Euclidean geometry had been regarded as unique and inevitable—the sole true description of space. The fact that one could choose the axioms and obtain an equally valid description turned the entire concept on its ear. The certain, carefully constructed deductive scheme suddenly became more similar to a game, in which the axioms simply played the role of the rules. You could change the axioms and play a different game. The impact of this realization on the understanding of the nature of mathematics cannot be overemphasized.
Figure 40
Quite a few creative mathematicians prepared the ground for the final assault on Euclidean geometry. Particularly notable among them were the Jesuit priest Girolamo Saccheri (1667–1733), who investigated the consequences of replacing the fifth postulate by a different statement, and the German mathematicians Georg Klügel (1739–1812) and Johann Heinrich Lambert (1728–1777), who were the first to realize that alternative geometries to the Euclidean could exist. Still, somebody had to put the last nail in the coffin of the idea of Euclidean geometry being the one and only representation of space. That honor was shared by three mathematicians, one from Russia, one from Hungary, and one from Germany.
Strange New Worlds
The first to publish an entire treatise on a new type of geometry—one that could be constructed on a surface shaped like a curved saddle (figure 41a)—was the Russian Nikolai Ivanovich Lobachevsky (1792–1856; figure 42). In this kind of geometry (now known as hyperbolic geometry), Euclid’s fifth postulate is replaced by the statement that given a line in a plane and a point not on this line, there are at least two lines through the point parallel to the given line. Another important difference between Lobachevskian geometry and Euclidean geometry is that while in the latter the angles in a triangle always add up to 180 degrees (figure 41b), in the former the sum is always less than 180 degrees. Because Lobachevsky’s work appeared in the rather obscure Kazan Bulletin, it went almost entirely unnoticed until French and German translations started to appear in the late 1830s. Unaware of Lobachevsky’s work, a young Hungarian mathematician, János Bolyai (1802–60), formulated a similar geometry during the 1820s. Bursting with youthful enthusiasm, he wrote in 1823 to his father (the mathematician Farkas Bolyai; figure 43): “I have found things so magnificent that I was astounded…I have created a different new world out of nothing.” By 1825, János was already able to present to the elder Bolyai the first draft of his new geometry. The manuscript was entitled The Science Absolute of Space. In spite of the young man’s exuberance, the father was not entirely convinced of the soundness of János’s ideas. Nevertheless, he decided to publish the new geometry as an appendix to his own two-volume treatise on the foundations of geometry, algebra, and analysis (the supposedly inviting title of which read Essay on the Elements of Mathematics for Studious Youths). A copy of the book was sent in June 1831 to Farkas’s friend Carl Friedrich Gauss (1777–1855; figure 44), who was not only the most prominent mathematician of the time, but who is also considered by many, along with Archimedes and Newton, to be one of the three greatest of all time. That book was somehow lost in the chaos created by a cholera epidemic, and Farkas had to send a second copy. Gauss sent out a reply on March 6, 1832, and his comments were not exactly what the young János expected:
Figure 41
Figure 42
Figure 43
If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years. So I remained quite stupefied. So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime.
Let me parenthetically note that apparently Gauss feared that the radically new geometry would be regarded by the Kantian philosophers, to whom he referred as “the Boetians” (synonymous with “stupid” for the ancient Greeks), as philosophical heresy. Gauss then continued:
Figure 44
On the other hand it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise with me that I am spared this trouble, and I am very glad that it is the son of my old friend, who takes the precedence of me in such a remarkable manner.
While Farkas was quite pleased with Gauss’s praise, which he took to be “very fine,” János was absolutely devastated. For almost a decade he refused to believe that Gauss’s claim to priority was not false, and his relationship with his father (whom he suspected of pre
maturely communicating the results to Gauss) was seriously strained. When he finally realized that Gauss had actually started working on the problem as early as 1799, János became deeply embittered, and his subsequent mathematical work (he left some twenty thousand pages of manuscript when he died) became rather lackluster by comparison.
There is very little doubt, however, that Gauss had indeed given considerable thought to non-Euclidean geometry. In a diary entry from September 1799 he wrote: “In principiis geometriae egregios progressus fecimus” (“About the principles of geometry we obtained wonderful achievements”). Then, in 1813, he noted: “In the theory of parallel lines we are now no further than Euclid was. This is the partie honteuse [shameful part] of mathematics, which sooner or later must get a very different form.” A few years later, in a letter written on April 28, 1817, he stated: “I am coming more and more to the conviction that the necessity of our [Euclidean] geometry cannot be proved.” Finally, and contrary to Kant’s views, Gauss concluded that Euclidean geometry could not be viewed as a universal truth, and that rather “one would have to rank [Euclidean] geometry not with arithmetic, which stands a priori, but approximately with mechanics.” Similar conclusions were reached independently by Ferdinand Schweikart (1780–1859), a professor of jurisprudence, and the latter informed Gauss of his work sometime in 1818 or 1819. Since neither Gauss nor Schweikart actually published his results, however, the priority of first publication is traditionally credited to Lobachevsky and Bolyai, even though the two can hardly be regarded as the sole “creators” of non-Euclidean geometry.