Is God a Mathematician?

by Mario Livio

  The lesson here is clear. The mathematical tools were not chosen arbitrarily, but rather precisely on the basis of their ability to correctly predict the results of the relevant experiments or observations. So at least for this very simple case, their effectiveness was essentially guaranteed. Humans did not have to guess in advance what the correct mathematics would be. Nature afforded them the luxury of trial and error to determine what worked. They also did not have to stick with the same tools for all circumstances. Sometimes the appropriate mathematical formalism for a given problem did not exist, and someone had to invent it (as in the case of Newton inventing calculus, or modern mathematicians inventing various topological/geometric ideas in the context of the current efforts in string theory). In other cases, the formalism had already existed, but someone had to discover that this was a solution awaiting the right problem (as in the case of Einstein using Riemannian geometry, or particle physicists using group theory). The point is that through a burning curiosity, stubborn persistence, creative imagination, and fierce determination, humans were able to find the relevant mathematical formalisms for modeling a large number of physical phenomena.

  One characteristic of mathematics that was absolutely crucial for what I dubbed the “passive” effectiveness was its essentially eternal validity. Euclidean geometry remains as correct today as it was in 300 BC. We understand now that its axioms are not inevitable, and rather than representing absolute truths about space, they represent truths within a particular, human-perceived universe and its associated human-invented formalism. Nevertheless, once we comprehend the more limited context, all the theorems hold true. In other words, branches of mathematics get to be incorporated into larger, more comprehensive branches (e.g., Euclidean geometry is only one possible version of geometry), but the correctness within each branch persists. It is this indefinite longevity that has allowed scientists at any given time to search for adequate mathematical tools in the entire arsenal of developed formalisms.

  The simple example of the pebbles in the vase still does not address two elements of Wigner’s enigma. First, there is the question why in some cases do we seem to get more accuracy out of the theory than we have put into it? In the experiment with the pebbles, the accuracy of the “predicted” results (the aggregation of other numbers of pebbles) is not any better than the accuracy of the experiments that had led to the formulation of the “theory” (arithmetic addition) in the first place. On the other hand, in Newton’s theory of gravity, for instance, the accuracy of its predictions proved to far exceed that of the observational results that motivated the theory. Why? A brief re-examination of the history of Newton’s theory may provide some insight.

  Ptolemy’s geocentric model reigned supreme for about fifteen centuries. While the model did not claim any universality—the motion of each planet was treated individually—and there was no mention of physical causes (e.g., forces; acceleration), the agreement with observations was reasonable. Nicolaus Copernicus (1473–1543) published his heliocentric model in 1543, and Galileo put it on solid ground, so to speak. Galileo also established the foundations for the laws of motion. But it was Kepler who deduced from observations the first mathematical (albeit only phenomenological) laws of planetary motion. Kepler used a huge body of data left by the astronomer Tycho Brahe to determine the orbit of Mars. He referred to the ensuing hundreds of sheets of calculations as “my warfare with Mars.” Except for two discrepancies, a circular orbit matched all the observations. Still, Kepler was not satisfied with this solution, and he later described his thought process: “If I had believed that we could ignore these eight minutes [of arc; about a quarter of the diameter of a full moon], I would have patched up my hypothesis…accordingly. Now, since it was not permissible to disregard, those eight minutes alone pointed the path to a complete reformation in astronomy.” The consequences of this meticulousness were dramatic. Kepler inferred that the orbits of the planets are not circular but elliptical, and he formulated two additional, quantitative laws that applied to all the planets. When these laws were coupled with Newton’s laws of motion, they served as the basis for Newton’s law of universal gravitation. Recall, however, that along the way Descartes proposed his theory of vortices, in which planets were carried around the Sun by vortices of circularly moving particles. This theory could not get very far, even before Newton showed it to be inconsistent, because Descartes never developed a systematic mathematical treatment of his vortices.

  What do we learn from this concise history? There can be no doubt that Newton’s law of gravitation was the work of a genius. But this genius was not operating in a vacuum. Some of the foundations had been painstakingly laid down by previous scientists. As I noted in chapter 4, even much lesser mathematicians than Newton, such as the architect Christopher Wren and the physicist Robert Hooke, independently suggested the inverse square law of attraction. Newton’s greatness showed in his unique ability to put it all together in the form of a unifying theory, and in his insistence on providing a mathematical proof of the consequences of his theory. Why was this formalism as accurate as it was? Partly because it treated the most fundamental problem—the forces between two gravitating bodies and the resulting motion. No other complicating factors were involved. It was for this problem and this problem alone that Newton obtained a complete solution. Hence, the fundamental theory was extremely accurate, but its implications had to undergo continuous refinement. The solar system is composed of more than two bodies. When the effects of the other planets are included (still according to the inverse square law), the orbits are no longer simple ellipses. For instance, the Earth’s orbit is found to slowly change its orientation in space, in a motion known as precession, similar to that exhibited by the axis of a rotating top. In fact, modern studies have shown that, contrary to Laplace’s expectations, the orbits of the planets may eventually even become chaotic. Newton’s fundamental theory itself, of course, was later subsumed by Einstein’s general relativity. And the emergence of that theory also followed a series of false starts and near misses. So the accuracy of a theory cannot be anticipated. The proof of the pudding is in the eating—modifications and amendments continue to be made until the desired accuracy is obtained. Those few cases in which a superior accuracy is achieved in a single step have the appearance of miracles.

  There is, clearly, one crucial fact in the background that makes the search for fundamental laws worthwhile. This is the fact that nature has been kind to us by being governed by universal laws, rather than by mere parochial bylaws. A hydrogen atom on Earth, at the other edge of the Milky Way galaxy, or even in a galaxy that is ten billion light-years away, behaves in precisely the same manner. And this is true in any direction we look and at any time. Mathematicians and physicists have invented a mathematical term to refer to such properties; they are called symmetries and they reflect immunity to changes in location, orientation, or the time you start your clock. If not for these (and other) symmetries, any hope of ever deciphering nature’s grand design would have been lost, since experiments would have had to be continuously repeated in every point in space (if life could emerge at all in such a universe). Another feature of the cosmos that lurks in the background of mathematical theories has become known as locality. This reflects our ability to construct the “big picture” like a jigsaw puzzle, starting with a description of the most basic interactions among elementary particles.

  We now come to the last element in Wigner’s puzzle: What is it that guarantees that a mathematical theory should exist at all? In other words, why is there, for instance, a theory of general relativity? Could it not be that there is no mathematical theory of gravity?

  The answer is actually simpler than you might think. There are indeed no guarantees! There exists a multitude of phenomena for which no precise predictions are possible, even in principle. This category includes, for example, a variety of dynamic systems that develop chaos, where the tiniest change in the initial conditions may produce entirely different end r
esults. Phenomena that may exhibit such behavior include the stock market, the weather pattern above the Rocky Mountains, a ball bouncing in a roulette wheel, the smoke rising from a cigarette, and indeed the orbits of the planets in the solar system. This is not to say that mathematicians have not developed ingenious formalisms that can address some important aspects of these problems, but no deterministic predictive theory exists. The entire fields of probability and statistics have been created precisely to tackle those areas in which one does not have a theory that yields much more than what has been put in. Similarly, a concept dubbed computational complexity delineates limits to our ability to solve problems by practical algorithms, and Gödel’s incompleteness theorems mark certain limitations of mathematics even within itself. So mathematics is indeed extraordinarily effective for some descriptions, especially those dealing with fundamental science, but it cannot describe our universe in all its dimensions. To some extent, scientists have selected what problems to work on based on those problems being amenable to a mathematical treatment.

  Have we then solved the mystery of the effectiveness of mathematics once and for all? I have certainly given it my best shot, but I doubt very much that everybody would be utterly convinced by the arguments that I have articulated in this book. I can, however, cite Bertrand Russell in The Problems of Philosophy:

  Thus, to sum up our discussion of the value of philosophy; Philosophy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good.

  NOTES

  Chapter 1. A Mystery

  As the British physicist James Jeans: Jeans 1930.

  Einstein once wondered: Einstein 1934.

  he singled out geometry as the paradigm: Hobbes 1651.

  Penrose identifies three different: Penrose beautifully discusses these “three worlds” in Emperor’s New Mind and Road to Reality.

  Physics Nobel laureate Eugene Wigner: Wigner 1960. We shall return to this article many times in this book.

  that he emphatically declared: Hardy 1940.

  One of his works was reincarnated: For a discussion of the Hardy-Weinberg law in context see for example Hedrick 2004.

  the British mathematician Clifford Cocks: Cocks invented in 1973 what has become known as the RSA encryption algorithm, but at the time it was classified. The algorithm was independently invented a few years later by R. Rivest, A. Shamir, and L. Adleman at MIT. See Rivest, Shamir, and Adleman 1978.

  to describe all the symmetries of the world: A popular description of symmetry, group theory, and their intertwined history is given in The Equation That Couldn’t Be Solved (Livio 2005), Stewart 2007, Ronan 2006, and Du Sautoy 2008.

  He noticed that a sequence of numbers: A wonderful popular description of the emergence of chaos theory can be found in Gleick 1987.

  Black-Scholes option pricing formula: Black and Scholes 1973.

  The traveling salesman problem was solved: A superb but technical description of the problem and its solutions can be found in Applegate et al. 2007.

  expressed his views very clearly: Changeux and Connes 1995.

  He once wittily remarked: Gardner 2003.

  While reviewing a book: Atiyah 1995.

  In the words of the French neuroscientist: Changeux and Connes 1995.

  In one place she complains: A brief biography of Marjory Fleming can be found, for instance, at Wallechinsky and Wallace 1975–81.

  author Ian Stewart once put it: Stewart 2004.

  Chapter 2. Mystics: The Numerologist and the Philosopher

  Descartes was one of the principal architects: A more detailed description of Descartes’ contributions is presented in chapter 4.

  “I recognize no matter”: Descartes 1644.

  credited with introducing the words: Iamblichus ca. 300 ADa, b; discussed in Guthrie 1987.

  biographies of Pythagoras from the third century: Laertius ca. 250 AD; Porphyry ca. 270 AD; Iamblichus ca. 300 ADa, b.

  finds it difficult to identify: Aristotle ca. 350 BC; discussed in Burkert 1972.

  The Greek historian Herodotus: Herodotus 440 BC.

  Empedocles (ca. 492–432 BC) added in admiration: Porphyry ca. 270 AD.

  For instance, the monad: A clear discussion of the Pythagorean perspective can be found in Strohmeier and Westbrook 1999.

  The English historian of philosophy: Stanley 1687.

  The fact that someone would find numbers: For a fascinating compilation of properties of numbers see Wells 1986.

  Pythagoras asks someone to count: Cited in Heath 1921.

  “I swear by the discoverer”: Iamblichus ca. 300 ADa; discussed in Guthrie 1987.

  When two similar strings: Strohmeier and Westbrook 1999; Stanley 1687.

  The word “gnomon” (a “marker”): T. L. Heath gives a detailed discussion of the term and what it meant at different times (Heath 1921). The mathematician Theon of Smyrna (ca. 70–135 AD) used the term in relation to the figurative expression of numbers described in the text in Mathematics, Useful for Understanding Plato (Theon of Smyrna ca. 130 AD).

  “If we listen to those who wish”: You will notice that in his comment Proclus does not state specifically what he himself believes with respect to the question of whether Pythagoras was the first to formulate the theorem. The story about the ox appears in the writings of Laertius, Porphyry, and the historian Plutarch (ca. 46–120 AD). It is based on verses by Apollodorus. However, the verses only talk about “that famous proposition” without stating which proposition this was. See Laertius ca. 250 AD, Plutarch ca. 75 AD.

  These constructions were clearly known: Renon and Felliozat 1947, van der Waerden 1983.

  The basic philosophy expressed by the table: This cosmology was based on the notion that reality emerges from the fact that Matter (considered indefinite) is shaped by Form (considered the limit).

  The book Philosophy for Dummies: Morris 1999.

  The oldest surviving story: Joost-Gaugier 2006.

  From the perspective of the questions: Good discussions of the Pythagorean contributions and their influence can be found in Huffman 1999, Riedweg 2005, Joost-Gaugier 2006, and Huffman 2006 in the Stanford Encyclopedia of Philosophy.

  One of the Pythagoreans: Fritz 1945.

  the recognition of the existence of “countable”: I do not discuss topics such as transfinite numbers and the works of Cantor and Dedekind in the present book. Excellent popular accounts can be found in Aczel 2000, Barrow 2005, Devlin 2000, Rucker 1995, and Wallace 2003.

  the philosopher Iamblichus reports: Iamblichus ca. 300 ADa, b.

  to the Pythagoreans, God was not: See discussion in Netz 2005.

  “the safest generalization that can be made”: Whitehead 1929.

  Who was this relentless seeker: The titles of texts about Plato and his ideas can, of course, by themselves fill an entire volume. Here are just a few texts that I found to be very helpful. On Plato in general: Hamilton and Cairns 1961, Havelock 1963, Gosling 1973, Ross 1951, Kraut 1992. On mathematics: Heath 1921, Cherniss 1951, Mueller 1991, Fowler 1999, Herz-Fischler 1998.

  According to an oration by the fourth century: The oration was written in 362 AD, but it did not give any details on the contents of the inscription. The words of the inscription come from a marginal note in a manuscript of Aelius Aristides. The note may have been written by the fourth century orator Sopatros, and it reads (in a translation by Andrew Barker): “There had been inscribed at the front of the School of Plato, ‘Let no one who is not a geometer enter.’ [That is] in place of ‘unfair’ or ‘unjust’: for geometry
pursues fairness and justice.” The note seems to imply that Plato’s inscription replaced “unfair or unjust person” in a sign that was common in sacred places (“Let no unfair or unjust person enter”) with the phrase “one who is not a geometer.” This story was later repeated by no fewer than five sixth century Alexandrian philosophers, and it eventually made its way into the book Chiliades, by the twelfth century polymath Johannes Tzetzes (ca. 1110–80). For a detailed discussion see Fowler 1999.

  I was disappointed to discover: A summary of many unsuccessful archaeological attempts can be found in Glucker 1978.

  The first century philosopher and historian: Discussed in Cherniss 1945, Mekler 1902.

  To which the Neoplatonic philosopher: Cherniss 1945, Proclus ca. 450.

  “What we require is that those who take”: Plato ca. 360 BC.

  “The science of figures, to a certain degree”: Washington 1788.

  is no more real than shadows projected: An interesting discussion of the allegory can be found in Stewart 1905.

  Plato’s views formed the basis: For interesting discussions of Platonism and its place in the philosophy of mathematics, see Tiles 1996, Mueller 1992, White 1992, Russell 1945, Tait 1996. For excellent presentations in popular texts, see Davis and Hersh 1981, Barrow 1992.

  mathematics becomes closely associated with the divine: For a discussion of this topic see Mueller 2005.

  He argued that in true astronomy: Plato’s comments on astronomy and planetary motion appear in the Republic (Plato ca. 360 BC), in Timaeus, and in Laws. G. Vlostos and I. Mueller discuss the implications of Plato’s position (Vlostos 1975, Mueller 1992).