After hearing about Eugene Wigner’s article, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” for many years, I recently decided to read and digest the article for myself. (If you’re so inclined, you can read about the results here.) Continuing this exploration of the philosophy of mathematics led me to Mario Livio’s intriguing book, Is God a Mathematician?, selected by the Washington Post as one of the best books of 2009. Here, I present a summary of that book. My research on this book in turn led me to Livio’s Scientific American article, “Why Math Works,” an interview with Krista Tippett, and a Nova special that features him prominently. I encourage you to explore these sources as well as the book itself.

Themes

Livio does not intend the title as a theological question. He is instead referring to two questions concerning the mysterious effectiveness of mathematics in the physical world.

The first question is “Does mathematics have an existence independent of the human mind?” or put another way, is mathematics discovered, extant in an abstract realm, independent of the world we perceive with our senses, or is it an invention, that is, a creation by the human mind? Those who belong to the “discovered” camp are known as “Platonists,” and include (or have included) such notables as G. H. Hardy, Roger Pendrose, Kurt Gödel, and Martin Gardner. On the “formalist” side of the argument lies Albert Einstein, David Hilbert, Georg Cantor, the Bourbaki group, and others.

The second question is: “Why do mathematical concepts have applicability far beyond the context in which they have originally been developed?” This is the question that Eugene Wigner asked. As he put it, “It is difficult to avoid the impression that a miracle confronts us here, comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions.”

Livio clarifies and focuses this second question by making a distinction between active and passive applicability. By active applicability, he means mathematics that was created as a tool for a purpose. Although no one is surprised that active mathematics is effective — it was developed specifically for the purpose, and the theories were tailored to fit the observations — it may be surprising just howeffective it can be. For instance, the theoretical calculation of the magnetic moment of an electron in quantum electrodynamics matches the actual value to within a few parts per trillion. Passive applicability is more surprising. “Passive” in this context refers to applicability of mathematics beyond the intent of the original theory. For instance, entire subfields of mathematics are created with no physical application in mind, “math for math’s sake,” but then decades or even centuries later, these theories fit real world situations with astounding exactitude. For instance, group theory was developed in the early 1800s by Évariste Galois to study the solvability of polynomial equations, yet worked beautifully in the 20th century to categorize elementary particles. As another example, so-called “non-Euclidean geometries,” which defy the parallel postulate of traditional Euclidean geometry (more below), came as a shock to the mathematical community in the mid-1800s and were even derided by many as not being “true” mathematics. Yet Einstein formulated his general theory of relativity on Riemannian geometry.

Early Platonists

The first few chapters of the book concentrate on some famous early names in mathematics: Pythagoras, Plato, Archimedes, Galileo, Descartes, and Newton. These men helped to simultaneously formalize mathematics and effect its role as the language of science. Each of these mathematicians were Platonists, who viewed mathematics as pre-existing, either from some mysterious realm of thought, or as arising from the mind of God.

The Pythagoreans

While their predecessors were primarily interested in mathematics as a practical tool, for instance, for measuring out land, or predicting flood cycles, the followers of Pythagoras were the first “pure mathematicians,” interested in mathematics for its own sake. As such, they pioneered the use of rigorous proof, the practice of starting with self evident postulates and utilizing logic to prove conclusions from them. As to the question of whether mathematics was discovered or invented, they fell firmly into the discovered camp. Livio says it this way: “On the question of whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt — mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the feeble human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician — mathematics was God!”

They also set the stage for Plato. “The importance of the Pythagorean philosophy lies not only in its actual, intrinsic value. By setting the stage, and to some extent the agenda, for the next generation of philosophers — Plato in particular — the Pythagoreans established a commanding position in Western thought.”

Archimedes

Although today Archimedes is mostly known for his clever engineering inventions, famously including the hydraulic screw for raising water and various engines of war, his primary interest was in pure mathematics. In geometry, he presented general methods for areas and volumes of a variety of figures, in number theory, he approximated lower and upper bounds of π as 3 10/71 and 3 1/7, respectively. He also invented a system to denote and manipulate numbers of any magnitude.

Legend has it that he died because he wouldn’t move from a mathematical diagram during a battle, and a Roman soldier killed him. Alfred North Whitehead wrote, “The death of Archimedes at the hands of a Roman soldier is symbolical of a world change of the first magnitude. The Romans were a great race, but they were cursed by the sterility which waits upon practicality. They were not dreamers enough to arrive at new points of view, which could give more fundamental control over the forces of nature. No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.”

Plato

Although Plato was not a mathematician per se, he was a great philosopher of mathematics, influencing the field for the following millennia. To Plato, mathematical truths did not refer to geometric figures represented on papyrus or marked in the sand, but to abstract objects that exist in an idealized non-physical world of true forms. This view that mathematics is extant in an abstract realm independent of the world of our senses became known as “Platonism.”

Galileo

Galileo enrolled in the faculty of arts of the University of Pisa to study medicine but changed to mathematics. He became a huge fan of Archimedes. He said, “Those who read his works realize only too clearly how inferior are all other minds compared with Archimedes’ and what small hope is left of ever discovering things similar to the ones he discovered.”

Galileo is, of course, known for his challenges to the geocentric model. Although he did not invent the telescope, he greatly improved it, and made many discoveries that supported the heliocentric view of the universe. This view, the scientifically correct one, resulting in his eventually being tried by the inquisition and subjected to house arrest for the rest of his days.

More importantly for our purposes is the impact he brought to science. Livo says, “The philosopher of science Alexandre Koyré (1892–1964) remarked once that Galileo’s revolution in scientific thinking can be distilled to one essential element: the discovery that mathematics is the grammar of science. While the Aristotelians were happy with a qualitative description of nature, and even for that they appealed to Aristotle’s authority, Galileo insisted that scientists should listen to nature itself, and that the keys to deciphering the universe’s parlance were mathematical relations and geometrical models… Centuries before the question of why mathematics was so effective in explaining nature was even asked, Galileo thought he already knew the answer! To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician.”

Descartes

Descartes, of “cogito, ergo sum” fame, introduced several mathematically revolutionary concepts. In particular, he developed the use of functions in mathematics. Such a function encodes a “rule” that a physical phenomena follows. For instance, Newton’s gravitation states that the gravitational attraction between two masses is proportional to their masses and inversely proportional to the square of the distance between them. He also proposed the Cartesian coordinate system, which married the fields of algebra and geometry into a new field, analytic geometry.

Descartes also argued that the natural world is best described through mathematics, and not through the our often misleading perceptions. “I recognize no matter in corporeal things apart from that which the geometers call quantity, and take as the object of their demonstrations…And since all natural phenomena can be explained in this way, I do not think that any other principles are either admissible or desirable in physics.” He went even further, suggesting that nearly all of human knowledge could follow from systematic application of mathematics.

Newton

Newton, undoubtedly one of the greatest mathematicians of all time, took Descartes’ idea that the cosmos can be described in terms of mathematics and made it reality. He formulated the fundamental laws of mechanics, deciphered the laws describing planetary motion, and erected the theoretical basis for the phenomena of light and color. All of these ideas were written in the language of mathematics. He also, of course, developed differential and integral calculus.

Non-Euclidean Geometry

Thus far, we would be forgiven for either concluding that the Platonist view is the obvious perspective or that the whole debate is a moot point. However, in the mid-1800s, the mathematical world was in for a shock. This bombshell came in the form of so-called “non-Euclidean” geometries.

Euclidean geometry

Until the beginning of the 19th century, Euclidean geometry was considered the epitome of truth and certainty. The Platonic ideals of Euclidean geometry were considered to be abstractions of their real-world counterparts. Even Kant, who in his Critique of Pure Reason asserted that we know what we know by sensory perception processed by the brain, made an exception for geometry, stating, “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…” Or put more simply, it was taken more or less as a given that absolute truths about the universe exist. Non-Euclidean geometries were about to challenge this view.

To understand non-Euclidean geometries, we must first understand what is meant by Euclidean geometry.

Circa 300 BC, Euclid laid out geometry in his thirteen volume work The Elements. He started with ten axioms assumed to be self-evidently true and constructed geometry through logic applied to these postulates. The first four axioms, such as the first one, which states that a straight line may be drawn through any two points, are obvious. The fifth postulate, the so-called “parallel postulate” read as follows. “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.”

So if angles α and β together are less than 180 degrees, the lines must cross.

Non-Euclidean Geometry

The fifth postulate seems convoluted and artificial. For centuries, attempts had been made to “clean up” Euclid’s geometry by proving the fifth postulate as a consequence of the other four postulates with no success.

In the mid-1800s, it was shown that one could create a new kind of geometry by choosing a different axiom than the fifth postulate. Furthermore, the resulting geometries described space just as accurately as Euclidean geometry. This is the key turning point in our story. The implication is that mathematics is not a preordained truth, but determined by choices that one makes. This is why the questions we’re talking about even make sense to discuss. Up until now, no one looked at mathematics in this way — it was taken as a given.

Livio says, “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.”

The stage for alternate geometries was set by Girolamo Saccheri, who investigated the consequences of replacing the fifth postulate by another statement, and Georg Klügel and Johann Heinrich Lambert, who realized that alternative geometries could exist. Three mathematicians in particular created non-Euclidean geometries, once and for all showing that Euclidean geometry was not the only representation of space: Nikolai Ivanovich Lobachevsky, János Bolyai, and Carl Friedrich Gauss, who presented what is now known as hyperbolic geometry. Bernhard Riemann furthered the field of non-Euclidean geometry with “elliptical geometry.” These developments led Henri Poincaré to declare that Kant was wrong. Space is not intrinsically Euclidean before we perceive it, but rather is a learned framework. In his words, “…to conclude that the axioms of geometry are ‘neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts, but it remains free.” Even more geometries would follow, such as projective and differential geometry.

Reactions

The reactions to non-Euclidean geometries were varied. Many mathematicians did not consider these alternative geometries to be legitimate types of mathematics. One objection was consistency. Would these geometries ultimately lead to self contradictions? By the 1870s, Eugenio Beltrami and Felix Klein had shown that if Euclidean geometry is consistent, then so are the non-Euclidean geometries. Some mathematicians turned to arithmetic (number theory), connecting geometry with numbers in the spirit of Descartes analytic geometry. David Hilbert demonstrated that if arithmetic is consistent, then so is geometry. This observation doesn’t really answer the question though, but only pushes it downstream.

Another objection was relevance. Initially, the non-Euclidean geometries were seen as curiosities, with no connection to the three dimensional space of the real world that we live in. Many mathematicians, including Gerolamo Cardano and John Wallis, refused to consider any mathematics that dimensions higher than three. Jean D’Alembert and Joseph Lagrance considered 4D mathematics, with only with time as the fourth dimension.

The “Freedom of Mathematics”

Hermann Grüssman explored geometries in any number of dimensions, which originated the branch of mathematics we now call linear algebra. To him, mathematics was an abstract creation of the mind that does not necessarily have any application to the real world. At this point, mathematics was no longer restrained to describing the three dimensional observable world.

In the 1700s, Euler had declared, “mathematics, in general, is the science of quantity; or, the science that investigates the means of measuring quantity.” The introduction of these abstract geometries, along with the notion of infinity, had distorted the meaning of “measurement” or “quantity” beyond recognition. In addition, mathematics had become the study of abstractions, far removed from physical reality. Georg Cantor characterized the new spirit of mathematics by a “declaration of independence”: “Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.” Dedekind added, “I consider the number concept entirely independent of the notions or intuitions of space and time…Numbers are free creations of the human mind.” Cantor continued, saying “The essence of mathematics lies entirely in its freedom.” By the close of the 1800s, most mathematicians considered mathematics to not be about the explanation of the physical world, but rather as the logical consequences of given axioms.

Foundations of Mathematics

Non-Euclidean geometries and their aftermath led mathematicians to seriously investigate the foundations of mathematics. This investigation was pursued via logic.

Logicism

Logicism is the point of view that mathematics can be reduced to logic. Bertrand Russell explained it this way. “Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now [in 1919] become wholly impossible to draw a line between the two; in fact the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic.”

Both camps, formalists and Platonists, embraced logicism. Formalists were happy to see the “games” of mathematics coalesce into one übergame, whereas Platonists hoped that logic would eventually reveal a single metaphysical origin to mathematics.

Logic and Mathematics

Logic has long concerned itself with concepts and propositions, and by making valid inferences from the relationships between propositions. For example, a syllogism is a form a reasoning in which a conclusion is drawn from two given premises of a certain form. For instance, all cats are mammals, all mammals have four legs, therefore all cats have four legs is a syllogism. Symbolically, this line of reasoning can be denoted as “every X is a Y; every Y is a Z; therefore, all Xs are Zs.” One cannot help but notice the parallels between symbolic logic and mathematics.

In the 1700s, Gottfried Wilhelm Leibniz, famous for formulating calculus independently of Newton, and the ensuing bitter dispute over priority, envisioned a characteristica universalis, which was a system to compute the truth or falsity of any statement symbolically. However, his formulation had issues — the vagueness of what might constitute an “alphabet of thought” and the difficulty of determining when thoughts are “equal” — that doomed this experiment to never really get off the ground.

In the mid-1800s, Augustus DeMorgan began to make strides with the algebraization of logic, including quantifying the predicate. George Boole, who corresponded with DeMorgan since 1842, published two influential books, The Mathematical Analysis of Logic, and The Laws of Thought, which together transformed logic into a type of algebra.

Friedrich Frege wrote two more notable books, the Begriffsschrift and the Grundgesetze der Arithmetic, in which he set out to prove the truths of arithmetic from a few axioms of logic. He wanted to show that even the natural numbers could be reduced to logical constructs, and build from these concepts to show the truth of arithmetic. Although it was a brilliant idea to prove arithmetic from basic concepts, his execution was fatally flawed — one of his axioms, known as Basic Law V, was shown by Bertrand Russell to lead to a contradiction.

Despite this flaw, Russell believed that deriving arithmetic from logic was the correct approach. He wrote, along with Alfred North Whitehead, the Principia Mathematica, still considered as one of the most influential books in the history of logic. Russell indeed successfully fixed Basic Law V, but in a highly artificial and convoluted way.

Eventually the paradoxes were more elegantly eliminated by Ernst Zermelo and Abraham Fraenkel by axiomatizing set theory in a self-consistent way. This seemed to give mathematics a kind of objective certainty, giving the Platonists a long sought after victory. However, once again, one of the axioms, this time the so-called axiom of choice, proved to be problematic.

The axiom of choice states that if X is a collection of nonempty sets, then we can choose one element in each set in X to form a new set Y. There is no controversy when considering finite sets since one can easily see, for instance, that for a hundred boxes, each containing at least one marble, one can take one marble from each box to form a new set of a hundred marbles. Extending this notion, we can see that the axiom must be true even for countably infinite sets. However, in the case of uncountably infinite sets, we cannot even define how such a choice could be made.

Similar to the days of Euclid’s fifth axiom, mathematicians attempted to either prove or refute the axiom of choice from the other axioms. In the 1930s, Kurt Gödel showed that the axiom of choice can neither be proved nor disproved from the other Z-F axioms.

The next phase of the debate was initiated by David Hilbert, the Formalist, who believed mathematics was no more than a collection of meaningless formulae. His ambitious program was to show the correctness of mathematics using mathematical methods (so-called “metamathematics”). As he said, “My investigations in the new grounding of mathematics have as their goal nothing less than this: to eliminate, once and for all, the general doubt about the reliability of mathematical inference…Everything that previously made up mathematics is to be rigorously formalized, so that mathematics proper or mathematics in the strict sense becomes a stock of formulae…In addition to this formalized mathematics proper, we have a mathematics that is to some extent new: a metamathematics that is necessary for securing mathematics, and in which — in contrast to the purely formal modes of inference in mathematics proper — one applies contextual inference, but only to prove the consistency of the axioms…Thus the development of mathematical science as a whole takes place in two ways that constantly alternate: on the one hand we derive provable formulae from the axioms by formal inference; on the other, we adjoin new axioms and prove their consistency by contextual inference.”

Once again, enter Gödel. As a teenager, he had become fascinated by Russell and Whitehead’s Principia Mathematica and by Hilbert’s program. For his dissertation, he chose to determine whether Hilbert’s formal approach was sufficient to produce mathematics. A year after being awarded his dissertation, he dropped the bomb in the form of Gödel’s “Incompleteness Theorem.” From Livio: “…the incompleteness theorems proved that Hilbert’s formalist program was essentially doomed from the start. Gödel showed that any formal system that is powerful enough to be of any interest is inherently either incomplete or inconsistent. That is, in the best case, there will always be assertions that the formal system can neither prove nor disprove. In the worst, the system would yield contradictions.”

For a technical explanation of the Gödel’s Incompleteness Theorem, please refer to this article by Jørgen Veisdal.

Modern Views

One suspects that most professional mathematicians go about their daily work without becoming mired in philosophical meta-discussions of their favorite subject. As Philip Davis and Reuben Hersch say in The Mathematical Experience, “the typical mathematician is a Platonist on weekdays and a formalist on Sundays.”

Some twentieth century mathematicians did indeed have strong views in one camp or the other. For instance, G. H. Hardy gives a strong Platonist persective in his A Mathematician’s Apology, whereas Edward Kasner and James Newman describe the opposite Formalist view in Mathematics and the Imagination.

On the Platonist side, some people believe that mathematics exists not in some kind of metaphysical realm, but is baked into the physical world. For instance, Max Tegmark, the MIT astrophysicist, takes the view that “our universe is not just described by mathematics — it is mathematics.” He argues that external physical reality exists independently of humans, then examines what the nature of the “theory of everything” of such a reality, states that such a description must be free of any human “baggage” (such as language), then concludes that such a final theory cannot consist of any human constructs, and therefore the only possible description of the cosmos is one that only involves abstract concepts and their relations, i.e., mathematics. Livio fails to be impressed by this chain of reasoning, claiming that Tegmark in effect assumes his conclusion.

Livio also mentions a number of neuroscientists, such as Jean-Pierre Changeux, and Lakoff and Núñez, that take the view that mathematics is a human construction of the human brain.

So what does the author think? Livio concludes the book by stating that the question of whether mathematics is invented or discovered is misleading, since it implies that mathematics must be either one or the other. His view is that mathematics is both invented and discovered. The mathematical concepts and definitions are invented, whereas the logical consequences of these concepts are discovered.

Conclusion

In this article, I’ve necessarily given you only the highlights of the book. I have skipped over much of the biographical material and the more technical aspects of the debates, which are well worth delving into. I encourage you to read it for yourself. Enjoy!