The City Summer 2016

Three-Part Harmony: The Remarkable Relationship between Mind, Matter, and Mathematics

by Melissa Cain Travis

The conviction that the cosmos is orderly and that this order is discernible through human reason was a major philosophical engine of the scientific revolution. Johannes Kepler, Galileo Galilei, and Isaac Newton harnessed the power of mathematics for scientific discovery and did so from a broadly Christian perspective on the universe and mankind. However, since their time, naturalism has enjoyed increasing influence, and a common perception seems to be that advancement in the sciences has made the “God hypothesis” unnecessary for explaining the fundamental nature of the world. On the contrary, this hypothesis is as viable today as ever before; Christian theism offers a needed overarching explanatory coherence concerning the applicability of mathematics to the natural sciences, the objectivity of mathematical truth, and mankind’s aptitude for higher mathematics.

Johannes Kepler was committed to the notion that the heavens are elegantly ordered by God in such a way that they can be comprehended by the human intellect. In a letter to the Baron von Herberstein dated May 15, 1596, Kepler wrote that “God, like a human architect, approached the founding of the world according to order and rule and measured everything in such a manner, that one might think not art took nature for an example but God Himself, in the course of His creation took the art of man as an example.” To the mathematician Johann Georg Herwart von Hohenburg, Kepler wrote:

To God there are, in the whole material world, material laws, figures and relations of special excellency and of the most appropriate order…Those laws are within the grasp of the human mind; God wanted us to recognize them by creating us after his own image so that we could share in his own thoughts. For what is there in the human mind besides figures and magnitudes? It is only these which we can apprehend in the right way, and if piety allows us to say so, our understanding is in this respect of the same kind as the divine, at least as far as we are able to grasp something of it in our mortal life. Only fools fear that we make man godlike in doing so; for the divine counsels are impenetrable, but not his material creation.[1]

In his Conversation with Galileo’s Sidereal Messenger, Kepler says that Geometry “shines in the mind of God” and that a “share of it which has been granted to man is one of the reasons why he is in the image of God.”

In his 1623 work, The Assayer, Galileo Galilei poetically articulated the same idea:

Philosophy is written in this all-encompassing book that is constantly open before our eyes, that is the universe; but it cannot be understood unless one first learns to understand

the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders around pointlessly in a dark labyrinth.[2]

The creation is open to the observation and analysis of man, who is able to comprehend the structure of the universe if he first learns mathematics. Galileo argues that it is through the mathematical book of nature that man has some perception of God. In a 1615 letter to the Grand Duchess Christina, he says that “God reveals Himself to us no less excellently in the effects of nature than in the sacred words of Scripture.”

Like Kepler and Galileo, Sir Isaac Newton recognized the fundamentally mathematical nature of the world and saw a connection between this fact and the intellect of mankind. Rob Iliffe has pointed out that Newton, in some of his unpublished scientific papers and letters, “alluded to the fact that human beings possessed an intellect that resembled—to a finite degree—the infinite capacity of God’s understanding,” and that Newton considered his own work “a test case of how far the divine mind could be accessed by the human brain.”[3] In correspondence with Richard Bentley in 1692 and 1693, Newton expressed his conviction that God had mathematically ordered the material creation. His letters offered Bentley, who was a scholarly opponent of atheism, “genuinely novel and powerful additions to the battery of arguments used by scholars who, working in the tradition of natural theology, inferred the existence and attributes of the deity from the present order of the world.”[4]

Since the scientific revolution, progress in physics has further demonstrated the acutely mathematical structure of the universe. For example, all fundamental properties of nature that have ever been measured can be computed from a set of 32 particle physics parameter measurements. Moreover, the intrinsic properties of the fabric of space—numbers specifying curvature, topology, and dimensionality—and the intrinsic properties of all elementary particles are mathematical. Even the relationship between energy and mass are beautifully expressed in Einstein’s stunningly simple and elegant equation, E=mc², where c represents the speed of light. Significantly, the mathematical nature of the universe even allows physicists to predict the existence of various entities. For example, Peter Higgs’ calculations in the 1960s indicated the existence of an invisible field that endowed elementary particles with their mass. The reality of the field was required to preserve mathematical harmony, symmetry, and consistency in quantum theory equations. Higgs’ paper, in which he explained his prediction, was published in 1965, and became accepted as part of the Standard Model of particle physics, despite having no experimental support. Over half a century later, in 2012, experimental confirmation of the Higgs field was achieved, vindicating the purely mathematical evidence.

Sir Arthur Eddington, an astronomer and physicist of the early 20th century, also remarked upon the predictive power of mathematics in cosmology. In his work, The Expanding Universe, he discusses the determination of the rate at which spiral nebulae are receding from the earthly observer’s perspective, a feat which is accomplished by combining quantum theory and wave mechanics. “By combining the two theories,” he explains, “we can make the desired theoretical calculation of the speed of recession” and then after giving the mathematical range he remarks, “No astronomical observations of any kind are used in this calculation, all the data being found in the laboratory. Therefore when we turn our telescopes and spectroscopes on the distant nebulae and find them to be receding at a speed within these limits the confirmation is striking.”

In his famous essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” physicist and Nobel Prize winner Eugene Wigner discussed the sovereign role mathematics plays in physics. By the time of the essay’s publication in the 1960s, enormous strides had been made in physics, and he echoed Galileo when he affirmed that “the laws of nature must have been formulated in the language of mathematics to be an object for the use of applied mathematics.” Wigner recognized that the astounding level of applicability of mathematics to nature and the human mind’s aptitude for such mathematics demand explanation:

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions or to the two miracles of the existence of the laws of nature and of the human mind’s capacity to divine them.[5]

About the first problem he said, “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and…there is no rational explanation for it.” The latter problem, the human intellect’s fitness for mathematics, will be further discussed momentarily.

It is an extraordinary fact that when physicists are developing their mathematical theories about the material world, the mathematical concepts upon which they arrive often turn out to have been previously conceived by pure mathematicians. Contemporary atheist physicist Steven Weinberg has even admitted, “It is positively spooky how the physicist finds the mathematician has been there before him or her.”[6] A classic example is the ancient Greek mathematicians’ explication of the curves of conic sections, which were pressed into service to physics well over one thousand years later when Kepler used the ellipse to describe planetary motion and Galileo used parabolas in his analysis of terrestrial projectile motion.

In response to this grand enigma, contemporary physicists and philosophers have postulated various explanations for the mathematical cosmos and its accessibility for the human intellect. Is mathematics something that has simply been invented by man and then applied descriptively to the material world, sort of like sewing a glove to fit a hand? Or rather, is mathematics something that transcends nature, an independently existing reality that man has discovered? From what has been seen in terms of the “unreasonable effectiveness” of mathematics and its eerie ability to predict physical reality, the latter view seems much more plausible. Yet, this raises further questions: If mathematical truths are indeed self-existent, universal, and eternal rather than artificial constructs of the human intellect, whence did they come? Furthermore, how do we explain the tripartite harmony between the metaphysically distinct worlds of mathematics, matter, and mind?

Mathematical physicist and philosopher of science Roger Penrose (a non-theist) is convinced that 1) some degree of mathematical Platonism must be the case, in light of the objectivity of many mathematical truths and 2) the abstract realm of mathematics must be related to the material world in some manner, because “operations of the physical world are now known to be in accord with elegant mathematical theory to an enormous precision.”[7] He recognizes an “extraordinary concurrence” between the mechanics of nature and sophisticated mathematical theory: “It makes no sense to me,” he argues, “that this concurrence is merely the result of our trying to fit the observational facts into some organizational scheme that we can comprehend; the concurrence between Nature and sophisticated beautiful mathematics is something that is ‘out there’ and has been so since times far earlier than the dawn of humanity….”[8]

            Intuitively, it seems that mathematical propositions are objectively and timelessly true. Even if no humans existed to think of them, their truth would hold. Thus, perhaps it is right to say that numbers and their relationships are discovered rather than invented. Yet, in affirming any degree of naturalistic Platonism, as Penrose and some other physicists and mathematicians do, mathematical truths are left with no ontological grounding—they simply are, end of story. As Alvin Plantinga explains:

Platonism with respect to these objects is the position that they do exist…in such a way as to be independent of mind; even if there were no minds at all, they would still exist. But there have been very few real Platonists, perhaps none besides Plato and Frege, if indeed Plato and Frege were real Platonists (and even Frege, that alleged arch-Platonist, referred to propositions as gedanken, thoughts). It is therefore extremely tempting to think of abstract objects as ontologically dependent upon mental or intellectual activity in such a way that either they just are thoughts, or else at any rate couldn’t exist if not thought of.[9]

Plantinga also points out that that if an idea is necessarily true, as any mathematical truth seems to be, then it exists necessarily, thus eternally. Perhaps the only resolution, he suggests, is that mathematical truths are thoughts in the divine mind, an idea known as Divine Conceptualism. Another relevant advantage of this view is that it is also able to explain the applicability of mathematics in the natural sciences. If abstract objects and the material realm have the same rational source, the mystery dissolves.

Some naturalists may be content to call objective mathematical truth brute fact and call the applicability of mathematics to nature a happy, though mysterious, accident. However, when it comes to human aptitude for higher mathematics, biology enters the picture, and this unavoidably brings up the issue of man’s neurological evolution. From a naturalistic perspective, the human mind and all of its capacities are the result of non-teleological natural selection acting upon random variation. Yet, this view is confronted with at least two problems where mathematical aptitude is concerned.

First, the ability to perform the highly complex mathematical operations involved in sciences such as quantum theory and astrophysics is far beyond what could be conceivably required for our ancestors’ survival and reproduction. Agnostic physicist Paul Davies has articulated this difficulty well:

One of the oddities of human intelligence is that its level of advancement seems like a case of overkill. While a modicum of intelligence does have a good survival value, it is far from clear how such qualities as the ability to do advanced mathematics…ever evolved by natural selection. These higher intellectual functions are a world away from survival “in the jungle.” Many of them were manifested explicitly only recently, long after man had become the dominant mammal and had secured a stable ecological niche.[10]

Plantinga has also recognized the problem in attributing mathematical aptitude to survival-of-the-fittest:

Current physics with its ubiquitous partial differential equations (not to mention relativity theory with its tensors, quantum mechanics with its non-Abelian group theory, and current set theory with its daunting complexities) involves mathematics of great depth, requiring cognitive powers going enormously beyond what is required for survival and reproduction. Indeed, it is only the occasional assistant professor of mathematics or logic who needs to be able to prove Godel’s first incompleteness theorem in order to survive and reproduce.[11]

The assumption that such abilities are simply evolutionary “spandrels,” side effects which had no adaptive use at the time they were genetically acquired as neutral accompaniments to genuinely adaptive powers, is implausible at best, argues Plantinga. Any naturalistic explanation along this vein is essentially speculative.

Second, the mental deliberation required for even intermediate mathematical operations is directly dependent upon free, goal-directed mental agency; yet, if the mind is nothing more than the material brain or at least wholly dependent upon it, some form of physical determinism must be true, and there is no such thing as genuinely free mental agency. When presented with a string of mathematical facts from which we set out to draw correct conclusions, we must be able to make free choices along the pathway of reasoning. This rationality requires an agent with true free will to consciously deliberate and direct the reasoning process according to mathematical content and rules of mathematical logic.

            Not all non-theistic scholars have remained unmoved by the foregoing points, and have postulated various non-theist avenues for making sense of them. In his 2012 book, Mind and Cosmos: Why the Neo-Darwinian Conception of Nature is Almost Certainly False, atheist Thomas Nagel argued, “The intelligibility of the world is no accident. Mind, in this view, is doubly related to the natural order. Nature is such as to give rise to conscious beings with minds; and it is such as to be comprehensible to such beings.” Nagel is convinced that the naturalistic, reductive approach to explaining these things is gravely insufficient: “There are things that science as presently conceived does not help us to understand, and which we can see, from the internal features of physical science, that it is not going to explain.” However, Nagel rejects theistic explanations in favor of a “natural teleology,” a cosmic predisposition for the evolution of higher mental faculties and mathematical order. He is in accord with Plantinga with his assessment that “the judgment that our senses are reliable because their reliability contributes to fitness is legitimate, but the judgment that our reason is reliable because its reliability contributes to fitness is incoherent.” Ultimately, he admits that an understanding of why the cosmos is the way it is will involve “a much more radical departure from the familiar forms of naturalistic explanation,” if it turns out to even be within the reach of human rationality. In his earlier work, The Last Word, where he explored many of the same ideas, Nagel admits that his rejection of theistic explanations of the world are his a priori philosophy: “I don’t want there to be a God; I don’t want the universe to be like that.”

Davies also recognizes that “there is no logical reason why nature should have a mathematical subtext in the first place, and even if it does, there is no obvious reason why humans should be capable of comprehending it.”[12]  Davies takes a somewhat similar tack to Nagel’s by speculating about some as-yet-to-be-discovered—or undiscoverable—natural principle behind the cosmos: “Somehow, the universe has engineered, not just its own awareness, but also its own comprehension. Mindless, blundering atoms have conspired to make not just life, not just mind, but understanding. The evolving cosmos has spawned beings who are able not merely to watch the show, but to unravel the plot.” He reaches no conclusion other than to deem current naturalistic explanations as well as theistic ones “either ridiculous or hopelessly inadequate.” Yet, his reasons for rejecting theism seem to be based exclusively on philosophical preference.

Unlike naturalism, Christian theism provides a comprehensive explanation; it posits a rational Creator who formed the physical world according to a preconceived, orderly structure and formed mankind in his own image, making man a rational being capable of understanding, to some extent, the fundamental structure of the natural world. Moreover, the mystery of mankind’s aptitude for, and recognition of, the mathematical orderliness of nature is solved if man is the free-willed, image-bearing creation of the same source. As Christian mathematician John Lennox has succinctly put it, “It is, therefore, not surprising when the mathematical theories spun by human minds created in the image of God’s Mind, find ready application in a universe whose architect was that same creative Mind.”[13]

Scientific progress over the past three centuries has shown the cosmos to be mathematically precise to an extent Kepler, Galileo, and Newton could never have imagined; yet, naturalism has come to dominate many scientific circles. Why is this so?  Naturalism faces major obstacles on every point, while Christian theism bestows an all-encompassing and elegant coherence to the observed reality. The existence of an eternal mind behind all things is a more intellectually satisfying explanation for the mathematics—matter—mind conundrum than simply saying there is no ultimate explanation, and can be none, for this fortuitous reality in which we find ourselves.

About the Author
MELISSA TRAVIS, MA, is an Assistant Professor of Apologetics at Houston Baptist University. Travis is the author of How Do We Know God is Really There? (Apologia Press, 2013), How Do We Know God Created Life? (2014), How Do We Know Jesus is Alive? (2015), and How Do We Know Right and Wrong? (2016). These comprise the Young Defenders series, illustrated storybooks that teach the fundamentals of Christian apologetics to young children.

[1] Carola Baumgardt, Johannes Kepler, Life and Letters (New York: Philosophical Library, 1951), 50.

[2] Galileo, The Assayer, reprinted in The Essential Galileo, (Indianapolis: Hackett Publishing, 2008), 183.

[3] Snezana Lawrence and Mark McCartney, ed., Mathematicians and their Gods (Oxford: Oxford University Press, 2015), 124.

[4] Ibid., 126.

[5] Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in Communications in Pure and Applied Mathematics, vol. 13, No. 1 (February 1960). Reprinted in The World Treasury of Physics, Astronomy, and Mathematics, edited by Timothy Ferris (Camp Hill, PA: Little, Brown & Co., 1991), 533.

[6] Steven Weinberg, “Lecture on the Applicability of Mathematics.”  Notices of the American Mathematical Society 33.5 (Oct), quoted in Mark Steiner, The Applicability of Mathematics as a Philosophical Problem (Cambridge: Harvard University Press, 1998).

[7] John Polkinghorne ed., Meaning in Mathematics (New York: Oxford University Press, 2011), 44.

[8] Ibid. 44-45.

[9] Alvin Plantinga, Where the Conflict Really Lies (New York: Oxford University Press, 2011), 288.

[10] Paul Davies, Are We Alone? (New York: Orion Productions, 1995), 85.

[11] Plantinga, 286.

[12] Paul Davies, The Goldilocks Enigma (Boston: Houghton Mifflin, 2008), Kindle loc 218.

[13] John Lennox, God’s Undertaker: Has Science Buried God? (Oxford: Lion Books, 2009), 62.

© 2017 News and Events