Mathematics and the Mind of God:
Implications of the New Cosmology of the 17th Century
Robert L. Shearer
Florida Institute of Technology
The Inverse Square Law of Newton, the law of gravitation, has been called “the most stupendous single achievement of the human mind”[1] in all of intellectual history. Even Alexander Pope, despite the worries about science he expressed in his “Essay on Man,” composed a very significant couplet:
Nature and nature’s laws lay hid in night; God said, “Let Newton be,” and all was light.[2]
That is, whatever purely religious sense—what Hegel was to refer to as the “pictorial” account of truth—had prevailed in accounting for the world, from the ancient Hebrews onward through its Christian addition and throughout history, had now shifted to what essentially was the Hellenic notion of Being: a cosmos intelligible, rational, and knowable through mathematics.
Of course, Newton’s work did not take place in a vacuum; in a rare moment of humility he acknowledged that his having seen farther than anyone into the workings of the universe was owed to his having “stood on the shoulders of giants”—Copernicus, Kepler, Galileo. And indeed, these giants had themselves inhabited the universe of mathematics; but it was Newton, breathing its rarified air at that altitude, who turned it to greatest account.
The results for the West were revolutionary. As Greer and Lewis put it, in linking Galileo to Kepler, Newton “eliminated the barrier between forces acting in space and those on earth, and established by empirical and mathematical proof the existence of universal laws.”[3] It is these universal laws that replace the presence of God. The Enlightenment of the 18th century owes its beginnings in very large part to the mathematical science that had overturned Aristotle’s account of the physical cosmos.
As well, in addition to the rising validity of mathematical thinking, there are other events that seem to have conspired to bring in the acceptance of universal mathematical laws, events unrelated to mathematics as such.
It may seem odd that the Protestant Revolution of 1520 would have any connection to science, but William Barrett has made the case that even though Martin Luther considered reason “the whore,” his, and in general Protestantism’s, view of nature is opposite that of the Catholic Church: whereas in the medieval universe of a Dante all of nature is religiously symbolic, Protestantism empties nature of all of its symbolic connections to Christianity; faith must be so pure as not to have the “props” of symbols.[4] For that matter, Luther wanted even to rid the religious service of the Eucharist, and would have, had his followers not protested. This is to suggest that nature simply becomes neutral—or, if anything, as Barrett points out, hostile to human purposes—and it is this same neutrality, this drained and chastened view, that science must assume toward nature in accordance with “scientific method.”
The second event affecting the ground of the new science is the rejection of the Church by Henry the Eighth. This will not seem obvious until we recall that, owing to his desire for a male heir, Henry wanted a divorce from Catherine of Aragon the Church never got around to granting; Henry then kicked the Church out of England and set up, not a Protestant ceremony, but what mimicked to a very large extent the Catholic service—yet the result was that the Church had no lasting influence after 1534, despite the attempts of Henry’s daughter, Mary, to reestablish it, and could not get to Isaac Newton in the later 1600’s as it had Galileo earlier. Interesting to reflect that if not for the monomania of the eighth Henry, Newton might have been silenced and the true battle for the future would have taken place in the theater of religious warfare. As it turned out, science triumphs especially in the eighteenth century, making (as it did for Voltaire) religious wars objects of ridicule and disgust to scientific minds.
Of course, science and the scientific method were around as early as the twelfth century with Robert Grosseteste and Roger Bacon, and the later Bacon of the sixteenth and early seventeenth centuries. But in these cases science is more empirical than mathematical.
Because mathematics completes the methodology of science, let us look especially at the mathematical as the abstract chronicle of the shift into the eighteenth century. We need only notice its prevalence in times long before that of Newton to appreciate its gathering force. And If we allow that “the mathematical” need not necessarily refer to numbers, but is best understood as a rigorous abstract conception (perhaps bearing the marks of Kant’s a priori, necessity and universality), then Copernicus is thinking mathematically when he rejects the data of his senses—not to mention the views of Aristotle, Ptolemy and the Church—in favor of a simplifying conception for the solar system. But the Ethics alone of Baruch Spinoza, an application of Euclid to the relations obtaining between human and human, and human and God, bear out this thinking; surely Hobbes’ love of geometry, so that even human thought was composed of “bodies in motion,” is relevant here. In art, the mathematics of foreshortening and perspective, worked out by Brunelleschi in the early Renaissance, bear testimony here as well. One might also cite the growing mathematical methodology for musical composition, observable as early as the late medieval period and the beginning of the Renaissance—canons, fugues, imitative counterpoint, even a palindromic composition entitled “My End Is My Beginning,” in which the second half of the piece is literally the backwards version of its first half—by way of suggesting that mathematical spores were carried on the cultural currents of Western Europe and England long before the advent of Kepler, Galileo and Newton.
The Mathematical
It is with this shift to something like an unquestionable basis—for how could mathematics possibly present a false picture?—that the foundation for the Age of Reason is laid. That this period would come to be called “The Enlightenment” shows to what extent the mathematical had replaced what heretofore could only be called, by comparison, “the mythical”; of course, this latter term includes for the most part revealed religion. Not that the eighteenth century was irreligious; rather, deism reinterpreted God in less pictorial ways, more like a principle of exact creation. Terms like “the Great Mathematician,” and “the Great Engineer” came, in deism, to replace the God of the Hebrews and Christianity. Most significantly, deism held that once God had put his mathematically coherent creation in place he retired, leaving it to run on its own perfection; with this, God becomes something of a creative principle, whose method is mathematical. One way to appreciate this new notion of God is in the orderliness of Kepler’s second and third laws, and the universality Newton proclaims for gravitation. With regard to the former, Kepler states that the planets, traveling in elliptical orbits (rather than the perfect circles of Aristotle and the Church) and at varying speeds, yet sweep out equal areas in equal times, and that “the square of the time a planet takes to complete its orbit is proportional to the cube of its mean distance from the sun.”[5] That this should be so was surely proof of divine order.
Perhaps one can conclude that the divine mysteries which the doctors of the Church interpreted—so that truth had been doctrina—were stood open by the mathematical; indeed, perhaps now the mind of God could be read in the mathematical order of revealed astrophysics.
But what, then, is truth in the Newtonian and enlightened era? Descartes supplies something of an answer in his Meditationes de prima philosophia. I can doubt everything, he notes, except that I am doubting: the “I” here, in its indubitability, is so clear and distinct that this very clarity and distinctness ought to be the marks of any true proposition, according to Descartes. What sort of propositions have this indubitable clarity and distinctness? That two times two is four, that a triangle has three sides, are examples; truth then becomes something like mathematical certainty. If Descartes is the “father of modern philosophy” it is not because he refutes Aristotle—indeed, he uses Aristotle’s term (translated as) “substance” to characterize thought, while a separate “substance” characterizes matter—but because he is in step with an age in which mathematical certainty is well on the way to replacing traditional theology.
But what is the essence of “the mathematical,” then? And why does it go so well with Descartes’ self-founding proposition, cogito ergo sum?
Martin Heidegger addresses just this question in an essay entitled “Modern Science, Metaphysics, and Mathematics.”[6] According to the Greek origins of the term mathesis, as Heidegger explicates them, the mathematical is something we already have with us, and comes to knowledge only because it was already there. Plato tries to show this in the Meno dialogue by having Socrates question a slave boy about the areas of various squares—a boy who has had no training in mathematics or geometry. Plato considered true knowledge to be memory of the Forms which the soul beheld before falling into the body, and takes the boy’s answers as proof of the prior existence of the soul in a realm of truth.
Of course, Heidegger is no Platonist, but he recognizes that mathematical thinking is something that precedes the accounting of phenomena; it is a prearranged way of accepting data. An illustration of this is in the experiment of Galileo in which he dropped objects of different weights from the tower in Pisa in an attempt to refute the Aristotelian notion that the heavier object should fall faster than the lighter one. Usually, the outcome of this experiment is reported to have been that the objects landed at exactly the same time, as his observers supposedly confirmed. Heidegger writes that there was a slight difference in the arrival times, and that arguments between Galileo and his observers broke out when the former insisted that the times were close enough. The witnesses to the experiment “persisted the more obstinately in their former view. By reason of this experiment the opposition toward Galileo increased to such an extent that he had to give up his professorship and leave Pisa.”[7]
In other words, a mathematical mind-set had prearranged what was to be significant and what would not. But more than this is what the mathematical needs in order to be a pre-set standard for the calculation and interpretation of phenomena: its absolute self-foundation. According to Heidegger, this foundation is not some elaborate derivation from a higher standard; rather, the mathematical is self-founded only in its self-positing. “The positing, the proposition, has only itself as that which can be posited. Only where thinking thinks itself is it absolutely mathematical, i.e., a taking cognizance of that which we already have.”[8] This is never more clearly given than in the “I think, therefore I am” of Descartes. With him, the mathematical arrives as a criterion for truth.
Heidegger comments on the famous cogito ergo sum, pointing out that there is no question here of any deduction from thinking to being. He writes:
Descartes himself emphasizes that no inference is present. The sum is not a consequence of the thinking, but vice versa; it is the ground of thinking, the fundamentum. In the essence of positing lies the proposition: I posit. That is a proposition which does not depend upon something beforehand, but only gives to itself what lies within it.[9]
With the arrival of the mathematical as self-founded in its self-positing, “There is not only a liberation in the mathematical project, but also a new experience and formation of freedom itself, i.e., a binding with obligations which are self-imposed.”[10] That is, the mathematical is a “detachment from revelation as the first source for truth and the rejection of tradition as the authoritative means of knowledge.”[11]
Historical Implications
With this way of looking at the mathematical, we can perhaps see the overriding significance of the Renaissance for intellectual history. That period was certainly a break with the medieval world, although not an abrupt one. This is to suggest that the roots of the mathematical, in its self-positing, lie in not taking the Church as the ultimate authority on “the good life,” and in the acceptance of the “here and now”; to the extent humankind could come into the egoism of the Renaissance (where, we are reminded by various historians, humility and self-abnegation were reserved only for saints) the emergence of the mathematical was being prepared: the “I think” accompanying any new account of phenomena. When Galileo can write that he conceives “in my mind” of a body not affected by forces, he has engaged the mathematical, for nowhere in actual existence is there such a body; likewise, for Galileo to have conceived (“in my mind”) a frictionless plane on which a ball would roll on to infinity is to have posited the mathematical over the physical: not even the Smithsonian contains such a plane, perfectly frictionless and infinitely extended, nor could it.
Whence this “I think in my mind” long before Descartes? The question is unnecessary if we appreciate the rise of humanity as such in its break with the all-encompassing theological interpretation of the world that was the medieval period. The Renaissance, a break with ancient authority and the concomitant appreciation of the human individual—though obviously not on the scale of the individualism of the nineteenth century—was the gestation period for the mathematical; by the end of the Renaissance it is borne forth as the ground of the new principle: science. Surely, for any period to serve authentically as a transition—as the Renaissance was—it must carry within it the seed that will bloom into a new era. That seed was the egoism of humanity—humanism in its starkest outline—which accepts this world as the worthy arena for human striving and development of talents; by the end of the Renaissance this egoism had refined itself into the “I posit,” or the self-grounding of the mathematical mind-set, by which the new principle of truth, mathematical science—the truth of this world—is informed.
Let us look to some of the ways the arrival of mathematical thinking affects history. Certainly, as mentioned above, Copernicus is thinking mathematically when he embraces an abstraction over the physical appearance of the heavens in relation to earth. And by the time of Galileo, what is left of the medieval Church is under attack, both on the ground of religion, with Protestantism, and that of science. With Newton, the argument is ended about whether Copernicus or the Church is correct about the positions of earth and sun, although it took the latter centuries to admit that Galileo was right about Copernicus, who had had the good sense to die before he might have been burned at either the Catholic or the Protestant stake. But it is not a case where there is the sudden abandonment of the theological in favor of mathematical astronomy.
Rather, as hinted at above, the move to the mathematical is a radical theological change. When we consider how closely allied Platonic philosophy was with theology, and that Plato placed mathematics in his Analogy of the Line near the eternal realm, just below the highest section—the Forms united under The Idea of The Good—the move to the mathematical over the absolute truths of the Church is not a change outside the theological; rather, it is a reinterpretation of it. It really is not surprising that Newton sought the density of angelic matter, or that he dabbled in Biblical chronology. Least of all was he an atheist.
Here then is the unity of the shift from Church doctrina to the mathematical: “God’s mind” became accessible in the orderliness and rigor mathematics embraces. The Church lost out to Protestantism in the same way it lost out to mathematical science: the Church became unnecessary to religious truth, as Luther believed, and, if mathematics is itself a divine revelation, then doubly unnecessary. In this way Protestantism indeed goes “hand in hand” with the new science. For that matter, Luther embraces something of the “I posit” of the mathematical when he arrives at the notion that the individual cannot compromise on what each believes to be true, and frees worshippers effectually to be their own priests in a direct relation to God. And before Luther, these notions had their intuitive beginnings as early as the early 1400s with John Wyclif.
The human consequences of this shift out of medieval faith into rigorous science are too staggering to calculate in one article. Suffice it to say, when the earth lost its status as the center of the cosmos—thereby diminishing by an infinite power the significance of humankind—the very seeds of postmodernism were sown. It would take centuries for a sense of desolation and despair—groundlessness—to become the theme of an era in the West, and in the meantime a profound optimism would arise and decline: the eighteenth century’s Enlightenment that came to fail by the nineteenth century.
An intuition of the sense of groundlessness that is the subtlest implications of mathematical science—for the mere self-positing of mathematics fails, as Kurt Gödel’s two Incompleteness Theorems showed in 1931—was had by Blaise Pascal, who died about the same time Newton was writing his Principia Mathematica:
When I consider the brief span of my life, swallowed up in the eternity before and after, the little space which I fill, and even can see, engulfed in the infinite immensity of spaces of which I am ignorant, and which knows me not, I am frightened, and am astonished at being here rather than there; for there is no reason why here rather than there, now rather than then.[12]
It is in Newton’s work that place becomes the same throughout the cosmos, as opposed to the medieval/Aristotelian understanding that place is a category of being to which objects are uniquely related; “place is no longer where the body belongs according to its inner nature, but only a position in relation to other positions,” Heidegger observes.[13] Pascal is frightened at the silence of this homogenized, uncharged space. Jorge Luis Borges has pointed out that instead of a remark Pascal made as it remains today—”Nature is an infinite sphere, whose center is everywhere and whose circumference is nowhere”—the proto-postmodernist had first used the word “fearful” (effroyable) instead of “infinite” in reference to the sphere of nature circumscribed by the new science.[14]
On the other hand, the grand optimism for the advancement of man—the “heaven on earth” of utopian society, rather than the heaven of traditional theology—comes logically out of the new science. To the extent this science was the new cosmology of the here and now, it seemed to promise transcendence for humankind to the Enlightenment thinkers. Of course, by the nineteenth century and writers like Dostoevsky, especially in his Notes from Underground, the whole business had only provided humanity with a ghastly distortion of itself.
Reason is at the core of the Enlightenment, and just as the essence of the mathematical is its self-positing as ground, Reason becomes its own ground. It is this that informs the Age of Reason in its rejection of the tenets of traditional religion and, as well, in its finalization of the rejection of a monarchy that had originally arisen as God’s lieutenants on earth; it is the latter to which the two great revolutions of the eighteenth century testify.
Weakened in the twentieth century by the revelations of a new physics in which Einstein makes Newton’s absolute space/time relative to the speed of the observer, and Heisenberg’s Uncertainty Principle shatters the hope of Laplace for a perfectly calculable universe; relegated to a secondary and derived status in an existential philosophy that follows Heidegger’s reopening of the question of Being; put into grave question with the irrationality of World War I, Reason, borne into modernity on the current of mathematics, becomes in postmodernity the legacy of a demystified God.
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[1]Quoted in Lerner, Meacham, Burns, Western Civilizations, Thirteenth Edition, vol. 2 (New York, W.W. Norton & Co, New York,1998), p. 640
[2]Quoted in Chambers, Grew, Herlihy, Rabb, Woloch, The Western Experience, Fourth Edition (New York, Alfred A. Knopf, 1987), p. 632.
[3]Greer and Lewis, A Brief History of the Western World, Sixth Edition (New York, Harcourt, Brace, Jovanovich, 1992), p. 417.
[4]William Barrett, Irrational Man (New York, Anchor Doubleday,1990). This thesis is defended in his Chapter Two. See also Merton’s “Science, Technology and Society in Seventeenth Century England” in Osiris 4.
[5]Greer and Lewis, op. cit., p. 413.
[6]Martin Heidegger, Basic Writings (New York, Harper and Row,1977).
[7]Heidegger, Ibid., p. 266.
[8]Ibid., p. 278.
[9]Ibid., p. 279.
[10]Ibid., p. 272
[11]Ibid., p. 272
[12]Blaise Pascal, Pensees, No. 205. Quoted in Diane Barsoum Raymond’s Existentialism and the Philosophical Tradition (New Jersey, Prentiss Hall, 1991).
[13]Heidegger, op. cit., p. 263.
[14]Jorge Luis Borges, Labyrinths (New York, New Directions Books, 1990), p. 192.