Can We Know What We Know?

Mathematics and Religion:
Our Languages of Sign and Symbol
by Javier Leach
Templeton, 188 pages, $19.95

A man is like a carpenter, mused Tevye the milkman; for a man dies, and so does a carpenter. Fr. Javier Leach, S.J., informs us that mathematics is pluralistic, and so are religion and philosophy. All three fields are ways of not arriving at the truth, and thus are like each other. And since the twentieth-century revolution in logic, in particular Kurt Gödel’s Incompleteness Theorems, proves that we cannot know the full truth, we must content ourselves with whatever partial truth seems best to us. Leach is Catholic (“Jesus is my companion in the journey between the Absolute and nothing”), but he insists that others are just as entitled to their own truths.

Mainstream research has long abandoned the conceit of the 1960s that Gödel showed that we cannot know anything for certain. On the contrary, Gödel”a deeply religious man”set out to demonstrate we cannot prove with formal logic everything that we know to be true in mathematics. The nub of his Incompleteness Theorems is that there are things that we can define to be true in a mathematical system, but which we cannot capture in a formal proof. If we concede that mathematical objects are in some way real, Gödel requires us to concede as well the existence of some higher source for the intuition which sees a mathematical truth that transcends formal logic.

Leach willfully interprets Gödel’s results to mean quite the opposite. “One other way to think of incompleteness,” he declares, “is the fact [sic] that the human mind is creating many of our mathematical systems ad hoc. Mathematics is not outside the mind, as Platonism and the classical tradition [in mathematics] have held.” That is not a “fact” at all, but an opinion, and one he shares with the positivists and the New Atheists.

Philosophers have debated whether ideas in our mind have a “real” existence simply because we can define them mentally, or whether we must first instantiate such ideas in order to establish their reality, since fifth-century b.c. Athens. But classical mathematics did not simply rake over Plato’s ancient arguments. The problem is more difficult, and more disconcerting, than Leach concedes.

Leach engages in a bit of sleight-of-hand. Gödel sought to show that our capacity to know truth transcends mere formal logic. But Leach does not mention this; instead, he reduces Gödel’s Incompleteness Theorems to a mere demonstration of the limits of logic. He then makes a case for pluralism from the poverty of formal mathematical language itself.

Half the book is taken up with a trivial and misleading example. Leach offers a New-Age motto: “The Spirit sleeps in the river / dreams in the flowers / and feels in the birds, / but the Spirit only knows that it feels in men and women.” He then imagines how a logician, mathematician, philosopher, and theologian might make sense of this. The logician translates this into formal language (“Let there be R ( x ) such that x is a river, and let there be SL ( x , y ) such that x sleeps in y ,” and so forth). The vapid phrase, reduced to the “universal language” of propositional logic, is then reinterpreted in quite different ways by the mathematician, theologian, and philosopher, which is supposed to illustrate how each of us creates a personal “truth.”

What Gödel sought to show, rather, is that there are truths that exist in our mind that we cannot “prove.” Leach seems to be aware that this is the underlying issue, for he devotes one of his preliminary chapters to the problem of the Infinite. Of all the ideas that exist in our mind but cannot be instantiated empirically, the Infinite is the oldest and least tractable.

What Aristotle called an “actual infinity” may be imagined, but cannot be comprehended empirically, for we never can finish counting it. The great nineteenth-century mathematician George Cantor introduced infinite collections, or sets. But Cantor, as Leach observers, failed to convince a minority of Aristotelian realists in the mathematics profession, the Constructivists, who “argue that the actual infinite cannot be constructed by finite means in a finite process.”

“Such disputes allow mathematicians to choose the system they prefer,” Leach concludes. “If we choose classical mathematics, then we accept the actual infinite. A constructivist in mathematics accepts only the potential infinite. Both camps have a series of argument to justify their choices in applying a mathematical approach to infinity.”

But it is misleading to suggest that some mathematicians sit around imagining such things as infinite sets, while other mathematicians wonder if they are “real” or not. On the contrary: Physical reality thrust the problem of the infinite upon the mathematicians. Aquinas dismissed Plato with the quip that knowing the essence of the mythical Phoenix does not mean that such a bird actually exists. Mathematics, though, presents problems of a different kind: objects that exist in the mind but cannot possibly derive from the senses, and yet have a manifest relationship to the real world.

That is why we distinguish “strong Platonism” (the claim that essence implies existence) from “weak Platonism,” that is, the far more restricted claim that certain kinds of ideas are real, notably well-ordered mathematical concepts. The ancients and the Scholastics fought about abstractions. After the fifteenth century, though, metaphysics was compelled to respond to physics.

The “actual infinite” that Aristotle eschewed forced itself upon the philosophers. Curiously, the first intimation of the actual infinite appeared in music. During the 1430s, musicians began tempering musical intervals, using string lengths that corresponded to irrational numbers. Because the results were audibly harmonious, Nicholas of Cusa asserted that irrational numbers therefore must be real, after the fashion of Augustine’s “numbers in the mind of God” that were “too simple” for us to grasp directly.

This provoked a crisis in mathematics as well as metaphysics. Aristotle knew that an irrational number could be represented as an infinite series of rational numbers, and that the irrationals therefore implied the existence of an actual infinite, which of course could not be grasped by the senses. He rejected the concept, and under his influence, fifteenth- and sixteenth-century mathematicians and music theorists agonized over whether to admit the irrationals into musical tuning.

The discovery of the calculus in the second half of the seventeenth century, though, introduced a new kind of “real” object in the mind that was not derived from the senses. The calculus gives us the exact sum of an infinite number of infinitesimal quantities, which by definition are imperceptible. We cannot perceive vanishingly small quantities, yet in the calculus their sum is a definite number. The physics that issued from the work of Newton and Leibniz transformed the world. That made it more difficult (if not quite impossible) to dismiss infinitesimals as the mere imaginings of mathematicians. At issue was not a mythical bird, but rather the precise calculation of ballistic trajectories and planetary orbits. Leibniz embraced the “actual infinite” that Aristotle abhorred.

If Leach misrepresents Gödel, he egregiously lampoons Leibniz, Gödel’s most important influence. Leibniz’ “pre-established harmony” appears in Leach’s account as a simple-minded determinism that “would deny all free will and all chance and creativity in nature.” It seems odd for a Catholic writer to attribute creativity to nature, but in any case, Leibniz had no such thing in mind.

Rather, he proposed an alternative to the pantheistic ontology of his older contemporary Baruch Spinoza, whose “single self-generating substance” erased all individuation in the universe. If God is nature, there can be nothing in nature except God, and individual objects cannot exist. Leibniz removed God from nature and re-situated Him outside it, where He creates an endless multiplicity of infinitesimal monads that comprise a coherent world through a pre-established harmony.

Leibniz did not “prove” the existence of an actual infinite in the form of infinitesimals, for there is no proof that mathematical objects “exist” in the same way that thistles and marmalade exist. As Leach points out, there still are dissenters among mathematicians. But the revolution in mathematical physics and physics made for a different sort of debate than had occurred among the ancients or the Schoolmen. Plato’s theory of species, with its borrowings (for example in Timaeus ) from Pythagorean mysticism, was speculation, not physics. The infinitesimals, by contrast, were not simply a new sort of Platonic number mysticism, but rather a working principle that transformed the world.

The new mathematics of the sixteenth century roused the philosophers to explain the existence of objects in the mind that were not in the senses. There is nothing entirely new under the sun, to be sure: Descartes’ “innate ideas” looked back to Augustine’s theory of Divine Illumination. Immanuel Kant, by contrast, proposed an inborn (“a priori”) capacity for transcendental reason”reason that transcends sense perception”in order to do what Augustine proposed without the inconvenient presence of God.

Leach gets this backwards. “In the modern age science itself has offered a metaphysical vision. This vision says that our physical senses can apprehend all there exists.” Quite the opposite is true: The philosophy that attended the first stirrings of modern science came from Descartes, Leibniz, and Kant, all of whom shared the premise that some faculty of the mind must transcend the senses.

That is why Kant triumphed over the Scholastics and empiricists: His followers quickly learned how to use his theory to explain Newtonian natural science. The neo-Kantian school that dominated Continental academic philosophy from the last quarter of the nineteenth century through the first quarter of the twentieth hung its hat on the problem of infinitesimals. “The infinitesimal magnitude, thought of as reality, becomes the idealistic lever of all knowledge of nature,” wrote its founder Hermann Cohen in 1883.

And it got stranger still. Leibniz’ troublesome infinitesimals turned out to be only one among an infinity of “actual infinities.” Georg Cantor’s theory of infinite sets responded to anomalies in the calculus. By the early-nineteenth century mathematicians had learned that “spiky” functions, for example functions that shift sign at arbitrarily small intervals, cannot be analyzed with Leibniz’ infinitesimals. Somehow the infinitely small intervals that the calculus integrates into a finite sum were not quite “small” enough to capture such functions. There was the infinitely small, and the infinitely smaller still”and that is what Cantor proved: Different orders, or densities, of infinity do in fact exist. That is why infinite sets became so important”not because mathematicians sat around starting at the ceiling and thinking of the infinite. Cantor named the different infinities “transfinite numbers.”

Since then mathematicians have proved that there exists an infinite number of transfinite numbers. Thanks to the work of Gödel and Paul Cohen on the independence of the Continuum Hypothesis, we know that we cannot know how dense they are, or in what order they should be arranged”not at least within any existing framework of mathematical logic. Whereas Aquinas had argued that there are things we can imagine, but do not exist, Gödel proved, rather, that there are things that exist that we cannot imagine. The latter seems rather more interesting.

What Gödel could not prove was the “reality” of mathematical objects; there is no decisive refutation of the view that mathematics is merely a man-made syntax. “Aristotelian realism,” the insistence that nothing is real that we cannot instantiate empirically, cannot be disproven. But this opinion (not “fact” as Leach tendentiously states) places us on a slippery slope. As Leach shows, it can be construed to imply that all truths are equally valid. And that is just what Leach concludes: “Metaphysical and religious formulations seem true to us when they offer an intuitive veracity and coherence in the context of the personal values that a group of people share.”

One is entitled, to be sure, to believe that truth is to be found neither in religion nor mathematics. Both realms exist, though, because our antecedents believed they were pursuing the truth. That much is a matter of fact, and to it is wrong to suppress it in an account of the relationship of religion and mathematics.

David P. Goldman is senior editor of First Things .