Reading Selections from Can We Know What We Know?

May 2, 2011


Chichen Itza's Pyramid of Kukulcan marks the Summer Solstice. Although its the light of the afternoon sun on the Spring Equinox that causes the shadowy appearance of a huge serpent descending from the sky on the Summer Solstice there is also some precise shadow play going on, as the South and West sides of the pyramid stay in total darkness between 7 and 7:30 AM. This pyramid is a stunning combination of function, religion, philosophy, mathematics and geometry.

A review of Mathematics and Religion Our Languages of Sign and Symbol by David P. Goldman who is senior editor of First Things.  The book purports to be a study of the historical development of mathematical language and its influence on the evolution of metaphysical and theological languages. Leach traces three historical moments of change in this evolution: the introduction of the deductive method in Greece, the use of mathematics as a language of science in modern times, and the formalization of mathematical languages in the nineteenth and twentieth centuries. As he unfolds this history, Leach notes the differences and interrelations between the two languages of science and religion. Until now there has been little reflection on these similarities and differences, or about how both languages can complement and enrich each other.

Dr. Goldman takes issue with a lot of Leach’s conclusions. Some reviews contain far more interesting stuff in their refutation of the book at hand than their introduction of the book itself. I captured some of Dr. Goldman’s asides here rather than his issues with Fr. Leach:

Kurt Gödel’s Incompleteness Theorems
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.

Debating “Real” Existence
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….

What Gödel sought to show is that there are truths that exist in our mind that we cannot “prove.” 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.”

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.

Aristotle’s “Actual Infinity”
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.

Leibniz is Gödel’s most important influence. 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… 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.

Philosophy And The Birth Of Modern Science
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.”

Aristotelian Realism
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.

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