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Reading Selections from The God of the Mathematicians by David P. Goldman

May 13, 2011

Gödel and Einstein, 1954


Some of the religious beliefs that guided Kurt Gödel’s revolutionary ideas.

David P. Goldman is a senior editor at First Things. The following are reading selections from a piece that appeared last August

The Human Mind Cannot Be Reduced To A Machine
Kurt Gödel was a believer — or, at least, a knower — whose engagement with God included a reworking of the ontological proof of God’s existence. Born in 1906, Gödel was arguably the great mathematician of his time. Certainly no twentieth-century thinker did more to show that the human mind cannot be reduced to a machine. At twenty-five he ruined the positivist hope of making mathematics into a self-contained formal system with his incompleteness theorems, implying, as he noted, that machines never will be able to think, and computer algorithms never will replace intuition. To Gödel this implies that we cannot give a credible account of reality without God. But Gödel’s God is not the well-behaved deity of the old natural theology, or the happy harmonizer of the intelligent-design subculture. Gödel’s God hides his countenance and can be glimpsed only in paradox and intuition. God is not an abstraction but “can act as a person,” as Gödel once wrote, confronting those who seek him with paradox, uplifting man through glorious insights while guarding his infinitude from human grasp. Gödel’s investigations in number theory and general relativity suggest a similar theological result: that God cannot be reduced to a mere principle of the natural world. Gödel may have seen himself as a successor to Leibniz, whose critique of Spinoza’s atheism set a precedent for much of Gödel’s work.

The Difficulties In Ascertaining His Theological Intent
When we try to ascertain the theological intent underlying Gödel’s mathematical investigations, though, several difficulties arise.The first is Gödel’s reticence. “Although he did not go to church,” his wife Adele told the logician Hao Wang shortly after Gödel’s death in 1978, he “was religious and read the Bible in bed every Sunday morning.” But fear of ridicule and professional isolation made him reluctant to talk about his faith. “Ninety percent of contemporary philosophers see their principal task to be that of beating religion out of men’s heads,” he wrote to his mother in 1961.

A two-page draft of an ontological proof for God’s existence forms the whole of Gödel’s explicitly theological output. He showed his paper only to close friends, but word got out, and the clever young things on campus giggled behind his back. His biographer Rebecca Goldstein, who was a graduate student at Princeton during Gödel’s last years, snickers that she and her peers “found it hilarious” that Gödel “deluded himself into believing that God’s existence could be proved a priori.” The ambient hostility drove some of his best students out of the profession and may have worsened the eating disorder that hastened his death.

Another difficulty is that Gödel’s work extends across several demanding fields, each with a high threshold of preparation.

An Irrationalist Who Proved That Nothing Can Be Proved?
There is also the problem that scavengers have been at work on his legacy for decades. The postmodernists have tried to claim him as an irrationalist who proved that nothing can be proved — just the opposite of what he intended. Rebecca Goldstein rightly debunks the postmodern claim, but her biography of the great mathematician makes no mention of his religious faith except to ridicule it, ignoring key facets of his work with theological implications.

Nonetheless, the ontological proof provides a point of entry into Gödel’s work, linking intuitive theology with his mathematical investigation. The proof, in one form or another, has been known at least since the eleventh century, when St. Anselm of Canterbury paused to ask: If God is greater than we possibly can conceive, then how could God not exist?

In the best-known version of the argument, Anselm noted:

1. The definition of the word God is “that than which nothing greater can be conceived.”
2. God exists in the understanding, since we understand the word with that definition.
3. To exist in reality and in the understanding is greater than to exist in the understanding alone.
4. Therefore, God must exist in reality.

Versions of the argument, with key twists and turns, have been introduced in Western philosophy through the centuries by thinkers from Descartes and Leibniz down to the twentieth-century American philosopher Charles Hartshorne. And, from the monk Gaunilo in Anselm’s own time down through St. Thomas Aquinas, Immanuel Kant, and Bertrand Russell, these same centuries have seen philosophers who vehemently reject any form of the argument.

In all versions there exists a tension among premises. If God is “that than which nothing greater can be conceived,” how can we understand God? Whatever we might conceive, God must be greater. If this is so, then how can God exist in the understanding?

Barth’s Reading Of Anselm
In his 1930 book on Anselm, Karl Barth offers a theological rather than a philosophical answer: We understand God by calling on him by his proper name, “that which is greater than anything that can be conceived.” “It does not say,” writes Barth, “that God is, nor what he is, but rather, in the form of a prohibition that man can understand, who he is.” In effect, Barth says that “that than which nothing greater can be conceived” is something of an alternative appellation for YHWH, God’s personal name in the Bible.

In Barth’s reading, Anselm’s statement is not only proof, but also prayer. By asking us to attempt to conceive of that than which nothing greater can be conceived, Anselm restrains our impulse to worship, instead, a projection of ourselves. This intent, Barth adds, makes irrelevant the usual critique of Anselm. Thomas Aquinas objects that simply thinking something exists does not necessarily mean that it does exist. Kant jokes that the essence of a hundred imaginary thalers is the same as the essence of a hundred real thalers in my pocket, but their spending power is quite different.

Such refutations do not apply, Barth insists, for Anselm’s exercise presumes faith. But that move, in turn, raises up a theological problem. As the Jewish philosopher Michael Wyschogrod has noted, “If the ontological proof is successful, then God would be one of the things that have being. It would then be the case that the Empire State Building is, and the Eiffel Tower is, and God is.” Being would be an umbrella concept, covering God and many other things — which would make being our ruling concept. Naming God as “that which is greater than anything that can be conceived,” or “TTWNGCBC” rather than YHWH, gives us not a more refined faith but only a more refined atheism.

Spinoza And Leibniz
And that is just how the secular Enlightenment proceeded. Atheism in its modern form began with a revision of the ontological argument at the outset of Spinoza’s Ethics. It could not have been otherwise, for to make sense of a world without God, atheism first had to usurp the attributes of God and assign them to nature. Something “than which nothing greater can be conceived” within the natural world gives us Spinoza’s “infinite substance.”

Spinoza begins his Ethics with an ontological argument: “God, or substance, consisting of infinite attributes, of which each expresses eternal and infinite essentiality, necessarily exists” because “if this be denied, conceive, if possible, that God does not exist: then his essence does not involve existence. But this is absurd. Therefore God necessarily exists.” In Spinoza’s words, “By God, I mean a being absolutely infinite — that is, a substance consisting in infinite attributes, of which each expresses eternal and infinite essentiality.” God is reduced to the “substance” of nature.

But the ontological argument turned out to be a cuckoo’s egg in the nest of atheism. Spinoza abducts Anselm’s TTWNGCBC from heaven and locks him up inside the natural world. The trouble is that once inside nature, TTWNGCBC consumes everything else and becomes all that there is in nature. If God is inside nature, then there can be nothing in nature outside of God. Spinoza concludes: “As God is a being absolutely infinite . . . and he necessarily exists; if any substance besides God were granted it would have to be explained by some attribute of God, and thus two substances with the same attribute would exist, which is absurd; therefore, besides God no substance can be granted, or consequently, be conceived.” If we actually can conceive of TTWNGCBC within the natural world, then we can conceive of nothing else at all.

Hegel quipped that the cause of Spinoza’s death “was consumption, from which he had long been a sufferer; this was in harmony with his system of philosophy, according to which all particularity and individuality pass away in the one substance.” Spinoza’s younger contemporary Leibniz pounced on this vulnerability, turning Spinoza’s system inside out: Instead of an “infinite substance,” Leibniz postulated a “pre-established harmony” controlling an infinite number of independent monads. Leibniz added a purely theistic premise: By the law of sufficient reason, he argued, God does not do anything superfluous and therefore does not create anything twice.

In a purely formal sense, the systems of Spinoza and Leibniz seem to be mirror images: Spinoza’s single substance cannot explain individuality, and Leibniz’ individual monads cannot communicate with each other; “pre-established harmony” has the same function as “infinite self-generating substance.”

Undergraduate courses misleadingly lump the two together under the rubric of “rationalism.” But there is a fundamental difference: By inverting Spinoza’s metaphysics, Leibniz makes room for God to return from his Babylonian captivity in natura naturans, [vocab:  Natura naturans is a Latin term coined during the Middle Ages, meaning "Nature naturing", or more loosely, "nature doing what nature does".] to lordship over being. “Sufficient reason” is a theistic premise, to be sure, but it explains the world as we perceive it, rather than the single monistic glob implied by Spinoza.

If Spinoza tries to capture the ontological proof for atheism, Leibniz sets out to restore it to theism, suitably corrected to answer the objections of Thomas Aquinas. If we suppose that God possesses all positive properties, Leibniz argues, then necessary existence is a positive property and must pertain to God. If we agree that it is logically possible for a perfect being to exist because “all perfections are compatible with each other,” then we must conclude that a being possessing all positive properties must exist: “There is, or can be understood, the subject of all perfections, or a most perfect being.”

Gödel And Leibniz
Kurt Gödel was a lifelong student of Leibniz; during his four decades in Princeton at the Institute for Advanced Study he checked out every book on Leibniz in the university library. In an answer to a questionnaire found in his posthumous papers, he wrote, “My belief is theistic, not pantheistic, following Leibniz rather than Spinoza.” He reworked Leibniz’ version of the ontological proof in 1941, using the notation of modern mathematical logic (although he showed it to no one until 1970). I doubt Gödel believed he had found the ultimate and irrefutable proof of the existence of God. His deep interest in the ontological proof, rather, was one facet of his commitment to defend Leibniz’ theism against the new Spinozans of mathematics and physics.

Gödel’s proof is written in logical notation. Like its Leibnizian model it turns on the notion of “necessary existence,” although it contains additional intermediate steps. Gödel begins by asserting that “positive” properties may be distinguished from properties in general. “Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world),” he explains. The property that conjoins all positive properties also must be a positive property; and because “necessary existence” is a positive property, this property-of-all-positive-properties must include necessary existence; Gödel defines it as the “God-like property.” God therefore possesses all positive properties, and, because non-positive properties are the negation of positive properties, God cannot have any non-positive properties. Moreover, because “necessary existence” is one of these positive properties, God must exist in all possible worlds.

The objection of circularity might be raised because “positive properties” may be called positive because they are Godlike to begin with. It is not clear either that all positive properties are logically compatible, as in the paradox of divine omniscience and omnipotence.

Objections To Gödel’s Argument
Other objections have been raised to Gödel’s argument. It seems to imply, for example, that every positive property also must exist, for all positive properties belong to the God whom the argument proves to exist. The modern philosopher Jonathan Sobel argues that we are thus driven back to Spinoza’s problem, in which God is in everything and everything is in God, because we cannot distinguish between necessary existence and contingent existence. (Dean Koons has suggested a possible if problematic repair of Gödel’s proof in which only the cosmos itself is considered to have necessary existence.)

Gödel’s proof is best understood as an exercise within his broader Leibnizian program. If we attempt to speak in purely natural terms of “that than which greater cannot be conceived,” we cannot help but refer to the mathematical concept of infinity. That is just how Spinoza thought of his infinite substance, and to embed the infinite, an attribute of God, within natura naturans was a pantheist credo. Leibniz, the co-inventor of calculus, knew that an infinite number of infinitely small quantities can have a finite sum, and his infinity of infinitesimals, therefore, does not upset the finite character of created nature. But already, with Leibniz’ refutation of Spinoza, mathematics was emerging as the laboratory for ontological investigation. Whether that should be the case is a source of perpetual debate, but it is hard to imagine how it could have happened otherwise in the seventeenth century.

To treat mathematical objects as the proper subject of ontological investigation requires a grand assumption, to be sure, that, in Gödel’s words, “mathematical objects exist independently of our constructions.” Gödel worked for years, for example, on an unpublished essay refuting Rudolf Carnap’s view that mathematics was no more than a syntax to manipulate man-made symbols.

The meta-mathematical debate continues to this day, and how it will be resolved has not yet come clear. Nonetheless, it is perhaps telling that Leibniz, the philosopher who offered the most logical rejoinder to Spinoza, was able to do so because, coincidentally, he was the mathematician who formulated the modern concept of infinitesimals.

Leibniz’ infinitesimals led to another Gödel discovery with deep theological implications. In the nineteenth century, mathematicians learned that the calculus discovered by Leibniz and Newton could not integrate or differentiate some classes of functions. The calculus began with the insight that an infinite series whose terms grew infinitely small might have a positive sum. But some functions resisted the calculus. These include “spiky” functions in which changes in sign, for example, occurred in arbitrarily small intervals. From the study of such functions came the disturbing insight that some infinities are “bigger,” that is, more densely packed with numbers, than other infinities. And this inspired one of the nineteenth century’s greatest mathematicians to attempt to treat the different orders of infinity as if they were just another kind of number — the “transfinite numbers” — and thus to domesticate infinity.

Georg Cantor
Georg Cantor was the discoverer of these transfinite numbers in the early 1870s, when he showed that some infinite collections were “larger” than others. There is a one-to-one correspondence between the integers and the rational numbers, such that the rational numbers may be thought of as a “countable” infinity. But there is no such correspondence between the integers and the real numbers, conceived of as a continuum line.

That was an insight of historic importance. But it did not satisfy Cantor, who believed that he had solved the problem that had eluded Spinoza — namely, to preserve the differentiation of individual objects within natura naturans. By bringing God’s infinitude into the natural world, Spinoza was left with nothing but his single substance. At least in the realm of numbers, Cantor believed, infinity itself could be ordered with a new series of “transfinite numbers,” each representing a different order of infinity. He envisioned a new kind of cardinal numbers denoting infinite sets of numbers.

The infinitesimal monads of Leibniz thus would no longer require God and the principle of sufficient reason to differentiate themselves because infinite series of numbers would arrange themselves naturally into Cantor’s transfinite ordering. Cantor’s theology was confused — he also hoped to reconcile his views with those of Thomas Aquinas and the Church Fathers — but his belief that he had solved not only a mathematical problem but also an ontological mystery is well documented. A draft in his letter book states, “I have examined all objections that have ever been made against the infinite numbers, and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.”

Cantor asserted that the infinity of the integers and rational numbers was the first transfinite number, and he named it “Aleph-zero.” What, then, was the second transfinite number, or “Aleph-one”? He had proven that the infinity of the continuum of real numbers was “denser” than that of the integers; unlike that of the rational numbers, it could not be counted. He assumed that if the first transfinite number contained the integers, the second transfinite number would contain the continuum, and that no other transfinite number could be discovered between these two.

That is Cantor’s “continuum hypothesis,” which attempts to identify a first and second transfinite cardinal number. From there, he believed, all the possible orders of infinity could be counted, the same way the integers count groups of one, two, three, and so forth. He not only recognized, but was driven by, the ontological implications of this assertion: If the continuum hypothesis turned out to be true, Spinoza would be vindicated because God’s infinity could be packaged into a neat series of numbers. Cantor spent the last thirty-five years of his life in a vain effort to prove this. He died in 1918 in a mental hospital.

Gödel And Cantor
It was Gödel and, later, Paul Cohen who demonstrated respectively that Cantor’s continuum hypothesis could be neither proved nor disproved within existing set theory. Indeed, Cantor’s hypothesis remains maddeningly undecidable. Intuition, added Gödel, strongly suggests that Cantor’s hypothesis is wrong: Among the infinite number of transfinite numbers, there are an infinite number of cardinalities between the integers and the points on the continuum line, and mathematical investigation of the infinite will remain infinitely fruitful. God’s infinitude remains safe in heaven. Mathematicians have proven that an infinite number of transfinite numbers exist but cannot tell what they are or in what order they should be arranged.

Gödel noted drily that this represents a problem for philosophy and epistemology rather than for mathematics, which can continue its investigations without ever exhausting the subject. Gödel’s result shows that not even in terms of numbers, the simplest objects we can specify, can natura naturans explain the individuality that we observe. The parallel between Gödel’s attack on the continuum hypothesis and Leibniz’ critique of Spinoza is very strong, and it is remarkable that both hinged on foundational insights into number theory.

Whether or not we believe, as did Gödel, in Leibniz’ God, we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel’s critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.

Gödel’s Critique Of General Relativity
There is yet a third place where Kurt Gödel’s mathematical work has theological purchase: in Einstein’s failure to reconcile the deterministic world of general relativity with the probabilistic world of quantum mechanics. Einstein famously declared his belief in “Spinoza’s God”: a god, that is, who is indistinguishable from nature and who reveals himself through natural harmonies. Einstein, we might say, was a “strong Platonist” who actually believed that if one discovers the eternal forms to which natural phenomena correspond, all the world’s mystery will yield itself up to science.

The often noted problem is that the intuitively intelligible world Einstein created with the deterministic equations of general relativity jars with the probabilistic world of modern quantum mechanics. Einstein and Gödel were close friends, but they disagreed profoundly on religious and philosophical matters. As Gödel told Hao Wang, “Einstein’s religion [was] more abstract, like Spinoza and Indian philosophy. Spinoza’s god is less than a person; mine is more than a person; because God can play the role of a person.”

Gödel’s personal God is under no obligation to behave in a predictable orderly fashion, and Gödel produced what may be the most damaging critique of general relativity. In a Festschrift for Einstein’s seventieth birthday in 1949, Gödel demonstrated the possibility of a special case in which, as Palle Yourgrau described the result, “the large-scale geometry of the world is so warped that there exist space-time curves that bend back on themselves so far that they close; that is, they return to their starting point.” This means that “a highly accelerated spaceship journey along such a closed path, or world line, could only be described as time travel.” In fact, “Gödel worked out the length and time for the journey, as well as the exact speed and fuel requirements.”

Gödel, of course, did not actually believe in time travel, but he understood his paper to undermine the Einsteinian worldview from within. Yourgrau observes, “The very fact that this inconceivably fast spaceship would return its passengers to the past demonstrated, by Gödel’s lights, that time itself — hence speed and motion — is but an illusion.” Stephen Hawking so abhorred the implications of Gödel’s demonstration that he proposed an ad hoc bylaw for general relativity, the “chronology protection conjecture,” simply to exclude it. Like Einstein, Hawking then believed that a grand theory of the universe would allow humankind to see into the “mind of God.” In recent years, though, Hawking has come closer to Gödel’s point of view, going so far as to conjecture that a sort of Gödelian “incompleteness principle” might exist in physics as well as in mathematics.

Summary
Gödel’s incompleteness theorems, critique of the continuum hypothesis, and examination of time in general relativity all have theological implications. After reviewing them, is it appropriate for us to speak of a theology? Gödel evidently thought so. In 1961 he made notes for a lecture in which he ranked the contending worldviews in contemporary science “according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion).” “Skepticism, materialism, and positivism stand on one side; spiritualism, idealism, and theology on the other.” He dismisses “idealism, e.g., in its pantheistic form,” as “a weakened form of theology in the proper sense.”

“One would, for example, say that apriorism belongs in principle on the right and empiricism on the left side.” But, he adds, “On the other hand, however, there are also such mixed forms as an empiristically grounded theology.” Gödel was at least a weak Platonist — he considered mathematical objects to be real and his research therefore to be empirical. He thought his theology thus to be an empirical one, founded on man’s experience of the infinite fecundity of the creator’s mind. That is why Gödel’s religious thinking is so rich and also why it is so challenging: One must actually follow the work to make sense of the conclusions, an unusual challenge for theologians, and one they have shirked for too long.

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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
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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Reconciling Science and Religion by Britton Johnston

April 6, 2011

William Blake, The Body of Abel Found by Adam and Eve

 

Another reading selection from the article titled “How Girard’s Mimetic Theory Can Help Us Understand the Relationship Between Science and Religion” by Britton Johnston, a Presbyterian minister who lives in Santa Fe, New Mexico. The complete article is at Metanexus: The Online Forum on Religion and Science.

It might be helpful here to give an example or two of how mimetic theory can reconcile the claims of science and religion. Let’s explore the issue of creation, and the question of the existence of spiritual beings.

The question of Creation is fundamentally a question of the distinction between culture and cosmos. Archaic cultures, being unable to distinguish the two, use material realities to express cultural ones. The creation story in the first chapter of Genesis, is not a text about physics or biology, but about cultural origins — one that stands in contrast to pagan creation myths.

For example, in the ancient Babylonian creation myth known as the Enuma Elish, the warrior God Marduk forms the world from the dead body of his mother Tiamat after he has slain her. He hews her body into two pieces, and with the upper half of her body he forms the heavens, and with the lower half he forms the earth. For the ancient Babylonians, this is the story of the foundation of the world.

From the point of view of Girard’s mimetic theory, this myth is a disguised account of an actual murder (or, more precisely, of a series of ritual murders). The murder would have taken place as Babylonian culture was forming for the first time. There was a mimetic crisis, which was resolved only when one or more people were killed by the mob, bringing order out of the social madness.

This originary murder rescued these Proto Babylonians from a state of acute mass psychosis. As they emerged from the psychosis, it appeared to them as though the cosmos itself had been re-created. They described the event as best they could, given that they were emerging from a condition of total delusion; so what amounted in actuality to a lynching, came to be described as a divine event, a divine drama playing itself out in the heavens. Historically, there were probably more than one of these collective murders. This lynching was reproduced in sacrificial rituals. These ritual sacrifices came to be understood as re-enactments of the original divine event, enriching and refining the narrative into a creation myth.

All creation myths seem on the surface to be about the creation of the material world; but they are really about the origin of human culture. It would be natural to ask at this point, why cultures don’t just describe their origin literally and factually? Why don’t the Babylonians simply say, “we were in a crisis and we saved ourselves by lynching a member of our community.” Why the elaborate narrative? Why the obfuscation?

There are two reasons for this: first of all, as I indicated above, these events are generated on the boundary between psychotic delusion and sane reality; therefore, myths have a dreamlike, semi-delusional quality. Secondly, the culture has a stake in disguising the original murder. Every culture knows that murder must not be spoken of approvingly, because murder invites revenge and revenge escalates, plunging the whole society into bloody madness. Therefore the society must pretend that it is innocent of murder. Yet still the original murder must be remembered, because it brought the benefits of social order. Myths have this dreamlike quality because they are the result of a double-bind: they must simultaneously recall and hide the crime they trace.

The creation story in the first chapter of Genesis is not a literal description of the origin of species or of the origin of the planets; rather, this is a story which uses the concept of species and the image of the planets as symbols to describe something which was much more pressing to the ancient Israelites than the matter of scientific explanations. This is a story of the origins of human culture and consciousness. In this way it is not unlike the creation myth of the Babylonians. Yet it also differs sharply from the Babylonian story (and virtually every other creation story from ancient culture): it contains no sign of a murder!

In fact, scholarship has revealed that the first chapter of Genesis was composed while the people of Judea were in exile in Babylon. The first chapter of Genesis was the Jews’ response to what their captors insisted was the origin of the world. In their refusal to go along with the idea of a founding murder, the Jews became the first culture in the history of the world to claim that violence is not essential to the cosmic order.

There is no murder in the first chapter of Genesis. There is only a powerful God working hard to establish a place for everything and to put everything in its place. The language of creation in the first chapter of Genesis is the language of establishing boundaries. There’s a boundary between the light and the dark. There is a boundary between the dry land and the sea, there is a boundary between the different kinds of plants and animals. The story has the quality of the storekeeper taking care of his inventory, or of a housekeeper picking up clutter. Instead of a warrior God carrying out the sacred execution, the Jews revealed to history a worker God who establishes order not by killing somebody, but by cleaning house.

The principle of creation in the beginning of the Bible is the principle of difference. In the first chapter of Genesis, we do not have a description of a creatio ex nihilo, but an account of the imposition of difference on the primordial waters of chaos. The day is separated from the night; boundaries are set between the sea and dry land; the plants and animals are separated out, “each according to its kind.” The original creation was the ordering of chaos through the establishment of differences.

Why is this important? According to Girard’s mimetic theory, the primordial waters are an archetypical image representing the pre-cultural mimetic sacrificial crisis. The (pre)human community is in a state of crisis brought about by the imitation of everyone by everyone. This has produced a confusing maelstrom of desire, rivalry, hostility and violence within which life is impossible. Like a swimmer tossed in the chaos of a riptide, everyone finds it impossible to distinguish up from down, left from right, light from dark.

According to Girard, the first strategy to resolve this crisis is human sacrifice. But in the Hebrew scriptures, humanity begins to move away from human sacrifice. In order to defer the sacrificial crisis, the Hebrew strategy is to put in place a strong system of sacred difference.

Difference defers or delays the sacrifice by blocking the development of mimetic rivalry. It works because when boundaries are drawn between people, they tend to imitate each other less strongly. We are most strongly mimetic toward those whom we perceive to be like us. If we see the other as different, we are less likely to want what they want, return their insults, and so forth. Thus we are less likely to come into mimetic rivalry with people who are different (violence against those who are of a different ethnicity or who are differently abled is not technically mimetic rivalry; it is a kind of scapegoating that discharges mimetic rivalry and unifies the community).

Consider an ice cube tray, the kind that has the removable dividers. If you fill it with water without the dividers and try to carry it across the kitchen, chances are that the water will spill. But if you insert the dividers into the water before carrying it, you find it is much easier to carry it without spilling. Differences in culture are analogous to this. They prevent the free flow of mimetic rivalry from building up to a chaotic loss of control.

The first chapter of Genesis is a projection onto nature of precisely this concern for difference. As a subtle anti-Babylonian polemic, Genesis 1 substitutes a structure of differences for the violent structures built on human sacrifice. This is an enormous advance in human consciousness. The fact that it retains the archaic confusion between culture and cosmos should not be grounds to dismiss it. After all, we owe our very awareness of that difference to documents such as this.

Science still has some things to learn from Genesis 1. The theory of natural selection itself depends on the notion of the selfish competition for survival as essentially a “creative principle of the cosmos.” Numerous critics have pointed out that this idea is far too much like the Malthusianism and “social Darwinism” (which is misnamed – Darwin borrowed it from Malthus and Spencer and applied it to biology) to be entirely independent of cultural bias. Even in our scientific endeavors, we may be too susceptible to the tendency to project our culture onto nature. Perhaps science should stay a little closer to the insights of the Bible after all.

A Theory of Spirits
Mimetic theory opens up a new category for describing reality that hasn’t been available until now: “mimetic forces.” Such forces are recognized by every culture, but they are not described, merely named – spirits, angels, ghosts, demons, etc. These are forces with real power but that are unseen and hard to measure. Mimetic theory gives us the means to actually describe them.

A “mimetic force” exists in the relationships between people. A simple desire is a mimetic force. According to Girard, if one person makes an acquisitive gesture toward an object, another person nearby will tend to focus on the same object, with an impulse to make a similar acquisitive gesture. The original gesture, by stimulating a mimetic response in the neighbor, could be said to be a kind of “force.” The force draws people under its influence. They in turn add their own energy to the mimetic force, causing it to strengthen. One person wants the object, generating a weak mimetic force in the next person, who likewise comes to desire the object; now the mimetic force is twice as strong, so that a third person will desire the object even more readily than the first two people. The force propagates through the population, gaining power to affect individuals as each additional individual is affected.

If the desire so propagated is a desire for the well-being of others, it could be called an “angel”; on the other hand, if the mimetic force is a spirit of resentment, it will be called a “demon” – that is, after its violent denoument is done.

The definition of demons, spirits and angels as mimetic forces accounts for all the characteristics attributed to them. They are invisible, more “felt” than seen. They are personal, yet not contained within a body; they affect people to the point of taking over the human will; and they can be invoked or exorcised by ritual and prayer.

Such mimetic forces doubtless play a huge role in illness and disease. They can affect the functioning of the body and mind in profound ways. The “science” of managing such forces exists almost exclusively within religious traditions. It could be an extremely important advance in medical technology if we were to begin to explore the means by which such forces can be managed. If scientists are to learn how to do this, however, they will have to become students of religion.

Mimetic theory, if it is correct, offers a fresh and clear path for us to understand how science and religion are radically interdependent. I hope that those who read this article will be motivated to explore this new and potentially fruitful avenue of inquiry into the relationship between the two.

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Reading Selections From Girard’s Mimetic Theory And The Relationship Between Science And Religion by Britton W. Johnston

April 1, 2011

René Girard

This is from an article titled “How Girard’s Mimetic Theory Can Help Us Understand the Relationship Between Science and Religion” by Britton Johnston, a Presbyterian minister who lives in Santa Fe, New Mexico. Rev. Johnson earned his Masters of Divinity at the McCormick Theological Seminary in Chicago, class of 1990 and organized the annual meeting of the Girardian Colloquium on Violence and Religion in June, 2004. The complete article is at Metanexus: The Online Forum on Religion and Science. I add this to my little collection of Girardian articles and summaries.

Culture And Truth
The question of the relationship between science and religion, like many other leading concerns of theologians today, has to do with the relationship between culture and truth. It therefore seems appropriate to approach these theological issues anthropologically. Unfortunately, the field of Anthropology tends to be dominated by a “politically correct” suspicion of religion in general, and of theology in particular.

Fortunately, there is a new anthropological theory emerging. This new theory is congenial to theology, promising to give us powerful new concepts and tools to finally resolve these vexing theological questions. This theory is the “mimetic theory” of René Girard. It has been around for about 30 years, though it has made little progress among theologians and anthropologists until recently. What I would like to do with this essay is to introduce the basics of Girard’s theory, and to suggest how this theory might supply us with a fruitful new approach for reconciling science and religion.

Who Is René Girard
René Girard is what you might call a “literary” anthropologist — this despite the fact that his formal education was in neither literature nor anthropology. His Ph.D. is in history. His doctoral dissertation was on the subject of Franco-American relations after World War II. Although his “outsider” status might lead us to question the validity of his theories, in fact a lack of official credentials is common among those who bring revolutionary new ideas to a field of study.

Girard was born in France in 1923. He came to the United States in 1947, working on his doctoral dissertation at Indiana University. They put him to work teaching French literature, something for which he had little training beyond the fact that he was a Frenchman. In fact, he was often just barely ahead of his students, reading some of the novels for the first time, two chapters ahead of the class assignments. In the process of teaching literature, he began to notice certain patterns in the great novels, in their treatment of human desire

Deceit, Desire, and the Novel
His first book, Deceit, Desire, and the Novel, was published in French (with the title Mensonge romantique et vérité romanesque) in 1961, after he had become a professor at Johns Hopkins University. In that book he argues that great literature reflects awareness that human beings get their desires from one another. We are “mimetic” creatures, meaning that we internalize one another through imitation. A crucial aspect of the mimetic process is that it is the means by which we acquire our desires. Human desire therefore is not innate; rather, we “borrow” our desires from those we imitate. This brings us into conflict with those others. The person who is our model also becomes for us the greatest obstacle to getting what we want. Great literature, Girard argued, depicts its protagonists’ entanglement in these mimetic webs of desire and rivalry — but often with liberation at the moment of the hero’s death or expulsion.

Patterns Of Expulsion
Girard continued to examine the theme of expulsion, in ancient literature and primal myths. He found that every ancient myth contained traces of a pattern of expulsion. Every ancient myth, that is, except for the Bible. In 1972, he published La Violence et le Sacré, in which he argued that all religious myths are disguised accounts of actual historical events, specifically expulsions, the sacrifice of scapegoats.Even the Bible follows this same pattern, but with one important difference: the Bible is the first narrative to present the expulsion from the point of view of the scapegoat.

Girard went on to develop his mimetic theory in subsequent books, such as Things Hidden Since the Foundation of the World (English edition, 1987), The Scapegoat (1986), To Double Business Bound (1978), and I See Satan Fall Like Lightning (English edition 2001). In these books and others, Girard and his followers have demonstrated that his theory has amazing power to clarify issues in anthropology, theology, biblical interpretation, psychology, political science, economics, linguistics, and on and on. It is truly a “grand theory,” simple yet powerful. Such theories are not in fashion in these postmodern and multicultural times; they are in fact regarded with suspicion. So far, there have been no grand theories that have worked. So far.

A Sketch Of Mimetic Theory
The preeminent characteristic of human beings is that we imitate each other (thus the term “Mimetic Theory”). This mimesis is not mere mimicry, but an instinctive and preconscious impulse. Even our desires–especially our desires–come from the imitation of others. Because we want the same things that others want, we come into conflict over who will possess the desired object. This rivalry is in turn imitated so that it escalates into violence. The rivalry does not remain limited to the first individuals involved, but others imitate it until it spreads to the entire community, generating a mimetic crisis. Violence threatens to destroy everyone involved, unless a solution is found.

The solution that our species stumbled upon was the mechanism of sacrifice. One individual is singled out by the community as the scapegoat whose death absorbs the violence in the community, delivering the community from this threat. The community mistakenly believes that the scapegoat was at once the cause as well as the all-powerful cure for the chaos of the mimetic crisis. The pagan concept of the gods emerges from this misrecognition. The deliverance brought about by sacrificial violence is the basis for the primitive sacred. It is also the basis of archaic religion and the foundation of human culture.

Human culture extends the power of sacrifice by creating myths and idols, which remind the people of the sacred event of the sacrifice, damping down the fires of the mimetic crisis. The function of a myth is to preserve and obscure the historical event of the sacrifice. By preserving the experience of the sacrifice, a myth reduces the need for frequent repetitions of it. But it is also important that myths obscure the murderous reality of the sacrifice, because to speak openly of murder is to risk triggering a new mimetic crisis.

Human culture inhibits the development of the mimetic crisis by also putting in place taboos, laws, and other forms of sacred differentiation so that the effects of mimesis are reduced, thus slowing the development of mimetic crises.

The biblical revelation (in both the Old and the New Testaments) breaks the power of this sacred violence by revealing it for what it is, the collective murder of an innocent victim. The voices of the prophets, and especially the revelation of Christ on the cross, demythologize human culture by forcing us to acknowledge our sacred sin. Because the sacred depends upon denial, the biblical revelation renders sacred murder unworkable. The Bible brings the workings of the sacred to an end. This is why Jesus is described in the Gospel of John as “the lamb of God who takes away the sin [singular] of the world.” (John 1:29)

The loss of the sacrificial mechanism would result in our self-destruction, if some alternate form of functioning were not provided. Fortunately, the Gospel also gives us new means to avoid mimetic rivalry, supplanting the old taboo systems, by calling us to imitate Christ. When we imitate Christ (“Set your minds on things that are above” Col 3:2), we are possessed by a desire for the well-being of our neighbors, in place of the old desire to have what the neighbor wants. This process of acquiring new desires transforms humanity and leads to a new and better non-sacrificial culture.

The Difficulty of the Sacred
One of the things that are hard to get used to with this theory is the idea that the “sacred” is a bad thing. It’s not as bad as the mimetic crisis, but it is nevertheless fundamentally bloody and violent. Violence seems to inhere in the sacred object like an electrical charge. Whoever draws too near runs the risk of inciting the crowd to attack.

Girard argues that without religion human beings could not exist. The greatest threat to our existence has never been starvation or predation, but our own violence. The origin and purpose of religion is to save us from this threat.

A Clear And Simple Scientific Hypothesis For The Origin Of Culture And Religion
The advantage of Girard’s theory is that it gives us a clear and simple scientific hypothesis for the origin of culture and religion. With this as an analytical tool, we can unpack theological problems in fresh ways, when they have to do with culture and violence. Most of the really difficult theological problems can now be taken apart in a few quick steps, like an encoded message that becomes easily readable once the key for the code is discovered.

Science and Religion
According to René Girard, the sacred is inseparable from the practice of sacrifice. In fact, the word “sacrifice” literally means to “make sacred.” This is “sacrifice” in the ancient sense, meaning taking someone (a person or an animal) and ritually killing them. The sacred comes into being with the spilling of reconciling blood.

For example, belief in witches is typical of the workings of the sacred in society. In virtually every primitive culture in the world, there is a belief in witches. Whenever things seem to be going wrong, when resentments build between people, and sickness seems to be everywhere, the primitive culture will posit that a witch is at work causing problems. The community sets to work identifying the witch. When they identify someone (usually whoever has the fewest friends in the community) in such a way that everyone believes the accusation, they put the witch to death. Upon the killing of the witch, the buildup of hostility in the community is discharged, and things seem to return to normal. It seems obvious that therefore the witch indeed was the cause of the problem. This in turn reinforces the belief in witches. This scenario could never function without a fundamental misrecognition of the situation. The “plague” that the witch supposedly caused was really a mimetic crisis. The witch was only a scapegoat, blamed and punished to help the community regain its harmony.

This cycle of crisis, execution, and renewal tends to reinforce the superstitious belief in witches, because experience seems to show that it works. People feel “deep down” that it is obviously true; that the world is filled with magical powers and that witches are a grave danger to society.

Jesus as Witch
The biblical narrative deconstructs these superstitions by presenting the familiar story of the witch from the point of view of the “witch.” Jesus occupies the same cultural location as the witch; but the narrative reveals that it is the crowd that is guilty, rather than the innocent – (and forgiving!) victim. As a result of this revelation, humanity begins (dimly at first) to realize that the founding “Truth” of culture is in fact a lie.

The historical and cultural project known as modernity, building on the influence of the gospel, is designed to demolish superstitious worldviews. Modernity begins with the assumption that what is purely cultural or purely a matter of what people feel “deep down” is not sufficiently trustworthy. Modernity applies principles of truth that it considers beyond culture, i.e., what one can observe in nature or what is consistent with the principles of logical reasoning.

Modern science is the result of the discovery that there is a difference between “culture” and “cosmos.” All archaic or “primal” cultures assume that the natural world is an extension or expression of their culture. They make no distinction between “culture” and “nature.” Animistic religions believe that every rock, tree and stream has its own “spirit” with its own will and power, and that this spirit must be treated with respect, even awe. The belief in spirits comes from cultural and religious experiences. These concepts are projected onto the natural world, so that the primal culture considers them intrinsic to nature. This is a confusion between culture and nature.

The reason that primal cultures have this confusion relates to what might be termed the “mythological imperative”: the sacrifice of the victim must be remembered (for its reconciling benefits), but it must also be forgotten (so that speaking directly of collective murder doesn’t generate violence). The description of the victim’s death is forgotten, but the spiritual power of the sacrifice is remembered, because the victim is said still to be present in the rocks, trees, or streams.

Or the stars. Many cultures, especially agrarian ones, put a lot of effort into the contemplation of the stars. This is useful because observation of the movements of celestial bodies is the best means of timing the changing of the seasons. The timing of the seasons is important especially in agricultural societies as the means of assuring a good harvest; an early thaw is less likely to tempt you to plant too early, if you know how to watch for the spring equinox.

It seemed as though the stars controlled the seasons. Did they control other things as well? The product of the sacrificial altar came to be projected onto the stars. The planets and constellations were said to contain the spirits of sacred beings — gods, monsters, and the hero-priests who killed them. These figures in the sky came to be seen as guiding life in society. The culture was written in the sky by people who believed that somehow the sky was writing itself into their culture. Thus did astrology — that entertaining but pathetic superstition — come into being. This confusion of culture with cosmos is common to all archaic cultures (and to a large extent it is found in Western modernity, even Western science, as well).

The biblical revelation is the force in human history that has made humanity aware that there is a difference between cosmos and culture. It has brought about this change by revealing that the sacrificial victim is not the cause of the society’s problems. Jesus, the crucified victim of the crowd, is revealed to be innocent. It is the crowd that is guilty. As the sacrificial myth is thereby demolished, the other myths and superstitions of the culture begin to follow one by one. We realize that we can’t trust ourselves to be right about what causes the rain to fall or how the stars influence our lives. So we begin to explore ways to know things apart from the influence of culture. Science, the effort to insulate our inquiry from cultural influence, is born. The rest of modernity emerges at the same time. Modernity challenges and tests our cultural assumptions about our world. Culture is regarded with a considerable amount of suspicion. Culture and cosmos begin to separate in our thinking. As René Girard has said, “We didn’t stop burning witches because we invented science; we invented science because we stopped burning witches.”

Biblical Revelation as the Source of Science
The apparent conflict between science and faith is the result of our discovery that culture and nature are not necessarily the same. Such an endeavor as modern science would be unthinkable without the insight that our culture may be a source of falsehood. This is precisely the insight that the biblical revelation brought into the world. Without the Bible, Western science would never have been possible.

Although science is the product of biblical faith, science in turn contributes to biblical faith, by accelerating the process of demythologization. Science acts as a powerful solvent to wash away the sacred superstitions that still cling to biblical religion. Science has put an end to our belief in the power of witches’ magic, for example. This is a good thing, because it removes one of the falsehoods that distract us from the message of mercy in the Bible.

Science has confirmed the biblical insight that illness is not necessarily a punishment from God, but a condition that has nothing to do with our moral standing. By helping to lift the moral stigma of disease, science has helped us to be more faithful to the revelation of Christ who calls us to be merciful toward those who are sick.

The scientific worldview made possible the “historical-critical” reading of the Bible, which in turn has liberated our reading of scripture from all sorts of violent superstition.

But science must be careful not to be arrogant in this. The insight that culture can be wrong is a tremendous advance. It has led us to find ways to explore the truth in things that are not influenced by cultural biases and superstitions. We know that an experiment well-constructed can lead us to solid insight. But we must be careful not to conclude therefore that religion is never to be trusted. The rituals, moral standards and narratives of religion contain real wisdom that has controlled human violence for millennia.

God As A Mimetic Force
Scientists should not assume that because the term “god” cannot be separated from its cultural fabric, then the notion of a god is purely false. Mimetic theory suggests that indeed gods are very real, along with demons, spirits, and souls. But mimetic theory would describe them as mimetic forces, rather than metaphysical or supernatural beings. Science should be working on ways to describe gods scientifically, rather than dismissing the notion of a god as superstitious.

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Michael Behe and Stephen Barr Debate

November 19, 2010
 
 
 

Steven Barr (on the left) and Michael Behe

 

 In the video/audio at the following links, you can watch Catholic scientists Michael Behe and Stephen Barr debate whether Intelligent Design should be taught in science classes. Barr is a physicist, and you will recognize his writings in First Things and such books as Modern Physics and Ancient Faith. Behe is a biochemist and a leading voice in ID, writing such books as Darwin’s Black Box and The Edge of Evolution.

 Audio/Video links here:  http://www.isi.org/lectures/lectures.aspx?SBy=lecture&SFor=18fdfd28-e682-421f-9acf-2940402af8e3

 The video is 72 minutes but captivating.

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The Sacred and the Human by Roger Scruton

September 30, 2010

Religion is not primarily about God, but about the sacred, and that the experience of the sacred can be suppressed, ignored and even desecrated but never destroyed.

This essay first appeared in Prospect Magazine, August 2007 and has some of the thoughts included in Professor Scruton’s 2010 Gifford Lectures. An overview of the role of religion and a meditation on Girardian theory, the essay gives us much to reflect on, particularly for those who might wish to dismiss Religion as if it were some disproven theory.

A System Of Unfounded Beliefs?
It is understandable that decent, skeptical people, observing the widespread revival in our time of superstitious cults, the emerging conflict between secular freedoms and religious edicts, and the murderous insanity of radical Islam, should be receptive to the anti-religious polemics of Dawkins, Dennett and Hitchens. The ‘sleep of reason’ has brought forth monsters, just as Goya foretold in his wonderful engraving. How are we to rectify this, except through a wake-up call to reason, of the kind that the evangelical atheists are now shouting from their pulpits?

Goya's The Sleep of Reason

Nor is it surprising that decent, skeptical people should regard last-ditch attempts to retain the belief in God’s temporal concern for us (such as the theory of ‘intelligent design’), as testifying merely to the miraculous ability to believe in the miraculous. Either we leave the field to science, or we take refuge in the inexplicable – which is no refuge from science. For the skeptical observer of the human scene, there is nothing that religion can add to scientific explanation other than the invocation of a transcendental causa sui which, by its very nature, eludes human comprehension.

Somewhat more surprising is the extent to which religion is caricatured by its current opponents, who seem to see in it nothing more than a system of unfounded beliefs about the cosmos – beliefs which, to the extent that they conflict with the scientific worldview, are heading straight for refutation. Thus Christopher Hitchens, in his relentlessly one-sided diatribe, writes as follows:

One must state it plainly. Religion comes from the period of human prehistory where nobody – not even the mighty Democritus who concluded that all matter was made of atoms – had the smallest idea what was going on. It comes from the bawling and fearful infancy of our species, and is a babyish attempt to meet our inescapable demand for knowledge (as well as comfort, reassurance, and other infantile needs). (God Is Not Great, p. 64)

Now Hitchens is an intelligent and widely read man, who recognizes that the arguments that are most useful to him were already well-known two hundred years ago. His book takes us through territory already charted by Hume, Voltaire, Diderot and Kant, and nobody who is familiar with the Enlightenment can really believe that anything has been added to its stance against religion by our contemporary imitators, whatever new examples they can add to the list of religiously-motivated crimes.

The Origin of Religion
However, the thinkers of the Enlightenment, having argued the claims of faith to be without rational foundation, did not then dismiss religion, as one might dismiss a refuted theory. Many of them went on to draw the conclusion that religion must therefore have some other origin than the pursuit of scientific knowledge, and some other psychic function than providing a world-view that consoles those who subscribe to it. The ease with which the common doctrines of religion could be refuted alerted thinkers like Jacobi, Schiller and Schelling to the thought that religion is not, in its essence, a matter of doctrine, but of something else. And they set out to discover what that something else might be.

Thus was born the anthropology of religion. For thinkers in the immediate aftermath of the Enlightenment it was not faith but faiths, in the plural, that composed the primary subject-matter of theology. Hence the appearance of books with titles like Origine de tous les cultes; ou, Religion universelle (C.F. Dupuis, 1795), and hence the busy decipherment of oriental religions by the Bengal Asiatic Society, whose proceedings began to appear in Calcutta in 1788. For post-Enlightenment thinkers the monotheistic belief-systems were not related to the ancient myths and rituals as science to superstition, or logic to magic. They were crystallizations of the emotional need which found equal expression in the myths and rituals of antiquity and in the Vedas and Upanishads of the Hindus.

Myth And The Human Psyche
This thought led Georg Creuzer, whose Symbolik und Mythologie der alten Völker appeared between 1810 and 1812, to represent myth as a distinctive operation of the human psyche. A myth does not describe what happened in some obscure period before human reckoning, but what happens always and repeatedly. It does not explain the causal origins of our world but rehearses its permanent spiritual significance. Myth is a way of understanding deep processes of the human psyche, which cannot be easily described except through imaginative stories.

If you look at ancient religion in that way then inevitably your vision of the Judaeo-Christian canon will change. The Genesis story of the creation is easily refuted as an account of historical events: how can there be days without a sun, man without a woman, life without death? Read it as a myth, however, which recounts the concealed and repeatable meaning of events that we live through every day, and this naïve-seeming text reveals itself as a profound study of the human condition.

The story of the fall is, Hegel wrote, ‘not just a contingent history but the eternal and necessary history of humanity’. (Lectures on the Philosophy of Religion, 1827.) It conveys truths about freedom, about guilt, about man, woman and their relationship; about our relation to nature and mortality. A skeptical and scientific mind, such as that of Leon Kass in his The Beginning of Wisdom: The Story of Genesis, can use this text to explore moral and psychological truths that are nowhere else so vividly or succinctly evoked.

Hermeneutics
Not surprisingly, therefore, among the first effects of the Enlightenment were new ways of reading scripture, and new attempts to square the scriptures with the demands of the rational intellect. Thus was born the science, or at any rate the art, of hermeneutics, whose first conscious proponent, Friedrich Schleiermacher, saw religion as a distinctive activity of the rational soul, rooted in feeling rather than intellect.

For Schleiermacher’s contemporary Hegel, the Biblical stories had to be cleared of their merely imagistic nature, and construed as ventures of the spirit, on the path to self-knowledge. Religion, as he put it, is spirit that realizes itself in consciousness – it is the spirit coming to know itself, through the successive forms of human worship. The religions of mankind (which Hegel, spurred on by Creuzer, avidly collected and brilliantly analyzed) represent ‘determinations’ of the abstract idea of divinity. But we approach this idea through the path of alienation, and among religious concepts we should include not only those of God, creation, and design, but also those of guilt, unhappiness, atonement and reconciliation – features of the human condition which lead us to see the world from a position outside it, and to search it for the places and times in which freedom can enter the otherwise incomprehensible flow of events.

For Hegel, myths and rituals are forms of self-discovery, through which we understand the place of the subject in a world of objects, and the inner freedom which conditions all that we do. The emergence of monotheism from the polytheistic religions of antiquity is not so much an observation of the world as a form of self-knowledge, through which the spirit learns to recognize itself in the whole of things, and to overcome its finitude.

That idealist approach inspired Feuerbach to give a materialist rejoinder. Gods, angels, devils and the rest are, he argued, human creations, projections of the moral life, whose doings in the stratosphere reflect the moral tensions which animate the world below. The only reality here is the ‘species being’ of humanity, which creates these figments out of the raw material of human need. And the downside of religion is that it encourages our complacency, enabling us to place our virtues at an impassable distance from ourselves, by projecting them into that higher, and illusory, realm from which they can never thereafter be recuperated.

Nietzsche and Wagner
Between those early ventures into the anthropology of religion and the studies of Sir James Frazer, Emile Durkheim and the Freudians, two thinkers stand out as setting the agenda for a new intellectual enterprise – an enterprise which seems not to have been noticed by Hitchens, Dawkins or Dennett, but which is nevertheless of some importance to us today. The thinkers to whom I refer are Nietzsche and Wagner, and the intellectual enterprise is that of showing the place of the sacred in human life, and the kind of knowledge and understanding that comes to us, through the encounter with sacred things.

Nietzsche in The Birth of Tragedy, and Wagner in Tristan, The Ring and Parsifal as well as in his writings on tragedy and religion, painted a picture that, while rooted in the post-Enlightenment tradition and owing much to Feuerbach, placed the concept of the sacred at the centre of the anthropology of religion. The lesson that both thinkers took from the Greeks was that you could subtract the gods and their stories from Greek religion, and still the most important thing would remain. This thing had its primary reality not in myths but in rituals, in moments that stand outside time, in which the deep loneliness and anxiety of the human individual is confronted and overcome.

By calling these moments ‘sacred’ we recognize both their complex social meaning and also the respite that they offer from alienation. Forget theology, forget doctrine and belief, forget all the ideas about an after-life – for none of these have the importance in Homer or in tragedy that attaches to the moment of ritual sacrifice, when the human world is suddenly irradiated from a point beyond it.

Nietzsche’s Birth of Tragedy was dismissed as an unscholarly fantasy, and cost him his career as a philologist. Wagner’s artistic proof of his own insights remained accessible only to those with ears. While most anthropologists recognized that religion belongs to another category of thought than science, and that it should not be dismissed (as Hitchens dismisses it) merely as a residue of animal fears and childish yearnings, the attempt to understand the concept of the sacred remained where Nietzsche and Wagner had left it.

Girardian Theory
It was not anthropologists but theologians and critics who took the matter forward – Rudolf Otto in Das Heilige, 1917, Georges Bataille in L’Érotisme, 1952, Mircea Eliade in The Sacred and the Profane, 1957, and, most explicitly and also shockingly, René Girard in La violence et le sacré, 1972. It is Girard’s theory, it seems to me, that most urgently needs to be debated, now that atheist triumphalism is sweeping all nuances away.

Girard begins from an observation that no impartial reader of the Hebrew Bible or the Koran can fail to make, which is that religion may promise peace, but is also deeply implicated in violence. The God presented in those writings is frequently angry, given to insane fits of destruction and seldom deserving of the epithets bestowed upon him in the Koran – al-raHmân al-raHîm, ‘the compassionate, the merciful’. He makes outrageous and bloodthirsty demands – such as the demand that Abraham sacrifice his son Isaac. He is obsessed with the genitals and adamant that they should be mutilated in his honor – a theme that has been interestingly explored by Jack Miles in his riveting book God: A Biography.

Thinkers like Dawkins and Hitchens leap at once to the conclusion that religion is the cause of this violence and sexual obsession, and that all the crimes committed in the name of religion can be seen as the definitive disproof of it. Not so, argues Girard. Religion is not the cause of violence but the solution to it. The violence itself comes from another source, and there is no society without it since it is engendered by the very attempt of human beings to live together. The same can be said, too, of the obsession with sexuality: religion is not the cause of this, but an attempt to resolve it.

Girard’s theory is best understood as a kind of inversion of an idea of Nietzsche’s. In his later writings, Nietzsche expounded a kind of creation myth, by way of accounting for the structure of modern society. On the Genealogy of Morals (1887) envisages a primeval human society, reduced to near universal slavery by the ‘beasts of prey’ – namely the strong, self-affirming, healthy egoists who impose their desires on others by the force of their nature. The master race maintains its position by punishing all deviation on the part of the slaves – just as we punish a disobedient horse.

The slave, too timid and demoralized to rebel, receives this punishment as a retribution. Because he cannot exact revenge the slave expends his resentment on himself, coming to think of his condition as in some way deserved, a just recompense for his inner transgressions. Thus is born the sense of guilt and the idea of sin. The ressentiment of the slave explains, for Nietzsche, the entire theological and moral vision of Christianity. Christianity owes its power to the resentment upon which it feeds: resentment which, because it cannot express itself in violence, remains turned against itself. Thus arises the ethic of compassion, the mortification of the flesh, and the life-denying routines of the ‘slave morality’. Christianity is a form of self-directed violence, which conceals a deep resentment against every form of human mastery.

That ‘genealogy’ of Christian morals was effectively exploded by Max Scheler, in his book Ressentiment: as Scheler argues, the Christian ethic of agape and forgiveness is not an expression of resentment but rather the only known way of overcoming it. Nevertheless there is surely an important truth concealed within Nietzsche’s wild generalizations. Resentment is a fundamental component in our social emotions; it is widely prevalent in modern societies; and there is surely no way in which we might explain either the durability of egalitarian politics or such local phenomena as Islamist violence, if we do not see resentment as a major part of the cause.

We may suppose religion to make an input into social violence. But it is surely evident to any observer of the 20th century that you can take away religion, and the violence will usually remain. And the 20th century is the century of resentment. How else do you explain the mass murders of the communists and the Nazis, the seething animosities of Lenin and Hitler, the genocides of Mao and Pol Pot? The ideas and emotions behind the totalitarian movements of the 20th century are targeted: they identify a class of enemy, whose privileges and property have been unjustly acquired and at the expense of their victims. And this class must be destroyed. Religion plays no real part in the ensuing destruction, and indeed is usually included among the targets.

Girard’s theory, like Nietzsche’s, is expressed as a genealogy, or rather a ‘creation myth’: a fanciful description of the origins of human society, from which to derive an account of its present structure. (It is significant that Girard came to the anthropology of religion from his work as a literary critic, a student of Shakespeare, Cervantes and Stendhal.) And like Nietzsche Girard sees the primeval condition of society as one of conflict. It is in the effort to resolve this conflict that the experience of the sacred is born. This experience comes to us in many forms – in religious ritual, in prayer, in tragedy – but its true origin is in an act of communal violence.

Primitive societies are invaded by ‘mimetic desire’, as rivals struggle to match each other’s social and material acquisitions, so heightening antagonism and precipitating the cycle of revenge. The solution is to identify a victim, one marked by fate as ‘outside’ the community and therefore not entitled to vengeance against it, who can be the target of the accumulated blood-lust, and who can bring the chain of retribution to an end. Scapegoating is society’s way of recreating ‘difference’ and so restoring itself. By uniting against the scapegoat people are released from their rivalries and reconciled. Through his death the scapegoat purges society of its accumulated violence. His resulting sanctity is the long-term echo of the awe, relief and visceral re-attachment to the community that was experienced at his death.

According to Girard, therefore, the need for sacrificial scapegoating is deeply implanted in the human psyche, arising from the very attempt to form a durable community in which the moral life can be successfully pursued. One purpose of the theatre is to provide fictional substitutes for the original crime, and so to obtain the benefit of moral renewal without the horrific cost. In Girard’s view, we should see a tragedy like Sophocles’ Oedipus Tyrannus as a re-telling of what was originally a ritual sacrifice, in which the victim is chosen so as to focus and confine the need for violence. Through incest, kingship, or worldly hubris the victim marks himself out as the outsider, the one who is not with us, and whom we can therefore sacrifice without renewing the cycle of revenge. The victim is thus both sacrificed and sacred, the source of the city’s plagues and their cure.

In many of the Old Testament stories we see the ancient Israelites wrestling with this sacrificial urge. The stories of Cain and Abel, of Abraham and Isaac, of Sodom and Gomorrah, are residues of extended conflicts, by which ritual was diverted from the human victim, and attached first to animal sacrifices, and finally to sacred words. By this process a viable morality emerged from competition and conflict, and from the visceral rivalries of sexual predation. Religion is not the source of violence but the solution to it – the overcoming of mimetic desire and the transcending of the resentments and jealousies into which human communities are tempted by their competitive dynamic.

And it is in just this way, Girard argues, that we should see the achievement of the Christian religion. In his study of the scapegoat (Le Bouc émissaire, 1982) Girard identifies Christ as a new kind of victim – one who offers himself for sacrifice, and who, in doing so, shows that he understands what is going on. The words ‘Father, forgive them, for they know not what they do’ are pivotal for Girard. They involve both a recognition of the necessity for sacrifice, if the guilt and resentment of community is to be appeased and transcended, and the added recognition that this function must be concealed. Only those who are ignorant of the source of their hatred can be healed by its expression, for only they can proceed with a clear conscience towards the tragic climax.

The climax, however, is not the death of the scapegoat, but the experience of sacred awe, as the victim, at the moment of death, forgives his tormentors. This is the moment of transcendence, in which even the most cruel of persecutors can learn both to humble himself and to renounce his vengeful passion. Through his willing acceptance of his sacrificial role Christ made the ‘love of neighbor’ – which had featured from the oldest books of the Hebrew bible as the standard to which humanity should aspire – into a reality in the hearts of those who rehearse or meditate upon his gesture. Christ’s submission purified religion of the need for sacrificial murder: his conscious self-sacrifice is therefore, Girard suggests, rightly thought of as a redemption, and we should not be surprised if, when we turn away from our Christian legacy as the Nazis and the Communists did, the hecatombs of victims reappear.

Girard’s account of the Passion is amplified by many learned asides, by a vigorous and ongoing engagement with Freud and Lévi-Strauss, and by a conviction that religion and tragedy are (as Nietzsche argued) adjacent in the human psyche, comparable receptacles for the experience of sacred awe. The experience of the sacred is not an irrational residue of primitive fears, nor is it a form of superstition that will one day be chased away by science. According to Girard, it is a solution to the accumulated aggression which lies in the heart of human communities. That is how he explains the peace and celebration that attends the ritual of communion – the sense of renewal which must always itself be renewed.

Girard’s Vision Of The Eucharist
Girard’s vision of the Eucharist is anticipated in Parsifal, and in particular in the sublimely tranquil Good Friday music of Act III. It is anticipated too by Hegel, who writes that ‘in the sacraments reconciliation is brought into feeling, into the here and now of present and sensible consciousness; and all the manifold actions are embraced under the aspect of sacrifice.’ (Lectures on the Philosophy of Religion.) Girard, like Hegel, takes himself to be describing deep features of the human condition, which can be observed as well in the mystery cults of antiquity and the local shrines of Hinduism as in the everyday ‘miracle’ of the Eucharist.

Girard Theory Criticisms
There are many criticisms that might be leveled against Girard’s theory – not least, against the idea that human institutions can be explained through genealogies and creation myths. The alleged ‘mimetic’ nature of human competition is underdescribed and underjustified; there are other and more plausible explanations of the ancient ritual of animal sacrifice than that offered by Girard; and the success of the Christian ethic has many other causes besides the mystical reversal that allegedly occurred on the Cross. But those criticisms do not, it seems to me, account for the comparative neglect of Girard’s ideas. Girard’s thesis has been received with the same dismissive indifference as Nietzsche’s in The Birth of Tragedy, and though he has been honored with a siège at the Académie Française, the honor has come only now, as Girard approaches his ninetieth year.

I suspect that, like Nietzsche, Girard has reminded us of truths that we would rather forget – in particular, the truth, which is anathema to the evangelical atheists, that religion is not primarily about God, but about the sacred, and that the experience of the sacred can be suppressed, ignored and even desecrated (which is the routine tribute paid to it in modern societies) but never destroyed. Always the need for it will arise, for it is in the nature of rational beings like us to live at the edge of things, experiencing our alienation and longing for the sudden reversal that will once again join us to the centre. For Girard that sudden reversal is a kind of self-forgiveness, as the concealed aggressions of our social life are abruptly transcended – washed in the blood of the lamb.

Girard’s genealogy casts an anthropological light on the Christian ethic and on the meaning of the Eucharist; but it is not just an anthropological theory. Girard himself treats it as a piece of theology. For him the theory is a kind of proof of the Christian religion and of the divinity of Jesus. And in a striking article in the Stanford Italian Review (1986) he suggests that the path that has led him from the inner meaning of the Eucharist to the truth of Christianity was one followed by Wagner in Parsifal, and one along which even Nietzsche reluctantly strayed, under the influence of Wagner’s masterpiece.

Of course, you don’t have to follow Girard into those obscure and controversial regions in order to endorse his view of the sacred as a human universal. Nor do you have to accept the cosmology of monotheism in order to understand why it is that this experience should attach itself to the three great transitions – the three rites of passage – which mark the cyclical continuity of human societies. Birth, copulation and death are the moments when time stands still, when we look on the world from a point at its edge, when we experience our dependence and contingency, and when we are apt to be filled with an entirely reasonable awe. It is from such moments, replete with emotional knowledge, that religion begins, and the rational person is not the one who scoffs at all religions, but the one who tries to discover which of them, if any, can make sense of those things, and, while doing so, draw the poison of resentment.

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Selections From Dismissing God by Donald D. Hoffman

September 29, 2010

Donald D. Hoffman is a Professor in the Department of Cognitive Science, University of California Irvine, California. If you are like me, you may be a little tired of the Steven Pinker neuroscientists and their broad claims at having discovered the Soul or God in the human brain. Dr. Hoffman makes a clear case here as to what neuroscience knows and doesn’t know.

DEBATES BETWEEN THEISTS AND ATHEISTS often hinge, naturally enough, on advances in cognitive neuroscience and evolutionary biology. Here I contend that such advances, though relevant to the debate, cannot license deductively valid arguments for or against theism. I contend further that the central role of probability in evolutionary theory grants no inductive strength to arguments for or against theism. The Kolmogorov axioms of probability and the mathematical definition of a stochastic process suitably model mutation and selection; using this fact to conclude for or against theism requires, in either case, a leap of faith.

Neuroscience and God
In 1961 the Soviet cosmonaut Yuri Gagarin, while making history as the first person to orbit the earth, also became the first person to discuss theology from space with the famous comment, “I don’t see any god up here.” Nothing was settled, of course, by this observation, and substantive debate between science and religion continues to this day. But Gagarin’s comment raises a wider question. As science advances it probes, with an increasingly powerful array of tools, all aspects of nature from the submicroscopic to the cosmological. As new vistas of nature open to the advances of science, it appears each time that the scientists exploring the new vistas can say, with Gagarin, “I don’t see any god up here.” The unexplored gaps in nature where God might be hiding are rapidly vanishing. Will God suffer the same extinction as species whose habitat vanishes?  

Or has neuroscience already dealt the extinction blow? Normal activity of the human cerebral cortex can be altered by transcranial magnetic stimulation (TMS), a magnetic field that can be applied directly and noninvasively outside of the human skull. Michael Persinger of Laurentian University in Canada has found that appropriate application of TMS to the temporal lobes of the brain will cause many people to experience the presence of God. Andrew Newberg of the University of Pennsylvania in Philadelphia, using single photon emission computed tomography (SPECT) to image the brain activity of nuns and monks who meditate to experience oneness with God, found that when the meditation reached its goal, certain regions in the parietal lobe decreased their activity. Is the experience of God simply an artifact of brain activity?  

And didn’t evolutionary theory deal the extinction blow long ago? God is no longer needed to explain the origin of species. Chance operating with natural selection seems to do just fine. I will not try here to argue for or against the existence of God. I will simply observe that the three dismissals of God just scouted, despite their psychological appeal, do not survive a sober understanding of the scope and limits of science, the nature of human perception, and the modern theory of chance.

The Nature Of Human Perception
I begin with the nature of human perception, and in particular human visual perception, which will illuminate the scope and limits of science. Most of us think pretheoretically that human vision operates much like a camera. There is an objective physical world out there that exists independently of whether we perceive it or not, and our eyes, like a camera, faithfully record this world. In part this is true. Our eyes do focus an image, as does a camera, and the retinas of our eyes record this image, as does the film or CCD (charge-coupled device) chip of a camera.

But our eyes are just the first stage of visual processing. Behind the eyes the optic nerves transport filtered versions of the retinal images to the brain’s cortex. And here there is a big surprise: Roughly half of the brain’s cortex is engaged in vision. About 50 billion neurons, and tens of trillions of synapses, are engaged each time you simply open your eyes and look around. This is far more computational power than is necessary to simply record an image. What is going on?

Research in the cognitive and neural sciences has made clear that our visual systems are not simply passive recorders of objective reality, but instead are active constructors of the visual realities we perceive. Each of us has within us a reality engine, which takes the images at the eyes and constructs three-dimensional worlds of objects, colors, textures, motions, and depth. What we see with each glance is not the world as it is objectively and as it would be even if there were no observers. Instead what we see is entirely our own construction. Our process of construction proceeds so rapidly and confidently that we are misled by our own prowess into thinking that we are not constructing at all, but simply reporting what is there independent of us. In short, our belief that we see the world as it objectively is, unadorned, is an illusion made possible by the very brilliance and efficiency of our reality creating process.

Reality As We Construct It According To Our Rules
What we see at any moment is the best theory our visual system can come up with to explain the images at the eyes. The visual system is much like a scientist, in creating theoretical explanations for the evidence at hand. The big difference is that the theory building process of the scientist is usually conscious, while the theory-building process of our visual systems is for the most part conducted without our conscious awareness. The visual system does not just create its theories at random, but instead is guided by many rules of visual construction, rules that are the subject of much current investigation by vision researchers. Rules have been uncovered for our constructions of color, depth, motion, objects, shapes, and edges. A visual example of our constructive processes at work is the “subjective Necker cube” first devised in 1977 by psychologists Bradley and Petry:

 

Perhaps you see a cube floating in front of black disks when you view this figure. If you look for a while you might notice that the cube flips, and that a corner of the cube that was in front suddenly is behind, and vice versa. So you actually construct two different cubes floating in front of the black disks. You might feel that you see the edges of the cubes quite clearly, even where they pass between the black disks. But if you cover up the black disks with your hands, you’ll see that there is no edge between the disks. You construct the edge you see, just like you construct the two cubes.

But you can do even more. Imagine that the black disks are holes in a sheet of paper, and that you are looking through the holes, and behind the paper you see a cube. Notice that now you see the cube not floating in front of the black disks, but sitting behind them. And the edges of the cube, that look ghostly when the cube floats in front, now look solid when the cube is behind. If you keep looking at the cube behind the holes, you’ll again see that it can flip, so that you can actually see two different cubes behind. In total, then, you construct four different three-dimensional cubes from this flat drawing, and you construct illusory edges which you make to be either ghostly or solid. That is a lot of construction, and just a hint of what your visual system is doing all the time. Space here does not permit going into more examples, but I have placed some interactive visual demonstrations online, where you can explore for yourself how you create color, motion, and objects.

The demonstrations are at this URL:  http://www.cogsci.uci.edu/~ddhoff/Applets/index.html 

What is true for vision is true for all of our senses, including touch, smell, taste, and hearing. In each of these senses, what we perceive is not reality unadorned, but reality as we construct it according to our own internal set of rules. We are adept creators of all the sensory realities we experience. 

Insight Into The Nature Of Objective Reality
Philosophers studying perception distinguish two senses of perceiving: the phenomenal and relational. The phenomenal sense of perceiving refers to our visual experience, the way that things seem to us. If I am dreaming about an elephant, the elephant I am experiencing in the dream is being perceived in the phenomenal sense. The relational sense of perceiving refers to the objective reality that we interact with in an act of perception. For me to perceive something in the relational sense, that thing must exist independent of whether I perceive it or not.

Now clearly none of my sensory experiences exist independent of whether I perceive them or not. Therefore objects in the relational sense are not in my sensory experience, but must be inferred from my sensory experience. The situation, then, is that the world we experience as our perceptual reality is in fact an elaborate construction on our part. It is something we perceive in the phenomenal sense, not the relational sense. And what we construct is critically dependent on the rules we employ in the reality creation process. Realities that are not licensed by our rules are realities that we are not equipped to experience. What can we say then about an objective reality that does not depend on our sensory experiences for its existence? Do our sensory experiences give us secure grounds to make inferences about this reality, about the objects we might be perceiving in the relational sense?

One might be tempted to say this is so based on an evolutionary argument: Creatures whose perceptions in the phenomenal sense were too divergent from reality in the relational sense were at a competitive disadvantage, and natural selection has made sure that those of us who have survived have a good match between our phenomenal perceptions and the relational reality.

But this is not a valid argument within the structure of evolutionary theory. What natural selection secures, according to this theory, is survival to reproduction, not perceptual truth. Roaches, like humans, are the result of natural selection. But we have little confidence that roaches have deep insights into objective reality. They don’t need such insights in order to survive just fine. The same may be true of us. We have cognitive and perceptual apparatuses that allow us to survive long enough to reproduce, but we have no guarantees on evolutionary grounds that these apparatuses give us deep insight into the nature of objective reality.

Phenomenal Worlds And The Relational Realm
Indeed it is highly unlikely that objective reality resembles in any way the worlds of our phenomenal construction. It would be luck beyond belief to find that the human species, of the millions of species on earth, happens to be the one whose phenomenal worlds resemble the relational realm. It is a certain anthropocentrism that would lead us to assert otherwise, the same anthropocentrism that led us to assert that the earth is the center of the universe, about which all else revolves. What may be unique to humans as a species is a perceptual and cognitive apparatus which, for the first time in evolutionary history, can rise above the assumption, whether tacit or explicit, that our perceptions in some way resemble objective reality.

What view does this give of the scientific enterprise? Science walks on two legs: observation and logic. The success of science has been its care in arranging detailed observations, and its care in the logical interpretation of the results of these observations. But what the study of perception has uncovered is that, no matter how careful our observations are, we will always be limited to observing only what our internal rules of construction allow us to perceive. Even if we extend our senses with telescopes, microscopes, and various high-tech devices, we can never step outside our senses and see reality unadorned. We cannot get perceptual data that is independent of our own rules of perceptual construction.

The very rules that enable us to see also blind us to the infinity of other possibilities that do not conform to our rules. Evolution is not done yet. There is no reason to believe that we have arrived at the set of rules of construction that give deep insight into the nature of objective reality. There is every reason to believe that we are simply another species, like spiders and termites, that has developed an idiosyncratic perceptual system to fit the idiosyncrasies of the niches we happen to inhabit. This is, of course, no denigration of science. Science may be the best our species can do given the limits of its perceptual and cognitive endowments.

What this does make clear is that the ability of science to understand objective reality is limited by the perceptual and cognitive endowments of our species. Those endowments have not evolved, according to neo-Darwinian evolutionary theory, to give us truth, just to give to us, as also to the roach, survival to reproduction. We can point to the many successes of science to suggest that our species might be special, that our perceptual apparatus might just give us true insights into the nature of reality independent of our perceptions. But we can also point to these same successes to tell the opposite story.

The Matter And Energy We Can Perceive Is A Mere 4% Of The Total
One stunning success of science is the discovery of dark energy and dark matter, which together constitute something like 96% of the energy and matter in the universe. The matter and energy we can perceive is a mere 4% of the total, the light frosting on the cake. We have no current way to discover any properties of this dark matter and energy. We can only postulate its existence because without it the behavior of the 4% we can see and measure would not make sense. So our best science tells us that there are serious limits to how deeply our perceptual and cognitive endowments allow us to penetrate the nature of objective reality. The same message appears repeated many times elsewhere in science, for instance in the uncertainty principle and the measurement problem of quantum theory.So the story outlined above, in which science is systematically uncovering all the secrets of nature, and leaving less and less room for God to hide, is not only immodest, but a complete misunderstanding of the scientific enterprise.

Unprovable Truths
Science is a species specific enterprise, which proceeds under the restrictions of the cognitive and perceptual endowments of one species among millions on earth. The most striking results of this enterprise appear to inform that species of some of its own limitations. These results crop up not only in science but also in mathematics, where we have discovered hard limits to our methods of proof: there are unprovable truths.

If science isn’t eliminating places where God might hide, hasn’t it at least made God unnecessary, replacing the creative role once assigned to God with the creative power of chance? This is a common assumption, but one that fails to understand the modern theory of chance. This theory is modeled by a series of axioms, among them the Kolmogorov axioms of probability theory, and various axioms for stochastic systems. In the case of probability theory, for instance, these axioms define the properties of a probability measure: It must be an additive function on events whose maximum sum is 1, and so on. Any process in nature that can be modeled by these axioms is taken to be a  probabilistic process, a work of chance. But this leaves completely open the interpretation of these axioms. Subjectivists claim that the indeterminacies modeled in probability theory simply reflect our own epistemological limitations; objectivists claim that the indeterminacies are not merely in our heads, but in the objective world itself. Neither interpretation precludes a God orchestrating the probabilistic process. All the mathematics can do is describe the essential properties of such a process, regardless of its origin, and without constraining the nature of its origin.

Indifferent Neural Facts
If modern theories of chance do not preclude the possible agency of God behind random processes, surely at least the recent brain imaging and TMS studies show that God is simply a figment of our brains, not to be taken seriously. But this conclusion is by no means dictated by the neural facts. Every one of our perceptions, not just our perceptions of God, can be correlated with neural activity. Surely it is a mistake to take none of our perceptions seriously. To do so would lead to quick and certain death. We must be careful, then, in sorting through which perceptions to take seriously and which not. And the neural facts don’t a priori tell us which way to treat God. If there were no God, and God was simply a figment of our imagination, then we might expect to find the neural correlates of God perception that we do. On the other hand, if there were a God, and God wanted us to perceive God, then one might equally expect to find the neural correlates of God perception that we do. The neural facts are indifferent to the conclusion we should draw here.

This indifference of facts holds more generally. There is no evidence from the sciences or elsewhere that logically compels belief or disbelief in God. It is elementary in the philosophy of science that no matter how much data one collects, there will always be infinitely many theories compatible with that data, and that make contradictory predictions about the outcomes of new experiments. It is because the theories of science are not logically dictated (although surely influenced) by the facts that scientific theory building is such an interesting and nontrivial enterprise. The atheist, then, can marshal an array of evidence that there is no God, and the theist that there is. In neither case can the evidence logically prove the claim. Both choices are, equally, a step of faith.

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Memo to Hawking: There’s Still Room for God By Roger Scruton

September 28, 2010

Roger Scruton

Mr. Scruton, a philosopher, a visiting scholar at the American Enterprise Institute in Washington, is a writer, philosopher and public commentator. He has specialized in aesthetics with particular attention to music and architecture. He engages in contemporary political and cultural debates from the standpoint of a conservative thinker and is well known as a powerful polemicist.  He has written widely in the press on political and cultural issues. This is a short article that was recently featured in the Wall Street Journal.

Neither Kant nor Einstein thought physics explained everything. I have another scientist who will weigh in on that thought for tomorrow’s post. In the meantime, Roger Scruton’s wonderful reply to Stephen Hawking…

How did the universe begin? Some think the question has no answer — that it lies beyond the limits of human reason. Others think the question has an answer, but that the answer depends not on reason but on faith.

What almost no one believes is that there is a single, rational scientific theory that tells us how the universe emerged from the primeval nothingness. How could there be such a thing?

When Isaac Newton proposed his laws of gravity, he did so in a spirit of awe and reverence before the simplicity and beauty of the physical world. He did not doubt that so perfect a design implied a yet more perfect designer.

Immanuel Kant, who believed that Newton’s laws of gravity are not merely true but necessarily true, argued that we humans lack the ability to comprehend the universe as a whole, and thus that we can never construct a valid argument for a designer. Our thinking can take us from one point to another along the chain of events. But it cannot take us to a point outside the chain, from which we can pose the question of an original cause.

Indeed the question of how the universe began does not make sense. The concept of cause applies to the objects of experience, linking past to future through universal laws. When we ask about the universe as a whole we are attempting to go beyond possible experience into a realm where the concept of cause has no purchase, and where the writ of reason does not run.

All physicists since Kant have been influenced by this argument. Some admit the point, like Albert Einstein, Others, like Stephen Hawking, express the point in a language of their own.

But Mr. Hawking now wishes to break with this consensus and to argue that science actually does have an answer to the question of origins. We can know how the universe was created, he suggests, since the laws of physics imply that there are limiting conditions, in which universes come into being by the operation of those very laws. There is no room for the creator, since there is no need for Him. The laws of physics do it all by themselves.

Mr. Hawking, of course, dazzles us with his scientific discoveries. Einstein broke with the common-sense view of the world when he decided to treat time as a fourth dimension, on a par with the three dimensions of space. Mr. Hawking gives us dimension upon dimension, assuming that because every continuum can be squeezed into the axioms of a geometry there is no limit to the number of dimensions in which we humans find ourselves suspended. Nor is there a limit to the number of universes, even though we happen to inhabit only one of them and the others may be forever inaccessible to us.

The laws of physics are fast ceasing to be laws of the universe and are becoming laws of a “multiverse” instead. By the time people absorb all of these shifts, they have little strength left to dissent from the view that “the laws of gravity and quantum theory allow universes to appear spontaneously from nothing” or to question Mr. Hawking’s conclusion that therefore there is no need for God.

But what exactly has changed? Have we really moved on from the position that Kant presented? Have we really lifted ourselves outside of everything and everywhere, and achieved the view from nowhere that tells us how things began?

If Mr. Hawking is right, the answer to the question “What created the universe?” is “The laws of physics.” But what created the laws of physics? How is it that these strange and powerful laws, and these laws alone, apply to the world?

There are those who will say that the question has no answer —that it lies at or beyond the limits of human thought. And there are those who will say that the question has an answer, but that it is answered not by reason but by faith.

I say that perhaps, in the end, they are the same position. That is what Kant believed. You find out the limits of scientific understanding, he said. And beyond those limits lies the realm of morality, commitment and trust.

Kant, who destroyed all the systems of metaphysics and dug a grave for theology, was also a believer, who, as he put it, “attacked the claims of reason in order to make room for those of faith.” It seems to me that he was right.

Posted in Science And Religion | Tagged , , | 2 Comments »

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Reading Selections from “Fearful Symmetries” by Stephen M. Barr

September 17, 2010

Dr. Steven M. Barr

Science seeks the elegant, elusive simplicity of the universe itself but suffers from the propensity of many of its adherents to flatten and trivialize the world by reductionism. In yesterday’s post C.S. Lewis wrote:

Christianity claims to give an account of facts — to tell you what the real universe is like. Its account of the universe may be true, or it may not, and once the question is really before you, then your natural inquisitiveness must make you want to know the answer. If Christianity is untrue, then no honest man will want to believe it, however helpful it might be: if it is true, every honest man will want to believe it, even if it gives him no help at all.

I wonder which facts would affect the atheist more. I think of the atheists who deny the historicity of the New Testament or retreat to the narrowness of the scientific method, for whom the inability to pull God from a test tube is prima facie evidence of his lack of existence. I think Steven Barr’s latest writings might come close.  Dr. Stephen M. Barr is professor of physics at the University of Delaware and author of Modern Physics and Ancient Faith and A Student’s Guide to Natural Science. Reading selections from Fearful Symmetries here:

Reductionism
Since the time of Newton, science has advanced by a strategy rightly called “reductionism.” This method, which explains things by analyzing them into smaller and simpler parts, has yielded a rich harvest of discoveries about the natural world. As a means of analysis, then, reductionism has certainly proven its value. But many wonder whether science is reductive in a more radical and disturbing way — by flattening, collapsing, and trivializing the world. For all its intellectual accomplishments, does science end up taking our sense of reality down several notches? One could well get that impression from perusing the writings of certain scientists. Francis Crick famously asserted that human life is “no more than the behavior of . . . nerve cells and their associated molecules.” Marvin Minsky, a pioneer in the field of artificial intelligence, once described people as “machines made of meat.” Neuroscientist Giulio Giorelli announced that “we have a soul, but it is made up of many tiny robots.” And biologist Charles Zuker has concluded that “in essence, we are nothing but a big fly.”

A Metaphysical Tendency Accompanying Reductionism
This tendency to downgrade and diminish reflects a metaphysical prejudice that equates explanatory reduction with a grim slide down the ladder of being. Powerful explanatory schemes reveal things to be simpler than they appear. What simpler means in science is much discussed among philosophers—it is not at all a simple question. But to many materialists it seems to mean lower, cruder, and more trivial. By this way of thinking, the further we push toward a more basic understanding of things, the more we are im-mersed in meaningless, brutish bits of matter.

The philosopher Georges Rey has written, for example, that “any ultimate explanation of mental phenomena will have to be in non-mental [i.e., sub-mental or material] terms or else it won’t be an explanation of it.” Of course, the logic of this could be turned around. One could just as well say that any ultimate explanation of the material world must be in nonmaterial terms. But for materialists the lower explains the higher; and lower does not just mean more fundamental but instead suggests a diminished ontological status. The presumption is that explanations move from evolved complexity to primitive stuff.

At first glance, the history of the cosmos seems to bear this out. Early on, the universe was filled with nearly featureless gas and dust, which eventually condensed to form galaxies, stars, and planets. In stars and supernovas, the simplest elements, hydrogen and helium, fused to make heavier ones, gradually building up the whole periodic table. In some primordial soup, or slime, or ooze on the early earth, atoms agglomerated into larger and more intricate molecules until self-replicating ones appeared and life began. From one-celled organisms, ever more complicated living things evolved, until sensation and thought appeared. In cosmic evolution the arrow apparently moves from chaos to order, formlessness to form, triviality to complexity, and matter to mind.

Dennett Theory Of Religion
And that is why, according to philosopher Daniel Dennett, religion has it exactly upside down. Believers think that God reached down to bring order and create, whereas in reality the world was built — or rather built itself — from the ground up. In Dennett’s metaphor, the world was constructed not by “skyhooks” reaching down from the heavens but by “cranes” supported by, and reaching up from, the solid ground.

The history to which the atheist points — of matter self-organizing and physical structures growing in complexity — is correct as far as it goes, but it is only part of the story. The lessons the atheist draws are naive. Yes, the world we experience is the result of processes that move upward. But Dennett and others overlook the hidden forces and principles that govern those processes. In short, they are not true reductionists because they don’t go all the way down to the most basic explanations of reality.

As we turn to the fundamental principles of physics, we discover that order does not really emerge from chaos, as we might naively assume; it always emerges from greater and more impressive order already present at a deeper level. It turns out that things are not more coarse or crude or unformed as one goes down into the foundations of the physical world but more subtle, sophisticated, and intricate the deeper one goes.

An Example
Let’s start with a simple but instructive example of how order can appear to emerge spontaneously from mere chaos through the operation of natural forces. Imagine a large number of identical marbles rolling around randomly in a shoe box. If the box is tilted, all the marbles will roll down into a corner and arrange themselves into what is called the “hexagonal closest packing” pattern. (This is the same pattern one sees in oranges stacked on a fruit stand or in cells in a beehive.) This orderly structure emerges as the result of blind physical forces and mathematical laws. There is no hand arranging it. Physics requires the marbles to lower their gravitational potential energy as much as possible by squeezing down into the corner, which leads to the geometry of hexagonal packing.

At this point it seems as though order has indeed sprung from mere chaos. To see why this is wrong, however, consider a genuinely chaotic situation: a typical teenager’s bedroom. Imagine a huge jack tilting the bedroom so that everything in it slides into a corner. The result would not be an orderly pattern but instead a jumbled heap of lamps, furniture, books, clothing, and what have you.

Why the difference? Part of the answer is that, unlike the objects in the bedroom, the marbles in the box all have the same size and shape. But there’s more to it. Put a number of spoons of the same size and shape into a box and tilt it, and the result will be a jumbled heap. Marbles differ from spoons because their shape is spherical. When spoons tumble into a corner, they end up pointing every which way, but marbles don’t point every which way, because no matter which way a sphere is turned it looks exactly the same.

These two crucial features of the marbles—having the same shape and having a spherical shape—should be understood as principles of order that are already present in the supposedly chaotic situation before the box was tilted. In fact, the more we reduce to deeper explanations, the higher we go. This is because, in a sense that can be made mathematically precise, the preexisting order inherent in the marbles is greater than the order that emerges after the marbles arrange themselves. This requires some expla-nation.

Both the preexisting order and the order that emerges involve symmetry, a concept of central importance in modern physics, as we’ll see. Mathematicians and physicists have a peculiar way of thinking about symmetry: A symmetry is something that is done. For example, if one rotates a square by 90 degrees, it looks the same, so rotating by 90 degrees is said to be a symme-try of the square. So is rotating by 180 degrees, 270 degrees, or a full 360 degrees. A square thus has exactly four symmetries.

Not surprisingly, the hexagonal pattern the marbles form has six symmetries (rotating by any multiple of 60 degrees: 60, 120, 180, 240, 300, and 360 degrees). A sphere, on the other hand, has an infinite number of symmetries—doubly infinite, in fact, since rotating a sphere by any angle about any axis leaves it looking the same. And, what’s more, the symmetries of a sphere include all the symmetries of a hexagon.

If we think this way about symmetry, careful analysis shows that, when marbles arrange themselves into the hexagonal pattern, just six of the infinite number of symmetries in the shape of the marbles are ex-pressed or manifested in their final arrangement. The rest of the symmetries are said, in the jargon of physics, to be spontaneously broken. So, in the simple example of marbles in a tilted box, we can see that symmetry isn’t popping out of nowhere. It is being distilled out of a greater symmetry already present within the spherical shape of the marbles.

Spontaneous Symmetry
The idea of spontaneous symmetry breaking is important in fundamental physics. The equations of electromagnetism have a mathematical structure that is dictated by a set of so-called gauge symmetries, discovered by the mathematician and physicist Hermann Weyl almost a century ago. For a long time it seemed that two other basic forces of nature, the weak force and the strong force, were not based on symmetries. But about forty years ago it was found that the weak force is actually based on an even larger set of gauge symmetries than those of electromagnetism.

Because the symmetries of the weak force are spontaneously broken, however, they do not manifest or express themselves in an obvious way, which is why it took so long to discover them. (The strong force is based on a yet larger set of gauge symmetries, but this fact was obscured by a quite different effect and also was not discovered for a long time.)

This history illustrates a general trend in modern physics: The more deeply it has probed the structure of matter, the greater the mathematical order it has found. The order we see in nature does not come from chaos; it is distilled out of a more fundamental order.

Symmetry is just one kind of order. In the case of the marbles in the box, other principles of order were also at work, such as the principle that caused the marbles to seek out the configuration of lowest energy. This is an aspect of a beautiful mathematical principle, called the principle of least action that underlies all of classical physics. When physicists investigated the subatomic realm, however, they discovered that the principle of least action is just a limiting case of the much more subtle and sophisticated path integral principle, which is the basis of quantum mechanics, as Richard Feynman showed in the 1940s. The lesson is the same: The deeper one looks, the more remarkable the mathematical structure one sees.

An Underlying Order
The mathematical order underlying physical phenomena is most easily observed in the motions of the heavenly bodies. Even primitive societies were aware of it, and it inspired not only feelings of religious awe (many expressions of which are found in the Bible itself) but also the earliest attempts at mathematical science. And when scientists began to study the solar system with more precision, they discovered unsuspected patterns even more beautiful than those known to the ancients.

Four hundred years ago, for example, Johannes Kepler discovered three marvelous geometrical laws that describe planetary motion. So impressed was he by the beauty of these laws that he wrote this prayer in his treatise Harmonices Mundi (The harmonies of the world): “I thank thee, Lord God our Creator, that thou hast allowed me to see the beauty in thy work of creation.” Decades later, Newton succeeded in explaining Kepler’s laws — but he did not explain them down, if by down we mean reducing what we observe and experience to something more trivial or brutish.

On the contrary, he explained them by deriving them from an underlying order that is more general and impressive, which we now call Newton’s laws of mechanics and gravity. Newton’s law of gravity was later explained, in turn, by Einstein, who showed that it followed from a more profound theory of gravity called general relativity. And it is now generally believed that Einstein’s theory is but the manifestation of a yet more fundamental theory, which many suspect to be superstring theory. Superstring theory has a mathematical structure so sophisticated that, after a quarter of a century of study by hundreds of the world’s most brilliant physicists and mathematicians, it is still not fully understood.

It is true that science seeks to simplify our picture of the world. An explanation should in some sense be simpler than the thing it explains. And, indeed, there is a sense in which Einstein’s theory of gravity is simpler than Newton’s, and Newton’s theory of planetary motion simpler than Kepler’s.

As physics Nobel laureate Frank Wilczek notes, however, Einstein’s theory is “not ‘simple’ in the usual sense of the word.” Whereas Kepler’s laws can be explained in a few minutes to a junior-high-school student, Newton’s laws cannot be fully explained without using calculus. And to explain Einstein’s theory requires four-dimensional, curved, non-Euclidean space-time and much else besides. And yet, once we know enough, Einstein’s theory does have a compelling simplicity greater than Newton’s theory. The simplicity to which scientific reductionism leads us, then, is of a very paradoxical kind. It is a simplicity that is by no means simpleminded. It is not at all jejune, but deeply interesting and intellectually rich.

Paradoxical Reductionism In Chess
The same paradox can be found in many fields. The chess world champion Capablanca was admired for the purity and simplicity of his style. But to understand his moves one must have an understanding of the game that can be acquired only by years of experience and study. A later world champion, Mikhail Botvinnik, wrote of him, “In this simplicity there was a unique beauty of genuine depth.” Another world champion, Emanuel Lasker, observed that “[in Capablanca’s games] there is nothing hidden, artificial, or labored. Although they are transparent, they are never banal and are often deep.” Wilczek had just the right term for this kind of simplicity, which is also found in the fundamental laws of physics: profound simplicity.

Profound Simplicity
Profound simplicity always impresses with its elegance, economy of means, harmony, and perfection. This perfection, as Wilczek notes, is such that one feels that the slightest alteration would be disastrous. He quotes Salieri’s envious description of Mozart’s music in the film Amadeus: “Displace one note and there would be diminishment. Displace one phrase and the structure would fall.” Applying this to physics, Wilczek says, “A theory begins to be perfect if any change makes it worse. . . . A theory becomes perfectly perfect if it’s impossible to change it without ruining it entirely.”

Symmetry is one of the factors that contribute to profound simplicity, both in the laws of physics and in works of art. Paint over one petal of the rose window of a cathedral, remove one column from a colonnade, and the symmetry is destroyed. Each part is necessary for the completion of the pattern.

The symmetries that characterize the deepest laws of physics are mathematically richer and stranger than the ones we encounter in everyday life. The gauge symmetries of the strong and weak forces, for example, involve rotations in abstract mathematical spaces with complex dimensions. In other words, the coordinates in those pecu-liar spaces are not ordinary numbers, as they are for the space in which we live, but complex numbers, which are numbers that contain the square root of minus one. Grand unified theories—which combine the strong, weak, and electromagnetic forces into a single mathe-matical structure—posit symmetries that involve rotations in abstract spaces of five or more complex dimensions.

Supersymmetries
Stranger and more profoundly simple are supersymmetries. There is much reason to think that supersymmetries are built into the laws of physics, and finding evidence of that is one of the main goals of the Large Hadron Collider outside Geneva, Switzerland, which has recently begun to take data. Supersymmetries involve so-called Grassmann numbers, which are utterly different from the ordinary numbers we use to count and measure things. Whereas ordinary numbers (and even complex numbers) have the common-sense property that a × b = b × a, Grassmann numbers have the bizarre property that a × b = -b × a. A simple enough formula, but hard indeed for the human mind to fathom.

Esoteric symmetries also lie at the heart of Einstein’s theory of relativity. These Lorentz symmetries involve rotations not just in three-dimensional space but in four-dimensional space-time. We can all visualize the symmetries of a sphere or a hexagonal pattern, but Lorentz symmetries, supersymmetries, and the gauge symmetries of the weak, strong, and grand unified forces lie far outside our experience and intuition. They can be grasped only with the tools of advanced mathematics.

Physicists have found beauty in the mathematical principles animating the physical world, from Kepler, who praised God for the elegant geometry of the planets’ orbits, to Hermann Weyl, for whom mathematical physics revealed a “flawless harmony that is in conformity with sublime Reason.”

Some might suspect that this beauty is in the eye of the beholder, or that scientists think their own theories beautiful simply out of vanity. But there is a remarkable fact that suggests otherwise. Again and again throughout history, what started as pure mathematics — ideas developed solely for the sake of their intrinsic interest and elegance — turned out later to be needed to express fundamental laws of physics.

Starting As Pure Mathematics Later Needed For  Fundamental Laws Of Physics
For example, complex numbers were invented and the theory of them deeply investigated by the early nine-teenth century, a mathematical development that seemed to have no relevance to physical reality. Only in the 1920s was it discovered that complex numbers were needed to write the equations of quantum mechanics. Or, in another instance, when the mathematician William Rowan Hamilton invented quaternions in the mid-nineteenth century, they were regarded as an ingenious but totally useless construct. Hamilton himself held this view.

When asked by an aristocratic lady whether quaternions were useful for anything, Hamilton joked, “Aye, madam, quaternions are very useful — for solving problems involving quaternions.” And yet, many decades later, quaternions were put to use to describe properties of subatomic particles such as the spin of electrons as well as the relation between neutrons and protons. Or again, Riemannian geometry was developed long before it was found to be needed for Einstein’s theory of gravity. And a branch of mathematics called the theory of Lie groups was developed before it was found to describe the gauge symmetries of the fundamental forces.

Indeed, mathematical beauty has become a guiding principle in the search for better theories in fundamental physics. Werner Heisenberg wrote, “In exact science, no less than in the arts, beauty is the most important source of illumina-tion and clarity.” Paul Dirac, one of the giants of twentieth-century physics, went so far as to say that it was more important to have “beauty in one’s equations” than to have them fit the experimental data.

At the roots of the physical world, therefore, one does not find mere inchoate slime or dust but instead a richness and perfection of form based on profound, subtle, and beautiful mathematical ideas. This is what the famous astrophysicist Sir James Jeans meant when he said many decades ago that “the universe begins to look more like a great thought than a great machine.” Benedict XVI expressed the same basic insight when in his Regensburg lecture he referred to “the mathematical structure of matter, its intrinsic rationality,…the Platonic element in the modern understanding of nature.”

A Greater And More Impressive Order
Modern science does not directly imply or require any particular metaphysical theory of reality, but it does suggest to us that the picture presented by Daniel Dennett and Richard Dawkins is false because the picture is only partial. In the terms of Dennett’s meta-phor of cranes constructing complexity, one sees what is built from the ground up; but delving beneath the surface, one finds an astonishing, hidden world — the underground mechanisms of the cranes, as it were.

It is true that the cosmos was at one point a swirling mass of gas and dust out of which has come the extraordinary complexity of life as we experience it. Yet, at every moment in this process of development, a greater and more impressive order operates within — an order that did not develop but was there from the beginning. In the upper world, mind, thought, and ideas make their appearance as fruit on the topmost branches of an evolutionary tree. Below the surface, we see the taproots of reality, the fundamental laws of physics that shimmer with ideas of profound simplicity.

To describe people as machines made of meat is as scientifically unsophisticated as to think of the sun as a heat-emitting machine made of swirl-ing gas. It ignores the reasons why the machines function as they do—reasons that the explanations of modern physics reduce to simplici-ties as elegant as they are elusive. Peering into the hidden depths, we see that matter itself is the expression of “a great thought,” of ideas that are, as Weyl said, “in conformity with sublime Reason.” And we begin to discover that matter, although mindless itself, is the product of a Mind of infinite profundity and infinite simplicity.

Posted in Aesthetics, Science And Religion | Tagged , , , , | 2 Comments »

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