Plato at the Googleplex Read online

  I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems that we prove, and which we describe grandiloquently as our ‘creations,’ are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.48

  But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.49

  I view the mathematical world as having an existence of its own, independent of us. It is timeless. I think, to be a working mathematician, it’s difficult to hold any other view. It’s not so much that the Platonic world has its own existence, but that the physical world accords with such precision, subtlety, and sophistication with aspects of the Platonic mathematical world. And this, of course, does go back to Plato, who was clear in distinguishing between notions of precise mathematics and the usually inexact ways in which one applies this mathematics to the physical world. It is the shadow of the pure mathematical world that you see in the physical world. This idea is central to the way we do science. Science is always exploring the way the world works in relation to certain proposed models, and these models are mathematical constructions.… And it’s not just precision. The mathematics one uses has a kind of life of its own.50

  As these three examples indicate, “Platonism” often expresses itself in the assertion that abstract truths are out there, waiting to be discovered, just as scientific truths are out there, waiting to be discovered. A Platonist asserts that the abstract is as real as the concrete, the general as realized as the particular. Perhaps the assertion of reality is clarified by contrasting it with the alternatives, what the Platonist is asserting mathematics is not. Mathematics is not about our own mental ideas, not about the structure of our cognitive equipment, not about our own implicative fictions. We don’t do mathematics by introspecting. And mathematics is not about axiomatic systems that have been constructed by stipulating a set of formal recursive rules, a kind of higher-order chess. Our systems are tools for discovery, not for creation. As Gottlob Frege, the mathematician who established modern symbolic logic, put it in his own classic statement of Platonism: “The mathematician can no more create anything than the geographer can; he, too can only discover what is there and give it a name.”51

  Platonism reifies the abstract—but there is reification and there is reification. Talk of the “world” of Platonic entities suggests a picture of some sort of separate place, sometimes lampooned as “Plato’s heaven.” Here in the perfection of eternity, beyond the reach of the corrosive tides of time, such things as numbers and non-numerical abstract universals shine forth. They are to be glimpsed not by the crude organs of the body but by the far more refined—and inequitably distributed—faculties of mind. Such are the eternal exemplars that “virtuous logicians” may hope to meet in the “hereafter,” in the derisory words of Bertrand Russell, describing the view of the “unadulterated Platonist,” Kurt Gödel.52 “Plato’s heaven” may be invoked—or mocked—as the place in which all concepts, not just those having to do with mathematics, reside. Such talk of a world of abstract things, parallel to our sensed world of concrete things, a kind of space beyond space, is one way of presenting Platonism, though it isn’t the only way, and to my mind it doesn’t do justice to the subtlety of contemporary Platonist views.

  Nor, to my mind, does it do justice to the subtlety of Plato’s Platonism—that is, his reification of the abstract—which kept evolving throughout his long philosophical life. Perhaps Plato once did have something like the view that Russell mocks; he himself subjects some such view to his barrage of criticism in the Parmenides, propounding such difficulties as convinced at least one of his students, Aristotle, to give up on Platonism altogether and start all over again on the problem of abstract universals. But the ways in which Plato continues to reify the abstract don’t fit this lampooned picture. Yes, he continues to assert, in such works as the Timaeus, that the intelligible forms can’t be reduced to the “stuff” of the spatiotemporal world, to the world of appearance that we sense. But the abstract doesn’t transcend the spatiotemporal world of stuff either; it can neither be reduced to it nor exist in isolation from it. Abstraction—most especially mathematical abstraction—is the permanence within the flux, the very permanence that provides the explanation for the flux, that provides the right form for rendering the intelligibility of nature that the Greek thinkers had been chasing ever since the protoscientists of the Ionian Enchantment intuited that there was intelligibility out there. But the out there of the rationally apprehended is immanent within the out there of the empirically given. It inheres in the structural features of the given, and these features are captured in mathematics. This is the far subtler view that Plato suggests clearly enough so that such thinkers as Galileo can, millennia later, pick up the thread again.

  So, at least under some interpretation, Plato appears to have held firm throughout his life to the “reification of the abstract.” Evidence for this comes not only from the dialogues but from the Academy he established. To his Academy, Plato gathered all the best mathematicians of his day and put them to work on what the eminent philosopher Myles Burnyeat has called his “research program,” which was to discover the mathematical structures immanent in nature. Plato’s assertion of the reality of mathematical structures found its practical realization in the study of plane and solid geometry, of astronomy, harmonics, and optics—all of which were pursued in his Academy. His search for mathematical proportions and “harmonies” even lent itself to medical theories, premised on the supposition that health is a matter of the correct mathematical proportions between the “opposing” constituents of the body, which in those early days were thought of in terms of the hot and the cold, the moist and the dry.

  Was Plato a Platonist? The question sounds as dopey as asking who’s buried in Grant’s tomb. But the non-dopey answer is “It depends on what you mean by Platonism.” Some version of maintaining the primacy of the abstract, including, most essentially, the abstraction that finds expression in mathematics, seems to be a view we can pin on Plato.53 It’s a commitment that seems to have persisted relentlessly, if restlessly, throughout his philosophical life. In that sense, we can, with some relief, affirm that Plato was a Platonist. But no matter what his precise attitude toward the issue of the existence of the abstract, there’s no question that it was he who raised the issue, and that, according to Aristotle, it was a topic of fierce debate within the Academy—and it is an issue that remains with us still, robustly philosophical and scientifically unresolvable. Do mathematicians discover mathematics, construct mathematics, introspect mathematics, imagine mathematics? Science makes use of mathematics, but it doesn’t tell us what mathematics is.

  Another doctrine (although closely connected to this one) to which Plato seems to have held firm through all the philosophical twists and turns with which he presents us is the intertwining of truth, beauty, and goodness. Call it the Sublime Braid: truth, beauty, and goodness are all bound up with one another, sublimely. This assertion appears, at first blush, like the worst kind of metaphysics, like a positivist’s parody of metaphysics. Truth! Beauty! Goodness! Together again! (Well, actually since forever.) And the metaphysics doesn’t end here. Entailed in the Sublime Braid are other doctrinal strands. For starters, beauty and goodness are as objective as truth itself is. “Beauty—be not caused—It Is—,” said the poet. Yes, Emily, Plato agrees. Beauty is. And because beauty is, the world is the way it is. If the world really is shot through with intelligibility, as the Ionians first supposed, then this intelligibility is itself beautiful, and the more intelligible it is, the more beautiful it is; and the m
ore beautiful it is, the more intelligible it is. Mathematics provides, in itself, the most perfect intelligibility. When we understand a mathematical truth, we understand that it will always be so: no changes of perspectives or of contexts will render it untrue.54 This invulnerability to perspectival distortions makes it unqualifiedly what is, and thus unqualifiedly knowable or intelligible (Republic 477a). So mathematics, being maximally intelligible, is maximally beautiful. And this is why mathematics supplies the right form for explaining the world, and it is how it is that our sense of beauty becomes our most sure-footed guide on the vertiginously steep path to truth. Given two empirically adequate scientific explanations of the same phenomenon, go for the more mathematically beautiful one and you’ll go for the truth.

  Is Plato’s metaphysics sounding a little more congenial to the scientifically oriented philosophy-jeerer? After all, Copernicus, Galileo, and Kepler all appealed to Platonic doctrines—Galileo and Kepler both referring to “the divine Plato”—in order to argue the superiority of Copernican heliocentrism over Ptolemaic geocentrism. Even though the Ptolemaic view was itself a product of the mathematically oriented doctrines of the Academy, switching the point of orientation from the earth to the sun made the mathematics so much more beautiful. Being led by the beauty of the mathematics was quite an important aspect of that evolution of “natural philosophy” into science applauded by certain philosophy-jeerers.

  Plato’s intuition—of the intertwining of (mathematical) beauty and truth—is unabashedly echoed by many modern-day physicists of unassailable caliber. The Nobel laureate Paul Dirac, for example, said, “It is more important to have beauty in one’s equations than to have them fit experiment.” Einstein, too, often made similar remarks, for example telling the philosopher and physicist Hans Reichenbach that he had been convinced that his theory of relativity was true even before the 1919 solar eclipse, which delivered the first confirming evidence, because of its mathematical beauty and elegance. In our day, the sovereignty of beauty—of the mathematical variety—has often been most vociferously proclaimed by champions of string theory, which has so far been unable to produce any testable predictions. “I don’t think it’s ever happened that a theory that has the kind of mathematical appeal that string theory has has turned out to be entirely wrong,” Steven Weinberg—the third Nobel laureate quoted in this paragraph—has said. “There have been theories that turned out to be right in a different context than the context for which they were invented. But I would find it hard to believe that that much elegance and mathematical beauty would simply be wasted.”55

  Physicists have long been helping themselves to Plato’s metaphysics, without going through any of the steps he took to arrive at it, rather like people who consume hot dogs and would rather not know how they are made.

  All of this metaphysics comes spilling out of Plato’s Sublime Braid, and we haven’t even considered goodness yet. We’ll be considering goodness all through this book. It’s always Plato’s major concern, no matter whether he’s doing moral philosophy, political philosophy, epistemology, metaphysics, or cosmology. It turns out, on Plato’s view, that our sense of beauty is more reliable than our sense of goodness. It’s our sense of beauty that is enlisted to lead us to the truth, whereas our sense of goodness has to undergo a major revision in the light of the truth.

  But what does Plato mean by goodness, and how does he entwine it with truth and beauty?

  Plato’s truth-entwined goodness can best be gotten to by way of “the best reason” that he sees lurking inside truth. The truth is as it is because “the best reason” is determining it to be so.56 His language is, at first blush, suspiciously teleological, even suggestive of intentionality. Did someone—Some One—implement this best reason, designing the world accordingly? Or is it rather that the best reason works all on its own, a self-starter, with nothing external to it required to put it into action? It was the latter possibility that Plato had in mind. If there is “mind” determining the truth, an idea put forth in the Phaedo and explored in greater depth in the Timaeus, the existence of this mind amounts to nothing over and above the assertion that the truth is determined by “the best reason.” In other words, the best and final scientific theory would work all on its own to create the world in accordance with itself. In the Timaeus he presents a creation myth, in which a demiurge, or divine Craftsman, is implementing “the best reason,” but his using a myth to dramatize the point is in itself an indication that it’s a more abstract metaphysical principle he has in mind: the best reason is, in itself, a self-starter, an explanation that explains itself, a causa sui, as Spinoza—who picked up this Platonic intuition and ran all the way with it—was to put it.

  The determining role of “the best reason” in making the world what it is is what the goodness in Truth-Beauty-Goodness consists in. Goodness is interwoven with truth because the explanation for the truth is that the truth is determined by the best reason, and the best reason works all on its own—which is as good as it gets. The truth, being determined by the best reason, is ultimately capable of explaining itself. This makes reality as intelligible as it could possibly be. It’s its very intelligibility that provides the reason for its existence. For intelligibility-craving minds, what could possibly be more sublime?

  And once again, as it was with beauty so it is with goodness: it is mathematics that largely foots the bill. The best reason is the reason that is thoroughly intelligible, that presents its own justification transparently to the mind, which is what mathematics does (Republic 511d, Timaeus passim). In the creation myth of the Timaeus, the divine Craftsman imposes as much mathematics on the material world as it can possibly hold, because mathematics is the most perfect expression of the good intentions—the best reasons—by which the mythical Craftsman works (29d–e). The mythical Craftsman doesn’t make the forms he imposes on the world the best by virtue of choosing them; rather he chooses them because they are, independent of him, the best of forms, and their being the best of forms in itself explains why they must be realized.

  The talk of “the best reason,” which sounds deceptively teleological, is not teleological at all. The causality is fueled by the mathematics. The causality is at one with the intelligibility. In fact, it was the return to this version of Platonism that managed to get the teleology out of physics, by displacing Aristotle’s final causes with Plato’s mathematical conception of causality. Spinoza, who, like other seminal thinkers of the seventeenth century, was rebelling against the Aristotelian-scholastic teleology that held sway, put the point this way: “Such a doctrine (teleology) might well have sufficed to conceal the truth from the human race for all eternity, if mathematics had not furnished another standard of truth … without regard to … final causes.”57

  So there you have it: truth, beauty, and goodness, all bound up with one another, providing the ontological structure of reality. Such a confluence of truth, beauty, and goodness suggests a notion like the sublime—not identical with truth or beauty or goodness but rather with the confluence of all three. Reality is shot through with a sublimity so sublime that it simply had to exist. Existence explodes out of the sublime.

  Notice that the goodness that we’re speaking about here isn’t a specifically human goodness. The point is not that the world has been created with our good in mind. I can’t think of a single place in the corpus where Plato even floats this idea. It’s entirely foreign to his conception of the world. (It’s pretty foreign to the entire Greek conception of the world, even non-philosophically speaking. Those gods and goddesses pursue their own ends and pleasures. We mortals are, at best, incidental to their purposes.) The goodness that’s woven into the Sublime Braid has no more of the human element in it than E=mc2 does.

  But Plato also seems to suggest, all through his dialogues, in one form or another, that there is also some goodness—in the way that we humans understand goodness, as it applies specifically to people, the lives we live, the actions we perform—to be gained from knowledge of
the way the world is, the way it has to be because of the Sublime Braid that furnishes its structure. Knowledge is not only of the good, but also makes us good, reforming us so that we become more virtuous—more inclined, because of our knowledge, toward justice, temperance, courage, and reverence. Metaphysics—understanding how the world is by understanding how it must be, understanding, for example, that it must be maximally intelligible58—is ethically reforming.

  The term “goodness” is a placeholder. It needs filling in. Yes, indeed, we ought to be good; so much is trivially true. But tell us what we must be—or do—in order to be good. For Plato, it’s knowledge that does the filling in of the placeholder “good.” Knowledge is ethically active, even when it’s knowledge of the most impersonal kind, as indifferent to the world of humans as pure mathematics.

  In fact, it’s the very impersonality of impersonal knowledge that renders such knowledge the most ethically potent of all. Simply to care enough about the impersonal truth, devote one’s life to trying to know it, requires disciplining one’s rebellious nature, which is always intent on having things its own way, on seeing the world in whatever light does most justice to one’s own petty ego so that the truth-as-one-sees-it will push one’s own self-serving, power-centric agenda along. So simply to allow oneself to be overtaken by the reality of Truth-Beauty-Goodness—to become embraided oneself in the Sublime Braid—is to exert discipline over one’s unruly nature, to call a halt to its self-enhancing fantasizing.

  But that is only the beginning. Reality is of such a kind as to do us ultimate good, and that because of the principles by which it has been fashioned. As we take in the Truth–Beauty–Goodness that structures reality, its rational order is replicated within our own minds in the act of knowing it—and we are made better for this replication. We are rationalized by nature’s own rational order, our minds’ constituents reconfigured in their ideal proportions to one another, just as in health the constituents of the body are configured in their ideal proportions to each other. We become structurally isomorphic to reality itself, and in that way our natural affinity to it is strengthened. We become more like it (Timaeus 47b–c). This, too, further removes us from the smallness of our own lives, the strengthened kinship with the cosmos expanding us outward to take it in. Our reality-enhanced minds can’t help but see their own small place in the grand scheme of things and will be appropriately humbled in the process, which is what this secular kind of piety consists in (as Spinoza thought: piety is humility before reality). Knowledge of impersonal truth drives all personal thoughts from the mind [Timaeus 90a–c]). Plato would say, about a physicist avidly awaiting that call from Stockholm, or thinking only of the fame she can acquire by writing one of those scientific blockbusters, that she never was earnestly in love with the beautiful, not so that it overtook her own love of herself. Such a scientist has been fueled by intelligence but not by wisdom, which must include an overwhelming love for that which isn’t oneself. The appropriate reaction to the beauty of the Sublime Braid can only be love.