I have written this essay in response to a skeptical critic of Intelligent Design, who denies that the cosmos is beautiful in any objective sense. My aim is to defend two propositions: (i) mathematical beauty is an objective reality; and (ii) the cosmos instantiates this kind of beauty, and can therefore be called objectively beautiful. I will then endeavor to show that Intelligent Design is the only hypothesis which satisfactorily explains these truths. In my essay, I shall be quoting liberally from the writings of a number of atheist mathematicians and physicists, some of whom are pictured above: Bertrand Russell, G.H. Hardy, Paul Erdos and Steven Weinberg (with acknowledgements to photographer Larry D. Moore for the image of Professor Weinberg).
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The background to this dispute
Recently, in response to a query by an Uncommon Descent contributor named rhampton7, I argued that “the fact that the laws of Nature display a deep underlying beauty and “hang together” very well (as I explained in my post, Beauty and the Multiverse), points to a single Intelligent Designer of the cosmos,” and I added: “I think that this would be a sensible default hypothesis for ID to adopt.”
An anonymous critic of Intelligent Design did not agree with what I wrote, and forthrightly asserted:
Beauty is not a “fact”. “Hang together” is meaningless. The two put together don’t point to squat. There’s nothing sensible in ID and never will be.
There are many different kinds of beauty, and some (e.g. physical attractiveness and artistic beauty) are influenced by individual likes and dislikes, as well as by cultural preferences. For these kinds of beauty, a case could be made that “Beauty is in the eye of the beholder” – a subjective evaluation rather than an objective fact.
Mathematical beauty, however, is altogether different. In this realm, we find no trace of the subjective. As I am not a mathematician or a scientist, I thought it best to make my point by quoting from the works of mathematicians and physicists. The vast majority of quotes given below are taken from scientists who were atheists or who had no religious beliefs. (The bold emphases are mine.) Nevertheless, as I shall argue in the final part of my essay, they help to establish a powerful case for an Intelligent Mind behind the natural world, which is responsible for the mathematical beauty we observe in the cosmos.
What is mathematical beauty?
The British philosopher, mathematician and outspoken atheist Bertrand Russell (1872-1970) expressed a firm belief in the reality of objective mathematical beauty in his essay, The Study of Mathematics (1902):
Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry…
Philosophers have commonly held that the laws of logic, which underlie mathematics, are laws of thought, laws regulating the operations of our minds. By this opinion the true dignity of reason is very greatly lowered: it ceases to be an investigation into the very heart and immutable essence of all things actual and possible, becoming, instead, an inquiry into something more or less human and subject to our limitations. The contemplation of what is non-human, the discovery that our minds are capable of dealing with material not created by them, above all, the realisation that beauty belongs to the outer world as to the inner, are the chief means of overcoming the terrible sense of impotence, of weakness, of exile amid hostile powers, which is too apt to result from acknowledging the all-but omnipotence of alien forces…. It is only when we thoroughly understand the entire independence of ourselves, which belongs to this world that reason finds, that we can adequately realise the profound importance of its beauty.
A 1940 essay by the British mathematician and atheist Godfrey Harold (“G. H.”) Hardy (1877 – 1947), entitled, A Mathematician’s Apology (Cambridge University Press, 1994; available in the public domain in Canada, courtesy of the University of Alberta Mathematical Science Society) defends the objective reality of mathematical beauty, which can be defined in terms of the harmony of its underlying ideas. Hardy likens the beauty of mathematics to that of a beautiful poem:
The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics….
It would be quite difficult now to find an educated man quite insensitive to the aesthetic appeal of mathematics. It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind – we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.
There are masses of chess-players in every civilized country – in Russia, almost the whole educated population; and every chess-player can recognize and appreciate a ‘beautiful’ game or problem. Yet a chess problem is simply an exercise in pure mathematics (a game not entirely, since psychology also plays a part), and everyone who calls a problem ‘beautiful’ is applauding mathematical beauty, even if it is a beauty of a comparatively lowly kind. Chess problems are the hymn-tunes of mathematics…
I might add that there is nothing in the world which pleases even famous men (and men who have used quite disparaging words about mathematics) quite so much as to discover, or rediscover, a genuine mathematical theorem. (Section 10).
The German historian and philosopher Oswald Spengler (1880-1936), who had a strong interest in mathematics and the sciences, approvingly cited an aphorism of Goethe on the beauty of mathematics in his essay, “Meaning of Numbers” (in James R. Newman, The World of Mathematics, Vol. 4, p. 2320, Simon & Schuster, 1956):
To Goethe again we owe the profound saying: “the mathematician is only complete in so far as he feels within himself the beauty of the true.”
More recently, the Hungarian mathematician Paul Erdos (1913-1996) expressed his views on the ineffable beauty of mathematics when he remarked, “Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.” (Gina Kolata, “Paul Erdos, a Math Wayfarer at Field’s Pinnacle, Dies at 83.” The New York Times, Sept. 24, 1996.)
Although he was an atheist, Erdos spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdos wanted to express particular appreciation of a proof, he would exclaim “This one’s from The Book!” (Philosophy of Mathematics by John Francis, Global Vision Publishing House, Delhi, 2008, p. 51.)
Finally, Jacob Bronowski (1908-1974), a mathematician, biologist, atheist and the writer of the 1973 BBC series , The Ascent of Man, wrote about the poetic beauty of pure mathematics in his book, Science and Human Values (Pelican, 1964):
Mathematics is in the first place a language in which we discuss those parts of the real world which can be described by numbers or by similar relations of order. But with the workaday business of translating the facts into this language there naturally goes, in those who are good at it, a pleasure in the activity itself. They find the language richer than its bare content; what is translated comes to mean less to them than the logic and the style of saying it; and from these overtones grows mathematics as a literature in its own right. Mathematics in this sense, pure mathematics, is a form of poetry, which has the same relation to the prose of practical mathematics as poetry has to prose in any other language. The element of poetry, the delight of exploring the medium for its own sake, is an essential ingredient in the creative process.
The foregoing quotations by leading mathematicians point to the conclusion that pure mathematics has an intellectual beauty of its own, which is utterly independent of human beings. The beauty of pure mathematics is indeed a “fact” – to use the terminology of my skeptical critic.
The beauty that scientists discover in Nature is a physical instantiation of the beauty that we encounter in mathematics
The French mathematician and philosopher of science, Jules Henri Poincare (1854-1912), described the beauty of science in his work, Science and method(1908), as translated by Francis Maitland (1914), and recently reprinted by Cosimo Classics, New York. Poincare, who as far as I can tell had no religious beliefs, wrote:
The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp… Intellectual beauty … is self-sufficing, and it is for it, more perhaps than for the good of humanity, that the scientist condemns himself to long and pitiful labours…
It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts; that we take delight, now in following the giant courses of the stars, now in scrutinizing with a microscope that microscopic smallness which is also a vastness, and now in seeking in geological ages the traces of a past that attracts us because of its remoteness. (Part I, Chapter 1: The Selection of Facts, pp. 22-23.)
In a similar vein, the theoretical physicist Freeman Dyson wrote an obituary of the 20-century mathematician Herman Weyl (1885-1955), in Nature, March 10, 1956, in which he relates the following conversation with the great mathematician, who was a former colleague of his at the Institute for Advanced Study:
Characteristic of Weyl was an aesthetic sense which dominated his thinking on all subjects. He once said to me, half-joking, “My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.” This remark sums up his personality perfectly. It shows his profound faith in an ultimate harmony of nature, in which laws should inevitably express themselves in a mathematically beautiful form. It also shows his recognition of human frailty, and his humour, which always stopped him short of being pompous.
(Weyl’s Spinozan philosophical views are discussed here by Peter Pesic.)
Most strikingly of all, the Nobel Prize winner, theoretical physicist and atheist Steve Weinberg recently expressed his thoughts on the importance of mathematical beauty when adjudicating between theories in physics, in an interview with the PBS series NOVA for the 2003 Emmy award-winning TV program, The Elegant Universe:
NOVA: Do you think that string theory could turn out to be just plain wrong?
Weinberg: 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. 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. And in any case I don’t see any alternative to string theory. I don’t see any other way of bringing gravity into the same general theoretical framework as all the other forces of nature. Yes, it could be entirely wrong. I don’t think it’s likely at all. I think it’s best to assume it’s not and take it very seriously and work on it.
NOVA: What is beauty to a theoretical physicist?
Weinberg: It may seem wacky that a physicist looking at a theory says, “That’s a beautiful theory,” and therefore takes it seriously as a possible theory of nature. What does beauty have to do with it? I like to make an analogy with a horse breeder who looks at a horse and says, “That’s a beautiful horse.” While he or she may be expressing a purely aesthetic emotion, I think there’s more to it than that. The horse breeder has seen lots of horses and from experience with horses knows that that’s the kind of horse that wins races.
So it’s an aesthetic sense that’s been beaten into us by centuries of interaction with nature. We’ve learned that certain kinds of theories—the kind that win races—actually succeed in accounting for natural phenomena. The kind of beauty we look for is a kind of rigidity, a sense that the theory is the way it is because if you change anything in it, it would make no sense.
String theories in particular have gotten much more rigid as time has passed, which is good. You don’t want a theory that accounts for any conceivable set of data; you want a theory that predicts that the data must be just so, because then you will have explained why the world is the way it is. That’s a kind of beauty that you also see in works of art, perhaps in a sonata of Chopin, for example. You have the sense that a note has been struck wrong even if you’ve never heard the piece before. The kind of beauty that we search for in physics really does work as a guide, and it is a large part of what attracts people to string theory. And I’m betting that they’re right.
Well, this is surprising! Not only is mathematical beauty wholly independent of us, but it appears to be “out there” in the cosmos too. What’s more, the intellectual beauty of a scientific theory is a good scientific guide to whether or not it is actually true, in the real world. In the words of Nobel Prize-winning physicist Steven Weinberg, “The kind of beauty that we search for in physics really does work as a guide.”
The kind of beauty that characterizes a good scientific theory can be described in terms of its underlying simplicity and the “harmonious order of its parts” as Poincare described it – which is what I meant when I wrote that the laws of Nature “hang together” very well. Weinberg goes even further: for him, a good scientific theory is one which hangs together so well that if even a single part is changed, the whole edifice comes crashing down. He argues that string theory is just such a theory.
How do we account for all this?
Dr. Robin Collins, who is a noted proponent of the cosmological fine-tuning argument, has argued in a lecture he gave at Stanford University entitled, Universe or Multiverse? A Theistic Perspective. that theism is the only satisfactory way of accounting for the surprising degree of beauty that we see in the underlying principles of physics:
Further, this “fine-tuning” for simplicity and elegance cannot be explained either by the universe-generator multiverse hypothesis or the metaphysical multiverse hypothesis, since there is no reason to think that intelligent life could only arise in a universe with simple, elegant underlying physical principles. Certainly a somewhat orderly macroscopic world is necessary for intelligent life, but there is no reason to think this requires a simple and elegant underlying set of physical principles.
One way of putting the argument is in terms of the “surprise principle” we invoked in the argument for the fine-tuning of the constants of intelligent life. Specifically, as applied to this case, one could argue that the fact that the phenomena and laws of physics are fine-tuned for simplicity with variety is highly surprising under the non-design hypothesis, but not highly surprising under theism. Thus, the existence of such fine-tuned laws provides significant evidence for theism over the non-design hypothesis. Another way one could explicate this argument is as follows. Atheism seems to offer no explanation for the apparent fine-tuning of the laws of nature for beauty and elegance (or simplicity with variety). Theism, on the other hand, seems to offer such a natural explanation: for example, given the classical theistic conception of God as the greatest possible being, and hence a being with a perfect aesthetic sensibility, it is not surprising that such a God would create a world of great subtlety and beauty at the fundamental level. Given the rule of inference that, everything else being equal, a natural non-ad hoc explanation of a phenomenon x is always better than no explanation at all, it follows that everything else being equal, we should prefer the theistic explanation to the claim that the elegance and beauty of the laws of nature is just a brute fact.
So in response to my critic: yes, I really do believe that the mathematical beauty of the laws of nature offers eloquent testimony to the existence of a Mind Who designed Nature. If you’ve got a better explanation, then I’d love to hear it.
My critic may retort that the evil we see in this world makes any talk of the beauty of Nature ring hollow. I have to say that my intellectual intuitions are precisely the opposite. The beauty we find in physics extends to every nook and cranny of the cosmos: wherever the laws of Nature hold, there is beauty. The evil we find in the cosmos is confined to one planet: Earth. What’s more, it’s confined to sentient beings, which make up a tiny fraction of the millions of species which inhabit our Earth. Evil is local, while beauty is universal. Which fact should we pay more attention to? I’ll let my readers decide.