History of the Argument
The argument from beauty has two aspects. The first is connected with the independent existence of what philosophers term a 'universal', see Universal (metaphysics) and also Problem of universals. Plato argued that particular examples of, say a circle, all fall short of the perfect exemplar of a circle that exists outside the realm of the senses as an eternal Idea. Beauty for Plato is a particularly important type of universal. Perfect beauty exists only in the eternal Form of beauty, see Platonic epistemology. For Plato the argument for a timeless idea of beauty does not involve so much whether the gods exist (Plato was not a monotheist) but rather whether there is an immaterial realm independent and superior to the imperfect world of sense. Later Greek thinkers such as Plotinus (ca. 204/5–270 CE) expanded Plato’s argument to support the existence of a totally transcendent "One", containing no parts. Plotinus identified this "One" with the concept of 'Good' and the principle of 'Beauty'. Christianity adopted this Neo-Platonic conception and saw it as a strong argument for the existence of a supreme God. In the early fifth century, for example, Augustine of Hippo discusses the many beautiful things in nature and asks "Who made these beautiful changeable things, if not one who is beautiful and unchangeable?" This second aspect is what most people today understand as the argument from beauty.
A contemporary British philosopher of religion, Richard Swinburne, known for philosophical arguments about the existence of God, advocates a variation of the argument from beauty:
"God has reason to make a basically beautiful world, although also reason to leave some of the beauty or ugliness of the world within the power of creatures to determine; but he would seem to have overriding reason not to make a basically ugly world beyond the powers of creatures to improve. Hence, if there is a God there is more reason to expect a basically beautiful world than a basically ugly one. A priori, however, there is no particular reason for expecting a basically beautiful rather than a basically ugly world. In consequence, if the world is beautiful, that fact would be evidence for God's existence. For, in this case, if we let k be 'there is an orderly physical universe', e be 'there is a beautiful universe', and h be 'there is a God', P(e/h.k) will be greater than P(e/k)... Few, however, would deny that our universe (apart from its animal and human inhabitants, and aspects subject to their immediate control) has that beauty. Poets and painters and ordinary men down the centuries have long admired the beauty of the orderly procession of the heavenly bodies, the scattering of the galaxies through the heavens (in some ways random, in some ways orderly), and the rocks, sea, and wind interacting on earth, 'The spacious firmament on high, and all the blue ethereal sky', the water lapping against 'the old eternal rocks', and the plants of the jungle and of temperate climates, contrasting with the desert and the Arctic wastes. Who in his senses would deny that here is beauty in abundance? If we confine ourselves to the argument from the beauty of the inanimate and plant worlds, the argument surely works." 
Art as a Route To God
The most frequent invocation of the argument from beauty today involves the aesthetic experience one obtains from great literature, music or art. In the concert hall or museum one can easily feel carried away from the mundane. For many people this feeling of transcendence approaches the religious in intensity. It is a commonplace to regard concert halls and museums as the cathedrals of the modern age because they seem to translate beauty into meaning and transcendence.
Dostoevsky was a proponent of the transcendent nature of beauty. His enigmatic statement: "Beauty will save the world" is frequently cited. Aleksandr Solzhenitsyn in his Nobel Prize lecture reflected upon this phrase:
And so perhaps that old trinity of Truth and Good and Beauty is not just the formal outworn formula it used to seem to us during our heady, materialistic youth. If the crests of these three trees join together, as the investigators and explorers used to affirm, and if the too obvious, too straight branches of Truth and Good are crushed or amputated and cannot reach the light—yet perhaps the whimsical, unpredictable, unexpected branches of Beauty will make their way through and soar up to that very place and in this way perform the work of all three. And in that case it was not a slip of the tongue for Dostoyevsky to say that "Beauty will save the world" but a prophecy. After all, he was given the gift of seeing much, he was extraordinarily illumined. And consequently perhaps art, literature, can in actual fact help the world of today.
Another character in the Aldous Huxley novel just mentioned proved the existence of God by playing Beethoven’s string quartet no. 15 in A minor ('heiliger Dankgesang') on a gramophone. Unconvincing as that sounds, it does represent a popular strand of argument. I have given up counting the number of times I receive the more or less truculent challenge: 'How do you account for Shakespeare, then? ' (Substitute Schubert, Michelangelo, etc. to taste.) The argument will be so familiar, I needn’t document it further. But the logic behind it is never spelled out, and the more you think about it the more vacuous you realize it to be. Obviously Beethoven’s late quartets are sublime. So are Shakespeare's sonnets. They are sublime if God is there and they are sublime if he isn't. They do not prove the existence of God; they prove the existence of Beethoven and of Shakespeare. A great conductor is credited with saying: 'If you have Mozart to listen to, why would you need God? ' 
Philosophical Basis of Science and Mathematics
Exactly what role to attribute to beauty in science and mathematics is hotly contested, see 
When developing hypotheses, scientists use beauty and elegance as valuable selective criteria. The more beautiful a theory, the more likely is it to be true. Interestingly, the more rarefied the study such as higher forms of mathematics and quantum physics, the more likely is beauty found to be a reliable guide. The famous mathematical physicist Hermann Weyl said with evident amusement, "My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful."  The quantum physicist Werner Heisenberg wrote to Einstein, "You may object that by speaking of simplicity and beauty I am introducing aesthetic criteria of truth, and I frankly admit that I am strongly attracted by the simplicity and beauty of the mathematical schemes which nature presents us." 
The argument implies beauty is something immaterial instead of being a subjective neurological response to stimuli. Philosophers since Immanuel Kant increasingly argue that beauty is an artifact of individual human minds. A 'beautiful' sunset is according to this perspective aesthetically neutral in itself. It is our cognitive response that interprets it as 'beautiful.' Others would argue that this cognitive response has been developed through the evolutionary development of the brain and its exposure to particular stimuli over long ages. Others point to the existence of evil and various types of ugliness as invalidating the argument. Joseph McCabe, a freethought writer of the early 20th century, questioned the argument in The Existence of God, when he asked whether God also created parasitic microbes.
 However, he also wrote: "My conclusion is that there is no reason to believe any of the dogmas of traditional theology and, further, that there is no reason to wish that they were true. Man, in so far as he is not subject to natural forces, is free to work out his own destiny. The responsibility is his, and so is the opportunity."  H. L. Mencken stated that humans have created things of greater beauty when he wrote, "I also pass over the relatively crude contrivances of this Creator in the aesthetic field, wherein He has been far surpassed by man, as, for example, for adroitness of design, for complexity or for beauty, the sounds of an orchestra."
Notes and references
- Sermons of St. Augustine, 241, Easter: c.411 CE
- Swinburne, Richard, The Existence of God, OUP, 2nd Edition, 2004, ISBN 0199271682
- Fyodor Dostoevsky, The Idiot
- "Beauty Will Save the World: The Nobel Lecture on Literature" by Aleksandr Solzhenitsyn, 1970
- Dawkins, Richard, The God Delusion, (Mariner Books:2008), p. 110
- Dawkins, Richard, The God Delusion, (Mariner Books:2008), p. 110
- "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960). John Wiley & Sons.
- Quoted in Graham Farmelo, It Must be Beautiful: Great Equations of Modern Science (Granta Books, 2002), p. xii. Farmelo provides an extensive discussion of this topic and gives numerous examples from the history of science.
- Quoted in Ian Stewart (mathematician), Why Beauty is Truth (Basic Books, 2007), p. 278.
- quoted in Ian Stewart (mathematician), Why Beauty is Truth (Basic Books, 2007), p. 278
- quoted in Ian Stewart (mathematician), Why Beauty is Truth, (Basic Books, 2007), p. 278
- Joseph McCabe (1933), The Existence of God, p. 75
- Russell, Bertrand (1919). "The Study of Mathematics". Mysticism and Logic: And Other Essays. Longman. p. 60
- Russell, Bertrand "Is There a God?" (1952: repr. The Collected Papers of Bertrand Russell, Volume 11: Last Philosophical Testament, 1943-68, ed. John G Slater and Peter Köllner (London: Routledge, 1997), pp. 543–48
- Minority Report, H. L. Mencken's Notebooks, Knopf, 1956