Beauty speaks of our purpose and hence points to the future, while science speaks of the past. Each is an analysis that studies the pattern of the cosmos but does so in different ways. As such they are distinct. Both are valuable and are complementary but incomplete descriptions of reality.
Beauty is teleological
When we behold something that is beautiful it bears the mark of what we yearn for, but it is not itself what we yearn for, or not fully. It is like when we smell food cooking in the kitchen. We want to go in and ask, 'What's cooking?' The aroma tells us of the good food that is on the stove, but it doesn't matter how deeply we inhale we will not be satiated, only the food itself can satisfy the hunger. Similarly, the beauty of the cosmos and of the culture of man (if inspired by God) is a participation in the beauty of God, but neither is God. Perceptible beauty, therefore, draws us to itself and then beyond to God. We are always partially dissatisfied, left wanting more. Pope emeritus Benedict XVI talks of the way that beauty touches us and increases our desire for God as a benign wound. For example in his general audience of 31 August 2011, in a reference to the power of sacred art to inspire prayer he asked that, 'we may pause to contemplate the ray of beauty that strikes us to the quick, that almost “wounds” us, and that invites us to rise toward God'.
One of the qualities of beauty described by St Thomas Aquinas is integritas. This is the aspect by which it communicates the purpose of what is beheld and if that purpose is good, the object will be beautiful. The purpose derives its goodness from the part it plays in God's grand plan for the redemption of all things, and so it speaks therefore not only of its own purpose but of God and by our relation to both, of the meaning of our lives in the deepest sense.
God made all of us to respond to beauty in this way. It stimulates our desire for our personal telos - our ultimate end - and directs us to it, which is the Final Cause, God. If we reject or are blind to its message, then all is meaningless. This is the desolation of ugliness, which distorts or hides true meaning. We love beauty precisely for this reason - because it gives an authentic color and purpose to our lives that would be gray and empty otherwise, and connects us harmoniously with the environment in which we live.
Furthermore, we cannot see a purpose to an object unless we recognize that what we are looking at is an entity - something that is complete in itself. This might be an artifact, for example, that corresponds to the organizing principle that was an idea of the artist who created it; or a 'substance' with an organizing principle, a form, which is within it (such as the soul). These principles of organization are 'formal causes' that arrange the parts in relation to each other such a way that the whole is in harmony with its purpose. This is another quality of beauty described by St Thomas, and which he called due proportion. The object has due proportion when all its parts are in right relation with each other so that the whole can serve its purpose well.
The mathematics of beauty is the study of the patterns that connect all beautiful things, which we might call the science of harmony and due proportion. It begins with beautiful entities, identified by human consensus, then analyses the common patterns to characterize what we are responding to. So for example, nearly everyone recognizes the beauty of the cosmos, and we can consider what is it about the cosmos that we are all responding to? We might read of this mathematics in the works of Plato, Aristotle or Boethius or in the manuals of architects such as Vitruvius, Alberti or Palladio. These studies are as relevant today as they were when they were written. We might describe the ratio 2:1 as the formal cause of the octave for example. It is manifested in two pipes of equal diameter that produce sounds an octave apart, with one twice the length of the other. The purpose of creating such a thing is harmonious sound. We might then incorporate this ratio into the proportions of a beautiful building in such a way that fulfills its purpose well. The proximate end of both the organ and the building is good when it is in harmony with the ultimate end of all things, God, the Final Cause. As J.S.Bach once said: 'The final aim and reason of all music is nothing other than the glorification of God and the refreshment of the spirit.'!
Science needs help
Natural philosophy, what we call 'science' today, focusses on material and efficient causes. It is limited, therefore to the study of the material world (and so can never prove the existence of God!). As such, it analyses the present so as to discern what past events, or what chain of events occurring simultaneously in the present, created the data observed. Then it attempts to predict the future by extrapolating from the pattern of the past. As such, it is a different sort of analysis to the mathematics of beauty but good and useful nevertheless. It is very good at predicting what happens in simple systems in the near future. From this man has received great benefits - as demonstrated in technology and medicine. But it is limited in its ability to predict the future in complex systems and in the long term.
It's limitations lie precisely in the fact that as primarily a study of the pattern of past and present events, it ignores causes for change which can also have an impact on the future,. So the more complex the system and the longer into the future it tries to look, the greater the chance of error because it is more likely that other forces for change - final and formal causes - will have an impact.
Natural science as a method of inquiry is blind to the possibilities of human creativity, inspiration, and the way in which each of us uses our naturally endowed freedom.
Natural science is weak, for example, in any field in which free will might have a part to play. Free will is a force for change that does not conform to the pattern of physical laws precisely and so the social sciences, such as population medicine or economics, are weak when treated strictly as natural sciences.
Scientific socialism, otherwise known as Marxism, claims to predict the future based upon a scientific analysis of history, but it has failed spectacularly to do so. The response of socialists and Marxists to the non-appearance of the workers' Utopia on earth is to give history a push and to try to make it happen as they predict. Some have done this by forcing people at gunpoint to do what the enlightened Marxist thinks they ought to do, hastening resolution by revolution. Others do so through the attempt to stimulate a tension between opposites - a Hegelian dialectic - from which they believe the resolution will emerge. In today's world, we see this engineered social conflict in the language of oppressor and oppressed which and which is played out in grievance politics of race, gender, sexual orientation. Despite all the effort, nowhere have we seen anything that even approaches the happy resolution they predict, yet they press on with their efforts regardless.
This is not the only field of endeavor in which experts predict with great confidence what will happen and seem to fail spectacularly without, apparently any dent in their confidence to continue to do so, or for the regard with which much of society seems to hold them. Time after time we see predictions relating to politics, economics, cultural changes fail to materialize because experts in these fields of the study of human society ignore the true basis of human decision making. Numerous Malthusian predictions of doom for the human race which, it is claimed, have a scientific basis.
One wonders if the methods of the mathematics of beauty, which begin with human perception, might complement natural science here and offer help. It has occurred to me, for example, that the insight of the observation of 'spontaneous order' in the pattern of human activity by the Austrian Economist, Frederick von Hayak is an intuitive lateral jump on his part which takes him away from the approach of natural science to one that is similar to that of the science of harmony and proportion. He reached his conclusion by recognizing free will as a force for change. His is a more synthetic, as opposed to analytical, style of thinking which looks at the pattern of the whole, and as such is more in line with the approach of the ancients.
Furthermore, as the analyses of natural scientists dig ever deeper down into the chain of events that cause motion, at some point they are going to hit the First Cause that cranks the engine of the universe, which will, in all liklihood fail to conform to the order which describes efficient causes. One wonders if we didn't hit this point early in the 20th century with quantum mechanics. Quantum mechanics presents natural scientists with a philosophical conundrum which it has still not, to my knowledge, really been settled satisfactorily. The difficulties are described in the book published in 2005, A Different Universe: Reinventing Physics from the Bottom Down by Robert B. Laughlin, a Stamford professor and winner of the Nobel Prize in physics. In this book, he is describing how the scientific method seems to be inadequate in accounting for what is observed and is in effect (although as a scientist he doesn't say it directly) arguing for a new approach that requires philosophical insight normally lacking in natural scientists.
Science, in effect, needs its own new Hayak (or perhaps Aristotle or Aquinas) who is capable of making that intuitive philosophical leap that recognizes the different order that arises from different sorts of causes. As Christians, we can see that all change begins and ends with God, and any intellectual field that ignores this truth will hit problems eventually.