"Do not all charms fly At the mere touch of cold philosophy? There was an awful rainbow once in heaven: We know her woof, her texture; she is given In the dull catalog of common things. Philosophy will clip an angel's wings, Conquer all mysteries by rule and line, Empty the haunted air, and gnomed mine ---- Unweave a rainbow.........." - John Keats On the notion of "depth" in a scientific theory - excerpt from "Longing for the Harmonies" by Frank Wilczek & Betsy Devine. What is depth in a scientific theory? There is no universally accepted definition; indeed, the concept seems rarely to be discussed in print. It is, however, often used informally, and most scientists would surely claim to "know it when they see it." There is much more to science than the traditional categories of truth and falsity; ideas can be true but trivial (90% of what appears in scientific journals) or strictly speaking false but deep and fruitful (for instance, Newton's theory of gravity). Let us try the following definition, inspired by recent work of Charles Bennett and others in computer science. A theory is said to be deep when it satisfies two requirements. First, it must have verifiable consequences that can be derived from the premises of the theory only by long chains of logical deduction. Second, there must not be another theory that leads more easily to the same consequences from fewer premises. In less formal language, the first sign of a deep idea is that its consequences are not immediately apparent. Einstein's theory of gravity is in this sense extremely deep, since it is stated in terms of rules for determining the curvature of space-time, which have no obvious connection to the familiar force of gravity at all. Indeed, it takes a good deal of work involving tensor calculus to make the connection. Having hidden consequences is not enough to make an idea deep, though---otherwise we could manufacture deep ideas as easily as anagrams. The second requirement is that the work we have to do to find the consequences is really necessary. If all Einstein's theory of gravity did was reproduce Newton's laws of gravity from a different and more obscure starting point, it would be considered not a deeper theory but the same theory written in the form of a puzzle. It is largely because a striking property of gravity --- its universality, the fact that all bodies acquire the same motions (accelerations) in a gravitational field --- is built into Einstein's theory but must be grafted onto Newton's that all the extra work is worthwhile. (We do not, of course, mean to deprecate the genius of Newton, whose theory of gravity is itself extremely deep. To derive the motion of planets and tides from the basic inverse square law of gravity is a nice exercise in calculus even today, and it required immense mathematical skill and ingenuity, given the tools available in Newton's time.) Deep theories are often found to possess a special sort of simplicity, what might be called radical simplicity. In the ideal case, the theory is formulated as a single equation. This single equation, however, will refer not to anything immediately observable but rather to inferred underlying structures. ............ The behavior of the underlying structures is in a radical sense simple--- summed up in the one equation. The observable features of the world, the features that our sense organs and measuring instruments happen to be well adapted to perceive, can, however, be elaborate combinations of these simple ingredients. The observable combinations may behave in complicated ways that require a lot of mathematical and logical work to derive, though ultimately they are consequences of the single equation of our deep theory. If we are to understand complicated appearances in terms of underlying simplicity, this is the price that must be paid. Depth is not at all the same as fruitfulness. Depth is like the root system that supports the visible tree. A barren tree may have deep roots; inversely, a tree with shallow roots may bear much fruit. Einstein's theory of gravity, at the time it was formulated, made very few testable predictions that could not be derived, much more easily, from Newton's theory. In the long run, however, the deeper roots will support new growth. ........ Einstein's theory has become a necessary, workaday tool in the description of extreme astrophysical objects like neutron stars and black holes ---- and the foundation of big bang cosmology. The search for depth is , we think, partly what can only be described as an artistic impulse: to make our theories more self-contained, intricate, inevitable ---- and therefore beautiful. As a bonus, we hope and expect it will yield new fruit, as it always has in the past. A visitor to the gaslit halls of the London Exhibition of 1851 asked Michael Faraday what possible use could arise from experiments on electricity. Faraday replied with a question of his own: "Of what use is a newborn baby?"

Geometry  David Henderson in his book 'Differential Geometry: a geometric introduction' Logic can only go so far ---- after that I must see-perceive-imagine. This geometry can help I may reason logically thru theorem and propositions galore, but only what I perceive is real. If after studying I am not changed --- if after studying I still see the same --- then all has gone for naught. Geometry is to open up my mind so I may see what has always been behind the illusions that time and space construct. Space isn't made of point and line the points and lines are in the mind. The physicists see space as curved with particles that are quite blurred. And, when I draw, everything is flat there are no points and that is that. The artists and the dreamer knows that space is where an image grows. For me it's a sea in which I swim a formless sea of hope and whim. Thru my fear of Infinity and One I structure space to confine my imagination away from the idea that all is One. But, I can from this trap escape --- I can see the geometry in which I wander as but a structure I made to ponder. I can dare to let go the structures and my fears and look beyond to see what is there to see. But, to let go, I must first grab on. Geometry is both the grabbing on and the letting go. It is a logical structure and a perceived meaning --- Q.E.D.'s and "Oh! I see"'s. It is formal abstractions and beautiful contraptions. It is talking precisely about that which we know only fuzzily. But, in the end, and, most of all, it is seeing-perceiving the meaning that             I AM. Everything should be made as simple as possible, but not simpler - Albert Einstein As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality - Albert Einstein Politics is for the moment; an equation is forever - Albert Einstein There are two kinds of mistakes. There are fatal mistakes that destroy a theory; but there are also contingent            ones, which are useful in testing the stability of a theory - Gian-Carlo Rota for further samples of Rota's scintillating writing on academics and academicians try http://www.rota.org/hotair/lesson.html and his wonderful book 'Indiscrete Thoughts'.