It's not so much that math is a useful tool but that elegant mathematics have been found to correspond extremely closely to the micro and macro behavior of the universe. Math turns out to be powerfully descriptive. Why this should be so is a deep mystery. Early man must have felt he created mathematics and later perhaps that he had discovered aspects of it but there was no reason, then or now, to suspect that it is so deeply entwined with the fundamental nature of the universe, from the Materialist/Objectivist point-of-view. It is nonetheless true that understanding the math has become increasingly important if we care to understand the universe. We aren't yet there.
I know this is how it seems to you and others, and I am not trying to dissuade you or remove from your experience something which is profound to you.
I do not see it that way, however. It would be a mystery to me if mathematics was not useful in describing and providing the shortcut for understanding many aspects of existence. In fact, however, there is more about the material universe that mathematics is either only fairly useful for, or not useful at all.
At one time it was believed every shape in the universe could be reduced to Descartes' analytic geometry. A lot of them can, but a lot more cannot. For those we had to develop new fields such as topology. Then came along fractals and strange attractors which have enabled us to understand some other kinds of shapes and behavior, but, so far, this "chaotic" math is useless in describing any particular shape or behavior. (The problem with fractals is, one can plug in numbers and create all sorts of interesting patterns, but one cannot find a pattern in nature and determine what numbers to plug in to produce it. The other problem is fractals and strange attractors are both "discrete" iterative functions, and even when patterns seem analogue, they are only "connect the dot" type analogue shapes.
Another place that mathematics can only deal with existence as an approximation, at best, is in relationships, two of which have been mentioned before, pi and the ratio of the hypotenuse of an isosceles triangle to either leg. This latter is greatly misunderstood. The pythagoreans were the first to suggest the apparent "mystical" relationship between numbers and existence. It was the discovery of incommensurables, and the limitations of mathematics at its heart that cured them.
The whole significance of incommensurables is, that there are relationships which can certainly be described for which there is no arithmetic means of describing, thus whole worlds of real things mathematics is totally irrelevant to.
The Proof:
"In a right-angled isosceles triangle, the square on the hypotenuse is double of the square on either side. Let us suppose each side an inch long; then how long is the hypotenuse? Let us suppose its length is m/n inches. Then m2/n2 = 2. If m and n have a common factor, divide it out; then either m or n must be odd. Now m2 = 2n2, therefore m2 is even, therefore m is even; therefore n is odd. Suppose m = 2p. Then 4p2 = 2n2, therefore n2 = 2p2 and therefore n is even, contra hyp. Therefore no fraction m/n will measure the hypotenuse. This proof is substantially that in Euclid, Book 10." (Bertand Russell, A History of Western Philosophy) In other words, there is no even or odd number that can be the measure of the hypotenuse relative to either leg, therefore, there is no such number at all, but there is certainly such a relationship which can be both defined and understood, but not with the help of mathematics.
Hank
So mathematics is a language with a built-in logic. But what is so striking about this language is that it seems to describe how the world works—not just sometimes, not just approximately, but invariably and with unfailing accuracy. All the fundamental sciences—physics, chemistry, and astronomy—are mathematical sciences. No phenomenon has ever been discovered in these subjects for which a mathematical description is not only possible but also beautifully appropriate. Yet one could still fail to be impressed. After the fact, perhaps, we can force any hand into some glove, and maybe we have chosen to pick the mathematical glove because it is the only one available. It is striking, however, that physicists so often find that some esoteric mathematical structure, invented by mathematicians in the dim and distant past only for the sake of its elegance and curiosity value, is precisely what is required to make sense of new observations of the world. In fact, confidence in mathematics has grown to such an extent that one now expects (and finds) interesting mathematical structures to be deployed in nature. Scientists look no further when they have found a mathematical explanation.
There are many striking examples of the unexpected and curious effectiveness of mathematics. In 1914, when Einstein was struggling to formulate a new description of gravity to supersede that of Newton, he wished to endow the universe with curved space and time, and to codify the laws of nature in a manner that would apply for any observers no matter what their state of motion. His old student friend, the mathematician Marcel Grossman, introduced him to a little-known branch of nineteenth-century mathematics, called tensor calculus, that was tailor-made for his purposes. Upon adopting this mathematical language, how Einstein would describe laws of nature became clear and (if one is that clever) obvious.
In modern times, particle physicists have discovered that symmetry dictates the way elementary particles behave. Particular collections of related particles can behave in any way they choose so long as a particular abstract pattern is preserved. The laws of nature are superficially the catalog of habitual things that occur in the world while yet preserving these patterns. With every such catalog of changes one can always find an unchanging pattern, though the pattern is often subtle and rather abstract.
Intelligibility of the Universe - Gregory Chaitin
Abstract: We discuss views about whether the universe can be rationally comprehended, starting with Plato, then Leibniz, and then the views of some distinguished scientists of the previous century. Based on this, we defend the thesis that comprehension is compression, i.e., explaining many facts using few theoretical assumptions, and that a theory may be viewed as a computer program for calculating observations. This provides motivation for defining the complexity of something to be the size of the simplest theory for it, in other words, the size of the smallest program for calculating it. This is the central idea of algorithmic information theory (AIT), a field of theoretical computer science. Using the mathematical concept of program-size complexity, we exhibit irreducible mathematical facts, mathematical facts that cannot be demonstrated using any mathematical theory simpler than they are. It follows that the world of mathematical ideas has infinite complexity and is therefore not fully comprehensible, at least not in a static fashion. Whether the physical world has finite or infinite complexity remains to be seen. Current science believes that the world contains randomness, and is therefore also infinitely complex, but a deterministic universe that simulates randomness via pseudo-randomness is also a possibility, at least according to recent highly speculative work of S. Wolfram. [Written for a meeting of the German Philosophical Society, Bonn, September 2002.]
Hank, with regard to your assertion of Pythagoras's Constant, I would like to add that "it is not known if Pythagoras's constant is normal to any base": Normal Numbers