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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.

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

Your comments seem to identify "number" with "integer." The use of the "real line" allows other objects (pi, Sqrt(2), e, etc.) to have the same "existence rights" as the integers or fractions. There are no problems doing so. Actually, arithmetic on the reals (with addition and multiplication) is catagorical; there's only one real line. Arithmetic on the integers (with addition and multiplication) is undecidable.

The above can be taken to mean that, although one can start with the integers, proceed through the rationals, and complete the system by various methods (Caucy sequeces, Dedekind cuts, etc.), and thus obtain the reals; (breath mark); there is no consistent method of starting with the reals and uniquely identifying the integers.