The statements made in mathematics are recognition statements, not statements of independent creation. The symbols of the language are man-made, but the symbols and syntax (reasoning applied) report on a patterned reality, a dance, that transcends the human mind. That is why algebraic operations applied to these statements can generate new statements that are also dance with the way things work in the universe.
For example, we can use Geometry and Algebra to derive measurements impossible to make otherwise. The radius of the earth is a measurement derived algebraically from the earth’s circumference, a measurement first approximated by Eratosthenes (3rd century BC) when he applied propositions of Euclidean geometry to the distance between two cities and shadow angles in a well. Knowing the radius of the earth, we can apply algebraic operations to the Pythagorean Theorem and generate results that will tell an observer at a certain height above the earth how many miles he can see to the horizon.
From “The Dance of Number” (forthcoming).