How Math Works

Math stands apart in discussions of Science, Technology, Engineering, and Mathematics (STEM). Other fields of endeavor rely on mathematics to create models of theories and work products. The usefulness of mathematics comes from being inspired by physical reality without depending on it. Much like Plato’s theory of Forms, in which physical reality is a shadow of a perfect, idealized existence, math looks at a system where a few postulates are taken to be true. Using a method of logical reasoning, mathematicians prove a host of theorems, all based on the assumed truth of these postulates, or axioms.

Euclidean geometry illustrates the essential mathematical approach. Euclid formulated five postulates (“Euclid’s Postulates,” 2022):

1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

A point in Euclid’s geometry does not have any dimension. It has no length, width, or breadth, unlike any point you might draw in the real world. Straight lines have length but no width, and they extend infinitely. Although there is no evidence that points and lines behave this way in the physical world, mathematics does not require any such verification. There are no experiments that can invalidate the theorems of geometry. If you accept the axioms, then everything else in geometry is proved using the methods of logical reasoning.

Non-Euclidean geometry replaces the 5th postulate with an alternative. In one version, parallel lines do not exist because every pair of lines intersects. The result is a different mathematical system that has been used successfully to model such things as shapes drawn on the surface of a sphere.

Physics moved past the three-dimensional model of space from Euclidean geometry. Current models combine space and time into a four-dimensional manifold. Some theories add even further dimensions. Even the continuous nature of space has been questioned. It’s possible that our universe is actually digital, although the resolution would be extremely high.

In fields that accept the axioms of the mathematics they use, the emphasis falls on examining the validity of the models within the field of endeavor. Isaac Newton developed both a system of mechanics for physics and the mathematics of calculus. Calculus was used to model the laws of motion within mechanics. Once the theorems of calculus were proved, the models for mechanics using the new math of calculus produced predictions that could be tested in experiments. Some models were shown to be inaccurate, failing to describe how motion actually works. While double-checking for math errors is reasonable, the validity of calculus itself remained solidly intact. This gave researchers confidence in their investigation of these natural phenomena. Again, the focus remains on the science, not the mathematics.

References

Euclid’s Postulates. (2022). Retrieved August 2, 2022, from Wolfram.com website: https://mathworld.wolfram.com/EuclidsPostulates.html