On the definition of mathematics

Can mathematics be defined?

Even mathematicians don't agree on a definition of mathematics. The original Greek meaning was 'learning, study, science'. But throughout history--and in today's online forums--the debate rages on without resolution. Some define mathematics as the study of theorems, proofs, and algorithms, but these are only techniques. Mathematics must be more than the sum of its techniques.

Morris Kline wrote that Greek mathematics died in Roman times. This implies that the Romans practiced something else. Kline explains that the Early Greeks honored the search for truth regardless of utilitarian goals, while the more utilitarian Romans reduced mathematics to a collection of useful facts and techniques. Others claim that Kline overstated this. They point out that Romans also contributed to the advancement of mathematics and that not all of their contributions were purely utilitarian.

In any case, we inherited these somewhat incompatible views of mathematics, i.e., the awesome wonder of universal patterns vs. the pragmatic search for effective calculations. We are left to integrate them as best we can.

Can mathematics be described?

Mathematics is many things to many people. It might be the cause of beauty and harmony. It might be music frozen into formulas. It might be invented or discovered. It might be humanity's highest and most original achievement or a silly game leading nowhere.

It may be only a language, one among many, or the source of all languages. It may be the language of God or the language of science, or both. It may be God's thoughts, or it the logical program of a universal, Godless machine. It may describe ultimate Truth or delude us into believing we see truth where none exists. It may help us predict the future, or create only the illusion of predictability . It can be an empty mirror, reflecting the meaningless images of our deepest dreams and desires.

Any "nothing but" or "is just" definition that attempts to define part of a whole (reality in this case) is reductionist, and therefore incomplete. On the other hand, without definitions complex communication is difficult. Without language, all we can do is smile, growl and grunt at each other. This might improve some human relations, but it's not very rich, specific or interesting.

To communicate complex ideas we create language by dividing and categorizing aspects of reality, and labeling them with words. Maybe the best we can do is to not lose sight of the fact that categories are somewhat false since they are arbitrary, incomplete, and can create an error of perspective--even when that error is useful in limited and important ways.

An example from physics may help to illustrate why incompleteness poses a problem in math. Newton revolutionized our understanding of the world by developing a mathematical model for something we call gravity. His model enables us to accurately predict everything from the acceleration of a falling apple to the orbit of planets.

In his day, Newton's model seemed accurate for all situations and was therefore thought to describe Truth. As our ability to measure astronomical phenomena grew, small errors were noticed in calculations that used Newton's formula.

Dissatisfied with such errors, Einstein eventually developed the Theory of Relativity, which has proven far more accurate at measuring large astronomical events. However, Einstein was never satisfied with his theory either because it was unable to measure very small phenomena, such as those indirectly observed using Quantum Physics.

Physicist are still searching for the Unified Field Theory (the theory of everything). They think that a theory that only works for some cases must be overlooking some essential part of reality. In other words, until our theories accurately describe everything, they are incomplete and therefore wrong.

If this makes sense, then mathematics will always be more than our narrow definitions. It might make more sense to define math in relation to particular use cases. For example, it is often treated as a language and each language describes a realm. For the Ancient Greeks, mathematics was the language of geometric shapes, for Pythagoras it was a religion, on Wall Street it's a tool for maximizing private profit. For others, it's symbols of music frozen on paper... Each definition may be 'correct' within its own context, but none are complete. This leaves us in the fog of relativity. Not a comfortable place for those with a high need for certainty.

What seems eternal within mathematics, no matter the era or culture, is the search for truth (whatever truth may be and however the search evolves).  It seems that no matter what we do or think, mathematics itself remains. Or does it?