From Counting to Abstraction: The Metamorphosis of Mathematics

How is it that mathematics transforms from a simple act of "counting" into a sophisticated science filled with complex expressions that, over time, seem to shed numbers altogether? At first glance, this evolution may appear peculiar. Most of us still equate mathematics with the four operations, digits, and the act of calculation. Yet, the saga of mathematics is far too long and multi-layered to be confined within such a narrow frame.

Historically speaking, mathematics did not begin with numbers, but with geometry. Humanity was faced with profoundly concrete challenges: surveying land, constructing edifices, and distributing harvests. To solve these problems, the need for calculation arose. In time, we realized we needed symbols to make these calculations more orderly and transferable. This is precisely where numbers entered the stage.

The Mirage of Reality: Numbers as Symbols

However, there is a crucial distinction to be made: numbers are not "real entities" in the way many believe them to be. They are symbols we employ to define reality. We can count five apples, but the "5" itself is not the apple; it is a mental representation of it. Mathematics, at its core, is a science that operates with these symbols. In this regard, it stands distinctly apart from other scientific disciplines.

In mathematics, we do not conduct experiments or enter laboratories to take measurements when discovering something new. Progressing in math means logically placing new building blocks upon what we already know. As we distinguish between theorem and theory, truth in mathematics is established not by experiment, but by proof.

The Hierarchical Cathedral

Consequently, mathematics possesses an intensely hierarchical structure. If you struggle with addition, you cannot fully grasp multiplication. Without mastering multiplication, exponents become nonsensical. This is a rudimentary example, but it summarizes the general logic of the field: to understand certain concepts, you must simultaneously master a multitude of different subjects.

But how did we transition from such basic needs to the realms of abstract algebra, topology, or category theory?

The shortest and most honest answer is necessity. Although some mathematical structures may appear "redundant" or "excessively abstract" from the outside, almost everything that emerges in mathematics has a counterpart in reality. If a structure were truly useless, humanity would not have developed it with such persistence. Moreover, "utility" does not merely signify direct application in daily life.

When the Universe Speaks Math

A significant portion of the mathematical theorems used today in physics and engineering was purely theoretical when first proposed. Some were even thought to "never have an application." Yet, as time passed, nature began to speak through these abstract constructs. Differential equations became the language of physics; linear algebra became the backbone of modern engineering. More ironically, certain mathematical theories that initially "contained no numbers" are now at the heart of calculations performed with numbers.

In short, while mathematics seems to have started with counting and arrived at number-independent structures, it is essentially doing the same thing: producing better symbols to make sense of the world. What has changed is not the purpose of mathematics, but the complexity of our needs.