When presented in textbooks as lengthy equations and abstract theorems, mathematics seems unapproachable, one may assume it came out of nowhere. Whereas the development of mathematics is so closely related to human activities. Human physiology, agricultural needs, religion and even certain celebrities may introduce pivotal points to the construction of mathematical systems.
There are numerous examples I could name. For instance, why is it that we have adopted a base ten counting systems (i.e. nine special symbols and one “zero” symbol to denote empty space), instead of, say a base fourteen or base two counting system?
The number ten may seem like an arbitrary choice, but it happens to be the number of our fingers. As one of the most important and certainly most fundamental mathematical tools, counting was needed in early civilization to facilitate hunting, food distribution, and later on, trade. And fingers were the most accessible tool for counting at the time. Of course, there were other counting systems that were used historically. The ancient Mayan calendar had a base twenty. There is evidence that this system followed women’s menstrual cycles, which is 20-28 days, another indication of how human physiology impacted the birth of mathematics.
And it’s not just counting systems. To give a more advanced instance of mathematical formulation, let’s take a look at Georg Cantor and his idea of finitude. It may seem intuitive now to claim that the set of all natural numbers and the set of all real numbers are infinite. However, in the mid 19th century, infinity was considered to be an identity that’s preserved only by God. Cantor was the first to be able to prove that there are more real numbers than natural numbers, the cardinalities of both being infinite. The idea that there can be infinity outside of God, and that these infinities differ in magnitude, was considered unorthodox. Being a very religious man, Cantor was concerned with the orthodoxy of the relationship between God and mathematics. He attempted to define two kinds of infinity, one is transfinite, which we seek in mathematics, these infinities can have different cardinalities; and the other is absolute infinity, which he identified with God, this infinity is above all infinities, and is non-increasing. Despite framing his proposition in this manner, some Christian theologians considered transfinitude a challenge to the uniqueness of absolute infinity in the nature of God, and Cantor was accused of blasphemy. This is one of many instances in history where propositions in mathematics had to comply with religious beliefs.
Now to answer your question: how was algebra invented? You may be disappointed by the generality of the answer I am about to give: it is scattered all throughout history. For instance, how did someone randomly come up with the equation y = mx + b? This is a question that I can answer specifically. Someone named René Descartes invented the idea of representing an unknown variable with the symbol ‘x’. Before that, y = mx + b would be equivalent to something like: solve for the variable that induces a constant linear increase for a known variable with some constant overlay. I’m sure you can come up with real-world problems that call for such a definition. However, symbolic representations offer us a much more clear and concise way to generalize and unpack these situations.
To give a more concrete example, let’s look at y = x2. This can be written in an approximate form of y = mx + b as y = 1 ⋅ x2 + 0. How did this equation come about? For ancient Mesopotamians (modern day Iraq), this would mean adding the token x to itself for x number of times. For ancient Egyptians, it would mean the area of a square, each side with a length of x. As early as 1800 BC, humans have developed algorithms to calculate the square root of a known variable. Of course, at the time the question was posed in an approximate form of: what is the number when multiplied by itself gives us the number of which we are square-rooting? Nevertheless, ancient humans possessed a pool of knowledge in mathematics, which we’ve taken for granted today.
And that’s the story of how mathematics was constructed into what it is today. We’ve taken different routes, had setbacks due to societal conventions, and invented and discarded a lot of rules and ideas. But whenever there is a problem we want to generalize and to solve, we develop mathematics. Looking back in time, a huge portion of mathematics was heavily influenced by human needs and human activities. Mathematics is a human discipline, however it hasn’t been taught in a humanistic way. Maybe instead of shoving formulas in textbooks, modern education could lead its students down to the root of mathematics, explaining when it was needed, how it was developed. If we do a successful job in this, I’m sure the discipline would be much more appreciated.