Why do we use mathematical formulae and where do they come from?

All of us had to handle mathematical formulae when we were in school and in college/university and most of us had a certain degree of trouble (some significantly more than others) dealing with them. But, did any of us actually thought of what the formulae are, what they represent and where they originate from? In this short article I want to present the compressed history of mathematical symbols and formulae, how they came to be and how they developed into that thing tormenting school and college students (and not only them) all across the world.

Mathematical texts in Ancient times

In ancient times people expressed mathematical concepts as they would do everything else was – only with words, as if they were speaking to someone. The mathematical formulae and symbols we know today were not a thing back then and numbers, while existing in the form of many numerical systems like the Greek, Roman and Israelite ones, were used only within the range of the pragmatical and not as commonly as nowadays.

This was completely fine for the ancient people even despite the scarcity and cost of paper and its alternatives because of the short and more compressed orthography of the languages employed by the mathematicians of the time. Hebrew, Aramaic, Phoenician and few other Levantine languages were written without vowels. The tongues of Mesopotamia were written in cuneiform which is a syllable-based script. Ancient Egyptian was written in hieroglyphics and while Greek and Latin used alphabets, their spelling was very different from the one of, to say, Modern French or Modern English.

The appearance of the mathematical formulae

In the Middle Ages the situation in Europe, the Middle East and North Africa changed irreversably. The regions were conquered by foreign peoples who brought their own tongues. Some of these peoples, namely the Germanic ones, adopted the Latin alphabet to their languages which eventually resulted in the gradual creation of impragmatically elongated orthographies like the German, French, Dutch and English ones.

This created numerous obstacles as the scholaristic mileaux of Europe started abandoning Latin as language of science and transitioned to their local tongues. Writing a book on mathematical themes became very costly due to the larger amount of paper required. Many authors did not have the resources to afford writing a draft for their works and even if they managed to do so, they were commonly turned out by publishers because the cost of printing their books was just too high.

Then many mathematicians started doing something that was more usual for the clergy of the time, they started abbreviating their texts more and more. The abbreviations became shorter and shorter with time. Whole words were replaced with letters from Latin, Greek, Hebrew, other languages or with symbols of non-linguistical origin. Eventually, entire sentences were reduced to single symbols. These symbols were then combined into formulae that took little space write and were easier to both write by hand and print with the technology of the epoch.

In Modern times

While the adoption of mathematical symbols and formulae and the processes mentioned here were not by themselves bad in any way, they laid the foundation for a problem that has became too common in modern times – the poor teaching on Mathematics to students. Sadly, many pedagogues just flood their students with formulae and expect them to memorize them without explaining the processes and concepts which lay behind the symbols. Thus, many students, while learning mathematical operations to a certain level, are never shown the true meaning and purpose of the science of Mathematics. Some educational institutions try to solve this problem but the results of their experiments are yet to be seen in the future.

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