Exponents, History, Civilization

I decided to make a video the other day on exponent properties. I like to think about mathematical concepts and to explore first principles. I spend a lot of my spare time working through proofs and also studying the history of mathematics and physics (as well as science in general). A natural extension of these interests is the idea of making videos, where I can talk about my studies and explore different concepts and problems.

In making the recent video on exponent properties, this also gave me the opportunity to briefly touch on the fascinating history of exponents, which also raises a wider discussion on human evolution and the history of civilization.

For instance, math historians have suggested that the concept of exponents could date as far back as the 23rd century BC. This depth of history when it comes to basic mathematical ideas is something that I find so astounding. There is something both beautiful and intriguing about the idea of ancient Babylonians thinking about various mathematical concepts, not least exponents.

Of course, the context in which ancient Babylonians were thinking about exponents was very different than how we think of exponents today. For starters, the numbering system was in Sumerian, which is now an extinct language. Secondly, and perhaps most interestingly, ancient Babylonians would use symbols to denote mathematical formulae (the image below was borrowed from this article).

One of the most famous tablets that we have discovered is a cuneiform tablet known as Plimpton 322. I believe current estimates suggest that this particular tablet was created around 1900–1600 BCE. Although it is one of thousands of clay tablets that date as far back as 4000 years ago, during the Babylonian period in Mesopotamia, what I enjoy about it most is that it represents some of the most advanced mathematics prior to our now widely cherished and celebrated ancient Greek mathematicians.

In short, the tablet lists what we now term Pythagorean Triples. Basically, what a Pythagorean Triple consists of is three positive integers a, b, and c, such that a^2 + b^2 = c^2. One might be immediately reminded of Pythagorean Theorem, and you would be right to think of this famous formula. The name Pythagorean Triple is derived from Pythagorean Theorem, which one will know states that every right triangle has side lengths satisfying the formula a^2 + b^2 = c^2.

What we see in the Plimpton 322 tablet is a very cool table of numbers, with four columns and fifteen rows.

In the image above, which was borrowed from this article, we see a very nice illustration of what Robson (2002) has most recently claimed to be “a list of regular reciprocal pairs.” Robson has suggested that, when one considers all of the historical, cultural and linguistic evidence, the author of the tablet was most likely “a teacher and Plimpton 322 a set of exercises.” This account of the tablet deepens my fascination, as one can imagine an ancient teacher going over lessons that consist of very early ideas of mathematics and the relationship of numbers.

All of that aside, the wider history of exponents is fascinating. If it is true that human beings were already thinking about and experimenting with exponents as far back as 23rd century BC, this fact would contribute to an already beautiful picture that is the history of mathematics. There is something objectively human about the development and evolution of our mathematical ideas – a history that is intimately entwined with human evolution and the development of human civilization.

The ancient Babylonians were known to be fantastic record keepers, compiling ledgerbooks that act as sort of Sumerian spreadsheets. If the development of agriculture actuated the birth of human civilization approximately 12,000 years ago, as many experts agree, the emergence of basic things that we now take for granted – such as writing, town planning, the division of labour, administration, law, commerce – were able to crystalize as future possibilities.  Free from the precariousness of sustenance living, people were allowed more free time, with greater access to resources. New technologies were eventually conceived, and human pursuit was existentially freed from basic survival to expand beyond that which was unavailable to hunter-gathers. With all of this came writing, and thus also things like Plimpton 322.

There are many detailed, well-research and richly scholarly books available today that explore this particular moment in history. In many ways, it is an important chapter within the broader history of human thought, science and technological pursuit.

To close: The earliest known form of writing, as we already touched on, was cuneiform – a Sumerian system thought to have emerged roughly five-thousand years ago. The cuneiform tablets, such as Plimpton 322, allowed for the ability of human beings to begin to harness the power of knowledge; because, in essence, the most basic idea of writing and written language allowed for the keeping of records. Human thought could be recorded, shared among generations, and thus concepts and ideas could develop and evolve. The Babylonians, at the start of human civilization, harnessed this power as much as possible: they were not only avid record keepers, but they developed lists in order to keep track of acreages of wheat and quantities of livestock. They also kept track of taxes and legal disputes. And from the basis of these very humble foundations, using soft clay tablets, early ideas in mathematics could be pursued.