Is Mathematics An Invention Or Discovery?

The Poetry of Logical Ideas

Mathematics, possibly more so than any other field of study, is an academic practice aimed at providing a “rationale” for the order that exists in the world. Patterns are everywhere, and mathematics is used to identify these patterns and develop “structures” that provide insight into nature’s phenomena. Many different branches of mathematics have taken shape throughout its development, and each branch is used to study certain types of phenomena. Geometry, for instance, is used to study space and shapes. Calculus, on the other hand, is used to study change. While these two branches are distinct in the way that their particular operations are used to investigate various topics, they are unquestionably related insofar as they share the fundamental axioms, theorems, and principles upon which all of mathematics rests. What, then, is responsible for the simultaneous uniqueness and interrelatedness within the many branches and areas of mathematics? Through an examination of mathematics’ ancient roots, I am going to provide good reasons for thinking that the answer to this question is an “invisible, intangible, creative ingredient” called invention (Dr. Ze-Li Dou, Lecture 7).

It follows from above that in this essay, I am going to hold the position that mathematics is an invention rather than discovery. Although we have covered several mathematical topics and learned about their development in a number of ancient cultures, I am going to discuss two particular topics that we have surveyed in class. I will begin with a discussion of the representation of numbers in ancient Egyptian and ancient Babylonian cultures. Then, I will outline and explain the methods that ancient Egyptian and ancient Chinese cultures used to approximate the mathematical constant, pi.

There is hardly any question but that numbers are absolutely fundamental in the study of mathematics. They are values that can be expressed by all different types of symbols and words, and they are the objects of everyday mathematical operations. However, ancient civilizations did not record numbers in the same manner that we record them in our number system today (i.e. the number/quantity seven has not always looked like “7”). Different cultures have developed different types of number systems throughout history, and one of the earliest known number systems was created by the Egyptians in around 3,000 BC (Lindberg, 12). The earliest Egyptian number system consisted of symbols that represented each power of 10 (i.e. base 10 system, called decimal), and no position value system was employed (Lecture 2). Without a number system in place, simple tasks such as recording the number of days since the last full moon or quantifying the distance from home to the field were out of the question. Upon the invention of their number system, however, the Egyptians were able to use these representative symbols to do basic mathematical operations such as adding and subtracting (multiplication and division were “extremely clumsy,” as Lindberg phrased it.) (Lindberg, 12).

The ancient Babylonians also created their own number system, and it was fully developed by around 2000 BC (Lindberg, 13). The Babylonians were considered to be superior to their Egyptian “counterparts” in regards to their level of mathematical achievement, and there were a number of differences between the Egyptian and Babylonian number systems. In contrast to the Egyptian number system, the Babylonians did employ a positional value system. Moreover, the Mesopotamian number system was based on the number 60 (sexagesimal – base 60). This made it much easier for the Babylonians to represent fractions (they could avoid many of the “repeating decimals” that plague the base 10 system), and large numbers could be expressed with ease (Lecture 3).

I would like to briefly comment on the two paragraphs above that discuss the formation of numerical systems in ancient Egypt and Mesopotamia. While it is true that

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