Mathematics is a concept that indicates the study and tradition of number, quantity and space, either abstractly or practically. The notion of mathematics has existed for as long as conscious thought has, though initially it only existed as a very basic notion. It evolved slowly over the years from recognition of numbers to recognition of number patterns as practical need created a demand for it. The Classical Age in Ancient Greece was the era in which the ideas of learning and accumulation of knowledge flourished in the studies of different fields in Ancient Greece. The circumstances that accompanied the Classical Age aligned to create an accommodating environment in which mathematics could be fostered. Mathematics became a topic of inquiry by the brightest minds of the age, such as that of Hippocrates of Chios. The work on mathematics by minds such as his had a crucial impact upon the development of the Greek civilisation.
Mathematics was originally more of a primitive instinct than a studied matter. Counting on fingers and toes was the earliest form of the subject, first applied approximately 300 000 years ago by Neanderthals. The entire decimal system of numbers can be credited to these people, as it is based on the ten number grouping system that they used because they were accustomed to counting on ten fingers and ten toes (Boyer & Merzbach, 1989, p. 3). It is difficult to determine whether the origins of mathematics as known today are in geometry or arithmetic, since both predate writing. Herodotus said that geometry originated in Egypt because of the practical need for resurveying after the annual flooding of the river valley.
“Whoever was a sufferer by the inundation of the Nile was permitted to make the King acquainted with his loss. Certain officers were appointed to inquire into the particulars of this injury, that no man might be taxed beyond his ability. It may not be improbably to suppose that this was the origin of geometry, and that the Greeks learned it from hence.” – (Herodotus, Book II, Euterpe)
However, Aristotle argued that it was the existence of a priestly leisure class in Egypt that had prompted the pursuit of geometry (Zhmud, 2006, p. 211). Their two arguments represent the two theories of how mathematics came to be: practical need or as a result of leisure and ritual (Eves, date, page). Either way, a need for any mathematics more advanced than whole number counting never arose until much later in history, when ancient civilisations began to explore the topic more thoroughly. Mathematics in Greece began to grow during the sixth century BC, the age of geometers Thales and Pythagoras (refer Appendix 1), who began a movement in the Hellenic people to learn more about the world around them. However it was not solely their initiative that drove the educational expansion, but also the conditions that accompanied the Classical period that enabled the era to become a milestone in both mathematical and Greek history.
The Classical period was ideal for mathematical advances for two main reasons. The first was that the geographical location of Greece allowed for easy travel and contact between many different civilisations, enabling those in the pursuit of knowledge to obtain firsthand information from centres of ancient learning (refer Appendix 2). In Egypt, geometry was said to be a major topic of learning; in Babylon, astronomy; and the Greeks learned and improved on the works of these places (Boyer & Merzbach, 1989, p. 53). The proximity of the countries and the bold and imaginative spirit typical of pioneers in the sea-bordering colonies inevitably meant that early mathematicians were not isolated but also were part of a competitive intellectual environment in Athens, Ionia and Italy (Knorr, n.d., online). The second condition that supported the progress was related to the economic situation in Greece during the fourth century BC. The Classical Era was also the era of the Athenian Empire, an era of war and conflict, and the root of democracy. Every city-state in Greece had agreed to contribute resources of money, men or ships to the Delian League, founded in 478BC, a military force dedicated to protecting Greece in the face of the Greco-Persian wars of the time (Sinha, 2014, online). However, once Persia had been defeated, the member states stopped paying in men and ships and instead just gave money to the treasury to ensure their protection in the event of another war. Athens kept training their men, becoming a strong military force, and then they moved the treasury of the League to their city, spending the money on themselves. A very large portion of the funds was spent on supporting artists and thinkers in Athens, such as architects and mathematicians. As such, geometers and arithmeticians were able to study their respective topics at their own leisure, and the democratic growth of the city-state meant that anyone, no matter their position in society, had access to learn about what they wished. These combined conditions of a convenient geographical location, an assembly of colonists with keen enthusiasm for learning and creating, and a prospering and supportive economy created the perfect situation for the Greeks to expand their mathematical understanding.
The Greeks’ first pivotal and arguably most important contribution to mathematics was their transition from practical mathematics to theoretical or pure mathematics. Before the Classical Era, mathematics had only ever been used basically, for general number notation and tasks such as recording produce and measuring lengths. Some went so far as to identify and understand some mathematical patterns and their applications. However, as aforementioned, the Greeks studied mathematics not only for practical reasons but also for their own leisure; in this way, they considered mathematics beyond what they could physically apply. This is why the Greeks were said to be the ‘creators’ of mathematics, because they were the first to explore the subject with such a perspective. The development of theoretical mathematics is also often partially attributed to the discovery by the Greeks of irrational numbers in the fifth century BC (Knorr, n.d., online), which are numbers that extend infinitively (Oxford Dictionary). This made Greek mathematicians realise that arithmetic with integers was insufficient for the purposes of geometry, casting all previously devised mathematical assumptions under suspicion. There was suddenly a need for more researchers and geometers to revise everything they thought they knew about mathematics and to prove the accuracy of their conclusions.
“At the least it became necessary to justify carefully all claims made about mathematics. Even more basically, it became necessary to establish what a reasoning has to be like to qualify as a proof.” – (Knorr, n.d., online)
This way of thinking also introduced the idea of recording educational progress for the first time because that way if something had been proved to the qualifying standard, other mathematicians could see the proof and knowledge could be shared. This was particularly important to the development of the Ancient Greek culture because before this time, comprehensive and accurate written records were scarce, and after it, philosophers, geometers, arithmeticians, politicians and many other Greek tradespeople began to more habitually compose detailed written copies of their work (Tuplin, C & Rihll, T, 2002, p. 19), fundamentally changing the Greek civilisation. Without written records to share, the growth of knowledge and understanding in Greece could never have reached the heights it did. Documented works provided the bridge between those who wanted information and the information itself, catalysing the development of Greek education.
There are many Ancient Greeks who were and are held in high regard for the work they did in their individual fields. The Classical Era is sometimes referred to as the “Heroic Age of Mathematics”, for “seldom either before or since have men with so little to work with tackled mathematical problems of such fundamental significance” (Boyer & Merzbach, 1989, p. 73). One such Greek who had a significant impact upon Greek mathematics was Hippocrates of Chios (refer Appendix 3). He was a Greek geometer who lived during the mid to late fifth century BC, in the middle of the Classical Age. From the island of Chios, Hippocrates travelled to Athens in approximately 450 BC. Accounts differ on whether he travelled there because lost his money through fraud in Byzantium or was robbed by pirates (Burton, 1995, p. 118). Regardless, either way he remained in Athens for he turned to the study of geometry during his time there and became the dominant figure in mathematics of the second half of the fifth century BC.
“Hippocrates of Chios was a merchant who fell in with a piraté ship and lost all his possessions. He came to Athens to prosecute the pirates and, staying a long time in Athens by reason of the indictment, consorted with philosophers, and reached such proficiency in geometry that he tried to effect the quadrature of the circle.” – (Philoponus, “Commentary on Aristotle’s Physics”, 3–9)
The ‘quadrature’ or ‘squaring’ of the circle was one of the three Classical problems devised in Ancient Greece. All three problems were found unsolvable in more recent history, however at the time Hippocrates managed to make major progress and partially ‘square the circle’ with his quadrature of lunes. While this was a momentous achievement, his work on the second Classical problem, ‘doubling the cube’, had more of a lasting impact on Greek civilisation. He observed that the problem of doubling the cube is reducible to that of finding two mean proportionals in continued proportion between two straight lines (Heath, 1981, p. 183). This was a novel and groundbreaking method of solving problems by simplifying questions to an easier equivalent, which not only made future mathematics easier to solve, but it also introduced the idea that problems can be approached from different angles. This impacted particularly on the political aspect of Ancient Greece, and politicians were encouraged to train in mathematics because the clear and logical way of thinking that is practised in the subject had become valued in Greek civilisation (Tuplin & Rihll, 2002, p. 22). Hippocrates is also credited with creating the first known systematic works on the elements of geography, nearly a century before Euclid’s prominent “Elements”; in fact, it is highly likely that Euclid’s work had roots in that of Hippocrates.
“Hippocrates of Chios, the discoverer of the quadrature of the lune, ... was the first of whom it is recorded that he actually compiled ‘Elements’.” – (Proclus, Commentary on Euclid’s Elements, 1. 66)
Being the first known systematic works on mathematics, its creation was of great significance to the Ancient Greek culture. As aforementioned, written works enabled more people to learn more easily, therefore it meant that Hippocrates had essentially catalysed the spread of routine education in Ancient Greece. Hippocrates of Chios left a legacy in Greece that effected the fundamental Greek civilisation in the areas of mathematics, human thought, politics and education.
The development of mathematics during the Classical Age was revolutionary in not only its own field but also in the growth of the Ancient Greek civilisation. The factors of geography and economy in Ancient Greece worked together to create an accommodating environment in which mathematics could prosper. Mathematics became a complex notion for the first time, and with this advance, perception on what that notion meant changed. It became theoretical, and there was a need for all previous mathematical concepts to be revised. Greek mathematicians such as Hippocrates of Chios made significant progress in the field that fundamentally changed Ancient Greek society. The Greeks were the creators of mathematics as known today, and without their work, the Greek civilisation and culture would not have been the rich centre of knowledge and understanding it was.
S O U R C E S R E F E R E N C E D
Unfortunately, this is an old essay from 4 or 5 years ago and I no longer have my bibliography. I was lucky to recover the content. However, my in-text references still exist so it is possible for the reader to find these sources if they are so inclined.
コメント