One of the greatest tools we have today in mathematics is the ability to use previously proven ideas such as theorems in order to keep discovering more mathematics. This is exemplified in the mathematical findings done by the Greeks. They used the ideas of others as a stepping stone to develop their own, and these ideas were later used by other mathematicians such as Euclid and myself, specifically their discover of the deductive proof method. Although the Greeks adapted many of their ideas from the Romans and Egyptians, they were also able to develop some of their own.
One of their most well known developments was in the field of Geometry. To begin, in roughly 600 BC Thales developed what is known as Thales Theorem which states that if one side of a triangle is the diameter of a circle, then that triangle is a right triangle. Thales also developed the Intercept Theorem which describes the similar areas created when parallel lines intersect to intersecting lines.
During this time, the most well-known Greek mathematician, Pythagoras, also made substantial contributions to Geometry in Greece. Unknown to many, he was the first to think of math as a system. He realized that geometry could represent numbers. His most well known work is the Pythagorean Theorem which he developed using a rope. Although Thales and Pythagoras both in some way discovered geometry and made contributions to the field, Pythagoras is more well known.
Later in the fifth century BC, a third important contributor to geometry, Hippocrates of Chios, surfaces. He worked on one of the three famous problems: constructing a square with an area equivalent to that of a given circle. His book entitled The Elements was written in 440 BC and highlighted the basics of geometry. Chios’ and The Elements were later used by Euclid.
Also in the fifth century BC, Zeno of Elea used a version of the famous Tortoise and Hare story to come to develop the idea of infinity. He considered the following situation: If the tortoise starts the race ahead of Achilles, and the Achilles always catches up to the tortoise, how can the Achilles beat the tortoise? In theory, Achilles should have gained a little more distance between himself and the tortoise during the race until he was eventually within an infinitely small distance away from Achilles, but he never passed Achilles.
This paradox was called the Dichotomy Paradox and it was used by Zeno to introduce the concept of infinity.
Similarly, Democritus is thought to have come up with the idea of breaking a whole into smaller parts and again, dipping into the theory (at this point) of infinity during the late fifth and early fourth century BC. Instead of looking at the Dichotomy Paradox, he worked with volume and divided matter into cross-sections.
Around 360 BC, Eudoxus of Cnidus, student of Plato whose focus was largely logic, developed a solution to find the volume of a prism by breaking it up into smaller sections. This may have also led to the idea of proportions he developed and irrational numbers.
Later in the third century BC, Euclid used many of the ideas stated previously such as the Pythagorean Theorem, number theory, ratios of divided sections, and plane geometry in the various books he wrote. Euclid formalized and did further research in these areas which were first developed by the Greeks. Although the Greeks laid much of the groundwork, Euclid is the most well-known the work done in them. For example, we have Euclidean geometry and Non-Euclidean geometry.
Even though the Greeks made many contributions to Euclid’s work, their greatest contribution to mathematics was developing a new method of proof. Previously, the Egyptians had used inductive reasoning and recurring patterns in developing truths in mathematics. The Greeks used rules of logic to prove and to disprove statements. This discovery is tremendous in the field of mathematics because it allows us to prove things that are not seemingly possible or imaginable.
One of our human limitations is time. If we solely rely on finding patterns and physically counting or measuring something in order to make a conclusion, we will run out of time. When we use proofs, we are able to use previously proven truths in order to prove more true things. We can generalize and be 100% sure in our findings without having to check the physical measure. This also allows us to prove things which are beyond our physical domain such as Euler’s famous equation
He did not prove this by guess and check, but proved it by using a series of true things.
The deductive proof method lets us consider what happens when a process occurs an infinite amount of times, better known as taking the limit of something. We don’t know what the limit of an expression is as it approaches infinity because it would take us to long to get there. But we can use proofs to find out.
In conclusion, the Greeks played a large role in the development of mathematics by discovering key groundwork ideas used by Euclid and other mathematicians. One of their greatest contributions was the deductive proof method which allows us to prove things beyond our physical domain.