How was Greek mathematics helped by using proofs How was it

How was Greek mathematics helped by using proofs?  How was it hindered?  Give an example of each using lots of mathematical detail.

Solution

The Ancient Greeks played a huge role in the development of mathematics. They started mathematics on a journey that has made it what it is today.The idea of proof in mathematics was brought about by the Ancient Greeks. This came at a time when it wasn’t enough just to know that something worked, but Thales, Pythagoras, Euclid and many others wanted to know why.

Developing proofs into theorems, especially in the area of geometry, strengthened the basis that was already intuitively clear. Now the equality of vertical angles formed by two intersecting lines was proven by logical reasons, rather than it just being obvious, or shown by repeated experiments.

Proof based mathematics differs greatly from the non proof based. One historical example is the Pythagorean Theorem. The Babylonians knew how to find Pythagorean triples, and even primitive triples, but they did not have a formal proof for this. The first general proof was given by Pythagoras, and it most likely was a dissection proof.

The non-proof based mathematics relies completely on intuition and experimentation. Proof based mathematics uses that intuition and develops it into a reason “why”, and proves it for all cases, not just the experiments.Greek mathematics was both helped and hindered by the proof. The Greeks’ introduction of proofs into the math world totally changed that world from then on. Not only did they construct many, many proofs in their time, but they solidified the foundation of mathematics. They also inspired mathematicians of the future to build on this foundation and explore mathematics in great depth and the proof became the basis of math. Ideas and hypotheses were no longer accepted just because they seemed to work, but they had to be proven that they worked, and why. Greek mathematics was hindered by the proof with discovery of irrational numbers. It was believed and practiced in the Pythagorean school that whole numbers caused people and things to have certain qualities, and that some numbers themselves had a mysticism about them. One of Euclid’s many contributions to Ancient Greek mathematics was a process called the Euclidean algorithm. It is used to find the greatest common divisor of two or more integers. It is one of the oldest algorithms and the reason it’s so important is because it doesn’t require factoring of the two integers. It begins with two integers,a and b. Start with dividing a by b to determine the remainder. Repeat the step, this time divide b by the remainder and find a new remainder. Repeat this step until the remainder is zero, and the previous remainder is the greatest common divisor.

Using proof to solve this problem is useful because it can greatly reduce the amount of time it takes to find the gcd. On the other hand, we learned at a very young age to factor numbers, and using that intuition can help in understanding the problem better.

For a typical student, solving a problem intuitively is definitely a step in the right direction. Knowing that something works gives you a reason to find out why. Being able to solve a problem using proof shows your maturity as a mathematician and strengthens your understanding of the problem at hand. The fact that the Ancient Greeks invented the proof displays their great abilities as mathematicians and we celebrate their contributions to the math world.