Greek Mathematical Proof
Mathematical is a subject with a great history. The major historical contributors to mathematics history are the Greeks. They were the first people to try and provide mathematical proof. It was based on mathematical reasoning. The first person behind this important historical achievements is the Thales of Mellitus (634-548 B.C.). He was behind the earliest discoveries in the mathematical fields.
Thales of Mellitus was the first person to provide mathematical proof through the concept of deductive reasoning. The Greeks applied deductive reasoning during this era to every aspect of their thoughts. The deductive reasoning aspect of mathematics functions based on assumptions that their certain factors that known or assumed to be true. And Hence deductive reasoning Is used to discover the new facts. According to Thales of Mellitus, through deductive reasoning, known premises are known, and there is an assumption from which a new fact is derived. The knowledge Thales used to give mathematical proofs was the foundation laid out by the Egyptians and the Babylonians (Bramlett, & Drake, pg. 21, 2013). Don't use plagiarised sources.Get your custom essay just from $11/page
One of the earliest proof concepts of Thales was proof that a circle is divided into two equal parts. Thales of Mellitus also proved that the base angles of an Isosceles triangle are equal. Another one was that the vertical angles formed by two intersecting lines are equal. Besides, two angles are said to be congruent if they have two angles. Out of the two angles, one side in each of the angles are equal, respectively. Besides, any angle that is inscribed in a semicircle is a right angle (Bramlett & Drake, pg. 23, 2013). The proof is as follows;
Angle CDA is a right-angle triangle.
One of the proof illustrations is as follows; Draw Triangle ABC. Draw a line from A to the point B.C. to bisect the line into two halves. If line AD=DC=BD, then a circle can be drawn such that D is the focal point, and A.D. will be the radius. The circle will touch points A, B, and C. This will create a rectangle ABEC and a diagonal D.E. Hence, triangle ABC and AEC are equal since they have A.C. in common. Sides AB and E.C. are equal. Angles BAC and ECA are right angles (Bramlett, & Drake, pg. 23. 2013).
The Pythagoreans theorem is another Greek mathematical proof. The origin of this proof started with showing that is irrational.
Assume that is rational, Then numbers A and B s.t
=A/B
=2
then is even.
So A=2a where a is the whole number
Rewriting it,
it gives
Other proofs are shown in the picture below.
The Pythagoreans are one of the earliest known proofs that were invented during the Greek age. It was taught to the Greek children and is still learned even today in schools. The earliest proof received some criticisms that it was not well done. However, Hippasus of mespontum devised another method, as shown above.
Another mathematic proof is the Euclidean geometry. Euclid’s works remain even today to be the earliest mathematical proofs that are still prominent for approximately over the past 2000 years. Euclid’s started his mathematical proofs of geometry by twenty-three definitions. The definitions were based on lines, points, circles as well as the several various concepts, ten assumptions including five postulates. He also applied the aspect of deductive reasoning to prove other concepts in mathematics. He used axioms that were based on logical beliefs with the assumption that they were true instead of proved. This method of proof was dubbed the Axiomatic method.
Difference between proof Based mathematics and non-Proof based Mathematics
Although mathematic proofs are based on truth, they have many more elements that what we know. Mathematical proofs are based on providing a logic behind a certain mathematical problem. In other words, mathematical proofs depict the use of logic to establish a path through the problems presented in mathematical concepts. Mathematical concepts are categorized into proof-based and non-proof based. The concepts are classified in such ways because mathematical is concrete as well as influenced by other external forces. Non-proof-based mathematics is said to have no application of deductive reasoning in providing solutions. It is based on observation than deductive reasoning. In such a form of non-proof based mathematics, the answers are right, but it is difficult to explore how or why such answers are claimed to the real ones or true. A good example of the ancient non-proof mathematics is the one used by the Egyptians. The Egyptians used the wrong formula to calculate the area of a quadrilateral. The formula is as follows (carrymo, Bellimo, n.d. 2017);
K=(a+b) (b+d)/4
From the review of the formula above, it looks like a guessing concept. The formula seems to be derived from visual observations. With the advancement in mathematics, the understanding of the formula above was questionable, and it is not applied today.
Proof-based mathematics, on the other hand, focuses on using deductive reasoning to prove a problem. It is opposite to that of the Egyptians. Proof-based mathematics was pioneered by the Greeks. The Greeks used this method based on philosophical and rationality. They applied a different approach or organizing and making sense of the mathematical problems and surrounding problems around them (carrymo, Bellimo, n.d. 2017). Through deductive reasoning, mathematics was transformed into proof-based aimed at establishing order on the world they were in. The proof-based mathematics accomplished many things and even proved that the non-proof based mathematics was not logical and not understandable. Hence, the Babylonians and the Egyptians were outshined in their non-proof based mathematics. A factor that made them successful was their ambition to know how and why something was behaving the way it is. The proof-based mathematics incorporated the use of philosophies to dig deep on the logic and reasoning behind equations and numbers (Bramlett & Drake, pg. 24. 2013). An example of hospital proof-based mathematics is the Pythagoreans theorem done by Hippasus.
Assume that is rational, Then numbers A and B s.t
=A/B
=2
then is even.
So A=2a where a is the whole number
Rewriting it, it gives
How was Greek mathematics helped by using proofs? How was it hindered? Give an example of each.
Various proofs helped Greek mathematics. They involved all the proofs that were invented by the earliest Greek scholars. Using the proofs, Greek mathematics developed into deductive reason mathematics. The Greeks applied deductive reason in various aspects of their life. Through deductive reasoning, they assumed that there are known factors that are assumed to be true. Them the assumed factors are used to identifying new facts which, on the other hand, derives conclusions. A good example of mathematical proof influence mathematical is the assumption that all men are mortals, hence Socrates is a man, and he is mortal.
The mathematical proof of Thales of Mellitus was the initial one of the major contributors to Greek mathematics through deductive reasoning. He provides various proofs such as a circle can be bisected by any diameter, the angles of an isosceles triangle base are equal, any angle that is inscribed in a semicircle is a right angle, among other. This helped in Greek mathematics in understanding the mathematics of angles.
Another proof that influenced Greek mathematics is the Pythagoreans theorem. The Hippasus of mespontum did this. He proved that the root of 2 is irrational; this helped in the calculation of the hypotenuse value. It created the hypotenuse calculation formula. Greeks used the assumption of rationality of numbers to calculate future mathematical problems. The mathematical geometry proofs further influenced Greek mathematics. He devised a mathematical method known as the axiomatic method. It was founded on the understanding of the twenty-three key elements; lines, circles, assumptions as well as postulates. He provided a foundation and a formal structure on which the future of mathematics would thrive.
Various factors hindered the Greek mathematics proofs. First of all, they were based on philosophy done on verbal proofs as well as constructive proofs. This was a challenge as it hindered them by restricting them to certain mathematic sections. Their mathematical proofs were mainly not based on the use of algebra. As a result of the lack of creating algebra symbols. The renaissance of mathematics and intuition was a challenge that removed them from the game. The rise of symbolic notation also hindered the growth of Greek mathematics.
As a student, deciding on the best method to apply in your mathematical problems is quite a challenge. Mathematical is a concrete subject with various elements behind it. Providing solutions to mathematical problems can be into two forms; intuition and proof. Intuition is similar to non-proof based mathematics. The Egyptians and Babylonians used the aspect of intuition.
A good example is their earliest formula for calculating the area of a quadrilateral. The formula is based on no proof as to why the answer is the way it is. Besides, it does not explain the logic behind the formula. Intuition Is like guessing. As a student, using this form in providing solutions is not the best for me. It becomes difficult to explain to someone who wants to know more beyond the answer.
The best option is the use of proof to solve a problem for a typical student. A typical problem provides more insights into a problem. By proof methodology, it provides a rational approach to a problem. Also, you tend to get the correct answers that are supported by facts. As a student using proof methodology can help you to understand why and how a problem behaves the way it is and whether the answer is correct or not. Using the intuition method is just a literal method that should not be applied to mathematical problems. It analyses the surface of aa problem instead of the deep insights of a problem. Moreover, a typical student should use the proof method as it helps the student to dig deep into the logic as well as the reasoning behind equations and other mathematical concepts. You can be able to justify your answer to the third parties. In most cases, people who use intuition find it difficult to explain and defend their self-proclaimed solutions.
Work Cited
Bramlett, D., and C. Drake. “A History of Mathematical Proof: Ancient Greece to the Computer Age.” Journal of Mathematical Sciences & Mathematics Education 8.2 (2013): 20-33.
carrymo, Bellimo. “Proof And Non-Proof Based Mathematics Essay – 779 Words”. Studymode, 2017, https://www.studymode.com/essays/Proof-And-Non-Proof-Based-Mathematics-1655604.html.