Theorems And Proofs In Geometry The History Of Their Appearance

Theorems and proofs both play a crucial role in mathematics as they serve as something more than the justification behind a result. Theorems are statements that can be demonstrated to be true by accepted mathematical operations and arguments. A theorem makes up some general principle, meaning it is part of a larger theory. The process of showing a theorem to be true is called a proof. The two simply tie together as they are both what geometry revolves around. With the essential role each plays, theorems and proofs have most likely been around before the dawn of recorded history.

Geometry can easily be traced back to being one of the oldest branches of mathematics. Humanity, presumably, has been incorporating geometrical techniques since before the beginning of history. Whether it be for the construction of their buildings, boats, or tools, humans to this day need to have some inbuilt mechanism and instinct for judging distances, angles, and height. It is known that, the Egyptians, Babylonians and Indus were among the first to incorporate and use many of such geometrical techniques however they never took interest in finding out the rules and axioms governing geometry. As years went by,“the Greeks insisted that geometric fact must be established, … by deductive reasoning; …” (Historical Topics, 171), meaning they believed that geometrical truth would should be found by studying rather than experimenting. The Greeks’ deductive approach has become one of the foundations of geometry, their persistent questioning and search for justification is lived on to today’s modern society as it’s still applied.

Why were the Greeks so persistent in expressing that geometric facts had to be establish by deductive reasoning? Well, deductive reasoning is the process by which a person makes conclusions based on previously known facts. An example of deductive reasoning might go something like this: “a person knows that all the men in a certain room are bakers, that all bakers get up early to bake bread in the morning, and that Jim is in that specific room. Knowing these statements to be true, a person could deductively reason that Jim gets up early in the morning. Such a method of reasoning is a step-by-step process of drawing conclusions based on previously known truths” (SparkNotes). It’s with this deductive reasoning in which geometric proofs; the process that proves theorems to be true, are written.

Deductive reasoning is the method by which conclusions are drawn in geometric proofs. In geometry, deductive reasoning is much like the situation with Jim described above, except it relates to geometric terms. For example, “given that a certain quadrilateral is a rectangle, and that all rectangles have equal diagonals, what can you deduce about the diagonals of this specific rectangle? They are equal, of course. The premises used in deductive reasoning, in many ways are the most important part of the entire deductive reasoning process” (SparkNotes). If any premises are incorrect, then the foundation of the entire line of reasoning becomes unreliable, and nothing can be reliably concluded. Even if just one conclusion is incorrect, every conclusion after that is undependable, so it might as well be incorrect too. All in all, deductive reasoning is fully effective when all of the propositions are true, and each step in the process of deductive reasoning follows the previous step logically. With such persistent in deductive reasoning being needed, the Greeks easily became the first to move mathematics into a realm of theory, reasoning, and deduction rather than of measurement.

Geometry evolved as it was shaped by many Greek Mathematicians, Thales of Miletus being one of them. As a student, Thales traveled in attempt to learn as much possible from the scholars and their trial and error methods for solving mathematical problems. Quickly Thales came to question why and how the numbers were fitted the way they were. He questioned the scholars and the lack of elegance and verification behind their trial and error methods. Reflecting on the scholars and their methods Thales believed that reasoning should take the place of experimentation and intuition. He then began to look for solid principles which he could build theorems from. Doing so introduced the idea of proofs in geometry. He proposed some axioms that he believed to be mathematical truths. After his traveling expedition Thales set up a school to teach others all he’d come to know, still trying to establish axioms (mathematical proofs). By gathering, applying, and passing forward all he believed in, Thales later came to earn the title “Father of Mathematics” since it was him who taught the many mathematicians that would follow and build upon his theories.

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Alongside Thales, Euclid is another very significant name in the history of Greek geometry, so relevant that he is recognized as the “Father of Geometry.” Euclid gathered the work crafter by prior mathematicians and developed his legendary creation, 'The Elements,' one of the most published, influential, non-religious texts in history. The manner in which he used logic and demanded verification (proof) for every theorem, shaped the ideas of western philosophers to ones present day. Euclid’s novel served as a primary source of geometric reasoning, theorems, and methods up until the advent of non-euclid geometry in the 19th century. He put together his methodology using ten axioms, statements, that could be accepted as truths. Euclid determined his ten axioms as his ‘postulates’ and separated them into equal groups, both of five. The first set was universal to all mathematics and the second explicit to geometry. His theorems may come off as self-explanatory to many individuals of this day, even so Euclid worked upon the rule that no aphorism could be acknowledged without verification. Although a significant number of Euclid's results had been expressed by earlier Greek mathematicians, Euclid was the first to demonstrates how these propositions could be incorporated together into a comprehensive deductive and logical system much like the ones used current day.

When Euclid published The Elements in Alexandria, Egypt circa 300 BC the theorems were not perceived as common knowledge then. Euclid felt that anybody who could read and understand words could understand his notions and postulates. To make his theorems easier to apprehend, he incorporated 23 definitions of common words, such as 'point' and 'line', to ensure that there could be no semantic errors. From this basis, he built his entire theory of plane geometry, that too shaped mathematics, science and philosophy for centuries.

The Greeks did not have resources like the ones accessible present day. The times of the Greeks was a time of discovery since there was much to be discovered. Greeks lived in an era in which discoveries were needed for the function of their daily lives. Unlike their generation, in this one, there have been things passed down from such influential individuals like Pythagoras, Archimedes, Thales, Euclid, and many more influencers who have left behind revolutionary discoveries, developments, systems, strategies, and theories. What is passed down has been taught and further developed, now, theorems to many come off as simple, basic, and just straight forward. It is thanks to such mathematicians and other philosophers who have left behind the things they have developed (novels) that the people of this generation are taught as soon as they enter school. As they learn in it, they become familiar with the concepts being passed down and naturally one incorporates concepts into their lives. Greek developments are still used in modern day.

Geometry many times gets overlooked, as people don’t come to see how it is already a part of their lives. It is what people are surrounded by, it serves as the answer to solutions, it is the explanation to why things are the way they are. Take a moment to reflect, look around you, you’re probably in a room. Possibly a room that is full of parallel lines, it’s walls are equidistant and make equal right angles with the floor, all when observing the properties and using Euclid’s 5th postulate are proved. The simplest of things surrounding us have some correlation to geometry have most likely have reasoning behind why they’re the way that they are. That reasoning is a proofs.

A proof is an argument, a justification, a reason that something is true. It has to be a particular kind of reasoning – logical – to be called a proof. A proof is the answer to the question “Why?”. In geometry it would sound something like, “Why is the area of a rectangle the product of its side lengths?”, and because of the previous theorems and postulates created one could provide the another with an answer that’s impossible to argue with. “If we agree on the premises – the definitions of the terminology being used – and we agree that, if A implies B, and A is true, then B is true, then we agree on the conclusion” (Cooper, 3). There’s no way around it. In fact, you use “proofs” all the time.

Many, real-life situations require reasoning that is, at least in part, something that would qualify as a mathematical proof. When reasoning with yourself that it will be more affordable to buy the larger cans of beans, you are proving something about the respective prices. At the very least, mathematical-type reasoning is a powerful addition to anyone’s critical thinking toolbox, applicable in a wide variety of settings. Mathematics is also a cornerstone of the sciences, which in turn provide a profoundly useful way to understand and interpret the world around us.