For this blog, I am going to complete the ‘doing math’ requirement.
Euclid didn’t just develop the concepts of proofs, he also developed a set of conjectures and propositions that connected all proofs together and allowed them to make sense while looking at them piece by piece. Euclid used the most general concepts to create these proofs that explain larger theories, in an attempt to prove the proposition or conjecture correct. In the process of explaining these theories, we started with proving theorems. Theories consist of a set of ideas that are used in order to explain why something is true, while a theorem is a result that can be proven to be true using axioms or postulates. We have proven our theories and shown that they are true through proving theorems and conjectures, showing that the invention of the proof is more important than we think. In the process, he uses complicated wording and bases most of his geometric proofs off of his five postulates.
Euclid’s five postulate, representations shown above, consist of: “1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The fifth postulate is known as the parallel postulate ” (Wolfram Alpha). Euclid did not just use these five postulates while creating the idea of the proof and proving different theories, he also had a set of common notions.
These common notions were used to guide the rules of completing proofs and what should commonly be followed by others when attempting to do so. Since Euclid has created the concept of a proof, things have changed. Euclid brought the idea of proofs into the world of mathematics, showing that there is a strategic and more sophisticated way of proving an idea. Euclid set the groundwork for bigger and better things that occurred later on in the mathematical world.
Since Euclid has created proofs using conjectures, postulates, and theorems, proofs and mathematics have come a long way. Since math has continued to develop, the list of ways to prove things has advanced. We have gone beyond the idea that geometry is the only thing that can be proven using axioms and postulates. We have advanced into more complicated areas of mathematics, like proving that an equation can be solved or proving that something equals something else. Euclid primarily focused on geometry, leaving so much more room for improvement. For example, the creation of the proof and Euclidean geometry lead to the creation of Hyperbolic and Elliptic geometry.
Although he only focused on geometry, he did get a lot of things right. The idea that there needed to be a basic list of axioms or postulates in order to start, set the framework for the concept of writing proofs and was the biggest improvement in the world of mathematics. Euclid even set up the perfect outline for writing proofs, starting with what you know and ending with what you wanted to prove. To this day, this is still the general outline for how proofs should be written. The idea of combining multiple conjectures in order to create one proof through valid reasoning is another thing that Euclid did right, the need for explanations is essential while writing a proof. If the reasoning is correct, there is no room for argument, therefore showing that the reasoning about why the conjecture is correct serves as a valid proof.
Now that all I’ve done is talk about Euclid and his works, now we should check out just how some of it works. Many of the conjectures and theorems could be proved just using the five postulates or the axioms, which was very interesting! I am going to prove a theorem doing exactly that:
Consider the following incidence geometry defined by these three axioms:
A1. There are exactly four lines in the geometry.
A2. Given any two distinct lines, there exists exactly one point that lies on both of them.
A3. Every point is on exactly two lines.
Theorem: Each line in this geometry contains exactly three points.
According to the first axiom there are exactly for lines in this geometry. Call these four lines l, m, n, and o. Take line l. By the second axiom, lines m, n, and o must each have a point in common with line l.
The intersection of these three lines (m, n, and o) with l, create three distinct points at their intersections. The reason that these points are distinct is because if they were not distinct they would violate the third axiom, creating a point on three lines.
Line l therefore contains three distinct points at the intersection with line m, n, and o.
Suppose we have a fourth point P on line l. P cannot be on line m, n, or o because doing so would violate the second axiom. But every point is on exactly two lines, thus we would need a fifth line in order for point P to exist. But this violates the first axiom that only allows for four lines in the geometry. Thus we cannot add a fourth point to line l.
Since we cannot add another point to line l and it must have at least three points at the intersections with m, n, and o, l must have exactly three points on it.
Similar arguments can be made for lines m, n, and o. Therefore, each line in this geometry contains exactly three points. QED.
Euclid also was able to prove the pythagorean theorem in many different ways using the proof! I am going to take the pythagorean theorem and prove it only one way but there are many!
Theorem: If triangle ABC is a right triangle, then the measure of angle c will equal 90º, b = |AC|, c = |AB|, and a = |CB|, then a² + b² = c².
One of the many ways of proving the pythagorean theorem includes a square inscribed within a square. We can prove this algebraically, which makes it much easier to see. First, we can separate this into the square contained on the inside, whose area is c². Next, we have 4 triangles located on the outside of the square, all with area ¹⁄2ab. Lastly, we have the area of the big square, which is (a+b)². Now, the are of the large square must equal the area of the small triangles and the smaller square contained within so we can set these numbers equal to each other…
c² + 4(1/2 ab) = (a+b)²
c² = (a+b)² – 4(1/2 ab)
c² = a² + 2ab + b² – 2ab
c² = a² + b²
This allows us to see why the pythagorean theorem works and it proves the pythagorean theorem at the same time!
I believed that proof mattered to Euclid because it was a solid base for the studies and work that he created for Euclidean geometry. He was working with geometry, attempting to expand on what he was finding. In order to expand, there needed to be a base, something to work up from. Proofs were important to Euclid because it shows that his work was expanding, and it shows that his work is valid. He created the framework for proofs, and this became more and more important to him as he began to make more and more discoveries about how complicated geometry is. But when you think about the history of math and the advancements that were made, you can ask yourself, why did Euclid create the proof? In my opinion, I think that he created the proof because there was no other formal way of completing mathematics. I think that Euclid wanted a concrete format for proving theorems. If these theorems were proved, they would be able to log them and build the world of mathematics on top of them instead of continuing to work with ideas that may or may not be true. To this day, proofs are the most important framework for figuring out whether ideas are true or not. Mathematicians and even scientists work to prove theories and new ideas through already proven ideas, if the framework was not set through these proofs then there would not be the advancements that we have today.