Axioms, Proofs, and the Delicacy of the Elements

Hey guys!

About 2300 years ago, an old dude with a beard (or possibly several old dudes with beards, there's still controversy over whether Euclid was one person or a whole team) published the biggest mathematical blockbuster ever written: a journal/textbook consisting of 13 books that changed the face of human knowledge forever. It was a collection of axioms, postulates, and proofs that essentially created the subject of geometry and is still the main content of every geometry course taken in school. The old dude, Euclid, christened his book 'Elements'.

Before we delve into the content of the book, let's spend a moment just grovelling over how the book almost literally created the subject of math. Besides from being one of the most published books ever (second only to the Bible), it was also a masterpiece in what we would now call logic. Elements was not entirely Euclid's creation, however. It was largely composed of proofs by earlier mathematicians and was a compilation of the knowledge of the ancient Greeks. Though most of it was about geometry, it also contained some proofs which kickstarted the subject of number theory.

Probably the single most influential mathematician of all time.

The most important thing about Elements is that it gave us a way to do math never thought of before: using well defined logical and structural arguments called proofs. What these did was establish a statement, not as a conditional truth, but as a universal law. Proofs give us a way to generalize conjectures and are necessary for the functioning of math today.

So without further ado, let's begin!

Note: If you're reading this, I think it's worth giving Elements a read. The book is really a remarkable work of art and will excite you in different ways.

Note 2: If you're versed in math, it's totally alright to skip this post. It's really just a Basics for Beginners which carries only the important stuff you have to know.

Definitions

I think it's good to sum up the definitions given by Euclid for a good understanding of the content. The definitions usually begin each sub-book, but by far the most important are those of the first book which I summarize here in simple words.

1) A point is that which has no part.

2) A line is a length which has no breadth (this includes curves).

3) A line is made of points. 

4) A straight line is one which lies evenly between two points.

5) A surface is that which has length and breadth.

6) A surface is made of lines.

7) A plane surface is one that lies evenly between two lines.

8) An angle is the inclination between two intersecting (read: cutting) lines. 

9) If the lines are straight, then the angle is a rectilinear angle.

10) If two angles cut straight through each other in such a way that the angles on either side of the lines are equal, then it's called a right angle.

11) Angles greater than a right angle are called obtuse.

12) Angles smaller than a right angle are called acute.

13) A boundary is the edge of something.

14) A figure is something which has boundaries.

15) A circle is a special figure which consists of all the points equidistant from a center. 

16) The center of a circle is the point from which the other points are equidistant.

17) The diameter of a circle is a line which connects two points on opposite sides of a circle.

18) A semicircle is half a circle.

19) Rectilinear figures are those which have a boundary of lines.

20) This one describes types of triangles:

     1) Equilateral- Having all 3 sides equal.

     2) Isosceles- Having 2 sides equal.

     3) Scalene- Having no equal sides.

21) Right angled triangles have a right angle, obtuse angled ones have an obtuse angle, and acute angled ones have only acute angles.

22) This one describes 4 sided figures or quadrilaterals:

     1) Square: All equal sides, all angles 90 degrees.

     2) Rectangle:  All angles 90 degrees. Opposite sides equal.

     3) Rhombus: All sides equal.

     4) The rest: Trapezia.

23) Lines which never meet are called parallel.

Phew! 

We're finally done with the definitions. They shouldn't be very hard to learn, even those completely new to math learn this in no time. Keep referring back to the definitions if you feel confused.

Axioms/Postulates

Now we're getting to really interesting things. Axioms or postulates are the building blocks of math; what they do is give a few common statements which are so obvious that they don't need to be proved. Once we have these clear, we can prove whatever we want.

Note: Okay, not really. An incredibly smart dude called Kurt Godel proved in the 1930s that we can't prove everything from our axioms, no matter how many we have. This literally broke math and is despised by many mathematicians even today. The proof is really neat and you can get the gist of it quite easily. But since we really haven't reached that point yet, I'm assuming that we can prove everything.

Let's look at the various axioms we have.

Axioms

1) Things equal to the same thing are equal to each other.

2) If you add equal things to equal things, the results are equal.

3) If you subtract equal things from equal things, the results are equal.

4) Things that coincide are equal.

5) A whole is greater than a part.

Really obvious (though tongue-twisting) statements. The postulates are much cooler. I'll be discussing each individually.

Postulates

1) You can draw a straight line from any point to any point.

This one is obvious. On a plane, you can draw exactly one line between two points. The line equation models this line if you know the point coordinates.

2) You can extend a line as much as you want.

This one is kind of vague, but makes sense. After all, if a line has only length and no breadth, it has infinite length, right? I can make it as long as I want.

3) You can draw a circle with any radius.

Also simple. I can mark as many points equally away from the center as I feel like.

4) All right angles are equal to each other.

Yep. In fact, all angles with equal magnitude have the exact same properties.

5)  If I have a line, and any point not on the line, I can draw exactly one and only one parallel line which passes through the point.

Aaaah. The parallel postulate. Arguably the most famous, this one has troubled mathematicians for centuries. Mathematicians used to believe that this postulate wasn't really a postulate, but could be proved from the other postulates and axioms. How wrong they were.

The postulate led us to create whole new types of geometries: A mathematician called Bernhard Riemann, arguably the greatest mathematician of the 19th century used loopholes and twists in this postulate to dream up crazy things. 

Now we have the stage set and the characters introduced. As the bells ring, the curtains unfold. It's time for the play to begin...

Proofs

Logical arguments for establishing conjectures to be generalized for any case are known as proofs. In other words, proofs are processes which demonstrate something to be universally true. Every mathematical statement must have a proof for the mathematical committee to accept it.

This doesn't mean that all proofs are short and sweet. Most can be long-winding and difficult to  relate to. Often, you seem to forget what the whole thing was about and what steps you need to take to reach the required goal. Equations as simple as a = b + c can have hidden secrets to them.

But let this not change the intrinsic beauty of a proof. Unlike the observations we make in science, proofs are universal. If it's true here, it's true in both Mordor and Naboo. 

There are tons of types of proofs, but I've highlighted a few that really have been game changers. I've also given an example with each, just to show the way we can use these to create truly startling conclusions.

Direct Proof

These proofs start with a statement A and use axioms and already established statements to prove B. The most common method of proof, a direct proof is perhaps the most aesthetically pleasing. It's truly wonderful, how a few statements can change the face of something.

Example

One of my favorites: The square of an odd number is also odd. It's really neat and concise. A few simple equations establish something great and profound.



Let any odd number be 2k + 1 for any integer k.

Where m is any integer. Thus, 2m + 1 is an odd number.

Mathematical Induction

This method basically uses that fact that the conjecture is true for the number 1, and extends it to apply for all other numbers. It's a special kind of proof that can be quite helpful in proving statements with no direct proof.

However, the proof by mathematical induction is kind of different, because instead of proving why a conjecture is universally true, it instead establishes itself as true for all numbers.

Example

Another classic, the proof that dictates the sum of first n numbers. It's slightly long, but gives great results.

Let the sum of the first n natural numbers be

Now, we need to prove that 

We know that this is true for 1. Let's assume that it's also true for any other natural number n.

Note: While this step may seem unreasonable as there may be no number n which satisfies the identity, actually since the number 1 can also be n and we have proved it for 1, technically we are proving it for all numbers.

Now, it's true for some number n and 1. Let's try to prove it for n + 1. If the conjecture is true for n + 1,

We know that

Our Assumption matches the result. QED.

Proof by Contradiction (AKA Reductio ad Absurdum) 

This kind of proof essentially says 'Hey! Since we can't prove this statement true, let's try to prove this statement false and fail!'

This is probably the single coolest proof invented in math, like, ever. The proof literally sweats logical mechanisms.

Example 1

The most beautiful, elegant, and wondrous proof of all time which shattered to pieces millennia of mathematical thought by creating numbers which we cannot hope of visualizing or creating in the real world: the irrationality of √2. 

Example 2

My favorite proof personally, the proof that there are an infinite amount of prime numbers never ceases to amaze.

Proof by Exhaustion

Although this is quite an important type of proof, mathematicians have argued for centuries that exhaustion is not really a kind of proof. It's basically a way of proving something to be true by individually checking every case that's possible. It's not very elegant, but it works, and that's mainly why it's on this list.

Example

The four color theorem which was proved by computer is a great example. It's not a very great proof though, because it still has around 600 cases.

Beyond Euclid

This probably seemed like a boring post, but it's really the only post which has so much info dumped into it. The main reason I did this was so that I could sum up all the basics for beginners into one single post. Don't be afraid to use this as a glossary in case you get confused; it was necessary for me to make this reference page. The really awesome things will start from the next geometry post.

Yay!

Thanks for reading!