Let's talk about numbers

In this post I intend to tell a concise and perfected story of how and why numbers came to be. Please don't expect this to be a factual account of all the trial and tribulations that accompanied the process. So, for instance, I won't dwell in all the misconceptions that plagued negative numbers for a long time, nor will I tell in a detailed way how complex numbers came to be (even though I'm saving this bit for a later post).

Here the story will have no false starts, no dead-ends and everything will flow perfectly and rationally.

— 1. Operations —

Let us first introduce the concept of an operation. For us an operation will be a process that will allows to act on a set of numbers to obtain another number:

• Addition, which is represented by the symbol ${+}$, as an operation that picks up two numbers and obtains a third one. In symbols: ${a+b=c}$.
  

• Subtraction is defined as the inverse operation to addition and is represented by the symbol ${-}$. So, we'll say that ${c-b=a}$ if it is ${a+b=c}$.
  

• Multiplication is represented by the symbol ${\times}$ or ${\cdot}$ and also associates two numbers to a third one: ${a \times b = c}$.
  

• Division is the inverse operation to multiplication and is represented by the symbol ${/}$. So we'll say that ${c/b=a}$ if it is ${ c = a \times b}$. Another way of representing ${c/b}$ is by the symbol ${\dfrac{c}{b}}$.

Fortunately for us as human beings these are not the only operations available but for now they are all that we need.

Some readers may be taken aback with the level of abstraction that I used and with the fact that we still don't know what these numbers really are. To those readers I say that in Mathematics this is the normal way to present things. Yes, I'll do things in a more digestible way from now on (cringe worthy to mathematicians I'd assume) but nevertheless I wanted to show how things work out in the real world.

— 2. Numbers —

Well, on to numbers we go. The first numbers that we need are what called the natural numbers. The symbol to denote them is ${ \mathbb{N} }$ and they appeared and evolved from the need to count how many objects of a given class we possess:

$\displaystyle 0,1,2,3,4,5,6,\ldots$

Where the symbols ${ \ldots }$ mean that that list doesn't have an end. Note that I included the number ${0}$ in this list. The history of the number ${0}$ is a fascinating one and I urge the reader to find out a little bit about it.

— 2.1. Operations with Natural numbers —

In order to break down the level of abstraction that we had while introducing mathematical operations we'll now see what they look like for natural numbers.

• Addition. The first thing I want to say about addition in the special case of natural numbers is that we can see it like this:

  $\displaystyle m+n=p$

  ${m}$ is the number from where we start and ${n}$ is the number of steps we'll walk till we stop in point ${p}$

  ${0}$ is giving no steps at all, ${1}$ is what we define as giving one step, and the natural numbers that follow are defined in relationship to ${1}$. For instance: we define ${2=1+1}$ and so we interpret ${n+2}$ as giving two steps starting from point ${n}$. All the natural numbers that follow have the definitions and interpretations that you expect them to have.
  

• Subtraction is the inverse operation to addition. We can see addition, ${m+n}$ as starting from ${m}$ and giving ${n}$ steps to the right, thus ${m-n}$ can be seen as starting from ${m}$ and giving ${n}$ steps to the left.

• Multiplication is defined to be consecutive additions. By that we mean that we interpret ${ 2 \times 3}$ as being ${2+2+2}$ and ${3 \times 2}$ as being ${3+3}$. As we can see it is ${2 \times 3 = 3 \times 2 = 6}$. In general we can say that for every natural number it is ${m \times n = n \times m = p}$
  

• Division is the inverse operation to multiplication and thus can be seen as consecutive subtractions.

  For example let's compute ${6/2}$.

  It is ${6-2=4}$. Now we take the result and go subtracting ${2}$ until we can.

  ${4-2=2}$, and ${2-2=0}$. Since we were allowed to make three subtractions and the end result was ${0}$ we say that ${6/2=3}$ with remainder ${0}$.

  Another example is ${7/3}$. This time it is ${7-3=4}$, ${4-3=1}$ and here we have to stop. This time we made two subtractions and ended up with ${1}$. So what we say is seven divided by three is two with remainder one.

The beautiful thing about numbers and these four operations is that now a whole new realm of nifty and useful things pop out almost for free. All that we need to do is to think and to be ambitious.

Let us analyze with a more watchful eye to the inverse operations and let's see what we can do with them.

— 3. Augmenting the available numbers —

— 3.1. Negative numbers —

First off let s take a look at subtraction and the numbers we already have. The numbers are ${ \mathbb{N} = \left\lbrace 0,1,2,3,4,\ldots \right\rbrace }$. For instance ${7-2=5}$ and ${10000-1000=9000}$. This notion of inverse operation to addition seems to work really well. But we don't need to go very far to encounter some possible problems ${3-7}$. In our current number system we can't go seven steps to the left starting in three. But naturally we'd like for subtraction to work all the time and so we'll take matter into our own hands.

We'll just increase the numbers available to us. We define ${-1=0-1}$ and ${-2=-1-1}$, ${-3=-1-1-1}$, and so on... Now subtraction is always possible. For instance the once problematic ${3-7}$ just equals ${-4}$ and the new number system that we end up with is ${ \mathbb{Z}=\ldots,-3,-2,-1,0,1,2,3,\ldots }$.

Besides providing for a way to subtraction being always possible negative numbers can also represent debts, heights below a reference level, etc.

— 3.2. Rational numbers —

After being provided with a whole new set of numbers via subtraction we will now construct a new subset of numbers in order to division always have a meaningful result.

Now the problematic cases are ${ \dfrac{m}{n} }$ with ${n>m}$. Imagine that you had seven breads to divide for 10 people. How would you do that? If you limited yourself to numbers in ${\mathbb{Z}}$ you couldn't share the bread. Hence to make division always possible (and remember that this problem certainly doesn't lack practical motivation) we have to further increase the scope of the numbers that are available to us. These new numbers will be called fractional numbers. To the set that is formed by the reunion of the set of fractional numbers and the set of integer numbers (${ \mathbb{Z} }$) we'll give the name of rational numbers ${ \mathbb{Q} }$.

So for instance imagine that you have one bread and you want to share it between you and two friends. You just break the bread in three equal parts and there you go. The number that represents this process is ${\frac{1}{3}}$. One of the disconcerting things about fractional numbers is that they allow an infinite number of representations:

$\displaystyle \frac{1}{3} = \frac{2}{6}= \frac{3}{9} = \ldots$

This problem is circumvented with the notion of equivalent classes, but in a more pedestrian way just think about sharing breads and you'll see why those fractions do have the same value.

— 4. Catching up —

At this point we can be very happy about ourselves. We started off with just the ordinary natural numbers, the addition operation, the notion of inverse operation, some brain power and a lot of ambition.

After deciding what addition meant in the context of the natural numbers we introduced the notion of subtraction as being the inverse operation to addition. Multiplication was defined as being a form of shorthand to repeated additions. Imagine that you had to add fifty times the number twenty. Instead of writing ${20+20+20+20+\ldots+20}$ one just writes ${20 \times 50}$ and everything is said.

After defining multiplication in this way once again we wanted to know what the inverse operation would look like. We defined division as being consecutive subtractions and so once again an operation appeared as a short hand for an already known operation.

After having arrived to these four operations (starting from just one) we allowed ourselves to apply them to the natural numbers without any reservations. In that way we came to the conclusion that either the numbers we were working with weren't enough or that we had to put limits in to the applications of the inverse operations. The second option wasn't compatible with our principle of ambition so we had to extend the amount of numbers in our disposal:

$\displaystyle \mathbb{N} \rightarrow \mathbb{Z} \rightarrow \mathbb{Q}$

— Glimpses of a near future —

In the next post we'll continue in our search of coherence and wider scope of application of inverse operations. Powers and radicals will be introduced and we'll see that the operation of extracting radicals forces us to once again widen the scope of our numbers, and this time not once, but twice!

Aristotelian physics

As a way to start things up in this blog I decided to recycle one of my old University essays. This essay was done for a discipline that versed on the history of the ideas in Physics. This a subject that is very fond to me because I always like to see how concepts evolve and how people react to that natural evolution.

The essay was written in Portuguese and had four pages. I won't feign definiteness, completeness, nor authority in the topic of Aristotelian (meta)physics but nevertheless it was an essay that gave me a lot of pleasure writing and that can be a very helpful first introduction to some people.

In the course of this blog I intend to write a lot more about this topic (and give references to my sources) and hopefully things will get as far as the Newtonian revolution.

Without further ado:

— A very short introduction to Aristotelian physics —

The sun dwell in the afternoon sky. A man of distinguished appearance was followed by a group of young people. He Gestured and spoke in a tone of completion and gives the lesson as over. All except one young student disperse. The student advances and questions the man:

- It's the first lesson of yours that I watch and I would like to better understand your ideas about the world that surrounds us.

- Well, young man, speak and let us see what we can do. If I can help you I will, if not forgive me... to me what surrounds us is a Cosmos. Something that can be rationally understood and explained in a rational way. It is a hierarchical cosmos in which each being seeks to realize its nature. It is finite and focused on our planet...

- But why our planet? Why not another heavenly body?

- Because it is logical. Everything in the Cosmos seeks to realize its nature. Be sure of this: it will be needed in the following discussion. What is the farthest thing you can see?

- The stars. Every night they are there to illuminate the sky ...

- You said it! "every night"! And how you know that?

- From the old records which are compiled... Nothing ever changes

- Nothing! Are you sure? Think about it.
 
- ... You're right. Nothing except their positions. They move as a whole in a cyclical way.

- To sum up: since the dawn of times we see the same stars cyclically running the same patterns in the heavens? - the student agreed - Then what will be the more reasonable inference about where they are? What is so perfect that they can always be there to move so regularly?

- As many have said before, they must be embedded in a spherical surface... but why is that the most perfect geometric configuration?

- Firstly, because all its points are the same distance from the center. It has no beginning nor end, the order of the cosmos is the same: it is eternal.

- But why is the Earth the center of the Cosmos ...

- Patience, young one, we'll get there ... the cosmos is hierarchical. But what is its hierarchy? Do you think that we can have mixtures of perfect beings with imperfect beings?

- Of course not!

- You already said that the stars are eternal and unchanging. Are these the attributes of what is perfect or not? In our world there is only corruption, generation, change... Is this perfect? No! What can we conclude about these two worlds?

- That they are radically different ...

- In that case do you think that two worlds so different are made by the same elements?

- No. In no way can our world be be made with the same substance of which the stars are made...

- Until now we only spoke of the stars, but in the heavens there is also the planets. They are also always there with cyclical movements. From what has been said they have to be perfect. Like the stars, the planets are also organized in their celestial spheres ...

- So if I understand, all these spheres are concentric, and the Moon should be the first one. And it marks the difference between our world and the celestial world ..

- Exactly! From the Moon upwards the bodies are composed of a special element called the aether. But what will be the elements of our world? As Empedocle, I say that is composed of four elements: earth, air, fire and water. All these elements are grouped in their places. The first is the sphere of the earth, above it is the water sphere, above it is the air sphere, and the last sphere of the sub-lunar world is the fire one.

- But by the way you describe it the sub-lunar world would be static and not have the attributes we know it has...

- Yes indeed, if it wasn't disturbed. But the movement of celestial bodies causes the fire to fall into the regions of the other spheres and this mixes the elements. And from the mixtures and perturbations change arises in our world. Now let us conclude that the Earth is spherical: when lunar eclipses occur the shadow that the Earth projects on the moon is always curved. From the top of a mountain you can see further than down below. And finally: the fact that the stars we see changes with our location. All this tells us that the Earth is spherical.

- Yes. A little knowledge of geometry suffices to this conclusion being necessary.

- Tell me, how will something that is mostly composed of earth behave when left alone.

- It will move naturally to the center of the sphere of the earth.

- Depending on your location, what would change if you dropped a stone?

- Nothing different would happen. The stone would always fall vertically.

- It always would fall to the center of the earth. But didn't you say earlier that something composed mostly by earth wold seek, when left alone, to meet the center of the sphere of the earth? - The student agreed - So what is your conclusion?

- ...That the two centers coincide. Or in other words: the center of the Earth is the center of the Cosmos!

- Bravo! But we also necessarily conclude that the Earth is at rest. If the Earth were to move the clouds, the stones, would be resting in their spheres. As we see, this doesn't happen so the Earth does not move. We will call this vertical motion that you talked about as the natural movement in the sub-lunar world. I the spheres above the moon one the natural movement is circular or is the result of the composition of circular movements. The movements that are not natural we'll call violent movements. These occur only in the sub-lunar world and need a motor.

- You are right! It just takes look at our world to see that the movement of all things when left alone is vertical: falling rocks, rain falling from the clouds, the fire moving up a bonfire ... But is there some kind of law that governs this movement?

- All we have to do is to think. Imagine two bodies suspended at a same height but with different weights. The heavier one has a greater desire to head towards the center of the earth. As it hits the ground, it will do so with greater speed. Consider now this same body being dropped in the water from the same height. Since water is more dense it opposes greater resistance to movement and thus the body will reach the end of the route with a lower speed.

- That is: the speed of a body, when it moves naturally, is directly proportional to body weight and inversely proportional to the resistance of the medium.

- Bravo! That's right! Or said in an equivalent way: a heavier body dropped from a given height reaches the ground first than a lighter one. And the more resistant the medium, the more time the body needs to fall and when it reaches the ground it does so with a lesser speed.

- So, master, this is where Mathematics enter your explanations?

- No! Mathematics does not enter into my explanations. Only logic does. Mathematics deals with pure forms and in our world form is always associated with matter. The later, as we have seen, is impure. If we want to know about what surrounds us we shouldn't follow the road of Mathematics, but by observing and using or reason. What do we have in the world of pure forms to prevent the existence of vacuum? Nothing! And yet in our world the concept of vacuum is absurd. What would be the speed of a body in a vacuum?

- The vacuum, if it existed, wouldn't offer any resistance to movement ... So the speed of a body would be infinite... This concept is disgusting...

- Thus vacuum does not exist. Space is always associated with matter. When one ends the other also ends.

- Ah! This is why the Cosmos is finite! Since matter ends in the fixed sphere, space has to have an end too! It makes no sense in even talking about beyond the fixed sphere!

- That's right! Unfortunately we have to stop our conversation. It is late and I have much to do. Goodbye! Until a next time!

- Goodbye and thank you master!

The Moon drifted in the starry sky. With a sense of awe the student contemplated the sky. This feeling was not for the show itself, but for the new found understanding.