0. The Very Beginning

To follow along, there are practically no prerequisites. You don't even need to know basic arithmetic, like addition or multiplication. The only things you need are to be able to read and understand English, as well as to know how to count, one number at the time (as natural numbers are embedded in the English language). If you don't know all the vocabulary of the English language, you'll be fine, as even most native speakers aren't perfect masters of their own language. If needed, check a dictionary for definitions.

The first topic of this project will be propositional logic. We will be defining the basic logical connectives, like and and or, which can create a new statement out of two others. But wait a moment, aren't and and or part of the English language, one of the only prerequisites that we already need to have? If you are knowledgeable about the English language, then and and or need no introduction, as you are already aware of their meaning! Heck, I myself have already used those two connectives in this very paragraph! What gives?

Here's what's happening: You need to distinguish the language we'll be studying, called the object language, from the language that we'll be using to talk about it, called the metalanguage. Thus, the object language's and the metalanguage's and connectives, for example, are to be treated as different entities, even though they may hold the same meaning. The contrast between object language and metalanguage is also present in the study of foreign languages, as one of the languages you are already familiar with is the metalanguage employed to learn the new one.

An important matter to address is how we'll be communicating. To talk about Mathematics, our metalanguage will be a natural language, namely English. The problem with natural languages though, is that they are often imprecise and ambiguous, as a given sentence can have different meanings. For example, consider the sentence "I saw her duck and her chest". The meaning of the words "duck", "chest" and even "and" is ambiguous. Did she duck or did she own a duck? Did I see her container or her breasts? And did I first see her duck and then her chest in that specific order or is order unimportant? When doing Mathematics, we cannot afford such ambiguities, as we aim to be as precise as possible. In our day-to-day life, when we aren't sure about the meaning of a sentence, we usually ask for examples to determine the context, by eliminating incompatible scenarios. I may do the same here to increase the likelihood that you understand a given idea.

Now that this is out of the way, we need to address the elephant in the room: what is Mathematics? A common misconception is that Mathematics is a science. Science is an empirical discipline that builds and organizes knowledge in the form of testable hypothesis and predictions about the world, whereas Mathematics is a field of study which involves the discovery of properties of abstract objects and the use of pure reason to prove them. Unlike science, it is completely independent from observations about the physical world. The study of Mathematics is carried out through the axiomatic method. This method consists of primitive objects/notions and axioms which define them by describing their behavior. For example, in planar geometry, the primitive objects are points and lines and the axioms may be Euclid's postulates. From the axioms and rules of inference, one then derives true statements called theorems, which can then be used in the same manner as the initial axioms to derive other theorems.

Notice that I've stated that primitive notions are defined by their properties. Some might say that a true definition consists instead of defining an object in terms of other objects that have already been defined, (i.e. we are what we are made of instead of the sum of our actions). If you are an adept of that meaning of definition, then primitive notions actually have no definition. How can that be? If you were to look for the word "point" in the dictionary, then you would surely find a definition, right? The thing is, dictionaries are flawed. Indeed, if a word can only be defined from previously defined words, then since dictionaries contain only a finite amount of words, this endeavour is impossible because circularity is inevitable, as there will necessarily be words whose definition requires other words that have previously been defined by them. Thus, there must exist primitive words possessing no definition. (You should then take the last sentence of the first paragraph with a grain of salt)

A similar idea applies to the axioms. Axioms are statements that are assumed to be true. You may view them as the rules of a game. If you decide to work with another set of axioms, then it's as if you were playing another game. The reason one needs to assume true statements is that one can't create something from nothing (with the possible exception of God or gods depending on your beliefs, but that's besides the point). In the past, mathematicians used to think that axioms were absolute truths, but in reality there's no such thing. In the 19th century, they began to seriously question that notion with the emergence on non-Eucledian geometries, in which Euclid's fifth postulate doesn't hold.

Some of the first questions I've asked myself when trying to figure out how to go about this project were "how did I learn what I already know in the first place?", "what was the first thing that I learned in mathematics?", "when did I learn it?". I've meditated on these questions for some time and some of the ideas I came up with were just shared with you, notably the following:

• You can't create something from nothing
• The distinction between the metalanguage and the object language
• Some of the mathematical concepts are already embedded in our natural language

With these ideas, I think I came up with an interesting answer to my questions. We may be unable to create something from nothing, but we definitely can learn from nothing, as that's what we've been doing ever since we were born. We've learned to walk, to speak and understand the world around us. And how did we do it? I believe it was by trial and error. When learning to speak, you may have heard someone uttering a word. You may then have tried to imitate it and then you may have realized that depending on the context, saying that word led to different outcomes, thus allowing you to grasp its meaning. There may be some words that you are able to define in terms of others and there may be others you aren't, but whose meanings you are aware of. And if someone makes a statement whose meaning you are not sure of, it's natural to ask for clarifications, namely by asking for examples and then picking the interpretation that satisfies them.

Some terminology

You may notice that I'll be using terms like "proposition", "lemma" and "corollary". All these are theorems, but one usually reserves "proposition" for theorems that aren't considered to be very important, "lemma" for theorems that are useful to prove other important theorems and "corollary" for theorems whose proof follows directly from another theorem. These distinctions not only are subjective, but they also aren't essential. However, making them may help you understand where each theorem stands in the grand scheme of things.

The end of proofs will be signaled by the use of the symbol ∎ on the right.