Mathematical Language
August 1, 2022

"The limits of my language are the limits of my world." — Wittgenstein

Introduction

Anyone who has tried learning university level mathematics knows that this kind of maths is very different from highschool maths. Highschool maths is mostly focused on computation and the problems vary only slightly — university maths puts an emphasis on concepts while computation remains a separate issue. Now, do you see a problem?

After spending four years on learning how to follow patterns, you’re now asked to appreciate mathematical rigour and use mathematical language to express your ideas. I certainly have (had) issues with this — but here’s what I’ve been working on so far and some revelations I’ve had in the meantime.

Another Way of Thinking: Logic

As with any game, in order to play it effectively you need to be familiar with the rules. Similarly, to learn the rules you must already be familiar with the language those rules are expressed in. In mathematics, mathematical logic is that language — mastering it means that you can begin to learn the rules — axioms, definitions, theorems and lemmas — all expressed using this language.

It's no surprise then that getting comfortable with logic is a fundamental step towards learning higher level mathematics. In mathematics, almost everything revolves around logic.

Implication

Conjuction and disjunction are relatively straight-forward and intuitive logical operations. Implication, however, is usually something that seems unnatural. You see, while the statement like $\phi \implies \psi$ reads as "$\phi$ implies $\psi$", it should be noted that mathematical implication differentiates causality from conditionality. In fact, mathematical implication cares only about conditionality, while the causality (that one event intuitively follows from the other) is not important. For this reason, mathematicians often call this operation the conditional.

Example

Let's illustrate this with the following example: $$\boxed{\text{Julius Ceasar is dead} \implies \text{Barack Obama was the US president}}$$ This statement is true because its conditionality is true, even though there's absolutely no causality here (for all we know). It's conditionally true since statements "Julius Ceasar is dead" and "Barack Obama was the US president" are both true.

Example

Now consider the next example: $$\boxed{\color{red} \text{A person is tall} \implies \text{A person is good at basketball}}$$ This statement is false because of its conditionality — it doesn't matter that it might be intuitive and has some potential causality. It's conditionally false since you can always find a person who is tall and not good at basketball. Because of that, this implication doesn't hold.

Beauty in Logic

One of the beautiful things about logic is that once you get the hang of implication, you can appreciate the way we've defined another logical operation: equivalence. In fact, equivalence just means that we have $\phi \implies \psi$ and $\psi \implies \phi$ at the same time (in conjuction). $$\begin{array}{|c|c|c|c|c|} \phi & \psi & I_1: \phi \implies \psi & I_2: \psi \implies \phi & I_1 \land I_2 \\ % Use & to separate the columns \hline % Put a horizontal line between the table header and the rest. T & T & T & T & T \\ T & F & F & T & F \\ F & T & T & F & F \\ F & F & T & T & T \\ \end{array}$$ Because both conditionals (from left to right and from right to left) hold in this case, we also call this the biconditional. We write the equivalence like $\phi \Leftrightarrow \psi$.

Example

Consider this as an example. For natural numbers $m, n$, saying that $mn$ is odd is equivalent to saying that $m$ is odd and $n$ is odd at the same time. In other words, $$\text{$mn$ is odd} \Leftrightarrow \text{$m$ is odd} \land \text{$n$ is odd}$$ If we continue to unpack this, we have two conditionals. First, we have $$I_1: \boxed{\text{$mn$ is odd} \implies \text{$m$ is odd} \land \text{$n$ is odd}}$$ and then $$I_2: \boxed{\text{$mn$ is odd} \impliedby \text{$m$ is odd} \land \text{$n$ is odd}}$$ We've denoted these conditionals as $I_1$ and $I_2$, respectively. Note that the direction of implication in $I_2$ is reversed in comparison with $I_1$. Then our equivalence statement can be expressed like $$\boxed{\text{$mn$ is odd} \Leftrightarrow \text{$m$ is odd} \land \text{$n$ is odd} \equiv I_1 \land I_2}$$

Limits of Our Language

I began this writing with a quote from a famous philosopher Ludwig Wittgenstein. He claims that the limits of our language are in fact the limits of our world. I believe this to be a very profound insight in general as well as a great mental note to keep when learning subjects like mathematics.

In other words: "If you become proficient at mathematical logic (the language of mathematics), then you also expand your mathematical world — provided you now learn the rules expressed in this language".

Some Advice

To wrap this up, here are some things/resources I would've appreciated someone telling me about when I started university. If you're a freshman or even if you're going off to university in a few months, I'd recommend checking these out.

Online Courses

Books