Axioms, Definitions, Theorems, and Proofs
Over the past two chapters, we’ve been building up a system of mathematical logic — what propositions are, how to determine their truth, when two propositions are equivalent, and how to use propositions in arguments to make valid deductions.
However, this isn’t how most of mathematics is communicated. Most of us understand math in terms of numbers and geometric shapes: arithmetic, algebra, trigonometry, lines and angles, polygons and circles, graphs and equations. What we’ve done so far looks very different. What gives?
The reason we spent so much time getting a grip on mathematical logic is that mathematics is fundamentally about using known facts (propositions) to deduce new facts (valid arguments). We’ve talked about the Pythagorean Theorem in numerous introductions, and asked how we know it’s true — it wasn’t inscribed on a stone tablet and sent down for all to see. Someone had to figure it out.
Moving forward, we’re going to be deducing new facts from old facts, all having some mathematical interest. Everything we do will rest on the logic we’ve learned so far, though the structured use of that logic will usually stay implicit rather than being explicitly pointed out — the tools are there to fall back on whenever we’re unsure about the soundness of a piece of reasoning. From this point forward, our examples will be mathematical in nature, rather than concerned with people and the situations they find themselves in.
Axioms, definitions, theorems, and proofs are the main tools of mathematics — they’re what we use, and what we produce. Let’s begin!
Defining Definitions
In everyday language, it’s common to speak with conditionals — but everyday language isn’t precise. People tend to use implications when speaking, even when what they mean is really a biconditional.
In Marina’s geometry class, her teacher stated: “If a quadrilateral has two pairs of parallel sides, then it is a parallelogram.” Marina translates this mathematically:
\[ \begin{array}{rl} \mathcal{U}\text{: } &\text{All planar quadrilaterals} \\ t(x)\text{: } &x \text{ has two pairs of parallel sides} \\ p(x)\text{: } &x \text{ is a parallelogram} \end{array} \]$$\forall x\ [t(x) \to p(x)]$$Later, as she’s studying with her friend Daisy, Daisy states: “If a quadrilateral is a parallelogram, then it has two pairs of parallel sides” — that is, $\forall x\ [p(x) \to t(x)]$.
Marina notes that Daisy isn’t wrong — parallelograms really do have two pairs of parallel sides. In fact, being a parallelogram goes hand in hand with having two pairs of parallel sides. So even though both statements use an implication, what’s really meant is a biconditional. Marina settles on: “A quadrilateral is a parallelogram if and only if it has two pairs of parallel sides” — that is, $\forall x\ [p(x) \leftrightarrow t(x)]$.
Would we ever be able to verify $p(q) \to t(q)$ for some given quadrilateral $q$? We’d have to know what a parallelogram is — the word itself doesn’t tell us much, since it’s just a man-made word. We could figure out what “has two pairs of parallel sides” means, since we understand the concepts it expresses. But determining whether a quadrilateral is a parallelogram requires knowing what idea the word “parallelogram” is being assigned to represent — and mathematicians have defined it to mean exactly “a quadrilateral having two pairs of parallel sides.” This is why $\forall x\ [t(x) \leftrightarrow p(x)]$ is the most appropriate statement — what the teacher and Daisy said were essentially defining the word parallelogram.
When writing mathematics, it’s imperative to be as precise as possible. If what you want to state involves a biconditional, you should use the double-ended arrow $\leftrightarrow$ — it’s almost never appropriate to use the implication arrow $\to$ when a biconditional is meant.
The exception is definitions. The purpose of a definition is to assign a meaning, idea, or concept to a single word — so it’s usually fine to state definitions in terms of implications. Suppose we want to define some new word, and assign it some definition. The most accurate way to convey that meaning is
$$\textit{word} \leftrightarrow \textit{definition}.$$Remember that for any two propositions $p$ and $q$, we have $p \leftrightarrow q \Longleftrightarrow (p \to q) \land (q \to p)$. So, instead of saying $\textit{word} \leftrightarrow \textit{definition}$, we’re fine saying $\textit{word} \to \textit{definition}$ or $\textit{definition} \to \textit{word}$ — even though an implication is used, what’s really meant is the biconditional.
Here, we present some mathematical words and definitions you’re likely already familiar with, presented just for completeness.
The numbers $0, 1, 2, 3, 4, \dots, 100, 101, \dots$ are collectively referred to as the whole numbers.
When we combine the whole numbers into a collection along with their negative counterparts — $0, 1, -1, 2, -2, 3, -3, \dots$ — the new collection is collectively referred to as the integers.
When the universe of discourse we’re using is the universe of all integers, we commonly use the symbol $\mathbb{Z}$ instead of $\mathcal{U}$.
An integer $n$ is called even if (and only if) there exists some integer $k$ such that $n = 2k$. Mathematically,
$$\forall n\ [n \text{ is even} \leftrightarrow \exists k\ [n = 2k]].$$An integer $n$ is called odd if (and only if) there exists some integer $k$ such that $n = 2k + 1$:
$$\forall n\ [n \text{ is odd} \leftrightarrow \exists k\ [n = 2k + 1]].$$Notice that in this definition, we wrote the propositions out in English rather than assigning them a single letter. Also note that we wrote “and only if” in parentheses — that’s to stress that a definition is really a biconditional. Going forward, it’s usually fine to say just the “if” part, since the biconditional is what’s meant either way.
Since $0$ is an integer, we can describe it as even or odd. $0$ is even because there’s an integer $k$ (namely $k = 0$) such that $0 = 2k$. Notice the definition of even doesn’t require $k$ to be different from $n$ — as long as $k$ is any integer, we say $n$ is even. There’s no integer $k$ such that $0 = 2k + 1$, so $0$ isn’t odd (the closest integers we could pick for $k$ are $-1$ and $0$).
$9$ is odd, since setting $k = 4$ gives $2(4) + 1 = 9$. There’s no integer we can substitute for $k$ so that $9 = 2k$ — the closest are $k = 4$ (giving $8$) and $k = 5$ (giving $10$).
Notice that when checking $4$ is even, we found $k = 2$, which is itself even — but when checking $6$ is even, we found $k = 3$, which is odd. Nowhere in the definition of even did we require $k$ to be even or odd; the only requirement is that $k$ be some integer.
The same holds for odd integers: for $11$, we have $11 = 2(5) + 1$, and $5$ is odd; for $13$, we have $13 = 2(6) + 1$, and $6$ is even. Again, $k$ doesn’t have to be the same type of integer as $n$, just an integer itself.
What about numbers other than integers? For $-8$, setting $k = -4$ gives $-8 = 2(-4)$, so $-8$ is even. For $-13$, setting $k = -7$ gives $-13 = 2(-7) + 1$, so $-13$ is odd. But what about $3.2$? Since our definitions of even and odd both require $n$ to be an integer, and $3.2$ isn’t one, neither definition applies — $3.2$ is neither even nor odd. The terms even and odd, as defined here, only apply to integers.
It may seem excessive to go over the details of how even and odd integers are defined — the point isn’t to teach what they are, but to show how much nuance even a simple definition can carry. Notice that our definition of even really has three requirements: $n$ must be an integer, $n = 2k$ for some number $k$, and $k$ must be an integer. If any one of these fails, we can’t describe $n$ as even. Three analogous requirements apply to odd. All definitions in mathematics work this way — they assert conditions that must be satisfied before the associated word can be applied.
Two integers are said to have the same parity if (and only if) they’re both even, or both odd. Two integers are said to have different parity if (and only if) one is even and the other is odd.
In order for the word “parity” to be applied, we need two integers to begin with — if either isn’t an integer, the word doesn’t apply. Using a more notational style: with $p(a, b)$ representing “$a$ and $b$ have the same parity,” $e(a)$ representing “$a$ is even,” and $o(a)$ representing “$a$ is odd,”
$$\forall m, n\ [p(m, n) \leftrightarrow (e(m) \land e(n)) \lor (o(m) \land o(n))].$$Going forward, we won’t be so pedantic about describing every aspect of a definition — it’s up to the reader to determine whether a definition can be applied. We’ll still show how to write a definition using mathematical notation.
The key is to carefully read every part of a definition, and understand what its requirements are. If even one condition fails to hold, the definition doesn’t apply.
An integer $n$ is called a perfect square if (and only if) there exists some integer $k$ such that $n = k^2$:
$$\forall n\ [n \text{ is a perfect square} \leftrightarrow \exists k\ [n = k^2]].$$In the previous two sections, we examined arguments with universally quantified premises, but were more concerned with the form their conclusions took — we never discussed how to get universally quantified premises in the first place. Appealing to a definition is one way to do so, as we’ll see in the next section.
Asserting Axioms
Whereas definitions are things that can simply be decreed, axioms represent something fundamentally different — statements of mathematical interest that are intuitively correct, but require no proof of correctness.
An axiom is a statement of mathematical interest that’s taken, or assumed, to be true without the need for proof, and is used as a premise in arguments — but never appears as the conclusion of an argument.
The word postulate is a synonym for axiom.
In arithmetic, we’re familiar with the associative law: $a + (b + c) = (a + b) + c$. It doesn’t matter whether we add $b$ and $c$ together first and then add $a$, or add $a$ and $b$ together first and then add $c$. For example, $1 + (2 + 3) = 1 + 5 = 6 = 3 + 3 = (1 + 2) + 3$, which is why we can simply write $1 + 2 + 3$ without parentheses at all.
But one example doesn’t prove anything — how do we know this always works? It’s impossible to check every combination of numbers, and it even seems to work for non-integers like $90.77$, $3.14159265$, and $2.718$. Trying to find a counter-example seems fruitless too. As such, we simply assert this as an axiom of basic arithmetic: we can always change the order in which numbers are added together, and always get the same result.
Most of us have taken a geometry class full of definitions and theorems about triangle congruence, parallel and perpendicular lines, angle measure, area, and volume. How do we know all of those theorems are true? Definitions can just be asserted, since we’re forcing ideas onto words — but theorems need to come from somewhere.
Roughly 2300 years ago, a Greek mathematician and philosopher named Euclid wrote a book known as The Elements, laying out a system of geometry based on five axioms, or as he called them, postulates:
- A straight line may be drawn through any two points.
- Any terminated straight line may be extended indefinitely.
- A circle may be drawn with any given point as its center and any given radius.
- All right angles are equal.
- For a given line, and a point not on that line, a second line can be drawn through the point that never intersects the first line.
From these five axioms, most of what we know about plane (Euclidean) geometry can be deduced using the rules of logic discussed previously — for example, that the angle measures in any triangle always add up to $180°$, no matter what kind of triangle it is.
We’re not trying to prove that these five statements are correct — we’re asserting them to be true, and deducing more true statements from them. When we assert some collection of axioms, we’re essentially creating a branch of mathematics. If even one axiom is altered, everything deduced from the originals no longer holds — instead, you have an all-new branch of mathematics, with new results and even more discoveries to be made!
As stated above, altering even one axiom defining a branch of mathematics gives you an entirely new branch, where the old results don’t necessarily apply.
For a long time, many mathematicians and philosophers tried to prove Euclid’s fifth postulate using the first four, believing it didn’t need to be asserted as an axiom at all. As time went on, it was eventually shown — using very sophisticated logic well beyond the scope of this book — that the fifth postulate can’t be deduced from the other four.
As a result, variations on the fifth postulate were asserted by many people over a long time. Using the postulate as originally stated gives “Euclidean” geometry, applicable to an infinitely long flat surface (a plane). There are geometries where the Euclidean fifth postulate is eschewed entirely — the two most commonly known are elliptic geometry and hyperbolic geometry. There’s also spherical geometry, distinct from all three, concerned with geometric figures on the surface of a sphere.
Working with such geometries requires sophisticated tools explored in later books. In short, elliptic geometry says there are no parallel lines at all, while hyperbolic geometry says there are infinitely many distinct parallel lines through a point not on some given line.
Theorizing Theorems
Whereas axioms are asserted and define an entirely new branch of mathematics, theorems are always deducible from axioms and definitions.
A theorem is a proposition of mathematical interest that’s derived, or deduced, from a set of axioms, definitions, or other theorems.
Axioms can only ever appear as premises in arguments; theorems can be premises or conclusions. Typically, we start with a collection of axioms and definitions, deduce some initial round of theorems, then deduce a second round using the previous theorems along with the axioms and definitions — and we can repeat this process to yield ever more theorems. During all of this theorem-proving, we may even come up with new definitions along the way.
Some theorems can be proven from other, previously deduced theorems, in addition to the given axioms and definitions. In other cases, some theorems are simply special cases of other theorems — and we have special names for those too.
A lemma is a type of theorem used to prove other theorems — that is, a theorem used as a premise in another argument.
A corollary is a type of theorem that results from considering special cases of some given theorem.
Often, the word “theorem” is reserved for major results. We could be pedantic about classifying various theorems as lemmas or corollaries, but we’ll mostly just stick to the word theorem — the important thing about all of them is that they’re deducible from axioms, definitions, and other theorems.
Providing Proofs
The final mathematical building block we’ll discuss is the proof.
A proof is a valid argument provided to show that an implication is a logical implication.
We’ve discussed arguments at length up to this point, especially how to determine whether a given argument is valid — and we’ve touched lightly on using the rules of inference to chain logical implications together into new logical implications. All a proof really is is a valid argument.
We typically describe a proof as being given in reference to a theorem — if someone proposes a statement of mathematical interest, and a proof can be given for it, that statement is henceforth called a theorem, because it can be deduced. Remember that an argument is simply an implication, with a conjunction of multiple propositions as the hypothesis, and a single proposition as the conclusion — any of these propositions, in the hypothesis or the conclusion, can be primitive or compound.
In the next section, we start learning specific methods of proof, and ways to devise them. This is the primary activity of mathematics.