What are properties, operators, and where do we get numbers from?

Question

I am a very confused person. My high school teachers told me division by 0 is undefined because "it just is. The mathematicians just did it that way." So when I found out for real why division by 0 is undefined, I realized that every basic thing I thought I learned couldn't be trusted.

I understand axioms as "something we make up to see what happens," postulates as "things we suspect are true, but haven't proven," and theorems as "things we've proven based on the axioms we've picked." Very tidy, I can dig it.

But I can't find anywhere in several textbooks up to Calc 1 that describes what a "property" is. Is it a particular kind of axiom? Or is it a simple theorem that's close to the axioms? I've found out about indicator functions, but it looks like they only describe properties, without being the properties themselves. How can I prove that $a \times 1 = a$?

Relations, I get them. Mappings from a set to a set, based on rules of any kind. And functions are just a kind of relation where we say each thing in the domain set only gets to have one mapping to the range set. And operators being functions being relations makes sense too, except, where do the operators come from? I'm used to functions being things we make up to study polynomials. But how do we "make up" addition? I can think of any number of algorithms to perform addition, and any number of plain English descriptions of addition, but I couldn't write the right-hand side of $(a, b) = ?$. All the functions I've ever seen are just compositions of addition and the other operators. And why do operators have to be functions, anyway? Couldn't we define division by 0 just by saying division is a relation but not a function? Then it could just return the set of all numbers for $a/0$.

And then numbers. There's an infinity of them, whatever kind you care to pick. How? I figure you could lay down the existence of 1 as an axiom, but how could you construct 2 from that without using addition? Another axiom? Then there'd have to be an infinity of axioms, and that can't be workable.

Yes. I am a very confused person. I hope I picked the right tags.

Answer 1

How many "big" questions... I'll try with a couple of them.

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For natural numbers, see Peano axioms : two basic "notions" are assumed :

• the existence of an "inatial" numebr : $0$

• the existence of a "basic" operation : the successor function $S$.

The first fact is "codified" by the first axiom :

$0$ is a natural number.

The second fact is established by the second axiom :

For every natural number $n, S(n)$ is a natural number.

These two "simple" axioms are the rules for the "number game" : start from the beginning and go one step after the other, i.e. counting.

We start from $0$ and apply the successor function $S$ to it, getting a new number : $S(0)$. We call it $1$. Then apply $S$ to $1$, i.e. to $S(0)$ and we get $2=S(S(0))$; and so on...

In order to make this basic machinery to work ad infinitum, we need some further axioms; the third one :

For all natural numbers $m, n, \ \ m = n$ if and only if $S(m) = S(n)$.

We want that every number has a unique successor.

Then we have :

For every natural number $n, S(n) \ne 0$.

This axiom is needed in order to avoid that, after a certain "amount" of numbers, we find a "loop" going back to $0$.

Finally, we have the Induction axiom.

With these axioms, we can define the addition operation.

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Regarding the "thorny" question of the division by $0$, the issue is quite simple.

We define subtraction as a "derived" operation starting from addition : if $a + b = c$, we want that $c - b = a$.

Thus, from $2 + 0 = 2$, we "derive" : $2 - 0 = 2$.

The same for division with respect to multiplication; from : $a \times b = c$, we "derive" $c/b=a$.

Unfortunately, we have $a \times 0 = 0$ for any $a$. Thus, what is the "expected value" of $n/0$ ? It must be a number $a$ such that, multiplied by $0$ will return $n$.

But no $a$, when multiplied by $0$ will gives us back the original $n$.

Thus, we are forced to agree with the awkward fact that : the division by $0$ is undefined.

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A (binary) relation is a way to associate things to other things; the world is plenty of them : "$x$ is father of $y$" defines a relation.

The way mathematics formalizes it is :

$Father = \{ (x,y) \mid x \ \ \text {is father of} \ \ y \}$.

Functions are relations that satisfy an additional condition, the "functionality" condition : for all $x$ there exists at most one $y$ such that ...

Thus, "father of" is not a function, because a father may have more than one son. The relation "son of" instead, is a function : every son has one father (and not two).

An operation is a function, and thus a relation. We can "describe" the sum as a relation in the following way :

$Sum = \{ ((n,m),k) \mid k=n+m \}$.

Of course this is not the "recipe" to perform additions; we already have to know how to add $n$ and $m$, but it is a way to "decide", for any triple $n,m,k$ if it satisfy the relation or not, because :

$((n,m),k) \in Sum$ iff $k=n+m$.

Thus, e.g. $((2,3),5) \in Sum$, while $((1,1),1) \notin Sum$.

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In mathematical "parlace", a property is something expressed by a theorem regarding an object or a colelction of objects.

Consider e.g. Euclidean geometry : it is a theory regarding objects (the "geometrical" ones) like points, line, circles, triangles, ...

If we consider the Pythagorean theorem, it states that :

the sum of the areas of the two squares on the legs of a right triangle equals the area of the square on the hypotenuse.

This theorem states that right triangles have the property that : "the sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse".

Answer 2

Formally axioms are arbitrary (apart from the fact that they are not allowed to contradict themselves or each other). However actually the axioms are chosen so that they are useful.

For example, the natural numbers are defined by the Peano axioms. But the Peano axioms are not arbitrary, they actually codify the act of counting.

Let's assume you want to count your marbles. Well, before you actually begin with the counting procedure, you've counted no marbles yet. Now it may turn out that you now discover that you actually don't have any marbles, in which case your counting has to result in that very fact. Therefore there has to be a number that says "there are no marbles" (or generally, none of whatever objects you want to count). This number is called zero. Therefore we have the first Peano axiom:

$0$ is a natural number.

OK, so you actually find you have some marbles, so you have to go on with counting the marbles. Now how do you count those marbles? Well, you take one marble at a time, and for each of them, you say a number. But not an arbitrary number, but always the next number after the one you previously said. And it is obvious that there can only be one next number, not several of them to choose from. After "one" the next number is always and without exception "two".

But what if you eventually run out of numbers? Well, of course you never run out of numbers, because you can simply "make up" yet another number if you come to the end to the numbers you already know, by just stating "this is the next number after that one. So we arrive at the second Peano axiom:

Every natural number $n$ has an unique natural number as successor $S(n)$.

Note that "successor" is just a fancy way to say "next number", and $S(n)$ is nothing but a short way to write "successor of $n$".

But that's not all: When counting, you can do things wrong. Imagine you count like this: "one … two … three … four … one …" Clearly that doesn't work well. When you count, you may not repeat a number you've already used, just as you may not recount a marble you have already counted.

But what are the numbers you already used? Well, one of the numbers is the one you started with, zero. Thus the next Peano axiom states that you never return to zero:

$0$ is not the successor of any natural number.

But you are not only allowed to repeat zero, you are also not allowed to repeat any other number. But how can we specify an arbitrary other number? Well, any number other than $0$ that we used was the successor of another number. So to prevent going back to any other number already used, we get the fourth Peano axiom:

The successors of different numbers are different: If $n\ne m$, then $S(m)\ne S(n)$.

OK, now there's only one question left: If we continue counting, will we eventually reach any natural number, or are there natural numbers we can never reach by counting? Well, the whole point of our definition is that the natural numbers is the numbers we use for counting. Therefore if we start with zero and go on counting, then we will indeed reach all natural numbers, because that's what we mean by "natural number". So we get the fifth and final Peano axiom:

Any set that contains $0$ and for every contained $n$ also contains its successor $S(n)$ contains all natural numbers.

Similarly, addition and multiplication get defined so that they match the corresponding real world operations.

Now, how do we know we've done it "right"? Well, we use the numbers we have defined that way in many real life situations, and it turns out they are useful. And whenever something occurring in the real world happens to match those axioms (like, for example, your marbles, but not necessarily water droplets — if you add a droplet to another droplet, you may get one bigger droplet) we also know that whatever we prove based on those axioms will also work out for those real world objects (for example, because we find that $5$ is not a multiple of $3$, you also know that you cannot fairly distribute your five marbles to three people).

Answer 3

It'd probably be better to ask several questions at a time, instead of a bundle such as this. Nevertheless, there's a rather common answer, so to speak, behind all of them, and that is set theory. If you view mathematics as being developed inside an axiomatic set theory (such as $\mathsf{ZFC}$), at least some of your questions will probably receive some kind of answer. I'll sketch a bit of this below.

But first, for starters, note that having infinitely many axioms is not a problem. $\mathsf{ZFC}$ itself, in its typical formulation, has infinitely many axioms (which are provably not reducible to finitely many), and it's perfectly workable. The trick is not having finitely many axioms, but having a recursive set of axioms, that is, a set of axioms such that there is an algorithm that tells you if a given formula is or is not an axiom of the theory. In other words, as long as you can decide for a given formula whether or not it is an axiom of the theory, it doesn't matter much (in terms of being workable) if you have finitely or infinitely many axioms.

Given that, it's possible to use set-theoretical resources, and a bit of logic, to answer your questions. In particular, most of the objects you're inquiring about (such as functions, relations, properties, and numbers) will be considered (or represented) as sets. Thus, relations will simply be sets of sequences of objects ($n$-tuples, if you're considering only finitary relations), in such a way that the relation "being the father of" and "being the male progenitor of" will actually denote the same relation (i.e. the set of all pairs such that the first coordinate of the pair is the father of the second coordinate). Functions will be a special kind of relation, that is, a relation in which it's impossible for two sequences in the relation to differ just in their last coordinate. As for properties, they will be just subsets of a given domain.

If the last definition sounds strange to you, the basic idea is this: properties are generally taken to be monadic relations. In terms of logic, that means that properties are defined by predicates which take just one argument. It follows that (almost) every such predicate determines a corresponding set, namely the set of things that satisfies it (also called the extension of the predicate). For instance, the property of "being a rational animal" determines the set of all humans, the property "being even" determines the set of all even numbers, etc. Of course, not all such properties are mathematically interesting; the interesting ones are generally those which we can somehow encode in a formula of some formal language, such as first- or higher-order logic. Notice that a property is not a theorem, nor does it need to be close to one: you can have the contradictory property defined as $x \not = x$, for instance. The closeness may be due to the fact that it's usually a theorem that a certain set has a certain property (thus it's a theorem of set theory that the set of all even numbers has the property of being equinumerous to the set of all natural numbers).

Notice that, under the above conception, the existence of a function or relation is not dependent on there being a description of this function or relation (though, usually, we take properties to be specifiable by formulas. There are uncountably many functions from the natural numbers to $\{0, 1\}$, but only countably many computable functions. So the vast majority of such functions are not algorithmic definable. The reason why most functions you've seen are defined in terms of addition and other things is that the class of such functions is highly interesting: it's the class of recursive functions, which is generally taken to be the class of computable functions (incidentally, addition itself may be defined using primitive recursion and the successor function very roughly, $+(0,x) = x$ and $+(n+1, x) = S(+(n, x))$). But there are many cases in which we may be interested in arbitrary functions, and set theory provides us with a way of dealing with those.

As for defining an operation as a relation, that's not very convenient, as it would mean that an application of the operation to a sequence could produce many different results. For instance, if addition were to be a relation, instead of a function, then we couldn't write things like $a+b=c$, for it could be the case that there were another number, say $d$, different from $c$, such that $a+b=d$ as well, whence $c=d$, contrary to the hypothesis that they were distinct. As a matter of convenience, it's better to take them as functions and deal with the few pathological cases that eventually show up (such as division by zero), than giving up the nice property that being a function has.

As for numbers, as I mentioned above, we also generally take them to be sets. In the case of the natural numbers, we generally construe them as von Neumann ordinals: $\varnothing = 0$ and, if $x$ is a natural number, then so is $x \cup \{x\}$, which is called the successor of $x$. So every number is the set of its predecessors. Without going into much detail, it's possible to show that there is a property which forces a set to be infinite (i.e. if a set contains $0$ and is closed under the successor function, that is, if $x$ is in the set, so is its successor). We can then postulate as an axiom that there is a set satisfying that property and take the set of all natural numbers to be the smallest such set. So we don't need to use infinitely many axioms to obtain infinitely many numbers (other kinds of numbers can be obtained by taking sets of natural numbers as representatives, e.g. the integers are defined as pairs of natural numbers such that the first coordinate is always $0$ or $1$; the idea is that $0$ codes being negative and $1$ codes being positive).

If introducing an axiom to deduce the infinity of the natural numbers sounds cheating to you, you may be pleased to know that it's possible to derive this proposition, and much more, (including basic properties of the successor function and the induction principle) from second order logic plus a very simple principle, called "Hume's Principle": the number of Fs is the same as the number of Gs iff there is a one-to-one correspondence between the Fs and the Gs. This surprising result is known as Frege's theorem and may be of interest to you.

Answer 4

But how do we "make up" addition? I can think of any number of algorithms to perform addition, and any number of plain English descriptions of addition, but I couldn't write the right-hand side of $(a, b) = ?$.

It's complicated.

You can start with Peano's Axioms that define the natural numbers. Here is the version I find most useful.

We define $\mathbb{N}, S, 0$ such that:

1. $0\in \mathbb{N}$

2. $S$ is a function mapping $\mathbb{N}$ to itself (the so-called successor function)

3. $S$ is injective.

4. For all $x\in \mathbb{N}$, we have $S(x)\ne 0$

5. For all subsets $P$ of $\mathbb{N}$, if $0\in P$ and for all $x\in P$, we also have $S(x)\in P$, then $P=\mathbb{N}$

Then, using the rules and axioms of logic and set theory, we can actually prove (with some difficulty) the existence of a unique addition function $+$ such that:

1. For all $x,y\in \mathbb{N}$, we have $x+y\in \mathbb{N}$
2. For all $x\in \mathbb{N}$, we have $x+1=S(x)$.
3. For all $x,y\in \mathbb{N}$, we have $x+S(y)=S(x+y)$

Then we can prove (also with some difficulty) the usual algebraic properties of addition: associativity, commutativity and cancelability.

The above development in its full detail would require hundreds of lines of formal proof. For obvious reasons, most authors don't bother with such a detailed development, and just "define" addition as above. Some simply include addition and multiplication in their definition of the natural numbers. So, you will have to forgive your high school teachers if they did not stop the class to explain all this.

I don't want to discourage your questions. Just be aware that a full explanation of some questions may be "beyond the scope" of the textbook or course.