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Three is a number, right? So are pi, the square root of two or *i*. But mathematics is all about abstraction, so how do we abstract the concept of a number? **How do we properly define numbers?**

## Starting off

**In school, youâve probably heard the âintuitiveâ definition of natural numbers, as ânumbers for things you can countâ.** They start off at one, then increase by one and go up to infinity. Itâs usually written like this: â = {1, 2, 3, âŚ}. Then you introduce the concept of whole numbers as a union of â with {0} and -â, writing it like â¤ = {…, -3, -2, -1, 0, 1, 2, 3, âŚ}. **Rationals get a better definition in high school**, where theyâre defined as fractions of whole numbers and natural numbers, like this: â = {m/n: m â â¤, n â â}, while **we really donât dive into how real numbers are defined** (usually as âanything that canât be written as a fractionâ, but that isnât even correct!), and complex numbers are just âthings you can make from the square root of -1â.

**But those definitions arenât particularly useful, and they sure arenât abstract enough to study on their own.**

## Why do we even *need* to define a number?

Sure, **we can just say that a number is âthe mathematical abstraction of the concept of the size of a set of objectsâ**, as dictionaries do, but a nice definition would allow us to **study the properties of the concept, not numbers themselves.** Furthermore, as mathematics is a highly formal field of study, just saying something

*is*âŚ isnât really acceptable.

**But numbers are a very complex concept. Like, think about it for a second. **Three, without any added context, has no meaning in day-to-day speech. âThree applesâ makes sense, but just âthreeâ doesnât. There isnât a âthreeâ anywhere in the world, but there could be âthree of somethingâ.

## Sets first

**So, to start defining what a number is, letâs look at the structure surrounding it**: in the same way that we donât really define a vector by itself, but only a vector space – and then define a vector as a âelement of a vector spaceâ. (A vector space is similarly defined using axioms, if you’re not familiar.) **Similarly, numbers also donât make a ton of sense to define on their own, so we start off by looking at the set of naturals.** We notice that thereâs a pattern: **every number has a successor (thereâs always a bigger fish number), but not every element is a successor**: we start with 1 and thereâs nothing smaller than 1: nothing preceding it.

But that isnât enough on its own: **these two properties of the set of natural numbers donât define just the set of natural numbers** – they also hold true in some other sets that donât have the same properties as naturals: for instance, they hold if we define a set like {0, 0.5, 1, 1.5, 2, 2.5, âŚ} and define that the successor of x is x+1. However, this isnât the set of natural numbers – and itâs not really alike (formally, theyâre not

*isomorfic*: we canât construct an order-preserving bijection whose inverse is order-preserving as well).

**So, we need something more.**

**Luckily, the principle of mathematical induction holds true in the naturals. **If youâre not familiar with induction, itâs the idea that if you can prove that something holds for n=1, and using only the assumption that it holds true for a general n, that it holds for n+1, then youâve proven that that statement is true for all natural numbers n.

**These three things are enough to define the natural numbers and were first formalized by Italian mathematician Giuseppe Peano and are called âPeanoâs axiomsâ**. He published them in a somewhat different form, but theyâre usually stated along these lines, with just a touch of mathematical formality to hold our ideas together in a more sensible notation đ

If youâre interested in how thatâs written down:

- For all natural numbers n, there exists a unique successor n
^{+ }in natural numbers and if n^{+ }= m^{+}, then n = m.^{ } - There is a first element in the set of naturals, i.e. 1 â â. It is the only* element of the set of naturals that is not a successor of any natural number.
- The principle of mathematical induction holds:

Let S be a subset of â such that:

- 1 â S
- For all n â â, n â S implies n
^{+}â S

Then S = â.

**Of course, these things hold true for some other sets as well. **If we define 1 = â
and n^{+} = {n}, we get the set {â
, {â
}, {{â
}}, âŚ} – and these axioms hold true there as well.

*Note: the word “only” in * prevents the {0, 0.5, 1, 1.5, …} example from working, but is usually left out from the Peano axioms. This depends on how Peano axioms are stated. See https://math.stackexchange.com/questions/132855/why-do-we-take-the-axiom-of-induction-for-natural-numbers-peano-arithmetic* *for more info.*

## Howâs that any better?

**It is: because that means that the same conclusions are valid **whether we think of 1 as an empty set (which lets us define operations and study the sets more closely in an easier to grasp manner (which I know sounds very counterintuitive)) or of 1 as âjust one thingâ intuitively.

**This leads us towards the conclusion that it isnât really about the number itself – itâs more about the structures those numbers build together.**Proving a property of 3 isnât really interesting (itâs easy to prove 3 is a prime: youâve done it at some point in your life), but proving properties that hold for sets of numbers is more useful (as itâs a

*more general*statement) and more fun, to be honest đ

## We can go further!

**How high can you count?** If youâre a child (congratulations for reading this blog in that case! :D), maybe up to a million? If youâre an adult, youâll probably sayâŚ well, up to infinity. **But, as anyone whoâs read The Fault in Our Stars knows, not all infinities are the same. So, up to which infinity could you count? **

Could we abstract natural numbers even further? Would this give a somehow *better *definition? How would we define reals? Are these actually nontrivial mathematical questions?

**The answer to the first, second and last question is yes, and the answers to them (and the ways we got there) are quite interesting.** Along with the question âwhatâs actually a set?â, which is something we havenât touched upon here but took for granted as something that we

*just know*(and itâs a really good question to ask!),

**these questions have built entire fields of study in mathematics and gave some shocking discoveries along the way.**

Perhaps itâs a story for a different post, perhaps this post will get a sequel if you like it, but to leave you pondering,** Iâll end on a riddle.**

Thereâs a village in which a barber shaves every person that doesnât shave themselves. Who shaves the barber?

Let us know what you think in the comments below.

## One reply on “What’s a number?”

An addendum for the very interested: the question of whether Peano’s axioms are

enoughto define N uniquely is very interesting. You can read more about it on Math Stack Exchange, but long story short: we need something a bit better than that to define N (up to algebraic structure isomorphism) uniquely. The point of this post was to explain why mathematics doesn’t bother defining concepts like numbers, vectors etc., how that’s worked around and what happens when you try to precisely define things you take for granted, which I believe still stands. Unfortunately, this post would be way too long if we were to be really precise, so some things were omitted for the sake of the message (and to avoid having to write a crash course in set theory)