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If someone asks me right now, what a number is, I would say 1,2,3,-1,1+i,2.13, etc. But what I have essentially stated are just symbols. I tried to explore a bit what a number exactly is. What are these symbols pointing to?

As I got to know the answer is not that simple, we cannot have an all-encompassing definition of numbers. Accepting this, I can also not find peace with just treating them as symbols.

So I just wanted to ask, is there a way I can somehow construct natural numbers, then fractions, then integers, real numbers, and so on, so that I at least have some notion attached to these numbers, rather than treat them just as symbols with which I'm playing.

Asaf Karagila
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Harshit Rajput
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    It doesn’t matter so much as to “what” they are, rather than the properties they satisfy. For example, the biggest (in fact defining) property of the naturals is the inductive property. For real numbers, it is the (Cauchy/Dedekind) completeness. For complex numbers, its the algebraic completeness (on top of containing the reals), etc. So, numbers are just a set of objects which satisfy certain properties. The real question is why such sets with such and such properties exist. Well, partly you study some set theory, and partly you learn it in analysis/algebra. – peek-a-boo Dec 25 '22 at 08:20
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    Of course there is a way to build almost from $\varnothing$ , the numbers; many mathematicians have asked themselves the question and have provided answers over the millennia. This was probably one of the main driving forces behind mathematics. Also, providing a global answer here is impossible, just a partial answer. Maybe specify which construction you want us to show you. – Stéphane Jaouen Dec 25 '22 at 08:39
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    Let's say you build numbers from something. From the same sets, as is customary. But then the question arises, what is the set itself. In mathematics, as is known, this led to the understanding that the set itself is not such a simple and initial concept and created several different directions, each explaining in its own way what a set is. Where and how will you stop? – zkutch Dec 25 '22 at 08:40
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    On the philosophy side of things, this notion of, "is maths just symbols?" dates back to the ancient Greeks. For example, see: https://en.wikipedia.org/wiki/Aristotelian_realist_philosophy_of_mathematics – Adam Rubinson Dec 25 '22 at 10:00
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    Related: https://math.stackexchange.com/q/865409/96384, https://math.stackexchange.com/q/31826/96384, https://math.stackexchange.com/q/1259198/96384, and many linked from there. – Torsten Schoeneberg Dec 25 '22 at 17:34

2 Answers2

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This question dives into philosophy, but I'll give you two glimpses of perspectives on what numbers can be seen as:

Abstract algebra

As user peak-a-boo put it, it's not really about what numbers are, but about what properties they satisfy.

In abstract algebra, you want to use general symbols and combine them with general operators, because when writing "$x \cdot y$" you may mean:

  • The multiplication of two natural numbers
  • The composition of two moves on a Rubik's cube
  • The concatenation of two strings
  • The multiplication of two matrices
  • The composition of two rotations
  • ...

So sets with operations with common properties are classified as algebraic structures (monoids, groups, rings, fields, vector spaces...)

The sets $\mathbb N$, $\mathbb Q$, $\mathbb R$, $\mathbb C$ etc. are just sets with operations, and so they're individually no different from the other sets whose elements behave in the same way (defining "same" in a very special way that you'll learn once you get into abstract algebra).

Set theory

Another way to look at this is from a set-theoretical point of view. If you want to define the natural numbers in an intuitive way you may want to write $0$ as $Z$, $1$ as $SZ$, $2$ as $SSZ$ and so on, adding an $S$ for each natural number. In this way there is no such thing as "symbols pointing to meaning", as the symbols themselves encapsulate the meaning directly (whereas base representations rely on the definition of addition and multiplication, so they're more "circular").

Now you can define bijective mappings (which, you'll learn are a form of "sameness") from the natural numbers to every other countable set, hence you can treat, say, the couples of numbers $(a,b) \in \mathbb N ^2$ as "basically the naturals", and you can do the same thing with other sets such as the rationals, or (since I see you've been studying relations) the kleene star (transitive and reflexive closure) of any finite or countable alphabet.

So, in set theory, you can transform the intuitive notion of the natural numbers to that of any other countable set.

Conclusion

There sure are other, more practical intuitions for what numbers are: the naturals are the numbers one uses to count, the integers are the numbers used to count in two distinct directions, and I won't get into the engineer-ish intuitions of what the real numbers are, but let's say they're used for "real world measures". Other than that, there are many things in mathematics that need not point to any real world thing. You won't meet any infinity-categories at the supermarket, but by learning how mathematics is about connections and generalizations you'll start to appreciate how symbols are enough.

mell_o_tron
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  • $\mathbb Q, \mathbb R $ and $\mathbb C $ are $3$ fields. But they are very different from each other. The only algebraic point of view (even Abstract algebra) is not sufficient here to differentiate them, is it? – Stéphane Jaouen Dec 25 '22 at 11:00
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    @StéphaneJaouen In general abstract algebra is more concerned with structure, so I'd say it need not differentiate them (although through algebra we can say that, e.g. the algebraic closure of the real numbers is the field of complex numbers, so it sort of does). More than focusing on the differences, I was putting an emphasis on the fact that "numbers are not special", and that there are many things that behave "just like" numbers, if we observe a subset of their properties. – mell_o_tron Dec 25 '22 at 11:16
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Come to think of it, it's a pretty fun and impossible challenge to answer this question. Let's try to get clues for OP. We start from the more or less naive set theory, from the assemblages of characters to which the mathematician alone grants some value:

  1. Axiom: there exists a set noted $\varnothing$;

  2. Definitions : $0:= \varnothing ; 1:= \{\varnothing\}; 2:=\{0,1\}; 3:=\{0,1,2\}; ...$

  3. We finish building $(\mathbb N,+,\times,\leq)$ ;

  4. On $\mathbb N \times \mathbb N$, we consider the relation $\sim$ defined by $(n,m)\sim (n',m') \iff m+n'=m'+n; $ it is an equivalence relation whose classes are the integers; for example, $-1:=\overline{(1,0)}$; we finish building the ring $(\mathbb Z,+,\times,\leq)$ ;

  5. $\mathbb Q$ is the fraction field of the integral ring $\mathbb Z$; for example, $2.13:=\overline{(213,100)}$;

  6. $\mathbb R$ is defined with the Cauchy sequences of elements of $\mathbb Q$ correctly defined from $\mathbb Q$ exclusively;

  7. Finally, $\mathbb C$ is defined after studying the Euclidean real vector space $\mathbb R \times \mathbb R$. For example, $\mathrm{i}:=(0,1), 1+\mathrm{i}:=(1,1)$.

Stéphane Jaouen
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