Do we draw a distinction between a number as an element of the reals, and an element of the naturals?

Question

I see in some explanations of attempts to formalize numbers such as Von Neumann's ordinals like in this rather philosophical question that we can draw a distinction between a real number '1' and a natural number '1', obviously in mathematical contexts I have encountered, we describe these as one and the same object, is this distinction one of a simply philosophical nature, And does it affect the rigor of mathematical foundations to treat them as one and the same, as we tend to do?

Do we ever see the number '2' as a natural number differently than '2' as a real number? Are they considered different objects? Is there any definitive answer?

To what extent is it fine to treat there as being one 'number' that exists? As you may see in the answers below (and in this question) some seem to believe they are the same object with different set based representations, two different objects.

try to avoid going too much into advanced set theory formalism as they are far above my level

Answer 1

The short answer is "it depends, but most of the time you can get away with treating them as the same thing."

The medium answer is: there are many different subtleties we could get into here, but the simplest one concerns what is called abuse of notation. If we wanted to be totally precise we'd refer to $2 \in \mathbb{N}$ as something like $2_{\mathbb{N}}$ and $2 \in \mathbb{R}$ as something like $2_{\mathbb{R}}$. Most of the time we don't bother to do this and we refer to all of these objects (and similar objects such as $2 \in \mathbb{Q}$ and $2 \in \mathbb{C}$) by the same symbol $2$. To some extent, it's a specific cultural fact about a given community of mathematicians which abuses of notation are tolerated and reasonable and which are not; this particular one is common and turns out not to cause problems most of the time, but that's something you have to pick up by osmosis from a community of mathematicians doing it.

One reason (not the only reason) this abuse of notation is tolerated is that there are "canonical inclusions"

$$\mathbb{N} \hookrightarrow \mathbb{Z} \hookrightarrow \mathbb{Q} \hookrightarrow \mathbb{R} \hookrightarrow \mathbb{C}$$

and each of these inclusions sends the old $2$ to the new $2$. These inclusions are so canonical that nobody bothers to name them and people will often write them as subset inclusions $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$, even though, to be extremely pedantic about it, strictly speaking none of these sets are subsets of each other as usually constructed in set theory.

The long answer is too much to get into here but it involves increasingly sophisticated concepts of what it means to treat two things as "the same" in mathematics. For a delightful and accessible discussion of this you can see, for example, Mazur's When is one thing equal to some other thing?

Answer 2

is this distinction one of a simply philosophical nature, or one that we can make in mathematics, or one that even makes sense to make?

If you want to show the theory of non-negative integers has a model in set theory, you're likely to represent $0$ as the empty set and $n+1$ as $S\cup\{S\}$, where $S$ represents $n$. If you want to do something similar for the theory of real numbers, you need to do things a different way I won't spell out in full, and each real number is then represented with a specific set.

But while $1$ is represented differently in these two cases, it's not identified with such representations, which in both cases are by no means unique. It's much more helpful to say e.g. $\Bbb N\subseteq\Bbb Z\subseteq\Bbb Q\subseteq\Bbb R\subseteq\Bbb C$. For each such $\subseteq$, there's an overhaul in how we represent the numbers, but that's only because the theories of such sets of numbers are progressively more complicated, not because the "old" numbers aren't also "new" numbers.

And does it affect the rigor of mathematical foundations to treat them as one and the same, as we do?

If you're worried about that, proceed as follows:

• Start from the most inclusive set of numbers (or whichever mathematical objects) you wish to give representations.
• Fix a specific such representation, or presuppose one has been fixed. One will exist if the theory is consistent.
• Apply it to each subset of interest, e.g. $\Bbb N$ in $\Bbb R$. The way we represent real numbers would also work for naturals, but we don't normally use it because, for that purpose, it's needlessly complicated. Even just the $\Bbb N$-in-$\Bbb Z$ case requires all integers, including naturals, to be represented as e.g. certain infinite sets of ordered pairs of finite ordinals, rather than the finite ordinals themselves.

If you're not worried about it - indeed, most of the time most mathematicians aren't - you'd not bother with any such numbers-as-sets formalism. But to those who are insistent on such a formalism, I say this: why don't you demand a similar treatment of the sets themselves?

Answer 3

As you mentioned $\sqrt2$ in the question I want mention one subtlety that comes up with number systems you won't meet early in your studies, namely the $p$-adics. The reason I bring this up is because when we extend our number system from $\Bbb{Q}$, things really branch out. We can extend $\Bbb{Q}$ into infinitely many different directions. The most popular one is surely $\Bbb{R}$ because that's where physic, engineering and a number of other things take place. When construct the real number called $\sqrt2$ we do it (for example) as a limit $S$ (or the equivalence class of the Cauchy sequence if you want to be pedantic) of the sequence $1$, $14/10$, $141/100$, $\ldots$. By the carefully designed arithmetic, we can then show that the equation $S^2=2$ holds.

In this limiting process we measure how far two rational numbers $q_1$ and $q_2$ are from each other by calculating the absolute value of their difference $|q_1-q_2|$. As we advance in the above sequence the differences eventually go below any power of $1/10$, for example $|141/100-1414/1000|=4/1000$ is below $1/100$. This leads to a working concept of a limit. Hopefully you have seen that.

The other ways of measuring how far $q_1$ and $q_2$ are from each other is number theoretic. We start out by picking a prime number $p$. Then we, again, look at the difference $q_1-q_2$. This time we write the difference, using factorization into primes, in the form $q_1-q_2=p^a\cdot m/n$, where neither of the integers $m,n$ is divisible by $p$. The integer $a$ can be positive or negative as the case may be. We then define the $p$-adic distance $d_p(q_1,q_2)=p^{-a}$. In other words, the numbers are close to each other if a high power of $p$ divides their difference. The surprising fact is that this concept of a distance still leads to a similar extension of rational numbers by means of equivalence classes of Cauchy sequences, the field of $p$-adic numbers $\Bbb{Q}_p$.

Since I have been motivated by $\sqrt2$ let us select $p=7$. We see that $3^2=9$ differs from $2$ by seven, now a small difference, so according to the $7$-adic metric $3$ is close to being a square root of two. We then observe that $3+1\cdot7=10$ is an even better approximation of $\sqrt2$ as $10^2-2=98$ is divisible by $7^2$. $3+1\cdot7+2\cdot7^2=108$ is better still, as $108^2-2=2\cdot7^3\cdot17$ is divisible by $7^3$. We can keep going on forever (requires a proof, but it isn't difficult). Observe that my "approximate square roots" also differed from the preceding one by something that is divisible by a higher and higher power of seven. Therefore they form a $7$-adic Cauchy sequence. We can (the proof is similar to the real case) again show that the limit $$S=3+1\cdot7+2\cdot7^2+6\cdot7^3+\cdots$$ satisfies $S^2=2$.

At this point we can may be agree that $1.41421356\ldots$ and $S=3+1\cdot7+2\cdot7^2+6\cdot7^3+\cdots$ don't really have much in common. Yet they can both claim to be $\sqrt2$.

Closing with a few remarks

• In the real numbers we can single out $1.414\cdots$ as the square root of two for it is positive. However, in the $7$-adic domain there is no similar division to positive and negative numbers, and $S$ and $-S$ have equal claims to being called $\sqrt2$.
• You know that in the reals only positive numbers have square roots. As I said that there is no positive/negative dichotomy in the $p$-adic realm, the existence of square roots in $\Bbb{Q}_7$ is different. For example, it turns out that there is no $7$-adic $\sqrt3$. The above process leading to $\sqrt2$ fails at the first hurdle because $n^2-3$ is not divisible by seven for any integer $n$. For the same reason there is no $5$-adic $\sqrt2$.
• One reason $p$-adic numbers are not used (much) in natural sciences is that their size and addition behave very differently. In calculus we arrive at interesting stuff like integrals by the process of adding together more and more very tiny numbers. This won't work in the $p$-adics. No matter how many numbers divisible by $7^5$ you add together, the end result is still divisible by $7^5$. It never "grows" to be divisible only by a lower power of seven.

Summary: When we extend from $\Bbb{Q}$ to directions other than $\Bbb{R}$, the number systems really branch out, and things like roots of the equation $x^2=2$ (should they exist) have nothing else in common (an exaggeration, but I'm not gonna discuss how we might bring these different extension back together again).