In the Rational vs Real post I mentioned that real numbers were each “an infinitely tiny island separated from direct contact with all other numbers.” The metaphor of each real number as an island comes from how, given any real number, it’s not possible to name the next (or previous) real number.
It’s easy enough to name a particular real number. For instance 1.0 and 3.14159… real numbers. There are infinitely many more we can name, but given any one of them, there is no way to get to any other number other than by explicitly naming it, too.
This applies to a variety of numeric spaces.
The first one is the real number line itself. Each point on the line is an island isolated from direct contact with even its closest neighbors.
The two real numbers I named above (1.0 and 3.14159…) are examples of islands on the line. So is every other ordinary number. The integers and rational numbers are also real numbers, so they, too, are islands on the line.
But the integers and rationals are not isolated in their own realms. For instance, for integers it’s easy to determine the next (or previous) integer: just add (or subtract) one.
It’s a little more involved with the rationals because we need to pick a protocol, but once we’ve decided on one, we can always name the next rational number, too.
This ability to name the next (or previous) number is what separates countable infinity from uncountable infinity. That we can’t name the next number is basically why we can’t count the real numbers.
(From now on, whenever I refer to naming the next number, just assume naming the previous number is included.)
The real numbers form a continuum in which we can select (name) points, but in which the concept of an “adjacent” point loses meaning.
For any points a and b we name, there is always some point c between them. For instance, if we start with zero (0.0), no matter how small a number we name as the next number, there is always a smaller number between our number and zero. (If we pick that smaller number, there’s a yet smaller number, and so on no matter what number we pick.)
As Cantor showed, there is no process that can list (enumerate) all the real numbers. That’s because there is no way to name the next number. There also isn’t a protocol (as with the rational numbers) that doesn’t name the “next” but does (eventually) name them all.
I used to think a number like 0.999… was an exception. That was the one case where it was possible to name the next number: 1.0. The same applies to all such numbers. For instance, the next number after 2.999… is 3.0.
But it turns out that 0.999… is just another way to “spell” 1.0 — they have the same numeric value. (Mind-blowing as that seems.)
So there are no exceptions to the rule: Given any real number, it’s not possible to name the next (or previous) number.
Every real number is a tiny lonely island!
An X-Y graph (assuming we use real numbers for X and Y) is an example of a two-dimensional real number space.
Mathematicians call it ℝ² for short. The fancy-looking “R” refers to the real number set. With the “2” it means it takes two numbers to specify a unique point in the space.
[As an aside: The font used is called Blackboard Bold, and there are Unicode characters in most fonts for the common math ones: ℕ (naturals), ℤ (integers), ℚ (rationals), ℂ (complex), and a few others.]
The dimension idea generalizes. For example, ℝ³ is a formal term for the three-dimensional space that we inhabit. The number of dimensions can be whatever is needed mathematically.
For any given 2D point we select, there are two dimensions along which it’s impossible to name the next number. Certainly there are points above, below, left, and right (and allowed combinations thereof), but there is no way to name the next point.
As an example, what is the next point to the right (positive X direction) of the point [0.0, 0.0]? It’s impossible to say (just as it is impossible to say what comes after 0.0, which is all we’re really doing here).
As with 2D X-Y points, complex numbers also contain two real numbers and form a continuum of their own, the complex plane.
In all cases, each of these numbers is an infinitely tiny isolated island.
(Infinitely tiny because points have no size. The whole problem is we need an infinitely tiny jump to the “next” point, and there is no such thing. Which suggests there is no such thing as the infinitely tiny point itself, which again really makes one wonder about the real numbers. They introduce all sorts of problems.)
On the other hand, for contrast, consider a different kind of two-dimensional space: a checkerboard.
We might label it ℕ² — a two-dimensional natural number space. In this case, coordinate numbers are limited to 1–8, but the basic idea applies to, for instance, the pixels in an image where the numbers are much larger.
Given a coordinate, say [3,5], we can name the all the neighbor points (the “next” points in two-dimensions). In this case, the point to the right is [4,5]. The point to the left is [2,5], the point above is [3,6], and the point below is [3,4].
A numeric space like this gives us a taxicab geometry, which has a variety of interesting properties, especially having to do with distances (and hence the name).
More importantly, in such spaces numbers have direct neighbors. It’s possible to name the next point (as in general with integer spaces).
These kinds of numbers are not islands.
The real numbers and the continuum seem distinctly different from actual physical reality. They seem an abstraction of physical reality.
It’s possible reality doesn’t use actually real numbers. It might be restricted to rational numbers — to countable infinity. Maybe it’s actually true that “God made the integers, all else is the work of man.”
In any event, it strikes my fancy this idea that any real number is an infinitely tiny island with no direct neighbors in sight.
We can set sail from the island to find other islands, but they’re alone at sea as well. The only way to get to them is to navigate to them.
For instance, we can navigate by adding 0.125 to whatever island we’re on — that will take us to a new island every time (just not the “next” island). Or we can navigate the other way by subtracting the value.
It’s not that we can’t get to other islands. It’s that there’s just no such thing as the “next” island (other than by a navigation protocol that skips the infinite number of islands between). And no matter how close an island we jump to, there are always infinite islands in between.
And no map can ever list all the islands.
But the (pardon the pun) real question is: Are the islands real?
Or — as I’m starting to think — just something we made up?
The flies in that ointment are über numbers like pi and e that show up in so many real (that pun again) places. Can the processes behind those be rational and not real? Are the perfect abstractions chimeras?
If real numbers are real, reality is weird.
But we knew that.
Stay countable, my friends!