Why should I believe that the real numbers model distances along a line?

Question

Taking the real numbers to be a complete ordered field, why do we believe that they model distances along a line? How do we know (or why do we believe) that any length that can be drawn is a real number multiple of some unit length?

Answer 1

Completeness often feels a bit technical at first: we show that there is exactly one complete ordered field up to isomorphism, but why should that confluence of properties correspond to line-ness?

I think it's more intuitive to focus instead on connectedness. This is really the same thing in our context, but is a priori phrased in more convenient language: the idea is that if I "cut" the line into a "lower piece" and an "upper piece," then there is some point which captures this cut. Specifically:

The line is connected, in the sense that it cannot be written as the disjoint union of two nonempty sets $A$ and $B$ where $A$ is downwards closed, $B$ is upwards closed, $A$ has no greatest element, and $B$ has no least element.

This is - for me at least - a pretty fundamental piece of the intuition I have about the line. (Connectedness is equivalent to completeness in our context, but is in my opinion more obviously a fundamental line property.) The fact that $\mathbb{R}$ is the unique connected ordered field then tells me that $\mathbb{R}$ is the only possible way we could faithfully "model" the line by a field, that is, the only way we could sensibly add/subtract/multiply/divide lengths.

This whole line of attack was developed by Dedekind in his Essays on the Theory of Numbers.

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That said, we can critique this idea.

The line has three fundamental properties (to me anyways): it is a linear order without endpoints (duh), it is connected, and it is dense (= between any two elements is a third). These three properties are not enough to pin down a single linear order up to isomorphism: there are lots of connected dense linear orders without endpoints much larger than $(\mathbb{R};<)$. This does not contradict the above, since these "unreal" linear orders do not support a field structure.

So one way I could argue that the ordered field $(\mathbb{R};+,\times)$ does not faithfully model the intuitive line is if I gave up the assumption that lengths do in fact form a field! This raises a natural question:

What sort of algebraic structure can a connected dense linear order without endpoints other than $(\mathbb{R};<)$ support?

It turns out that the answer is very little: any two (nontrivial) connected ordered groups are in fact isomorphic,$^*$ and in particular up to isomorphism the only (nontrivial) connected ordered group is $(\mathbb{R};+)$. So assuming we want to be able to add and subtract lengths in a reasonable way, we're stuck with $\mathbb{R}$.

This is pretty much case-closed for me: the idea of "non-additive intervals" is sort of a non-starter as far as my own intuition is concerned.

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$^*$Here's a proof sketch:

Suppose $G$ is a nontrivial connected ordered group. First, note that by connectedness $G$ must be divisible: e.g. for each $x$ there must be some $y$ such that $y+y=x$, since otherwise we could partition $G$ into $\{y: y+y<x\}$ and $\{y: y+y>x\}$ contradicting connectedness.

Now - by nontriviality - fix some positive group element $g$. By divisibility we get an embedding $\eta$ of $(\mathbb{Q};+)$ into $G$ generated by sending $1$ to $g$. By connectedness, $ran(\eta)$ is cofinal in $G$ (otherwise look at the downwards closure of $ran(\eta)$); this gives us a homomorphism $h:G\rightarrow \mathbb{R}$ (think about sending $x\in G$ to the least upper bound in the sense of $\mathbb{R}$ of $\eta^{-1}(\{y: y<x\})$). And we can show that $h$ is a bijection - hence an isomorphism - by applying connectedness again.

Answer 2

In my opinion, this is a perfectly good question, and the answer is more about the humans who practice mathematics than about mathematics itself. (Disclaimer: I am a practicing mathematician but not a historian.)

Whenever a mathematical structure or object is created, its creation is in response to some phenomenon or pattern we have already observed. The positive integers (and addition and multiplication) were created when people counted things; negative integers and zero were created when we observed that they make subtraction a much nicer operation, and analogues (a bit more abstract than for positive integers) were found in the real world.

The phenomenon of a seemingly continuously varying spectrum of measurements is something that would have been observed in the real world long long ago (as would the fact that all these measurements can be found along a single straight line and come with a natural ordering). There was a time, apparently, when people thought that rational numbers were sufficient to capture this set of measurements (and then comes the story of the school of Pythagoras discovering to their dismay that $\sqrt2$ is irrational).

Formal definitions/constructions of the real numbers (Cauchy in the early 19th century C.E., Bertrand and Dedekind later in the same century) were specifically designed to capture our real-world experiences with this ability to continuously measure distances along a line. Completeness, in particular, is the property most closely aligned with the observation that any length can be expressed as a multiple of some fixed unit of measurement.

Later on, mathematicians (such as Hilbert and Hölder around the turn of the century) realized that the real numbers could be characterized by a very small number of axioms (those of a complete ordered field), which was progress in the abstract real of mathematics but which took the definition farther away from the earlier ones. As is the case with many mathematical topics, this process of refining, streamlining, and abstraction can indeed make it harder for us to connect the formal definition with the human intuitions that spurred the mathematics in the first place.

Answer 3

I think what you are talking about is called The Ruler Postulate. The existence of Nonstandard Analysis seems to imply that there exists geometries that support calculus but not the Ruler Postulate.

Answer 4

Firstly, we have some geometrical intuition about what an ideal line should be like, which is based on our experience of the physical world. This ideal line satisfies:

1. The points (i.e. possible positions) on the ideal line are closed under translation and scaling, and addition and multiplication is defined in terms of those transformations. That is, the translation that moves $0$ to $x$ moves $y$ to $x+y$, and the scaling with origin $0$ that moves $1$ to $x$ moves $y$ to $x·y$. Geometric intuition then tells us that the points on the line satisfy the field properties! See viewpoint 2 of this post for more details.

2. Note that the above implies that all rational numbers are points on the ideal line, since we can iterate suitable translations to get all integers and then use suitable scalings to get all rationals.

Now are there other points on the ideal line besides rationals? Well, is there a point $r$ such that the scaling that moves $1$ to $r$ also moves $r$ to $2$? This $r$ is of course what we now know as $\sqrt{2}$, which we can construct algebraically. But the point is that we can envision such a scaling geometrically! After all, we 'can' continuously zoom in, and watch the point $r$ where $1$ moves to, and watch the point where $r$ moves to, which is $r·r$, and 'observe' that $r·r$ moves continuously too, so at some point during the zoom it must coincide exactly with $2$.

Similar reasoning makes it highly desirable that the ideal line satisfies IVT (the intermediate value theorem): Given any continuous function $f$ on the line (i.e. $f$ maps every sequence of points that converges to a point $x$ to a sequence of points that converges to $f(x)$), and any points $a<b$ such that $f(a) < 0 < f(b)$, then $f(c) = 0$ for some point $c$.

If you push your intuition a bit further, you would want the ideal line to satisfy more than IVT:

3. Given any increasing function $f$ on the line (i.e. $f(a) ≤ f(b)$ for every points $a ≤ b$), and any points $a,b$ such that $f(a) < 0 < f(b)$, then there is some first point $c$ such that $f(x) ≥ 0$ for every $x > c$. It may not be that $f(c) ≥ 0$, since $f$ may become positive only after $c$. However, intuitively $L = \{ x : f(x) < 0 \}$ and $R = \{ x : f(x) ≥ 0 \}$ are separated only by a single point $c$, and that $c$ is what we now call the supremum of $L$.

It is a good exercise to check that this 3rd property (together with the other 2 properties) implies IVT. It is equivalent to Dedekind-completeness, but I purposely gave this formulation to invoke your geometric intuition about continuous transformations.

Note that ultimately the Dedekind-completeness of the reals still relies on an ambient foundational system (that dictates what are functions on reals or sets of reals), and you may be interested in the fact that the computable reals satisfy the same first-order sentences as the reals. You can read more here and here about that and also alternative formalizations of the reals that are 'more absolute' in the sense of being 'less sensitive' to the foundational system.

Answer 5

I think steven gregory's answer is probably the best so far: we simply choose to believe that real numbers describe distances - it has turned out to be a useful notion, but it is simply a choice we made, it isn't absolute truth.

This really about the fundamental nature of mathematics: there are certain things we choose to accept as true witout proof (called axioms), and when we that maths represents absolute truth, it means simply that the logical statements of the form "if the axioms are true, then ..." are true.