The standard treatment of continuity in mathematics textbooks in schools tends to be a bit crooked.
- The continuum is first assumed, but it is not stated what is assumed.
- For the real line, the lack of holes is a key property of continuity, but it is called by a word that students might have no affinity with (“completeness” rather than “wholeness”).
- When continuity is actually discussed in analysis (if at all), then this concerns the continuity of functions, which is rather a different subject.
- A discussion of the continuum brings us to topology, but do we really need to start with topology before we can do analysis ? Do you want to start your junior highschool class by stating: “In the mathematical field of point-set topology, a continuum (plural: “continua”) is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space.” ?
Our research question for today is: What might be a more logical exposition ? (Didactics would be second phase.)
I will not be telling anything new here, but some students might benefit from the more explicit and straightforward discussion.
Continuity as a primitive notion that cannot be defined
The basic notion of continuity is the real line. One might also think about 3D space or time. L.E.J. Brouwer wouldn’t trust space (Euclidean or non-Euclidean ?) and take time as his intuition, and hence speak about “intuitionism”. My impression is that space is more easy to communicate about (measuring rods are easier to make than clocks), whence I adopt the real line.
Definition w.r.t. human experience: Continuity is a primitive notion, that you might grasp by considering a line (section) in space.
Definition for mathematics: The set of real numbers R can be defined in a particular way. Personally I prefer the method by Timothy Gowers to develop the real numbers as infinite decimals.
Once the real numbers have been defined, then we can say that they also satisfy the notion of continuity. Thus, continuity is either a human experience or defined as the real numbers.
Once we have done this, then we can find the “Cantor-Dedekind Axiom“:
“In mathematical logic, the phrase Cantor–Dedekind axiom has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.” (PM. This wikipedia page should link directly to Tarski’s axioms for geometry.)
I find the term “axiom” a bit problematic.
- Given the two properties mentioned above, this isomorphism rather expresses human experience from modeling practice. The real numbers would be a model for the line (section) in actual space.
- Likely though, the identification of R with space is best seen as a definition of what we regard as Euclidean space. It is a question whether actual space would be Euclidean. It is a question whether we can actually imagine space being non-Euclidean, since we imagine e.g. a sphere still in 3D Euclidean space.
But the isomorphism is explained rather easily – and for didactics we would likely begin with this. For the numbers we can look at a development of binary decimals between 0 and 1. The next decimal is 0 or 1, and again, and so on. For space we can make a cut and have left and right parts, make a cut again with new left and right parts, and so on. Thus this is the same structure. But these also are different realms: numbers and space. Thus it is not quite an identity but an isomorphism. Interestingly: when cutting in this manner, we will never meet a hole.
Continuity can be explained only for subsets
Subsequently, the key properties of continuity can be formulated w.r.t. subsets S of R, rather than w.r.t. R itself.
Definition: A subset S of R is called continuous, if between two elements (values) in S there is always another one (i.e. it is dense), and when there are no holes. Or in formulas:
(a) For each x, y in S ⊂ R with x < y, there exists z in S such that x < z < y
(b) For each x, y in S ⊂ R with x < y, there exists no z in R \ S such that x < z < y
The last property shows the difficulty for R. If one would want to specify that R has a hole, then one would have to specify what that hole belongs to. To some X ? What is X ? In the past, people had a problem imagining what a vacuum was: the horror vacui. For them, space could only exist if something occupied that space. Nowadays, mathematical space is understood merely as a set of co-ordinates, and the issue what physical space would be is left to physics.
Also observe that this definition essentially depends upon the fact that the real numbers have been given, i.e. the earlier section. Thus, continuity is a basic notion given for R and there is only a “proper” (explicit) definition for subsets: which definition relies on the use of R.
If you don’t assume R, you get into problems. For example, if you were to take the set of rational numbers Q rather than R, then (a-Q) could be satisfied for some S, say S = [1/2, 3/2], and (b) would become:
(b-Q) For each x, y in S ⊂ Q with x < y, there exists no z in Q \ S such that x < z < y
In that case, one might say that Q becomes Q-continuous, but this is not the continuity that we want, since there are elements in R still in that interval. (Contiguity comes to mind as a label, but already has some use.)
Further developments
Property (b) is called “(Dedekind) completeness“. It is true that “complete” is a proper translation of German “vollständig” (German wiki), but I would rather prefer “(Dedekind) wholeness”, since this better indicates the lack of holes. But let me admit that I am used to the phrase “completeness” as well, for the chapter of ordering, and thus my preference is weak. Perhaps it is best to speak about “completeness (wholeness)”.
Subsequently, when we forget about the reliance on R, and try for a more abstract formulation, then the notion of supremum (least upper bound) comes into play. We can look at some S independent from R, as the “linear continuum“. This is not intuitive and not feasible in junior highschool. Potentially this approach actually captures continuity in a definition, so that it isn’t just primitive, and can be defined, but (for me, yet) there is no clear connection between the notion of continuity and the property of having suprema. The switch to topology comes into play, see G.H. Moore “The emergence of open sets, closed sets, and limit points in analysis and topology” (and thanks to Dag Oskar Madsen for the reference, in a discussion about open sets that is closed now).
Continuous function
Obviously it helps to have clarity on continuity in the reals first before speaking about continuous functions.
The notion of a continuous function f uses both domain S and range f[S]. At stackexchange, many readers liked the view by Qiaochu Yuan (answer 17):
“One abstract way to think about continuity (…) is that it is about error. A function f:X→Y is continuous at x precisely when f(x) can be “effectively measured” in the sense that, by measuring x closely enough, we can measure f(x) to any desired precision. (…) This is an abstract formulation of one of the most basic assumptions of science: that (most of) the quantities we try to measure (…) depend continuously on the parameters of our experiments (…). If they didn’t, science would be effectively impossible.”
For Dutch readers, Vredenduin has a nice exposition in Euclides 1969 on the notion too, partly containing this intuition on the error too, but not so explicitly. He speaks about a small change in the domain and no dramatic change in the range, but it is more enlightening to explicitly speak about (measurement) error. (And I would have a question on continuity of “f / g“, p14.)
Conclusion
The main argument is that this storyline is more straightforward for understanding continuity. All this suggests that school would benefit from a discussion of the reals. This would include issues like 0.9999…. = 1.0000…
I am supposing that junior highschool could manage the expressions of mathematical logic. The New Math tried and failed, but there should be more clarity why it failed.
PM. For completeness: there is always philosophy (and nonstandard analysis).
Horror vacui: no space without something (Wikimedia commons)
Boycott Holland 