PeterJ said:
That's how I imagine points are usually defined, as the end point of a never ending process.
Generally, "point" is never defined at all.
A typical version of the theory of Euclidean geometry, for example, takes "point" as one of its "primitive notions" (others tend to include "line", "lies on", "between" and "congruent") and never attempts to define it. Instead, it postulates the properties that points have, and studies the consequences of those properties.
A typical version of the theory of real numbers is similar. It takes "number" as a primitive notion along with 0, 1, +, *, and <, and postulates the complete ordered field axioms.
A definition of "point" only comes when you want to apply Euclidean geometry to some purpose. e.g. a physical theory might assert that there is a notion of "position" in the universe, which obeys all of the axioms of Euclidean geometry describing points.
Another example is that, to better study the arithmetic of real numbers, we might define a model of Euclidean geometry in which "point" is interpreted to mean an ordered pair of real numbers. Conversely, in order to better study Euclidean geometry, we might construct a number line -- a model of the theory of real numbers -- and work with coordinates. In this sense, the two theories are actually the same theory just presented differently.
Locale theory defines "point", but that's simply because, pedagogically, it doesn't seem useful to take it as a primitive notion. I'm sure that would change if it was shown to be useful.
As you might guess I'm a formalist. But only weakly -- I don't make any assertions on whether or not mathematics has meaning, I just assert that the meaning isn't part to the formalism.
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Now, I tend to get a little uneasy when people talk about "never-ending processes". It's a perfectly good idea, but there are a lot of misconceptions often associated with it.
The most common, I think, is of this form. When talking about things in this way, it is the
process itself that names the point. (maybe we have defined "point" to literally be the process, or maybe as a limit in some suitable sense, or some other way to associate the name to the point) However, sometimes people get in their heads that the point is supposed to be the
result of the process, either taken after some ill-defined number of steps, or after some mythical final step.
I can't actually tell if you have a bad idea in your head, I'm just a little uneasy about it.
I assume that they simply have to be defined in this sort of way. It's the issues this raises that interest me.
It's a typical approach to naming things. You identify objects by some property they have -- in this case, a point is identified by a collection of "regions" that would contain it, but you need infinitely many regions to uniquely pin down the point.
The definition I cited is actually the unrolling of a simpler definition -- there is a locale called "*", and a "point of the locale L" is really just a mapping from * to L. But when you unfold all the complexities of the locale of real numbers, the locale-theoretic meaning of "point" transforms into something similar to and equivalent to the one I stated.
Typically, definitions that apply in most or all cases of interest tend to require infinite amounts of information. Specific cases often require much less -- e.g. for the Euclidean line, I could instead fix an orientation and name points with intervals (with the idea that the point is to be the left endpoint of the interval). But this particular scheme is very specific to the Euclidean line.
The real numbers suffer from this too. General naming schemes (like decimals, or continued fractions) tend to require infinite amounts of data. However, specific numbers can often be named with much simpler methods, such as the positive square root of 2.
I see no possibility of constructing a coherent metaphysic for which points are not a convenient fiction, and I think this is a quite widespread view.
True or not, convenient fictions are, well, convenient. That's why we have them.
It's analogous to scaffolding -- in the end, all you care about is building a building, but it's much easier to do so if you build the scaffolding along with the building, then remove the scaffolding at the end.
This is ubiquitous in mathematics. e.g. if we decide to name rational numbers with names of the form x/y where x and y are integers, one of the first things we do is decide which names really do name rational numbers (1/0 does not), and decide when two different names name the same object (e.g. 2/3 and 4/6). This extends to mathematical structures, structures of structures, and so forth.
This is also one of the reasons physicists are so interested in symmetries. e.g. from the fact the laws of classical mechanics are symmetric under rotations and translations of Euclidean space, we deduce things like an absolute notion of "position" or "direction" have no physical meaning.
