Philosophy of Integration (geometric interpretation): the difference between dx & a point?

Question

This may sound like a stupid question, but if you're familiar with "Infinite Hotel Paradox", probably it won't be; So here we go:

Integration of a scalar function $f: \mathbb{R} \rightarrow \mathbb{R}$ is assumed/defined to be equal to the area under the curve of a function displayed on x-y co-ordinate system, with dx representing the infinitesimal small width of a ...

Okay, let's hold on a second there: what happens if we replace "infinitesimal small" with "point"? These two notions are closely related to each other and probably there should be a mathematical relation between them!

In geometry: we know that a plane is made from infinite [infinitely long parallel] lines, and a rectangle from infinite [finitely long parallel] lines; The same could be said about the graphical interpretation of a function; hence the idea of replacing the notion of "infinitesimal small" with "point" should make sense.

Hence, we should be able to replace Riemann's rectangles with line segments, though an infinite numbers of line segments, maybe!?

So, what I'm wondering is how correct these musing are, or where is the catch? Because I haven't found anyone writing or describing dx like this ‍♂️

PS. I think, basically what I'm looking for is the branch of the mathematics which works on the relationships, i.e. transition from one layer of infinity to another layer of infinity. In this case from a magnitude of a [single] value on the x axis to the d(x) to the f(x).

PS. PS. Probably such a concept should fall under the "transfinite numbers [theory]" but I couldn't find any reference when integration is thought!

Answer 1

The integration needs a measure. The measure of a set is not the same as the number of elements (cardinality).

The reason to have measure instead of “counting points” is that as soon as you break through the first infinity (countable), you find out that many different sets have the same cardinality (like the number of natural and odd numbers) and it's not useful to measure things.

For example, interval $[0,1]$ and $[0,2]$ have the same number of points. But the length (measure) of $[0,1]$ is half of the length of $[0,2]$.

In the same sense, $dx$ is different from the point. It shows that we are interested in a measure of a set.

Answer 2

We don't write about the differential in the integral that way for the same reason we don't pretend we can just take the points of intersection of a secant line to be coincident and magically obtain a tangent line.

Derivatives (of continuous functions) are limits of the indeterminate form $\left[ \frac{0}{0} \right]$. They are undefined if evaluated as if we could just set the displacement to zero. The ratio is defined (at least on a sufficiently small open set of displacements) as the displacement passes through every value near $0$, but there is a gap (an open circle -- recall that on a graph this represents a single missing point -- think of the graph of $y(x) = x/x$, which is constantly $1$ except for the missing point at $x = 0$) in the graph of the slopes of the secant lines when we try to make the displacement be $0$. So we have to take limits to get the value that would go where that open circle is.

When we define integrals as a limit, we again are able to make the widths of the rectangles be whatever positive value(s) we want and get an area for each choice of widths (and sample points). When we just try to make the widths all $0$, we get the indeterminate form $[\infty \cdot 0]$, corresponding to the sum of infinitely many copies of something with zero area. Unfortunately, it doesn't matter how many $0$s you add together, even uncountably many, your running and final sums are always zero. So we don't get anything useful this way. (One way of saying this is "Whatever area is, it is not carried by points. And integrals are about area.") We have to make another graph of the range of values of Riemann sums we can get and this graph looks like a wedge that would have one point on the line $\mathrm{d}x$ = 0, but that point is missing in the graph. Again, we have to use limits to get the vertical coordinate of this open circle on this graph of values of Riemann sums.

Area (and volume, and so on, whatever those are) are better captured by measure theory. Weirdness of area for set of points can be demonstrated by Cantor sets. This is a closed subset on the interval $[0,1]$. The set of endpoints is dense in the Cantor set and every point of the Cantor set is an accumulation point but not an interior point (so this set is perfect and nowhere dense). The Cantor set has (Lebesgue) measure zero. (Lebesgue measure is very good at capturing the notion of length/area/volume/et c.)

Answer 3

Usually the classroom presentation of Riemannian integration considers the limit of these rectangles as their width goes to zero. In that limit, each individual rectangle seems like it's losing a dimension, yes, and you could think about them as approaching line segments.

However! As Vasily pointed out, dx has a more specific meaning. You could say it's telling us how we're "measuring" the area, rather than literally representing an infinitesimal step in the x direction. Relating Vasily's point to the geometric interpretation that you like, every line has a measure of zero in two dimensions. This means, geometrically, that it has an area of zero in 2D. If you're thinking about the integral of a scalar function over $\mathbb{R}$, then a line segment, by itself, does not contribute any increase to the value of an integral. Similarly, a point does not contribute any increase to the length of the interval [0,1] in one dimension. If you collect a bunch of points though (uncountably infinitely many), then you can change the length of the interval to [0,2]. Similarly, if you collect enough line segments, yes, you could think of them as collectively having an area equal to the integral.

If you're interested in how to interpet dx, check out measure theory and Lebesgue integration. You'll also find that dx takes on a slightly different meaning when you begin to talk about differentiable manifolds, but that's a tangent (pun intended) for another day.

As for identifying an infinitely small line segment with a point, be careful! By itself, yes, a line segment that is contracted to a point has that point as its limit. In that regard, Edward makes a good point (another pun). The actual limit under consideration is indeterminate; geometrically, we are not considering the limit of the line segment dx in isolation, but as it relates to the limit of the sums of rectangles whose width is dx.

PS Side note, what's the length of the "line segment" at x? It's just f(x). Keep in mind that the area of that line segment is zero though.

Answer 4

There are sort of three philosophical approaches to calculus, each of which got formalized at a different point in history:

1. Infinitesimals, which were what Newton and Leibniz used, and were more thoroughly formalized ca. 1950 as non-standard analysis.

2. Limits, which were invented ca. 1900. Measure theory is the fanciest incarnation of integration in this approach.

3. Smooth infinitesimal analysis (SIA), which uses non-aristotelian logic.

Good books are:

Keisler for #1. Free online.

Bell, A primer of infinitesimal analysis, for #3.

SIA leads in a different philosophical direction than what you propose, in that it asserts that a line is not realizable as a set of points. However, if you read Bell, he lays out the philosophical issues pretty clearly, and I think you'll see that he's considering the same issues you have in mind.