Basic Geometric intuition, context is undergraduate mathematics

Question

At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object.

For me the first instance that comes to memory was in 7th grade in a inner city USA school district.

Getting to the point, my geometry teacher taught,

"a point has no length width or depth in any dimensions, if you take a string of points and line them up for "x" distance you have a line, the line has "x" length and zero height, when you stack the lines on top of each other for "y" distance you get a plane"

Meanwhile I'm experiencing cognitive dissonance, how can anything with zero length or width be stacked on top of itself and build itself into something with width of length?

I quit math.

Cut to a few years after high school, I'm deep in the math's.

I rationalized geometry with my own theory which didn't conflict with any of geometry or trigonometry.

I theorized that a point in space was infinitely small space in every dimension such that you can add them together to get a line, or add the lines to get a plane.

Now you can say that the line has infinitely small height approaching zero but not zero.

What really triggered me is a Linear Algebra professor at my school said that lines have zero height and didn't listen to my argument. . .

I don't know if my intuition is any better than hers . . . if I'm wrong, if she's wrong . . .

I would very much appreciate some advice on how to deal with these sorts of things.

Answer 1

The viewpoint you're groping towards is not crazy, and at least historically you're in excellent company -- e.g., Leibniz had similar ideas when he viewed an integral as a sum of the heights of infinitely many lines, weighted by their infinitesimal width, and this intuition is still the background for the notation $\int f(x)\,dx$ for integrals. However, from a modern perspective this viewpoint carries a large risk of accidentally concluding nonsense by stretching it beyond what it can do. Therefore it is not really in favor anymore.

A better answer to your misgivings (in the sense of being closer to the mainstream presentation) is probably simply to jump in with both feet and declare that a line is not really made of points.

Decide to think of a line as something that is in principle a different kind of thing from a bunch of points glued together. You can do this and still acknowledge that points exist and some of them are on the line while others are not.

It then turns out that all of the properties of a line segment (or a smooth curve in general) can be recovered from knowing which points lie on it and which don't. This doesn't necessarily mean that the points make up the line, but merely that the points tell us enough about the line.

It is technically convenient, then, to speak about the set of points on the line as a placeholder for the line itself, when we're formalizing our reasoning -- for the pragmatic reason that we have a well-developed common machinery and notation for speaking of sets of things, which means that we don't need to introduce a new formalism for an entirely different kind of things.

Some people are so comfortable with this representation that they happily declare that the line IS its set of points -- but nobody says you have to think of it that way. As long as you agree that the set of points determine the line, you can still communicate with people who prefer the other idea.

Answer 2

This is less an answer and more of an extended comment. You seem to be struggling with the idea of a point as contrasted with an infinitesimally thickened point, and it sounds to me like you want to do geometry with with infinitesimals. Whereas Omnomnomnom suggests looking at nonstandard analysis, I would suggest a different approach to infinitesimals, namely smooth infinitesimal analysis. It's pretty intuitive (no pun on intuitionistic logic intended) and easy to use. I personally think in terms of synthetic differential geometry and smooth infinitesimal analysis all the time when working with smooth manifolds and Lie groups/algebras. If you're interested, have a look at John L. Bell's A Primer of Infinitesimal Analysis. He's a prominent philosopher of mathematics and I'm sure you'll find something in common with his views.

Answer 3

I don't know if your post has much to do with "life advice", but the question of whether there should be an "infinitely small but non-zero width" is something that bears answering.

The way math is done (with the standard set of axioms), it is indeed taken as fact that a point has exactly zero length and that a line, which has non-zero length, has exactly zero thickness. With an understanding of measure theory, one can see that putting "uncountably many" "measure-zero" things together can produce something with non-zero measure, even if that seems a bit counterintuitive.

However, "non-standard analysis" (which, as its name suggests, is not standard) does allow for a notion of "infinitely small but non-zero" quantities. I don't know enough about this, however, to judge how well the usual formulation of those ideas lines up with your intuition.

As far as life-advice goes, I'd say don't be afraid of not knowing, even if you're the only one in the room who appears to be struggling with the idea. Most of the time, other people in your situation are struggling with the same idea, but these things are hard to express and potentially embarrassing to ask about.

You are certainly not the only one to struggle with the notion of "of course there are infinitely small things!". For example, you'll find that the question of why does $0.999\dots = 1$ has been asked an re-asked many times over on this site.

Answer 4

Viewpoint from measure theory:

The length/area/volume/hypervolume of a set $S \subseteq \mathbb{R}^n$ is merely its Lebesgue measure $\lambda(S)$.

Since $\lambda$ is a continuous measure, the measure of any single point is $0$, so $\lambda(\{x\}) = 0$ for all $x$, but uncountable unions of points, such as $S = \bigcup_{x \in S} \{x\}$, may have non-zero measure.

The important thing here is uncountability. If you can iterate through the points in $S$ one by one such that any point in $S$ will eventually be encountered, then $S$ must have $0$ measure.

Answer 5

After Abraham Robinson died in 1974 there was a bit of a pogrom against both the framework with infinitesimals that he developed, and against his students who had great trouble finding jobs. Most of the critics were less than well-informed, as richly illustrated in the current literature.

Thus, Alain Connes' flawed critique is analyzed in this article. Errett Bishop's flawed critique is analyzed in this article.

In recent years Robinson's framework got an eloquent advocate in the person of Fields medalist Terry Tao, who has been publishing books and writing blogs about this. Robinson's framework with infinitesimals remains the best approach to geometry, calculus, and analysis, and in particular answers your question about infinitesimal width.

Answer 6

Meanwhile I'm experiencing cognitive dissonance, how can anything with zero length or width be stacked on top of itself and build itself into something with width of length?

That's not what happens, at all, when going from zero to one and then on to n dimensions. You are thinking too literal. A mathematical point, line, or plane does not exist in the physical reality, it is simply a thought construct.

We have to separate the "dots" we make on paper from mathematical "points". The point your professor is likely talking about probably consists of a vector (a bunch of numbers) together with an associated vector base in some vector space, with no degree of freedom (0 dimensions). A line is the same, with one degree of freedom (1 dimension), which is achieved by adding a second vector, multiplied by a variable. A plane is the same, with two degrees of freedom (2 dimensions), achieved by adding two vectors multiplied by two independent variables and so on. Or, if vector spaces are not used, then the point is simply a collection of numbers describing its location.

At no point whatsoever does even the concept of "thickness" enter the picture. At no point whatsoever do we "stack points with zero length and end up with a line".

In general, pretty much anything can be a vector space; those are not limited to what we understand as points, lines, planes, volumes, at all. But in each vector space, the concepts are the same - you have objects which are spanned up by vectors, which have no other attributes except those that are explicitely given (a base and something spanning them up) and so on; and usually no easy/intuitive/naive correlation to "reality".

What really triggered me is a Linear Algebra professor at my school said that lines have zero height and didn't listen to my argument. . .

So, did you then listen to his argument?

EDIT: removed some stuff not really belonging here.