Can the area be understood as the "number of points"?

[jordi]

In Euclidean geometry (and even in measure theory, see for example Stein and Shakarchi's Real Analysis), distance in the real numbers is defined as the difference of the real numbers, and area of a square is understood as the product of the distances defining the given square (and the same for 3-dimensional volume).

I would like to be able to define area, volume, distance ... in a unique way (as opposed to defining each term separately). One reason for that is that the geometric proofs of the Pythagorean theorem depend on the definition of area.

And intuitively, there seems to be a way: define n-dimensional measure of an n-dimensional region as "the number of points contained in the region".

For example, for area, since for natural numbers the product of 2 times 3 is equal to summing 3 twice, the area can be understood as summing the horizontal region (the number of points in a given segment) as many times as there are points in the vertical region. This process is consistent with thinking about an area as the "amount of painting"needed to cover the region, i.e. the amount of points needed to be painted, irrespective if the region is one dimensional, or two or three.

As a consequence, if we could think about our space not as the cartesian product of twice the real numbers, but a two dimensional lattice in a very fine mesh (smaller than the error measurement committed by our measuring device), we could define distance, area, volume ... in a unifying way: the n-dimensional measure of a region is simply the number of points inside the region.

But in order to define this is the Euclidean space, we should take the limit to a mesh tending to zero, and "completing" with all rational and irrational numbers (in the real line).

Can this process be done? If so, is there any book where the process is defined?

 

[Gaussian97]

Well, the problem here is that there are infinitely many points and infinity is not easy to visualize. If you "count" the points in a 1x1 square you will find that there are exactly the same amount of points that in a line.
Of course, you can define the area as the number of infinitesimal regions (but with non-zero measure) that is basically what you do when you compute an area by integrating.

 

[jordi]

If we consider the real numbers, sure, the "number of points" in a line is the same as the "number of points" in a square (the corresponding ordinal, in set theoretic terms).

But in a lattice with a non-zero spacing, this is not true. By counting natural numbers, you see that if the width (number of points in a line) is say 7, and the height (the number of points in the vertical) is say 6, the number of points in the corresponding square area is 42. The number of points in the corresponding line is 7. So, they are not the same.

And please note that within the square, there are also 42 "unit square" units.

Please note that in the definition of a 2-dimensional Riemann integral, you take the limit of the sum of many "small" areas. But there is the underlying hypothesis that a "basic" area is width*height. And my intention is to try and avoid to make such a definition. I would like to avoid having to define, separately, length, area, volume ... and instead, having just a single "measure" definition (i.e. the amount of points inside the region).

So, everything is consistent ... in the lattice. We do not need to give three separate definitions for length, area and volume. We just give one definition (volume) and the others are theorems from the first.

For example, in the plane, the length of a segment is the area with width the length of the segment, and the height "tending to zero" (i.e. there is only one point in the vertical dimension).

So, if we could take the limit to zero spacing, and we could prove that this lattice, in the limit, is the Euclidean space, I would be happy. But I have never seen this done, and I do not know even if it is possible to do it.

But in the end, the QFT which are rigorously defined, I guess they do this process (one defines a cutoff, which defines a lattice, and the QFT is defined as the limit with lattice spacing tending to zero).

 

[BvU]

amount of points inside the region

You'd still have to deal with subtle differences between $\infty, \ \infty^2, \infty^3, ...$

My hunch is that the word 'amount' is meaningless in your context.

 

[jordi]

Probably you are right. But let me remind that those QFT that have been defined rigorously (in 2D and 3D), I believe the definition of the QFT is done in this Wilsonian way: define first the theory in a lattice, and take the limit of lattice spacing to 0. The QFT in the continuum is defined as this limit. So, in some sense, QFT rigorous definition generalizes what I want to do.

 

[FactChecker]

And intuitively, there seems to be a way: define n-dimensional measure of an n-dimensional region as "the number of points contained in the region".

This intuition is wrong. Points have no length, area or volume, so there are as many in there as you want. There are the same number of points in [0,1] as there are in [0,2] because the mapping $p_{[0,2]} = 2*p_{[0,1]}$ is a one-to-one mapping between the points.

 

[RPinPA]

There's a method of numerical integration using Monte Carlo techniques which uses this idea:
1. Enclose the area you are interested in, within a box of known area.
2. Generate some large number of points uniformly distributed in the box
3. Count the points which fall in the area of interest and compare to the total number of points.

So if you have a finite number of points with a uniform density, then clearly "counting the points" gives you an approximation to the area. Thus your idea could be captured as defining area of set $S$ in terms of that procedure as $\lim_{N->\infty} (n/N) A$ where $A$ is the area of the enclosing box, $N$ is the total number of uniformly distributed points in $A$ and $n$ is the number of points that fall inside $S$.

 

[Gaussian97]

There's a method of numerical integration using Monte Carlo techniques which uses this idea:
1. Enclose the area you are interested in, within a box of known area.
2. Generate some large number of points uniformly distributed in the box
3. Count the points which fall in the area of interest and compare to the total number of points.

So if you have a finite number of points with a uniform density, then clearly "counting the points" gives you an approximation to the area. Thus your idea could be captured as defining area of set $S$ in terms of that procedure as $\lim_{N->\infty} (n/N) A$ where $A$ is the area of the enclosing box, $N$ is the total number of uniformly distributed points in $A$ and $n$ is the number of points that fall inside $S$.

Well, I think this makes no much sense, you are defining the area of a surface using the area of a surface...

 

[FactChecker]

The OP attempts to use the quantum theory minimums as basic units for length, area, volume, etc. That may have some benefits in quantum theory, but IMHO, it makes the mathematical concepts unnecessarily complicated.

 

[RPinPA]

Well, I think this makes no much sense, you are defining the area of a surface using the area of a surface...

Good point. I wasn't thinking clearly. While it's true that OPs question reminded me of Monte Carlo integration, it's not actually useful as a definition.

 

[jordi]

This intuition is wrong. Points have no length, area or volume, so there are as many in there as you want. There are the same number of points in [0,1] as there are in [0,2] because the mapping $p_{[0,2]} = 2*p_{[0,1]}$ is a one-to-one mapping between the points.

Sure, this is a standard behaviour of infinite sets.

But discretizing the real line is a standard procedure in QFT, for example.

 

[jordi]

The OP attempts to use the quantum theory minimums as basic units for length, area, volume, etc. That may have some benefits in quantum theory, but IMHO, it makes the mathematical concepts unnecessarily complicated.

The idea of discretization does not necessarily come from QFT or QM. I speak about discretization because if I could treat space as discrete, then measure can be defined in a unifying way (counting points). But instead, without discretization, one needs to define length, area, volume ... independently.

In fact, I believe Euclid defined area and volume differently: volume was close to our Riemann integral, but area was defined without that concept.

 

[FactChecker]

without discretization, one needs to define length, area, volume ... independently.

I don't really agree with this. The concept of length is used to define the higher n-dimensional measurements.

 

[jordi]

I don't really agree with this. The concept of length is used to define the higher n-dimensional measurements.

I disagree. Maybe you can say that area is defined "as an analogy" of the concept of length. But the definition of area is different from the definition of length.

In particular in physics, Classical Mechanics can be understood as an "application" of the definition of length, through the Fermat principle.

But the definition of area is only used in String Theory, through another action (the area of the worldsheet).

Also, Euclid treated area (and volume) differently to length. The Euclid books dealing with area are different from the ones of length.

Instead, if the space were discrete, the same definition could encompass length and area.

If you disagree with me, please provide a single definition that encompasses both length and area.

 

[FactChecker]

I am talking about the mathematical concept, not about its application in any particular physical context.

 

[FactChecker]

If you disagree with me, please provide a single definition that encompasses both length and area.

I guess the most general definition that I can find is the norm of the exterior product of vectors. (see volume of a parallelotope and Exterior algebra)

 

[jordi]

But this is hiding the original question under the rug. My main point is that the area of a rectangle of sizes a and b is defined as a times b.

The question is: why?

Of course, the definition is reasonable. But in general, I would like to be able to define both length and area as the "amount of paint" needed to cover such regions.

Of course, defining area as a times b, accomplishes that. But why?

If we think about a discrete space, the answer is obvious: length is the amount of points in a row of points. Area is just the number of rows of points, times the length of such rows.

As a consequence, in this discrete space, if we double any of the two dimensions of a rectangle, the number of points doubles. As a consequence, it takes double paint to paint the new surface.

But defining the area as a times b, one is defining a new concept altogether. Of course, this definition satisfies the requirement that "if we double any of the two dimensions of a rectangle (...) it takes double paint to paint the new surface". But the area definition cannot be understood as an application of the definition of length, you need two different definitions, one for length and another one for area.

 

[Stephen Tashi]

But this is hiding the original question under the rug. My main point is that the area of a rectangle of sizes a and b is defined as a times b.

The question is: why?

A nice "measure" has the property that it is additive. If we subdivide a rectangle A into other rectangles , the area of A is equal to the sum of the areas of its sub-rectangles. The sub-rectangles may intersect each other on sets of "measure zero" - e.g. line segments.

Of course, the definition is reasonable. But in general, I would like to be able to define both length and area as the "amount of paint" needed to cover such regions.

Does it take more paint to paint a rectangle if we visualize it as being composed of two sub-rectangles? Do we paint the common line segment shared by the sub-rectangles twice?

 

[FactChecker]

But defining the area as a times b, one is defining a new concept altogether. Of course, this definition satisfies the requirement that "if we double any of the two dimensions of a rectangle (...) it takes double paint to paint the new surface". But the area definition cannot be understood as an application of the definition of length, you need two different definitions, one for length and another one for area.

One MUST multiply the length times width in order to talk about the amount of paint needed to paint a line. Otherwise, the line can be "painted" as thin as needed to get the amount of paint as close to zero as one wishes. No matter what you want, a multiplication by each dimension is absolutely necessary. Given that, the current mathematical definitions are as concise as possible. Your desire to simplify an already simple concept is asking too much. Your idea of discretizing the definition makes it much more complicated.

 

[Stephen Tashi]

I would like to be able to define area, volume, distance ... in a unique way (as opposed to defining each term separately).

Of course, mathematical definitions are technically arbitrary. If there exists a correct way to define a concept like "area", then there must already exist specifications that a correct definition will satisfy. Some properties of area must be already be defined before we can objectively debate whether a particular definition is correct.

Some people take the Platonic view that "area" refers to a specific thing that exists independently of attempts to define it. This is often a useful way of thinking, but it isn't rigorous by the standards of modern mathemaics.

And intuitively, there seems to be a way: define n-dimensional measure of an n-dimensional region as "the number of points contained in the region".

That provokes the predictable negative reaction from people familiar with current mathematical definitions. Whether you can invent a number-of-points approach to defining n-dimensional measures is a meta-mathematical question. It involves both intuitive preconceptions about how you (and your audience) want the defintion to behave and the mathematical question of whether a specific definition has the desired behavior.

I think a number-of-points approach must begin with the definition of a "point". By conventional definitions, a "point" in a 1-dimensional space is not a "point" in 2-dimensional space. So perhaps you need a definition of point that implements the intuitive idea that a "point on a line" is really a point on a subset of 2-dimensional space, which is really a subset of 3-dimensional space, etc. (In that approach, perhaps the coordinates of a "point" must be an infinite sequence of numbers.)

 

[jordi]

A private communication (thank you) has reminded of the hyperreals:

From the wikipedia: "One way of defining a definite integral in the hyperreal system is as the standard part of an infinite sum on a hyperfinite lattice defined as a, a + dx, a + 2dx, ... a + ndx, where dx is infinitesimal, n is an infinite hypernatural, and the lower and upper bounds of integration are a and b = a + n dx"

Then, if a 2D integral can be considered as an infinite sum on a 2D hyperfinite lattice, i.e. the points (a + jdx, b + kdy), everything makes sense to me, since under the hyperreals, a 2D integral is just an infinite sum of 2D points, as required.

Then, length could be understood as the division of the 2D integral, where the 2nd dimension has only height dy, divided by dy (in other words, length is a concept that can only be assigned to objects of height dy; if an object has a height higher than dy, length is not defined).

 

[FactChecker]

The volume element is a tensor concept of volume that transforms correctly under coordinate changes and so is coordinate-system agnostic.

 

[metastable]

As a consequence, if we could think about our space not as the cartesian product of twice the real numbers, but a two dimensional lattice in a very fine mesh (smaller than the error measurement committed by our measuring device), we could define distance, area, volume ... in a unifying way: the n-dimensional measure of a region is simply the number of points inside the region.

But in order to define this is the Euclidean space, we should take the limit to a mesh tending to zero, and "completing" with all rational and irrational numbers (in the real line).

Can this process be done? If so, is there any book where the process is defined?

An infinite subset of points within a 3 dimensional region, not along the same plane, can be defined using only 2 dimensions:

https://www.physicsforums.com/threads/defining-vectors.972018/

and:

https://www.physicsforums.com/threads/is-the-universe-a-3-sphere-or-a-4-sphere.971938/

 

[BvU]

An infinite subset of points within a 3 dimensional region, not along the same plane, can be defined using only 2 dimensions:

One dimension is sufficient. But then again: the claim is butter soft (old dutch expression )

 

[metastable]

https://www.physicsforums.com/threads/is-the-universe-a-3-sphere-or-a-4-sphere.971938/

One dimension is sufficient. But then again: the claim is butter soft (old dutch expression )

Whoops, the second link I previously posted, while interesting on its own, wasn't the intended link, this was:

https://www.physicsforums.com/threa...iverse-is-absolutely-flat.971984/post-6186231

 

[WWGD]

The only model of this I can think of is that of n-squares, a line broken into dx_i ,lengths a volume broken into cubes $dxdyf(x_k,y_k)$ assuming converfence,,etc. Volume is the number of such units/scaled cubes. Edit I remember a contest a while back on counting the number of gumballs in a jar. This was , essentially the volume in gumball units instead of cc.

 

[WWGD]

A problem is that both, e.g., a square of side 1, side 2, any finite size will have areas $1,4,.., n^2$ respectively but each will have uncountably-many points, so measure and cardinality are different concepts.

 

[meopemuk]

I recommend you to look into the field called "nonstandard analysis." They invented clever ways to think about infinities. Perhaps you can even define "the number of points in a line" using their methods? Not sure.

Eugene.