Any geometrical form, infinitesimal/fractal and then point

[wtf]

This is my understanding of the subject matter: Any mathematical category, like real number, is internally consistent if you can reach from one entity to another one. There are two geometrical entities in real number, point and segment. But you cannot reach a point to a segment, by this I mean that you cannot construct a segment from an assembly of points since point size is zero.

This is an interesting remark.

First, in modern set theory, the real numbers are a "set" of points. That means you can express the entire set of real numbers as a set of individual points.

This echoes Euclid's idea that a line is made up of points. Although when I looked for a reference, I could not find anyone claiming that Euclid actually said that. But it's commonly understood in math that a line is made of points, and that in modern math, the set of real numbers consists of all the individual real numbers; and we may, by a leap of imagination, take the set if real numbers as modeled by a straight line; and the individual real numbers represent the addresses, if you will, of locations on the line.

Now it turns out that there's a philosopher Charles Sanders Peirce, with exactly that spelling, who has pointed out that a continuum must have the property that its parts are the same as the whole; and that the set-theoretic model most definitely does NOT satisfy that property, for exactly the reason you mentioned. Points aren't anything like a continuum! So to Peirce, and to you, and also to some or many others these days, the standard set of real numbers is not entirely satisfactory as the conceptual model of a continuum.

If that's what you mean, I understand your point. As a "math guy" I am aware of these philosophical issues but nevertheless I will persist in calling the real line "the continuum" unless pressed by a Peircean.

This means that more geometrical entities than point and segment are needed for a mathematical category that is internally consistent, you need fractal for example.

I see what you're getting at. You want to construct space out of smaller parts, and points simply won't do; because they have zero size. Yes I agree that's a philosophical mystery. I'm not convinced fractals are the answer but I do see what you're getting at.

Fractal is made of points.

Yes good. I was about to say that so I'm glad we agree.

You have all sorts of fractals, one of them with a finite size. The question is how you can make a segment from the other geometrical entities.

Are you looking for something like the space-filling curve?

Then we have the problem of measurability which means that there must be constants within a healthy mathematical category.

Measurability is a subtle problem in math. There's a subject called measure theory in which we try to assign a number, or measure, to various sets of points in a way that generalizes length, area, volume, and so forth.

It can be proved that there are non-measurable sets; that is, sets of real numbers that can not possibly be assigned a sensible measure that's consistent with how we think of length/area/volume.

There is also a famous fractal called the Cantor set, which serves as an example of an uncountable set of measure zero.

Is any of this on point to what you're trying to say?

ps -- Also the infinitesimals of nonstandard analysis will not help you. They're non-Archimedean. That means that no matter how many infinitesimals you put next to each other you'll never get enough length to cover the real line. They don't give you the fractal properties you need.

[Skepdick]

Consider a geometrical entity. Divide it to infinite number of pieces to get infinitesimal. Infinitesimal is a fractal since it doesn't change no matter how many more times you divide it. The fractal is made of points though.

It occurred to me that another way to interpret what you are saying might be "scale invariance".

https://en.wikipedia.org/wiki/Scale_invariance

[bahman]

This is my understanding of the subject matter: Any mathematical category, like real number, is internally consistent if you can reach from one entity to another one. There are two geometrical entities in real number, point and segment. But you cannot reach a point to a segment, by this I mean that you cannot construct a segment from an assembly of points since point size is zero.

This is an interesting remark.

First, in modern set theory, the real numbers are a "set" of points. That means you can express the entire set of real numbers as a set of individual points.

This echoes Euclid's idea that a line is made up of points. Although when I looked for a reference, I could not find anyone claiming that Euclid actually said that. But it's commonly understood in math that a line is made of points, and that in modern math, the set of real numbers consists of all the individual real numbers; and we may, by a leap of imagination, take the set if real numbers as modeled by a straight line; and the individual real numbers represent the addresses, if you will, of locations on the line.

Now it turns out that there's a philosopher Charles Sanders Peirce, with exactly that spelling, who has pointed out that a continuum must have the property that its parts are the same as the whole; and that the set-theoretic model most definitely does NOT satisfy that property, for exactly the reason you mentioned. Points aren't anything like a continuum! So to Peirce, and to you, and also to some or many others these days, the standard set of real numbers is not entirely satisfactory as the conceptual model of a continuum.

If that's what you mean, I understand your point. As a "math guy" I am aware of these philosophical issues but nevertheless I will persist in calling the real line "the continuum" unless pressed by a Peircean.

Interesting. So we are on the same page.

This means that more geometrical entities than point and segment are needed for a mathematical category that is internally consistent, you need fractal for example.

I see what you're getting at. You want to construct space out of smaller parts, and points simply won't do; because they have zero size. Yes I agree that's a philosophical mystery. I'm not convinced fractals are the answer but I do see what you're getting at.

But I don't know another geometrical entity that can fill the gap between point and continuum.

You have all sorts of fractals, one of them with a finite size. The question is how you can make a segment from the other geometrical entities.

Are you looking for something like the space-filling curve?

I was looking for how you can fill line from points and other geometrical entities. The point doesn't do it. So what our next choice?

Then we have the problem of measurability which means that there must be constants within a healthy mathematical category.

Measurability is a subtle problem in math. There's a subject called measure theory in which we try to assign a number, or measure, to various sets of points in a way that generalizes length, area, volume, and so forth.

It can be proved that there are non-measurable sets; that is, sets of real numbers that can not possibly be assigned a sensible measure that's consistent with how we think of length/area/volume.

There is also a famous fractal called the Cantor set, which serves as an example of an uncountable set of measure zero.

Is any of this on point to what you're trying to say?

Yes. And thanks for further information.

ps -- Also the infinitesimals of nonstandard analysis will not help you. They're non-Archimedean. That means that no matter how many infinitesimals you put next to each other you'll never get enough length to cover the real line. They don't give you the fractal properties you need.

Infinitesimal, defined as the smallest geometrical entity, should exist otherwise we have measurability problem. The next question which bothers me is that what is the structure of infinitesimal?

[bahman]

Consider a geometrical entity. Divide it to infinite number of pieces to get infinitesimal. Infinitesimal is a fractal since it doesn't change no matter how many more times you divide it. The fractal is made of points though.

It occurred to me that another way to interpret what you are saying might be "scale invariance".

https://en.wikipedia.org/wiki/Scale_invariance

I think that infinitesimal is real but I don't know what is its structure. The fractal is the only option that comes to my mind.

[Skepdick]

I think that infinitesimal is real but I don't know what is its structure. The fractal is the only option that comes to my mind.

If that which you call "infinitesimal" is to be a true infinitesimal, then surely it needs to be irreducible? You are basically taking an Atomist perspective.

For any infinitesimal X/N, X/(N+1) is smaller yet.

What you are really battling against is the principle of (infinite?) induction.

[bahman]

I think that infinitesimal is real but I don't know what is its structure. The fractal is the only option that comes to my mind.

If that which you call "infinitesimal" is to be a true infinitesimal, then surely it needs to be irreducible? You are basically taking an Atomist perspective.

Yes, an infinitesimal is irreducible but it has non-zero size.

For any infinitesimal X/N, X/(N+1) is smaller yet.

If that is true then infinitesimal is reducible and that cannot be the case.

What you are really battling against is the principle of (infinite?) induction.

What is that?

[wtf]

Yes, an infinitesimal is irreducible but it has non-zero size.

What does irreducible mean? If epsilon is an infinitesimal in the hyperreal numbers, for example, then epsilon/2 is a smaller infinitesimal.

[Skepdick]

Yes, an infinitesimal is irreducible but it has non-zero size.

If that is true then infinitesimal is reducible and that cannot be the case.

Then whatever you are looking for is not a number-type, because it doesn't (can't?) support division.

It can be some other data-type though. Like a Boolean. A boolean is irreducible, but then it doesn't satisfy the "bigger than zero" property because comparing a Boolean to a Number is a meaningless thing.

What is that?

Proof by induction: http://comet.lehman.cuny.edu/sormani/te ... ction.html

If you can find an object which satisfies: X/1 = X/2, X != 0, then that's "irreducible". But you can't find such a number.

Atoms/irreducibles are not numbers.

[bahman]

Yes, an infinitesimal is irreducible but it has non-zero size.

What does irreducible mean? If epsilon is an infinitesimal in the hyperreal numbers, for example, then epsilon/2 is a smaller infinitesimal.

By irreducible in here I simply mean indivisible. My problem is if infinitesimal is divisible then one cannot define a unit of length. In other words, every segment has a relative length rather than absolute length. The reality is that any segment that we experience has an absolute length, for example, the length of your table. This indicates that there must exist an absolute length which is the smallest length.

[bahman]

Yes, an infinitesimal is irreducible but it has non-zero size.

If that is true then infinitesimal is reducible and that cannot be the case.

Then whatever you are looking for is not a number-type, because it doesn't (can't?) support division.

It can be some other data-type though. Like a Boolean. A boolean is irreducible, but then it doesn't satisfy the "bigger than zero" property because comparing a Boolean to a Number is a meaningless thing.

Does your table have an absolute length? If yes, then it means that the smallest length must exist.

What is that?

Proof by induction: http://comet.lehman.cuny.edu/sormani/te ... ction.html

If you can find an object which satisfies: X/1 = X/2, X != 0, then that's "irreducible". But you can't find such a number.

Atoms/irreducibles are not numbers.

In the context of the number, a point is an atom. The problem is that you cannot construct a line from it since a point has zero size.

[Skepdick]

Does your table have an absolute length? If yes, then it means that the smallest length must exist.

Your reasoning is unsound. The smallest length for a table is 0. Any other length you specify is not "the smallest length", because any length you specify can be divided by 2.

There is no way to resolve this mess in Mathematics.
There is a way in physics. Planck length.

But even then, Planck length is not prescriptive on reality. It's descriptive of the current human/technical limitations which prevent us from measuring anything smaller.

In the context of the number, a point is an atom. The problem is that you cannot construct a line from it since a point has zero size.

Obviously. But you can construct a line by joining two zero-size points.

Every axiomatic system has edge/corner cases. Deal with it.

https://en.wikipedia.org/wiki/Edge_case
https://en.wikipedia.org/wiki/Corner_case

[bahman]

Does your table have an absolute length? If yes, then it means that the smallest length must exist.

Your reasoning is unsound. The smallest length for a table is 0. Any other length you specify is not "the smallest length", because any length you specify can be divided by 2.

What I am saying is that you cannot build something with an extension from something that has size of zero. It has to be non-zero. That is, however, a constant of the reality since your table has a specific length.

There is no way to resolve this mess in Mathematics.
There is a way in physics. Planck length.

Physics is a mathematical description of reality.

But even then, Planck length is not prescriptive on reality. It's descriptive of the current human/technical limitations which prevent us from measuring anything smaller.

The smallest length should exist. If there is any structure between then it is not the smallest length.

In the context of the number, a point is an atom. The problem is that you cannot construct a line from it since a point has zero size.

Obviously. But you can construct a line by joining two zero-size points.

Every axiomatic system has edge/corner cases. Deal with it.

https://en.wikipedia.org/wiki/Edge_case
https://en.wikipedia.org/wiki/Corner_case

You cannot make. You can only abstract.

[Skepdick]

What I am saying is that you cannot build something with an extension from something that has size of zero. It has to be non-zero. That is, however, a constant of the reality since your table has a specific length.

You are confusing Physics with Mathematics now. Points don't exist in reality. Virtual particles with zero mass lead to infinities - which renders physics useless. That is why QFT cares immensely about Renormalization.

Physics is a mathematical description of reality.

The Mathematics of Physics is more constrained than pure Mathematics.

Physicists says ℓP (Planck length) is the shortest distance.
Mathematicians say: No it isn't. ℓP /2 is shorter.

The smallest length should exist. If there is any structure between then it is not the smallest length.

The structure "between" is created through the application of the "divide" operator!

Atom means "indivisible" - it was called that because it was thought that "there was no structure in between". It was the smallest building block of reality. Until it wasn't.

You cannot make. You can only abstract.

No. You have this backwards.

In the physical world you cannot make, you can only manipulate existing things.
In the Mathematical world you can make, abstract, invent, destroy - or whatever else your imagination allows for.

[bahman]

What I am saying is that you cannot build something with an extension from something that has size of zero. It has to be non-zero. That is, however, a constant of the reality since your table has a specific length.

You are confusing Physics with Mathematics now. Points don't exist in reality. Virtual particles with zero mass lead to infinities - which renders physics useless. That is why QFT cares immensely about Renormalization.

I am not confusing anything. What you said is also valid and mathematically explains that the reality is made of infinitesimal and not from zero. Graviton is the quantum space-time.

Physics is a mathematical description of reality.

The Mathematics of Physics is more constrained than pure Mathematics.

True.

Physicists says ℓP (Planck length) is the shortest distance.
Mathematicians say: No it isn't. ℓP /2 is shorter.

True. Physical reality is different from what we in general can abstract.

The smallest length should exist. If there is any structure between then it is not the smallest length.

The structure "between" is created through the application of the "divide" operator!

Atom means "indivisible" - it was called that because it was thought that "there was no structure in between". It was the smallest building block of reality. Until it wasn't.

So there is a gap between two states of affair. You cannot physically have something than infinitesimal.

You cannot make. You can only abstract.

No. You have this backwards.

In the physical world you cannot make, you can only manipulate existing things.
In the Mathematical world you can make, abstract, invent, destroy - or whatever else your imagination allows for.

I mean, it is physically impossible. You, however, can abstract it: 1 apple plus another one is equal to one apple.

[Skepdick]

I am not confusing anything. What you said is also valid and mathematically explains that the reality is made of infinitesimal and not from zero.

It explains nothing. For all we know the whole cannot be understood by chopping it up into parts. Let alone infinitesimal ones.

You are just over-indexing on reduction/decomposition/analysis.

Graviton is the quantum space-time.

If gravitons exist, then their respective quantum fields exist - ergo - distance between gravitons.

So there is a gap between two states of affair. You cannot physically have something than infinitesimal.

It's just a fencepost error. You keep switching between digital (discrete) and analog (continuous) perspectives.

I mean, it is physically impossible. You, however, can abstract it: 1 apple plus another one is equal to one apple.

No. You can't. I know what it means to add numbers. I have no idea what it means to add apples.

Apple + Apple = ?

AppleApple? (String concatenation)
[Apple, Apple] ? (List construction)
[Apple]? (Set construction)

The plus-operator is Overloaded