## Any geometrical form, infinitesimal/fractal and then point

### Any geometrical form, infinitesimal/fractal and then point

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.

### Re: Any geometrical form, infinitesimal/fractal and then point

Yeah, I have been saying this for quite awhile.

### Re: Any geometrical form, infinitesimal/fractal and then point

### Re: Any geometrical form, infinitesimal/fractal and then point

Here are a few:

Point Circle Paradox:

viewtopic.php?f=26&t=28159

Lines are sets, sets are fractals, fractals are fractions:

viewtopic.php?f=26&t=27660

Positive Numbers Move to Point Zero:

viewtopic.php?f=26&t=27650

Four Pointed Triangles, the Case for an Irrational Triangle:

viewtopic.php?f=26&t=27647

0=0 is foundation for Number Lines and Numbers as Empty Loops:

viewtopic.php?f=26&t=27591

### Re: Any geometrical form, infinitesimal/fractal and then point

Interesting. I will look at them later.Eodnhoj7 wrote: ↑Tue Mar 10, 2020 11:22 pmHere are a few:

Point Circle Paradox:

viewtopic.php?f=26&t=28159

Lines are sets, sets are fractals, fractals are fractions:

viewtopic.php?f=26&t=27660

Positive Numbers Move to Point Zero:

viewtopic.php?f=26&t=27650

Four Pointed Triangles, the Case for an Irrational Triangle:

viewtopic.php?f=26&t=27647

0=0 is foundation for Number Lines and Numbers as Empty Loops:

viewtopic.php?f=26&t=27591

### Re: Any geometrical form, infinitesimal/fractal and then point

You will have a difficult time formalizing that. Consider the unit interval on the real number line: the set of real numbers between 0 and 1. With or without the endpoints, it doesn't matter for this discussion.

Cut it in half and its length is 1/2, say from 0 to 1/2. Cut that left half in half and you get the interval from 0 to 1/4. No matter how many times you divide the interval in half, it will always consist of infinitely many real numbers and have a nonzero positive length.

When you say to "divide it into an infinite number of pieces", you have to explain what you mean. I can think of a couple of ways that don't produce an infinitesimal.

For example cut the unit interval into the infinitely many intervals 0 to 1/2 and 1/2 to 3/4 and 3/4 to 7/8 and 7/8 to 15/16 and so forth. There are infinitely many intervals and they add up to the original unit interval but none of them are infinitesimal.

Or you could just divide it up into all its individual points. But points have length zero, they are not infinitesimal.

So you have to say how you propose to construct an infinitesimal. And this will be very tricky because there are no infinitesimals in the real numbers. And even though there are various subjects in math in which there are infinitesimals, they're sophisticated ideas that require some technical math.

### Re: Any geometrical form, infinitesimal/fractal and then point

Even if you aren't comfortable with monads, start here: https://www.youtube.com/watch?v=BBp0bEczCNg

### Re: Any geometrical form, infinitesimal/fractal and then point

Point is the smallest substance with the size of zero. This means that 0*"any number" is zero by definition too. Then there is infinitesimal. We then have the continuum. We experience the continuum so we know what it is. We have never experienced a fractal as a whole but we know that it exists. There is only one place that it could exist, between zero and continuum. Therefore, fractal is infinitesimal.wtf wrote: ↑Wed Mar 11, 2020 1:37 amYou will have a difficult time formalizing that. Consider the unit interval on the real number line: the set of real numbers between 0 and 1. With or without the endpoints, it doesn't matter for this discussion.

Cut it in half and its length is 1/2, say from 0 to 1/2. Cut that left half in half and you get the interval from 0 to 1/4. No matter how many times you divide the interval in half, it will always consist of infinitely many real numbers and have a nonzero positive length.

When you say to "divide it into an infinite number of pieces", you have to explain what you mean. I can think of a couple of ways that don't produce an infinitesimal.

For example cut the unit interval into the infinitely many intervals 0 to 1/2 and 1/2 to 3/4 and 3/4 to 7/8 and 7/8 to 15/16 and so forth. There are infinitely many intervals and they add up to the original unit interval but none of them are infinitesimal.

Or you could just divide it up into all its individual points. But points have length zero, they are not infinitesimal.

So you have to say how you propose to construct an infinitesimal. And this will be very tricky because there are no infinitesimals in the real numbers. And even though there are various subjects in math in which there are infinitesimals, they're sophisticated ideas that require some technical math.

### Re: Any geometrical form, infinitesimal/fractal and then point

I think that monad/infinetisimal is fractal.Skepdick wrote: ↑Wed Mar 11, 2020 6:03 amEven if you aren't comfortable with monads, start here: https://www.youtube.com/watch?v=BBp0bEczCNg

### Re: Any geometrical form, infinitesimal/fractal and then point

What does that mean to you? What definition of fractal are you using? What definition of infinitesimal? Or do you mean all this metaphorically rather than mathematically?

In the real numbers there are no infinitesimals. Do you mean to be specifically referencing the hyperreal numbers of nonstandard analysis? Or am I taking you too literally?bahman wrote: ↑Wed Mar 11, 2020 11:57 pm Point is the smallest substance with the size of zero. This means that 0*"any number" is zero by definition too. Then there is infinitesimal. We then have the continuum. We experience the continuum so we know what it is. We have never experienced a fractal as a whole but we know that it exists. There is only one place that it could exist, between zero and continuum. Therefore, fractal is infinitesimal.

And what do you mean by fractal? The Mandelbrot set is a fractal but it's not self-similar and it doesn't exist between zero and the continuum; rather, it's a set of points in the complex plane.

### Re: Any geometrical form, infinitesimal/fractal and then point

I think the category of real number is an abstract that is internally incoherent. Just consider a line with unit of one. To have the unit in place you need a measurable constant to define a unit. This constant does not exist in real number since everything is divisible.wtf wrote: ↑Thu Mar 12, 2020 3:35 amWhat does that mean to you? What definition of fractal are you using? What definition of infinitesimal? Or do you mean all this metaphorically rather than mathematically?

In the real numbers there are no infinitesimals. Do you mean to be specifically referencing the hyperreal numbers of nonstandard analysis? Or am I taking you too literally?bahman wrote: ↑Wed Mar 11, 2020 11:57 pm Point is the smallest substance with the size of zero. This means that 0*"any number" is zero by definition too. Then there is infinitesimal. We then have the continuum. We experience the continuum so we know what it is. We have never experienced a fractal as a whole but we know that it exists. There is only one place that it could exist, between zero and continuum. Therefore, fractal is infinitesimal.

And what do you mean by fractal? The Mandelbrot set is a fractal but it's not self-similar and it doesn't exist between zero and the continuum; rather, it's a set of points in the complex plane.

By fractal/infinitesimal, I mean a geometrical entity made of point which is self-similar. I think fractals are between continuum and zero, otherwise, the things are not internally consistent. Therefore, I think hyper-number is the correct category.

### Re: Any geometrical form, infinitesimal/fractal and then point

Why the hyper-number? Why not the infinitesimal epsilon in surreal numbers?bahman wrote: ↑Thu Mar 12, 2020 10:38 pm I think the category of real number is an abstract that is internally incoherent. Just consider a line with unit of one. To have the unit in place you need a measurable constant to define a unit. This constant does not exist in real number since everything is divisible.

By fractal/infinitesimal, I mean a geometrical entity made of point which is self-similar. I think fractals are between continuum and zero, otherwise, the things are not internally consistent. Therefore, I think hyper-number is the correct category.

Bonus points: the surreal numbers field contains all the reals and hyper-numbers.

Double-bonus points - the surreals are recursively enumerable e.g computable.

### Re: Any geometrical form, infinitesimal/fractal and then point

I think surreal numbers is another acceptable category given the properties you mentioned.Skepdick wrote: ↑Thu Mar 12, 2020 11:11 pmWhy the hyper-number? Why not the infinitesimal epsilon in surreal numbers?bahman wrote: ↑Thu Mar 12, 2020 10:38 pm I think the category of real number is an abstract that is internally incoherent. Just consider a line with unit of one. To have the unit in place you need a measurable constant to define a unit. This constant does not exist in real number since everything is divisible.

By fractal/infinitesimal, I mean a geometrical entity made of point which is self-similar. I think fractals are between continuum and zero, otherwise, the things are not internally consistent. Therefore, I think hyper-number is the correct category.

Bonus points: the surreal numbers field contains all the reals and hyper-numbers.

Double-bonus points - the surreals are recursively enumerable e.g computable.

### Re: Any geometrical form, infinitesimal/fractal and then point

The Mandelbrot set is fractal but not self-similar, as I mentioned.

Secondly, why do you write fractal/infinitesimal? They're completely different things.

A fractal is a set of points with fractional Hausdorff dimension.

An infinitesimal is an element of an ordered field (a mathematical system in which you can add, subtract, multiply, and divide, and in which there's a notion of order that allows you to compare any two elements) that's larger than 0 but smaller than 1/n for any positive integer n.

They're two completely different concepts.

What inconsistencies do you see? And what do you mean "between continuum and zero?" The real numbers are a continuum, and zero is a particular real number. There are a lot of sets of real numbers that contain zero but are contained in the real numbers. They're not infinitesimals.

### Re: Any geometrical form, infinitesimal/fractal and then point

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 means that more geometrical entities than point and segment are needed for a mathematical category that is internally consistent, you need fractal for example. Fractal is made of points. 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. Then we have the problem of measurability which means that there must be constants within a healthy mathematical category.wtf wrote: ↑Fri Mar 13, 2020 2:12 amThe Mandelbrot set is fractal but not self-similar, as I mentioned.

Secondly, why do you write fractal/infinitesimal? They're completely different things.

A fractal is a set of points with fractional Hausdorff dimension.

An infinitesimal is an element of an ordered field (a mathematical system in which you can add, subtract, multiply, and divide, and in which there's a notion of order that allows you to compare any two elements) that's larger than 0 but smaller than 1/n for any positive integer n.

They're two completely different concepts.

What inconsistencies do you see? And what do you mean "between continuum and zero?" The real numbers are a continuum, and zero is a particular real number. There are a lot of sets of real numbers that contain zero but are contained in the real numbers. They're not infinitesimals.