Simulation of smoothness using discrete quantities

[steve_bank]

We need a sticky thread called Zeno's Paradox In All Forms to move all Zeno related derails to.

 

[Juma]

1+1/2+1/3+1/4+1/5... - [1/2 +1/3 +1/4 + 1/5...] = 1

Then I would have the same issue as om the infinity threads. The idea that an infinite summation of finite numbers can lead to a finite result. A staement, not opening debate, but I would ssay the series approches 1 as a limit.

Where I left off in my personal dealings with infinity is that there isn't really an infinite number of "halves" that the arrow has to travel to reach its end point. It goes through an arbitrarily large natural number of halves (whatever that means, and I believe this is the formal way of putting it).

So it seems as if it's something that isn't quite infinity but is larger than any fixed number.

That is how I sort of made peace with this - sort of.

You are confusing the process of calculating/defining limes with the actual limit.

 

[untermensche]

We need a sticky thread called Zeno's Paradox In All Forms to move all Zeno related derails to.

They are not derails.

They are what happens when you try to apply infinity to anything real.

They are not paradoxes except to those with the religion that says infinity is something that can directly be applied to the real world.

Infinity only exists with imaginary entities, like points.

 

[steve_bank]

We need a sticky thread called Zeno's Paradox In All Forms to move all Zeno related derails to.

They are not derails.

They are what happens when you try to apply infinity to anything real.

They are not paradoxes except to those with the religion that says infinity is something that can directly be applied to the real world.

Infinity only exists with imaginary entities, like points.

The mind numbing endless regugitation of Zeno's Paradox by the same people asking the same questions and getting the same answersn over years is a derail. The thread was not about infinities and halving distances and the like.

 

[untermensche]

We need a sticky thread called Zeno's Paradox In All Forms to move all Zeno related derails to.

They are not derails.

They are what happens when you try to apply infinity to anything real.

They are not paradoxes except to those with the religion that says infinity is something that can directly be applied to the real world.

Infinity only exists with imaginary entities, like points.

The mind numbing endless regugitation of Zeno's Paradox by the same people asking the same questions and getting the same answersn over years is a derail. The thread was not about infinities and halving distances and the like.

On my side I see the mind numbing belief in real infinities everywhere.

The so-called paradoxes, nothing paradoxical about them, they show the absolute absurdity of real infinities, are reminders to those who try to take infinity out of models and try to apply it naked to the real world.

 

[steve_bank]

The mind numbing endless regugitation of Zeno's Paradox by the same people asking the same questions and getting the same answersn over years is a derail. The thread was not about infinities and halving distances and the like.

On my side I see the mind numbing belief in real infinities everywhere.

The so-called paradoxes, nothing paradoxical about them, they show the absolute absurdity of real infinities, are reminders to those who try to take infinity out of models and try to apply it naked to the real world.

How many times have you been told infinities in math and science are not reachable or quantifiable, infinity is a symbol for a variable that grows without boubd. Nothing more than that. Infinity is a useful concept, like the concept of 0.

Yet again you come back with your tired mantra 'infinities are not real'...QED

 

[ryan]

Where I left off in my personal dealings with infinity is that there isn't really an infinite number of "halves" that the arrow has to travel to reach its end point. It goes through an arbitrarily large natural number of halves (whatever that means, and I believe this is the formal way of putting it).

So it seems as if it's something that isn't quite infinity but is larger than any fixed number.

That is how I sort of made peace with this - sort of.

You are confusing the process of calculating/defining limes with the actual limit.

Well, I feel like "goes to zero" and zero are two diffrent things, yet they are equivalent. Remember with integrals 1/n (as n goes to infinity) does not only have a limit of 0 but equals 0.

Unless we get into hyperreals, the infinitesimal is ignored in formal calculus.

 

[Juma]

Where I left off in my personal dealings with infinity is that there isn't really an infinite number of "halves" that the arrow has to travel to reach its end point. It goes through an arbitrarily large natural number of halves (whatever that means, and I believe this is the formal way of putting it).

So it seems as if it's something that isn't quite infinity but is larger than any fixed number.

That is how I sort of made peace with this - sort of.

You are confusing the process of calculating/defining limes with the actual limit.

Well, I feel like "goes to zero" and zero are two diffrent things, yet they are equivalent. Remember with integrals 1/n (as n goes to infinity) does not only have a limit of 0 but equals 0.

Unless we get into hyperreals, the infinitesimal is ignored in formal calculus.

”Goes to zero” and ”zero” is in no way equivalent. It is two totally different categories.
”Goes to zero” refers to a method to manipulate limits. Zero is a number.

 

[Kharakov]

I asked (Jeopardy style):

What are the primary purposes of infinite series in mathematics?

another Q:

How many changes in direction are on a smooth curve?

How, in a discrete natural system, would one end up with smoothness, or the illusion of smoothness?

 

[steve_bank]

I asked (Jeopardy style):

What are the primary purposes of infinite series in mathematics?

another Q:

How many changes in direction are on a smooth curve?

How, in a discrete natural system, would one end up with smoothness, or the illusion of smoothness?

That is actually a good question. In the early days of glass CRT disdplays algoriithms to plot graphics with limited resolution was a hot topic. Even a simple line.Visually I'd say it is the eye's resolution.Plot a straight line on a high res display and put a straight edge against it.

Mechanicaly there is no such thing as a true flat surface. Draw a line with a straight edge on paper. Look at it from 5 feet away, then look at it with a magnifying glass. The edge of the line will apear rough. Same general idea with quantization.

Digitize an audio signal the reconstruct it with digital to analog converters. Filter it with a low pass filter and the digital reconstruction appears smooth. Artifacts of the finite quantization, coarseness, exist put are very low. Digital audio is quantized.

 

[ryan]

Well, I feel like "goes to zero" and zero are two diffrent things, yet they are equivalent. Remember with integrals 1/n (as n goes to infinity) does not only have a limit of 0 but equals 0.

Unless we get into hyperreals, the infinitesimal is ignored in formal calculus.

”Goes to zero” and ”zero” is in no way equivalent. It is two totally different categories.
”Goes to zero” refers to a method to manipulate limits. Zero is a number.

I agree that they aren't equivalent; that's why I said so. They are equivalent in formal calculus, though. Again, Reimenn integration does equate them. c/n (as n goes to infinity) = 0.

 

[Juma]

Well, I feel like "goes to zero" and zero are two diffrent things, yet they are equivalent. Remember with integrals 1/n (as n goes to infinity) does not only have a limit of 0 but equals 0.

Unless we get into hyperreals, the infinitesimal is ignored in formal calculus.

”Goes to zero” and ”zero” is in no way equivalent. It is two totally different categories.
”Goes to zero” refers to a method to manipulate limits. Zero is a number.

I agree that they aren't equivalent; that's why I said so. They are equivalent in formal calculus, though. Again, Reimenn integration does equate them. c/n (as n goes to infinity) = 0.

They why did you write that they are equivalent?
And, no they are not equivalent in formal calculus..
Maybe you ment: THE LIMIT of y = c/n when n goes to infinity is equivalent to 0.

 

[ryan]

I agree that they aren't equivalent; that's why I said so. They are equivalent in formal calculus, though. Again, Reimenn integration does equate them. c/n (as n goes to infinity) = 0.

They why did you write that they are equivalent?
And, no they are not equivalent in formal calculus..
Maybe you ment: THE LIMIT of y = c/n when n goes to infinity is equivalent to 0.

The Reimenn integral, as you know, takes the sum of an area under a curve. Sometimes we are left with some c over n. The limit becomes 0, but the rest of the area also becomes 0. It is not just the limit that becomes 0, but also the area, volume etc. that becomes zero. We are not left with infinitesimal areas/volumes.

 

[Kharakov]

I asked (Jeopardy style):

What are the primary purposes of infinite series in mathematics?

another Q:

How many changes in direction are on a smooth curve?

How, in a discrete natural system, would one end up with smoothness, or the illusion of smoothness?

That is actually a good question. In the early days of glass CRT disdplays algoriithms to plot graphics with limited resolution was a hot topic. Even a simple line.Visually I'd say it is the eye's resolution.Plot a straight line on a high res display and put a straight edge against it.

Mechanicaly there is no such thing as a true flat surface. Draw a line with a straight edge on paper. Look at it from 5 feet away, then look at it with a magnifying glass. The edge of the line will apear rough. Same general idea with quantization.

Digitize an audio signal the reconstruct it with digital to analog converters. Filter it with a low pass filter and the digital reconstruction appears smooth. Artifacts of the finite quantization, coarseness, exist put are very low. Digital audio is quantized.

Not that the thread is going anywhere now that unter has ahold of it, but one point of the thread is that an infinite amount of discrete elements are required to describe something smooth, like nature.

A presumably natural consciousness would by its nature have a tendency towards smoothing, because of field strength drop off and the like.

 

[untermensche]

Nature is not smooth.

"Smooth" is a human constructed concept.

 

[steve_bank]

That is actually a good question. In the early days of glass CRT disdplays algoriithms to plot graphics with limited resolution was a hot topic. Even a simple line.Visually I'd say it is the eye's resolution.Plot a straight line on a high res display and put a straight edge against it.

Mechanicaly there is no such thing as a true flat surface. Draw a line with a straight edge on paper. Look at it from 5 feet away, then look at it with a magnifying glass. The edge of the line will apear rough. Same general idea with quantization.

Digitize an audio signal the reconstruct it with digital to analog converters. Filter it with a low pass filter and the digital reconstruction appears smooth. Artifacts of the finite quantization, coarseness, exist put are very low. Digital audio is quantized.

Not that the thread is going anywhere now that unter has ahold of it, but one point of the thread is that an infinite amount of discrete elements are required to describe something smooth, like nature.

A presumably natural consciousness would by its nature have a tendency towards smoothing, because of field strength drop off and the like.

Don't know what you mean. Our brains are quantized by brain cells. Brain cells are quantized by molecules. And so on.

Conceptually the models say reality is discrete particles interacting through fields. Smoothness is an illusion of perception. Solid is a relative term.

 

[steve_bank]

Nature is not smooth.

"Smooth" is a human constructed concept.

Smooth is a baby's butt.
Smooth is woman's thigh.
Smooth is good whiskey.
Smooth is a fast talking con artist.

Not smooth is random interjections.

 

[untermensche]

Nature is not smooth.

"Smooth" is a human constructed concept.

Smooth is a baby's butt.
Smooth is woman's thigh.
Smooth is good whiskey.
Smooth is a fast talking con artist.

Not smooth is random interjections.

I call them clarifications.

 

[Juma]

I agree that they aren't equivalent; that's why I said so. They are equivalent in formal calculus, though. Again, Reimenn integration does equate them. c/n (as n goes to infinity) = 0.

They why did you write that they are equivalent?
And, no they are not equivalent in formal calculus..
Maybe you ment: THE LIMIT of y = c/n when n goes to infinity is equivalent to 0.

The Reimenn integral, as you know, takes the sum of an area under a curve. Sometimes we are left with some c over n. The limit becomes 0, but the rest of the area also becomes 0. It is not just the limit that becomes 0, but also the area, volume etc. that becomes zero. We are not left with infinitesimal areas/volumes.

so you didnt bother to actually read my post...

 

[steve_bank]

How did Riemann integration come up? It can not get any simpler.

I don't know latex. limit as dx -> 0 sum between limites f(x)dx = area. Rectangular integration with infinitesimally small rectangles.