Continuous limit

[bahman]

The continuous regime is defined as the limit of the discrete regime when 1/N (infinitesimal) tends to zero (or N tends to infinity). N tends to infinity but never becomes infinity, so it is finite. But we are in the discrete regime when N is finite. So either the definition of the continuous regime is wrong or we are dealing with a dilemma or infinitesimal is neither zero nor finite.

[wtf]

The continuous regime is defined as the limit of the discrete regime when 1/N (infinitesimal) tends to zero (or N tends to infinity). N tends to infinity but never becomes infinity, so it is finite. But we are in the discrete regime when N is finite. So either the definition of the continuous regime is wrong or we are dealing with a dilemma or infinitesimal is neither zero nor finite.

There are no infinitesimals in the real numbers. Nor is the phrase "continuous regime" defined in mathematics anywhere that I know of. So until you carefully define your terms, you are just ... in the common parlance ... "making shit up."

It's true that the limit of 1/n as n -> infinity is zero. But there is no "discrete regime" or "continuous regime" involved. Nor is there anything called infinity in this context. Saying "as n -> infinity" is just shorthand for "as n becomes arbitrarily large but still finite."

I wonder if you are just a victim of freshman calculus, where rigor is abandoned in favor of cookbook recipes. Or worse, a victim of online discussions of limits, in which superstition and rumor replace rigor.

[Impenitent]

Continuous limit...

build the wall...

-Imp

[bahman]

The continuous regime is defined as the limit of the discrete regime when 1/N (infinitesimal) tends to zero (or N tends to infinity). N tends to infinity but never becomes infinity, so it is finite. But we are in the discrete regime when N is finite. So either the definition of the continuous regime is wrong or we are dealing with a dilemma or infinitesimal is neither zero nor finite.

There are no infinitesimals in the real numbers. Nor is the phrase "continuous regime" defined in mathematics anywhere that I know of. So until you carefully define your terms, you are just ... in the common parlance ... "making shit up."

Integral and derivative are continuous concepts.

It's true that the limit of 1/n as n -> infinity is zero. But there is no "discrete regime" or "continuous regime" involved. Nor is there anything called infinity in this context. Saying "as n -> infinity" is just shorthand for "as n becomes arbitrarily large but still finite."

The discrete regime is when N is finite. N never becomes infinite therefore we are always in the discrete regime.

[bahman]

Continuous limit...

build the wall...

-Imp

But the wall is made of breaks. And breaks are made of...

[Impenitent]

Continuous limit...

build the wall...

-Imp

But the wall is made of breaks. And breaks are made of...

pauses between tones

-Imp

[wtf]

Integral and derivative are continuous concepts.

Even that's not true. For integrals, consider a finite sum is a special case of an integral, as in for example the case of an integral with respect to counting measure. For derivatives, consider the calculus of finite differences.

https://math.stackexchange.com/question ... ng-measure

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

I agree with you that in many contexts we can view the continuous as the limit of a discrete process. I wonder if there's an interesting discussion or topic to be had around this. What did you have in mind when you posted?

The discrete regime is when N is finite. N never becomes infinite therefore we are always in the discrete regime.

That doesn't convey anything to me in terms of what point you are trying to make. It's not that what you say is wrong or bad in any way, it's just ... kind of an, "Ok whatever" kind of remark. It doesn't seem to lead anywhere and you haven't given us a clue.

[bahman]

Integral and derivative are continuous concepts.

Even that's not true. For integrals, consider a finite sum is a special case of an integral, as in for example the case of an integral with respect to counting measure. For derivatives, consider the calculus of finite differences.

https://math.stackexchange.com/question ... ng-measure

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

If the integral is the area under the curve and the derivative is the tangent of a curve at a given point then integral and derivative are continuous concepts.

I agree with you that in many contexts we can view the continuous as the limit of a discrete process.

I am wondering how such a thing is possible.

I wonder if there's an interesting discussion or topic to be had around this. What did you have in mind when you posted?

I am wondering how one can have a continuous regime from a limit of the discrete process.

The discrete regime is when N is finite. N never becomes infinite therefore we are always in the discrete regime.

That doesn't convey anything to me in terms of what point you are trying to make. It's not that what you say is wrong or bad in any way, it's just ... kind of an, "Ok whatever" kind of remark. It doesn't seem to lead anywhere and you haven't given us a clue.

I mean continuous regime cannot be reached from the limit of the discrete regime since N is always finite.

[wtf]

If the integral is the area under the curve and the derivative is the tangent of a curve at a given point then integral and derivative are continuous concepts.

Yes, that's the freshman calculus point of view. But there are more abstract and general viewpoints on integrals and derivatives that extend what they teach you in calculus class.

I agree with you that in many contexts we can view the continuous as the limit of a discrete process.

I am wondering how such a thing is possible.

I guess I don't know what you mean. The rational numbers have "holes" where the irrationals should be. If you take the limits of all the sequences of rationals, you fill in the holes. Continuity can arise from discontinuity by taking limits. But that's a kind of a handwavy point of view.

I am wondering how one can have a continuous regime from a limit of the discrete process.

That's sort of what taking limits does. Remember that even in freshman calculus, they define a continuous function as one that preserves limits.

I mean continuous regime cannot be reached from the limit of the discrete regime since N is always finite.

Yes, 1/n is always nonzero for any finite n. And the limit of 1/2, 1/3, 1/4, 1/5, ... is 0. Limits are the formalism we use to make sense of arbitrary smallness. It's a tricky concept and perhaps you're expressing unease with the formalism. Historically it took 200 years from the time of Newton's calculus to the late 19th century for the idea of continuity and limits to finally be nailed down rigorously. So there's some philosophical difficulty there, if that's what you mean.

[wtf]

ps -- I think what you're getting at is that calculus is a brilliant formalism for approaching the continuous from the realm of the discrete. And maybe the question is, what's really going on?

This is a very old question. In Newton's time people thought in terms of infinitesimals but it was very controversial, since nobody could logically define infinitesimals. Today we have the concept of limits, and the "epsilon-delta" formalism that bedevils all students exposed to it. Perhaps what you are saying is that the underlying metaphysical mystery of how the discrete becomes the continuous, is still a mystery. In which case I agree with you. I think you just convinced me of your point of view. We have no idea why calculus works.

[bahman]

If the integral is the area under the curve and the derivative is the tangent of a curve at a given point then integral and derivative are continuous concepts.

Yes, that's the freshman calculus point of view. But there are more abstract and general viewpoints on integrals and derivatives that extend what they teach you in calculus class.

I agree with you that in many contexts we can view the continuous as the limit of a discrete process.

I am wondering how such a thing is possible.

I guess I don't know what you mean. The rational numbers have "holes" where the irrationals should be. If you take the limits of all the sequences of rationals, you fill in the holes. Continuity can arise from discontinuity by taking limits. But that's a kind of a handwavy point of view.

I am wondering how one can have a continuous regime from a limit of the discrete process.

That's sort of what taking limits does. Remember that even in freshman calculus, they define a continuous function as one that preserves limits.

I mean continuous regime cannot be reached from the limit of the discrete regime since N is always finite.

Yes, 1/n is always nonzero for any finite n. And the limit of 1/2, 1/3, 1/4, 1/5, ... is 0. Limits are the formalism we use to make sense of arbitrary smallness. It's a tricky concept and perhaps you're expressing unease with the formalism. Historically it took 200 years from the time of Newton's calculus to the late 19th century for the idea of continuity and limits to finally be nailed down rigorously. So there's some philosophical difficulty there, if that's what you mean.

Is n in the definiton of limit is finite? If yes, we are dealing with discrete regime. I am wondering how you could have continuous regime when you are in discrete regime!?

[bahman]

ps -- I think what you're getting at is that calculus is a brilliant formalism for approaching the continuous from the realm of the discrete. And maybe the question is, what's really going on?

This is a very old question. In Newton's time people thought in terms of infinitesimals but it was very controversial, since nobody could logically define infinitesimals. Today we have the concept of limits, and the "epsilon-delta" formalism that bedevils all students exposed to it. Perhaps what you are saying is that the underlying metaphysical mystery of how the discrete becomes the continuous, is still a mystery. In which case I agree with you. I think you just convinced me of your point of view. We have no idea why calculus works.

Aren't epsilon and delta finite/non-zero?

[wtf]

Is n in the definiton of limit is finite? If yes, we are dealing with discrete regime. I am wondering how you could have continuous regime when you are in discrete regime!?

You are repeating yourself. I met you more than halfway in trying to understand your point. I agree that there is a very long history and some thorny metaphysical issues involved with the nature of limits.

Can you say where your questions are coming from? Are you a philosopher trying to understand the calculus of the 17th century? A calculus student trying to make sense of what your book or teacher or saying?

I did my best to put the issue in perspective and you don't seem to be engaging.

Aren't epsilon and delta finite/non-zero?

Same point. Yes epsilon is nonzero but arbitrary. How much of the formalism have you studied? Again, what's your background and where are you coming from?

I really explained this to the best of my ability. It was a mystery for 200 years, the limit formalism resolves the issue, it's all logically correct from first principles, and yes there are metaphysical issues regarding the nature of continuity.