Analyzing Infinitesimal Motion

[Mark44]

A line segment may contain an infinite amount of points, but an infinite amount of points does not make a line segment.

Perhaps an infinite amount of points can make a line segment depending on the distribution of points.

It gets even worse. The segment [0, 1] contains both an infinite number of points with rational coefficients (such as 1/2 and 2/3) and points with irrational coefficients (such as $\frac{\sqrt 2} 2$ and $\frac \pi 4$). In a field of mathematics called measure theory, the measure of the rationals in [0, 1] is 0, while the measure of the irrationals in the same interval is 1.

 

[Chenkel]

Yes, and in this case the calculations are the normal souvat equations of motion. The OP's calculations messing with infinities and infinitesimals are pointless. It's trying to use a screwdriver to turn a hex nut. Just the wrong tool for the job.

I appreciate any feedback, but I need to be honest here, I'm a fool just like everyone in the universe, but some fools are wiser than others.

Just FWIW I would never tell my apprentice to shut up and calculate, I feel it's rude, but maybe I'm too sensitive?

I really don't want to create animosity because it's unproductive to an intellectual conversation, but I am not the one who started with this language and additionally I see no value in silencing anyone, so just to ask for clarification, who should shut up exactly and why?

 

[russ_watters]

I appreciate any feedback, but I need to be honest here, I'm a fool just like everyone in the universe, but some fools are wiser than others.

Just FWIW I would never tell my apprentice to shut up and calculate, I feel it's rude, but maybe I'm too sensitive?

I really don't want to create animosity because it's unproductive to an intellectual conversation, but I am not the one who started with this language and additionally I see no value in silencing anyone, so just to ask for clarification, who should shut up exactly and why?

The context is that it's more of a slang saying - a catchphrase - than a order. It just means you should follow the math where it leads and not get bogged down in non-mathematical logic or philosophical questions about if/how the math relates to reality. Just do what works. Yes, the wording is aggressive, but it really isn't meant that way.
https://physicstoday.scitation.org/doi/10.1063/1.1768652?journalCode=pto#references-1

That being said, you seem pretty determined to stay on the course you are on, despite being repeatedly told it is the wrong one. So maybe aggressive correction is warranted?

I doubt even Zeno himself was as troubled by this issue 2000 years ago. I bet he recognized he was using the wrong math.

 

[Dale]

I created a "recap" with a more correct way to conceptualize the motion of the rocket with additional examples, perhaps you were unable to see it because a forum bug?

You mean post 22? I saw and responded to you about it. It suffers from the same problem as the OP. It is no more or less correct than the OP. You simply need to either use different numbers than the reals (if so which?) or express your questions in terms of limits.

Additionally tell me exactly what I am to not use in my examples until I understand it, I heard the word hyperreals, is there anything else?

You cannot treat infinity as a real number. Specifically, you cannot divide by an infinite number because there is no such thing as an infinite real number. Instead you can divide by a real number, $n$, and take the limit of that division as $n$ increases without bound. This is written as $$\lim_{n\rightarrow \infty} \frac{1}{n}$$ but it is important to recognize that in this notation $\infty$ is not a real number, but shorthand notation for "$n$ increases without bound".

Just FWIW I would never tell my apprentice to shut up and calculate, I feel it's rude, but maybe I'm too sensitive?

I am not telling you that. I am telling you to slow down (a lot) and think seriously about the question you are asking. It is not meaningful in the way that you asked it, so I am encouraging you to think about how you should ask it instead to produce a meaningful question. Either you need to change the form of the question itself of you need to choose an alternative representation where that form of question is meaningful.

You are trying to jump to answers before you have a meaningful question. Slow down.

 

[jbriggs444]

Using which number system? It is not infinitesimal in the real numbers because there are no infinitesimal real numbers.

In the reals. Zero is an infinitesimal in the real numbes. The only infinitesimal in the standard reals.

Edit: It has been forever since I was taught that. But I see that the definition on Wikipedia disagrees and requires that "infinitesimals" be distinct from zero.

 

[jbriggs444]

How does observation of a zero time event work?

It does not. All of our actual experimental observations of time or position are approximate. We often treat the measurements as if they were exact. Or as if they are approximations to an underlying exact reality. We normally model both space and time in that supposed reality as a continuum using the real numbers.

Reality is not actually required to be a continuum. We model it as such because we have no better model with an underlying granularity together with supporting experimental evidence. You do not want to put graininess into your model until you know what sort of graininess you will need.

Edit: Then we emulate the model with 64 bit floats that have graininess. But that's been discussed elsewhere.

 

[Herman Trivilino]

Since it is always changing position due to continuity, then for each point it must spend a very small amount of time in each position.

This is not true. The amount of time a moving particle spends at any given location is zero.

But if the object spends zero seconds at each position, then it will take zero seconds to get from position 0 to position 1000 which doesn't make sense.

What is this conclusion based on? Are you assuming that the total time of a trip is the sum of the times spent at each location along the way? That's not how it's done.

To get the total time of a trip you note the clock-readings at the beginning and ending of the trip, and then subtract the two.

 

[Drakkith]

Just FWIW I would never tell my apprentice to shut up and calculate, I feel it's rude, but maybe I'm too sensitive?

My apologies. I was paraphrasing Richard Feynman (I think) when he talks about people getting too caught up in philosophies and interpretations or whatnot. I wasn't really even talking to you, but more about my own post.

 

[Chenkel]

The context is that it's more of a slang saying - a catchphrase - than a order. It just means you should follow the math where it leads and not get bogged down in non-mathematical logic or philosophical questions about if/how the math relates to reality. Just do what works. Yes, the wording is aggressive, but it really isn't meant that way.
https://physicstoday.scitation.org/doi/10.1063/1.1768652?journalCode=pto#references-1

That being said, you seem pretty determined to stay on the course you are on, despite being repeatedly told it is the wrong one. So maybe aggressive correction is warranted?

I doubt even Zeno himself was as troubled by this issue 2000 years ago. I bet he recognized he was using the wrong math.

Thank you for the clarification, I suppose the phrase is meant to encourage a person to focus on a task and not get distracted by other things. Personally, for myself, I would try not use it because I feel it's too prone to misinterpretation, and can provoke feelings of superiority in the teacher, and inferiority in the student, Furthermore it can shut down conversation that needs to happen; there is an expression, "a fool who persists in his folly will become wise," in regards to the expression it's important to distinguish between foolish vice, and foolish virtue; we're all engaged in foolishness to some degree, but generally it's best to assume the good hearted nature of the student and allow for mistakes to be made so he can grow.

A famous quote popularized by Steve Jobs, which he also used as the title of his commencement speech comes to mind, "Stay Hungry Stay Foolish;" and I think it's an idea that the scientific community, and all people in pursuit of truth should acknowledge.

For those interested, Steve Jobs was sentimental about the expression which he discovered in the famous magazine the "Whole Earth Catalog." It was very meaningful to him, and it is something I practice whenever I'm trying to learn something new.

 

[hmmm27]

My apologies. I was paraphrasing Richard Feynman (I think) when he talks about people getting too caught up in philosophies and interpretations or whatnot.

David Mermin, actually.

In regards "wouldn't tell my apprentice that" : Really ? While the phrase's utility is not exclusive to a master-apprentice relationship, that is exactly what a master would tell an apprentice.

 

[Chenkel]

My apologies. I was paraphrasing Richard Feynman (I think) when he talks about people getting too caught up in philosophies and interpretations or whatnot. I wasn't really even talking to you, but more about my own post.

I have no hard feelings, thank you for the clarification!

 

[Chenkel]

David Mermin, actually.

In regards "wouldn't tell my apprentice that" : Really ? That is exactly what a master tells an apprentice, notwithstanding that the phrase's utility is not exclusive to a master-apprentice relationship.

I'm fine with criticism, as long as foolishness is allowed.

 

[Dale]

In the reals. Zero is an infinitesimal in the real numbes. The only infinitesimal in the standard reals.

Edit: It has been forever since I was taught that. But I see that the definition on Wikipedia disagrees and requires that "infinitesimals" be distinct from zero.

Yes, that is a source of confusion. Different number systems and even different authors will use different conventions.

In number systems with multiple infinitesimals it can make sense to include 0 in the set of infinitesimal numbers or it can make sense to exclude it. If you include it then you wind up with lots of statements like "non-zero infinitesimals". If you don't include it then you wind up with lots of statements like "infinitesimal or zero". Most infinitesimal-containing number systems that I have seen count 0 as an infinitesimal (including the hyperreals), but for example dual numbers do not (an infinitesimal dual-number is a number $\epsilon$ such that $\epsilon^2=0$ but $\epsilon \ne 0$).

In the real numbers if there are any infinitesimal numbers only 0 is infinitesimal. So real infinitesimals have no properties different from the properties of 0. So it is not a particularly useful designation. Thus my preference is not to introduce the label of "infinitesimal" for the real numbers at all. But I should have acknowledged that authors and systems vary.

 

[Chenkel]

You mean post 22? I saw and responded to you about it. It suffers from the same problem as the OP. It is no more or less correct than the OP. You simply need to either use different numbers than the reals (if so which?) or express your questions in terms of limits.

You cannot treat infinity as a real number. Specifically, you cannot divide by an infinite number because there is no such thing as an infinite real number. Instead you can divide by a real number, $n$, and take the limit of that division as $n$ increases without bound. This is written as $$\lim_{n\rightarrow \infty} \frac{1}{n}$$ but it is important to recognize that in this notation $\infty$ is not a real number, but shorthand notation for "$n$ increases without bound".

I am not telling you that. I am telling you to slow down (a lot) and think seriously about the question you are asking. It is not meaningful in the way that you asked it, so I am encouraging you to think about how you should ask it instead to produce a meaningful question. Either you need to change the form of the question itself of you need to choose an alternative representation where that form of question is meaningful.

You are trying to jump to answers before you have a meaningful question. Slow down.

The relevant post I'm referring to is post 14, post 16 also might be useful and interesting, let me know if they're visible.

I appreciate and thank you for your feedback, and I'm going to study this post you wrote in more detail and let you know if I have any questions.

 

[Mark44]

"a fool who persists in his folly will become wise,"

Not a saying that I've ever heard. In fact, it's probably not even true.

 

[Chenkel]

Not a saying that I've ever heard. In fact, it's probably not even true.

It's very logical, you only truly learn when you are given an opportunity to make mistakes so you can challenge your own misconceptions.

I'm not asking you to believe it, I'm asking you to challenge it.

 

[PeroK]

I'm an amateur, could you please explain what I need to do as if I am ignorant? (Which I am btw, just not willingly.)

Learn calculus. Modern calculus takes the mystery out of these questions. Yes, you can muddy the waters by considering non-standard analysis (hyperreals etc.), but standard analysis is fine, IMO.

There are no infinitesimals, as such, in standard calculus. Instead, there are differentials:

https://tutorial.math.lamar.edu/classes/calci/differentials.aspx

 

[Mark44]

"a fool who persists in his folly will become wise,"

It's very logical, you only truly learn when you are given an opportunity to make mistakes so you can challenge your own misconceptions.

It's logical only if the "fool" learns from his mistakes, not if he persists in his folly (i.e., continues making them).

 

[Chenkel]

Learn calculus. Modern calculus takes the mystery out of these questions. Yes, you can muddy the waters by considering non-standard analysis (hyperreals etc.), but standard analysis is fine, IMO.

There are no infinitesimals, as such, in standard calculus. Instead, there are differentials:

https://tutorial.math.lamar.edu/classes/calci/differentials.aspx

I'm used to standard calculus, I read and understood most of Thomas and Finney, but it's been a while since I read it so I might be rusty in the areas that I haven't applied to recent problems and related studies.

I do like the idea of infinitesimals, they seem to possibly explain a way of looking at motion as a continuous summation of infinitesimals, i.e the concept of integration possibly becomes more intuitive in certain cases.

While I can use standard analysis, I may have a certain kind of intermittent curiosity in non standard analysis that is potentially productive, but perhaps detrimental if too much time is dedicated.

That being said if there's something practical about learning something, be it a better understanding, or a new way to solve a problem, I am all for it.

 

[PeroK]

I'm used to standard calculus, I read and understood most of Thomas and Finney, but it's been a while since I read it so I might be rusty in the areas that I haven't applied to recent problems and related studies.

In that case, understanding acceleration should not be a problem.

I do like the idea of infinitesimals, they seem to possibly explain a way of looking at motion as a continuous summation of infinitesimals, i.e the concept of integration possibly becomes more intuitive in certain cases.

An infinitesimal is not a real number. It's a "number" that is smaller than any real number. Although initially this may seem more, mathematically it gets tricky.

 

[Dale]

The relevant post I'm referring to is post 14, post 16 also might be useful and interesting, let me know if they're visible.

It is not a more correct formulation. It has the exact same issues as the OP.

 

[Chenkel]

It's logical only if the "fool" learns from his mistakes, not if he persists in his folly (i.e., continues making them).

I might be able to illustrate my idea in this way: a person who persists in his folly is not wise, but he will be wise.

It's like if I do something painful, I cannot do the same painful thing for eternity, reality requires I learn from my mistake and correct the action.

There are many kinds of fools, willfully ignorant fools, malevolent fools, benevolent fools, the list goes on.

Just to clarify, I'm not claiming special knowledge to wisdom, I'm just telling you what I choose to believe that has given me a sense of equanimity in my own life.

 

[Chenkel]

It is not a more correct formulation. It has the exact same issues as the OP.

I want to discover my misconceptions, so I apologize if I made the mistake of adding to the confusion with my own lack of knowledge, I'm looking to know what it is that I don't know so I can be logically consistent.

If I find something that doesn't make sense to me in physics, I tend to illustrate where the potential misconception is, and in the process I might show my own misconceptions and learn which concepts explain reality better than the ones I currently have.

 

[PeterDonis]

I might be able to illustrate my idea in this way: a person who persists in his folly is not wise, but he will be wise.

It's like if I do something painful, I cannot do the same painful thing for eternity, reality requires I learn from my mistake and correct the action.

There are many kinds of fools, willfully ignorant fools, malevolent fools, benevolent fools, the list goes on.

Just to clarify, I'm not claiming special knowledge to wisdom, I'm just telling you what I choose to believe that has given me a sense of equanimity in my own life.

@Chenkel, none of this is relevant to the thread topic. The thread topic is not general philosophy of wisdom, but a specific question about how motion in a continuum is analyzed. @Dale has several times now pointed out that your formulations of the problem using real numbers (and that means all of them, not just the one in the OP) are not well-defined and thus cannot be answered, and has asked you to reformulate the problem and given good advice on how to do so. You have not responded to that advice at all. Other posters have pointed you at measure theory, and you have not responded to that suggestion either. Those are the sort of responses you need to give to move this discussion forward. Continuing to talk about your general ideas about knowledge and wisdom, as you have in several posts now, does not do that.

 

[PeterDonis]

I want to discover my misconceptions

@Dale already told you the misconception: that you can use real numbers to formulate your question using infinitesimals. You can't. There are no such things as "infinitesimals" in real numbers (except for the edge case of $0$ itself being called an "infinitesimal" in some sources, but that doesn't help for what you want to do). So you can't formulate a meaningful question that talks about "infinitesimals" using real numbers. As @Dale has already told you, you need to either pick a different number system, one that includes well-defined "infinitesimals" other than $0$, or reformulate your question in terms of limits, which would allow you to use real numbers.

 

[Dale]

I want to discover my misconceptions

OK, are you clear now? If so how do you want to fix it? Are you going to reformulate your question in terms of limits using real numbers or if not then what numbers do you want to use instead of the reals?

 

[PeterDonis]

I'm not asking you to believe it, I'm asking you to challenge it.

Again, this is off topic for this thread. Please keep discussion focused on the actual specific question at issue and avoid tangents into other topics.

 

[Chenkel]

@Chenkel, none of this is relevant to the thread topic. The thread topic is not general philosophy of wisdom, but a specific question about how motion in a continuum is analyzed. @Dale has several times now pointed out that your formulations of the problem using real numbers (and that means all of them, not just the one in the OP) are not well-defined and thus cannot be answered, and has asked you to reformulate the problem and given good advice on how to do so. You have not responded to that advice at all. Other posters have pointed you at measure theory, and you have not responded to that suggestion either. Those are the sort of responses you need to give to move this discussion forward. Continuing to talk about your general ideas about knowledge and wisdom, as you have in several posts now, does not do that.

Thank you for bringing this to my attention. I apologize if my previous posts have not been directly related to the specific question about motion in a continuum. I understand that this is the focus of the thread and I will do my best to stay on topic in future posts.

I appreciate the suggestions and advice offered by @Dale and others about how to reformulate my question and make it more well-defined. I will take this into consideration and try to address any issues with my formulations. I am also open to learning more about measure theory and how it might be relevant to this topic.

I understand that it is important to stay focused and stay on topic in a discussion forum, and I will make an effort to do so in the future. However, I do want to acknowledge that I feel there is a human side to physics that is sometimes overlooked. Science, and specifically physics, is not just about logic and numbers, but also about understanding the world and our place in it. I believe that considering the human element and the impact of our work on society and humanity is an important aspect of scientific inquiry.

 

[Mark44]

I understand that it is important to stay focused and stay on topic in a discussion forum, and I will make an effort to do so in the future.

Good.

However, I do want to acknowledge that I feel there is a human side to physics that is sometimes overlooked. Science, and specifically physics, is not just about logic and numbers, but also about understanding the world and our place in it. I believe that considering the human element and the impact of our work on society and humanity is an important aspect of scientific inquiry.

This part is off-topic relative to the question you've asked in this thread.

 

[Chenkel]

OK, are you clear now? If so how do you want to fix it? Are you going to reformulate your question in terms of limits using real numbers or if not then what numbers do you want to use instead of the reals?

Thank you for your help, Dale. I am still trying to fully understand the information and how it applies to my question. I appreciate your patience and understanding as I take the time to process this.

I understand the importance of reformulating my question in terms of real numbers or another suitable set of numbers, and I will consider the suggestions you and others have made. I am committed to finding a solution and moving forward with this discussion.

I also want to express my appreciation for your efforts to help me understand this topic better. I recognize that we are all human and that it is important to approach discussions with respect and consideration for one another. I hope we can continue this discussion in a collaborative and respectful manner.