Think of a body which is at rest but not in equilibrium

[sysprog]

Zero is an infinitesimal. The only infinitesimal in the standard reals. When I say "zero duration" I do not mean a non-zero infinitesimal duration. Nor do I mean that something never happened.

I think that this highlights an inadequacy of the 'standard reals'. In my view, $\epsilon \in {}^*\mathbb{R}: \forall r [(|\epsilon| < r)\wedge(r>0)]$. That entails that $\epsilon$ is not a real number in the standard reals, as $0$ is, and it also restricts the infinitesimal to occupying 'the leftmost place' to the right of zero on the number line. I know that real analysis will immediately raise the objection that there is no such place on the real number line; however, such a place is nameable, albeit naming something that does not exist among the standard reals, wherefore the resorting to hyperreals.

 

[jbriggs444]

I think that this highlights an inadequacy of the 'standard reals'. In my view, $\epsilon \in {}^*\mathbb{R}: \forall r [(|\epsilon| < r)\wedge(r>0)]$. That entails that $\epsilon$ is not a real number in the standard reals, as $0$ is, and it also restricts the infinitesimal to occupying 'the leftmost place' to the right of zero on the number line. I know that real analysis will immediately raise the objection that there is no such place on the real number line; however, such a place is nameable, albeit naming something that does not exist among the standard reals, wherefore the resorting to hyperreals.

But remember the point of the exercise.

If we allow for hyper-real intervals we should allow for hyper-real velocities. The hyper-real interval during which the velocity is infinitesimally different from zero is infinitesimal. But the hyper-real interval during which the velocity is zero is still exactly zero.

 

[sysprog]

If we allow for hyper-real intervals we should allow for hyper-real velocities. The hyper-real interval during which the velocity is infinitesimally different from zero is infinitesimal. But the hyper-real interval during which the velocity is zero is still exactly zero.

I model the zero-velocity interval to be infinitesimal. Calling it an interval is calling it non-zero. I like your succinct positure, and I see its appeal; however, a velocity is a ratio of a directed distance per duration, while a duration is a positive interval. If I model the zero velocity as having occurred for an infinitesimal duration, I avoid division by zero, leaving the velocity as zero, and the duration of the zero velocity as infinitesimal. In my view, an 'instant' cannot be of zero duration, but must be of minimally greater than zero duration.

 

[etotheipi]

I don't know much about the technicalities, but here I think @jbriggs444's approach seems more natural. If the body comes to rest instantaneously at $t_0$, then $v(t_0+\epsilon) \neq 0$ and $v(t_0-\epsilon) \neq 0$ no matter how arbitrarily small you make $\epsilon$. So I think it would make sense to say that the body is at rest for zero time.

The Baez article is really interesting, but from a pure maths perspective all I can say is that the trajectory intersects the $v=0$ axis at one point, and a "point" in Euclidian space doesn't have any "width" (indeed such a notion doesn't even make sense!).

 

[jbriggs444]

a velocity is a ratio of a directed distance per duration

A velocity is the limit of the ratio of a directed distance over a duration as the duration and distance approach zero. It is a limit, not a ratio.

If you are careful, you can use the transfer principle to turn this statement about velocity in the reals as a limit into a statement about a velocity in the reals as a ratio of infinitesimals. But that does not help make the point you need to be trying to make.

The set of points where velocity is zero is a degenerate interval consisting of a single point. The measure of that set is zero.

 

[sysprog]

I was about to reply to your post while I thought that it consisted of only its first sentence $-$ when I hit the Reply button, I got (presumably) the rest of the post $-$ here's a screen shot of the anomaly:

Perhaps @berkeman will take note.

Anyway, onward to the reply:

A velocity is the limit of the ratio of a directed distance over a duration as the duration and distance approach zero. It is a limit, not a ratio.

I would call the time limit the infinitesimal instead of calling it zero. The distance from a point to itself is zero; the distance from any point in a continuum to 'the nearest point therein that is not the same point' (a point that does not exist in the reals) is positive, and therefore is non-zero. In my view, the shortest possible existent duration or distance is infinitesimal, not zero, A single point has zero distance, because it is not an interval, but it has non-zero, infinitesimal size, because it occupies a location, and I regard an instant as having infinitesimal non-zero magnitude, wherefore I think that it is misleading to say that it has zero duration.

If you are careful, you can use the transfer principle to turn this statement about velocity in the reals as a limit into a statement about a velocity in the reals as a ratio of infinitesimals. But that does not help make the point you need to be trying to make.

I agree regarding the transfer principle, and disagree that it does not help, although I think that it does not suffice. Also necessary are the acceptance of incompleteness and the denial of the excluded middle axiom.

The set of points where velocity is zero is a degenerate interval consisting of a single point. The measure of that set is zero.

I think that only the empty set should be said to have measure zero. I know that standard measure theory is less impromiscuous than that regarding its use of the term 'zero'. I am not so much griping about the mathematics as I am deploring what I regard to be the misuses of the term 'zero' $-$ when I brought this up in school, I was told that I could use such substitutes as 'treated as zero', or 'negligibly different from zero', if I wanted to eschew using 'zero' for infinitesimals. I agree that velocity is the first derivative of position wrt time, the second being acceleration (that can be problematic regarding direction), the third being 'jerk', and the fourth being 'jounce' or 'snap', and when I am doing derivatives, I use delta epsilon limits just like everyone else, but when saying things about them, I don't speak or write as if the infinitesimal were exactly equal to zero.

 

[jbriggs444]

In mathematics, words mean what they are defined to mean. Not what we feel they should mean.

 

[sysprog]

In mathematics, words mean what they are defined to mean. Not what we feel they should mean.

I regard that to be a classic false dichotomy. I don't consult my feelings to determine the meanings of words. If you want to define the meaning of a word, in a manner that is peculiar to your argot and inconsistent with the commonly established meaning, it is incumbent on you to appropriately distinguish your special meaning from the established default.

For an example of this being done $-$ I think well and reasonably $-$ from https://en.wikipedia.org/wiki/Signed_zero:

The IEEE 754 standard for floating-point arithmetic (presently used by most computers and programming languages that support floating-point numbers) requires both +0 and −0. Real arithmetic with signed zeros can be considered a variant of the extended real number line such that 1/−0 = −∞ and 1/+0 = +∞; division is only undefined for ±0/±0 and ±∞/±∞.​

I would add $\omega=\frac 1 {+\infty}$ and $-\omega=\frac 1 {-\infty}$, or $+\omega=+\frac 1\infty$ and $-\omega=-\frac 1 \infty$, and I would accept +0, 0, and -0 as 3 different numbers, with +0 being used for the positive infinitesimal and for the Lebesgue measure of the integers and for the probability of picking .5 at random from the unit interval, with -0 being used for the difference between 1 and 0.999..., and with 0 being used for the sum of 1-1 and for the probability of picking 2 from the unit interval whether randomly or not$-$ I think that this would be consistent, perspicuous, and not procedurally onerous $-$ it's not just how I feel.

 

[Frodo]

" Think of a body which is at rest but not in equilibrium. Give explanation as well as figure/diagram. Relevant Equations: Σ F = x (x is not equal to 0)"

A sphere on a horizontal surface is at rest and in neutral equilibrium. Without a force it will remain at rest indefinitely. The slightest horizontal force will move it.

A pencil balanced on its point is in unstable equilibrium. It is at rest but the slightest force will move it.

If ΣF is not equal to 0 then ΣF must pass normally (at right angles) through the point of contact or it will cause it to move.

 

[hmmm27]

A sphere on a horizontal surface is at rest and in neutral equilibrium. Without a force it will remain at rest indefinitely. The slightest horizontal force will move it.

A pencil balanced on its point is in unstable equilibrium. It is at rest but the slightest force will move it.

Those two examples are identical.

 

[Frodo]

Those two examples are identical.

Not so.

The sphere is in neutral equilibrium. The pencil is in unstable equilibrium.

See wiki for an explanation of the difference between neutral and unstable equilibrium.