4 ms·

One thing that is strange is that we can easily imagine the first two derivatives: position we can just imagine a static point, velocity we can imagine a consta

One thing that is strange is that we can easily imagine the first two derivatives: position we can just imagine a static point, velocity we can imagine a constant speed i.e. a straight line on a position-time graph, acceleration we just imagine a parabola, but jerk is somehow conceptually indistinguishable. The difference between a point, a line, and a parabola are stark, the third order jerk is not so easy to distinguish, instead still just looking like the parabola. I've always wondered why this is, why curves in general are perceptually similar if scaled correctly, while a straight line is so clearly different. Perhaps it is because our perceptions developed to distinguish between inertial and non inertial reference frames?

xscott 2y ago
I like and agree with your observation. But I think you can use conceptual tricks to get just a little further: Acceleration is "due" to a force (F=ma), so you can think of jerk as a change in that force linearly increasing over time. That doesn't help me recognize a cubic from a quadratic when looking at a small piece of it, but I can imagine an elevator ramping up it's lifting power or similar. It kind of feels like the tricks to conceptualize 4D as 3D position plus a temperature at each spot.
- bowsamic 2y ago
  I agree the linearisation trick can be used and is often used in physics, but we must do that as a consequence of the thing I'm confused about the origin of, which is not why do we only greatly distinguish between the first few derivatives of position, but why do we only greatly distinguish between the first few derivatives of most functions? I.e. why do we have to do these tricks in the first place?
  - xscott 2y ago
    I don't have any answers, but I suspect it's because we're evolved from things that didn't need to know. Related, I sometimes wonder how many derivatives you need to go down in order to find the one that is discontinuous when you decide to make a motion. For instance: pressing the first key to type this reply, my finger didn't instantly jump from zero to non-zero acceleration (or jerk/snap) I assume. How many terms in the Taylor series for moving a muscle?
  - thaumasiotes 2y ago
    > the thing I'm confused about the origin of, which is not why do we only greatly distinguish between the first few derivatives of position, but why do we only greatly distinguish between the first few derivatives of most functions? Because higher derivatives are insignificant. That's the entire concept of Taylor series approximation. If you change a high derivative of a function, the value of the function won't noticeably change - why would you care?
core_dumped 2y ago
It’s seems analogous to our spatial dimensions. We can all easily visualize or describe up to a 3D object, but 4D is almost impossible to fathom for most people
pfortuny 2y ago
Tje difference is because we cannot easily tell between “curve of second order” and “curve of other order”. You can get an idea when you try to understand why the function y=0 for x<0 y=x^2 for x>=0 has two derivativea but not three. But the issue is infinitesimal, so very hard to tell. Jerk you can “linearise” if you think of a car (with no air friction) and its accelerator. Somehow…
- bowsamic 2y ago
  > Tje difference is because we cannot easily tell between “curve of second order” and “curve of other order”. Why not, though? Why does third order "look like" second order but second order is starkly different to first order?
  - Sharlin 2y ago
    Well, firstly, if you plot the first n degrees of monomials and keep the scale invariant, the visual difference between x^k and x^(k+1) literally gets smaller the higher up you go. Secondly, presumably the distinction of "straight" vs. "curved" is quite deeply programmed into the brain's pattern recognition machinery. The degree of curvature is a quantitative parameter on top of the qualitative categorization. This may or may not have something to do with the fact that a modern human sees straight lines everywhere (something that very much was not the case in the ancestral environment).
  - tomek_ycomb 2y ago
    UHHHHHhhhh, it's because the last A*b is the only one that becomes a linear constant. For other polynomials, your derivative is a polynomial still, just different one. These are mathematical derivatives, I think of them as the slope of the thing it's derived of, aka the change in the thing that it's a derivation of. I think I don't have a sophisticated mathematical understanding, but my basic mechanic understanding makes it feel simpler than your question is acting.
  - meindnoch 2y ago
    Because a line has an infinite radius, while a curve has a finite radius. The difference between infinite and finite is stark. The difference between two finite values is not.
rcxdude 2y ago
I think it's in part because differences in gradient are quite difficult to distinguish by eye, and the difference in the higher order curves is in just that (and scaling things to be visible means the difference in curvature gets smaller as you go to higher orders). Even nearly-straight lines can be quite difficult to tell from actually-straight without a nearby reference.

xscott 2y ago

I like and agree with your observation. But I think you can use conceptual tricks to get just a little further: Acceleration is "due" to a force (F=ma), so you can think of jerk as a change in that force linearly increasing over time. That doesn't help me recognize a cubic from a quadratic when looking at a small piece of it, but I can imagine an elevator ramping up it's lifting power or similar. It kind of feels like the tricks to conceptualize 4D as 3D position plus a temperature at each spot.

bowsamic 2y ago
I agree the linearisation trick can be used and is often used in physics, but we must do that as a consequence of the thing I'm confused about the origin of, which is not why do we only greatly distinguish between the first few derivatives of position, but why do we only greatly distinguish between the first few derivatives of most functions? I.e. why do we have to do these tricks in the first place?
- xscott 2y ago
  I don't have any answers, but I suspect it's because we're evolved from things that didn't need to know. Related, I sometimes wonder how many derivatives you need to go down in order to find the one that is discontinuous when you decide to make a motion. For instance: pressing the first key to type this reply, my finger didn't instantly jump from zero to non-zero acceleration (or jerk/snap) I assume. How many terms in the Taylor series for moving a muscle?
- thaumasiotes 2y ago
  > the thing I'm confused about the origin of, which is not why do we only greatly distinguish between the first few derivatives of position, but why do we only greatly distinguish between the first few derivatives of most functions? Because higher derivatives are insignificant. That's the entire concept of Taylor series approximation. If you change a high derivative of a function, the value of the function won't noticeably change - why would you care?

core_dumped 2y ago

It’s seems analogous to our spatial dimensions. We can all easily visualize or describe up to a 3D object, but 4D is almost impossible to fathom for most people

pfortuny 2y ago

Tje difference is because we cannot easily tell between “curve of second order” and “curve of other order”. You can get an idea when you try to understand why the function y=0 for x<0 y=x^2 for x>=0 has two derivativea but not three. But the issue is infinitesimal, so very hard to tell. Jerk you can “linearise” if you think of a car (with no air friction) and its accelerator. Somehow…

bowsamic 2y ago
> Tje difference is because we cannot easily tell between “curve of second order” and “curve of other order”. Why not, though? Why does third order "look like" second order but second order is starkly different to first order?
- Sharlin 2y ago
  Well, firstly, if you plot the first n degrees of monomials and keep the scale invariant, the visual difference between x^k and x^(k+1) literally gets smaller the higher up you go. Secondly, presumably the distinction of "straight" vs. "curved" is quite deeply programmed into the brain's pattern recognition machinery. The degree of curvature is a quantitative parameter on top of the qualitative categorization. This may or may not have something to do with the fact that a modern human sees straight lines everywhere (something that very much was not the case in the ancestral environment).
- tomek_ycomb 2y ago
  UHHHHHhhhh, it's because the last A*b is the only one that becomes a linear constant. For other polynomials, your derivative is a polynomial still, just different one. These are mathematical derivatives, I think of them as the slope of the thing it's derived of, aka the change in the thing that it's a derivation of. I think I don't have a sophisticated mathematical understanding, but my basic mechanic understanding makes it feel simpler than your question is acting.
- meindnoch 2y ago
  Because a line has an infinite radius, while a curve has a finite radius. The difference between infinite and finite is stark. The difference between two finite values is not.

rcxdude 2y ago

I think it's in part because differences in gradient are quite difficult to distinguish by eye, and the difference in the higher order curves is in just that (and scaling things to be visible means the difference in curvature gets smaller as you go to higher orders). Even nearly-straight lines can be quite difficult to tell from actually-straight without a nearby reference.