Wednesday, January 19, 2022

More on Newtonian velocity

Here is a big picture story about Newtonian mechanics: The state of the system at all times t > t0 is explained by the initial conditions of the system at t0 and the prevalent forces.

But what are the initial conditions? They include position and velocity. But now here is a problem. The standard definition of velocity is that it is the time-derivative of position. But the time-derivative of position at t0 logically depends not just on the position at t0 but also on the position at nearby times earlier and later than t0. That means that the evolution of the system at times t > t0 is explained by data that includes information on the state of the system at times later than t0. This seems explanatorily circular and unacceptable.

There is an easy mathematical fix for this. Instead of defining the velocity as the time-derivative position, we define the velocity as the left time-derivative of position: v(t)=limh → 0−(x(t + h)−x(t))/h. Now the initial conditions at t0 logically depend only on what happens at t0 and at earlier times.

This fixed Newtonian story still has a serious problem. Suppose that the system is created at time t0 so there are no earlier times. The time-derivative at t0 is then undefined, there is no velocity at t0, and Newtonian evolution cannot be explained any more.

Here’s another, more abstruse, problem with the fixed Newtonian story. Suppose I am in a region of space with no forces, and I have been sitting for an hour preceding noon in the same place. Then at noon God teleports me two meters to the right along the x-axis, so that at all times before noon my position is x0 and at noon it is x0 + 2. Suppose, further, that the teleportation is the only miracle God does. God doesn’t change any other properties of me besides position, and God lets nature take over at all times after noon.

What will happen to me after noon? Well, on the fixed Newtonian story, my velocity at noon is the left-derivative of position, i.e., limh → 0−(2 − 0)/(0 − h)= + ∞. Since there are no prevailing forces, my acceleration is zero, and so my velocity stays unchanged. Hence, at all times after noon, I have infinite velocity along the x-axis, and so at all times after noon I end up at distance infinity from where I was—which seems to make no sense at all!

So the left-derivative fix of the Newtonian story doesn’t seem right, either, at least in this miracle case.

My preference to both the original Newtonian story and the fixed story is to take velocity (or perhaps momentum) to be a fundamental physical quantity that is not defined as the derivative, or even left derivative, of position.

The rest is technicalities. Maybe we could now take Newton’s Second Law to be:

  1. t+v(t)=F/m,

where ∂t+ is the right (!) time-derivative, and add two new laws of nature:

  1. t+x(t)=v(t), and

  2. x(t) and v(t) are both left (!) continuous.

Now, (2) is an explicit law of nature about the interaction of velocity and position rather than a definition of velocity. On this picture, here’s what happens in the teleportation case. Before noon, my velocity is zero and my position is x0. Because I supposed that the only thing that God miraculously affects is my position, my velocity is still zero at noon, even though my position is now x0 + 2. And I think (by the answer to this), laws (1), (2) and (3) ensure that if there are no further miracles, I remain at x0 + 2 in the absence of external forces. The miraculous teleportation violates (2) and (3) at noon and at no other times.

But of course this is all on the false premise of Newtonian mechanics.

7 comments:

Zsolt Nagy said...

But of course this is all on the false premise of Newtonian mechanics.

Well, Newtonian mechanics might be and is actually false, but not because of your nonsensical blogpost here, Pruss.
Actually according to our best theories today (General Relativity Theory and Quantum Mechanics) Newtonian mechanics is approximately true - with certain conditions and simplifications our current and best theories imply Newtonian mechanics or Classical physics. On a day to day basis Newtonian mechanics describes most of the observed motions approximately well. Given this, to simply state, that "... this is all on the false premise of Newtonian mechanics." is a blasphemy.

The most important axiom from Newtonian mechanics/dynamics is the second law of motion: A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force: dp(t)/dt=d(mv(t))/dt=m·d²x(t)/dt²=F
(The other law's/axioms are just special cases from this law and axiom of motion.)
If the second law of motion applies to all and every and any arbitrary times, then the position x as a function of time t x(t) is at least a twice differentiable function of time t and a continuous function of time t, since according to mathematics any (arbitrary or to say all) differentiable functions are continuous functions.

So really, Alexander R Pruss, it's not that "of course this is all on the false premise of Newtonian mechanics", but it's rather, that your assumptions and propositions of position x to be a discontinuous function of time t are incompatible with Newtonian mechanics and with the assumptions of position x to be at least twice differentiable and continuous functions of time.
You appear to assume here, that oil is supposed to mix with water, since the both substances are liquids and liquids are supposed to mix well together, since if any liquid doesn't mix well together, then what else is supposed to mix and go well together.
Really, Alexander R Pruss, why are you considering yourself to be an "analytical" philosopher? Post after post I see signs after signs against, that notion of you being a proper "analytical" philosopher.

Aria Banazadeh said...

>why are you considering yourself to be an "analytical" philosopher? Post after post I see signs after signs against, that notion of you being a proper "analytical" philosopher

I am not him but the obvious answer is that he has a PhD in mathematic and a PhD in philosophy and his books are taken seriously by other prominent philosophers

Zsolt Nagy said...

Why are you considering yourself to be an "analytical" philosopher? Post after post I see signs after signs against that notion of you being a proper "analytical" philosopher.

"I am not him[, Alexander R. Pruss,] but the obvious answer is that he has a PhD in mathematic and a PhD in philosophy and his books are taken seriously by other prominent philosophers"

I know all of that, Unknown commentator. What, I do not know, Unknown commentator, is, why isn't that properly reflected in Alexander R. Pruss' work?
Other "prominent" philosophers might take his books and work seriously, because they are not better or because they are worse than him - not just not being incapable of recognising such absurdities as proposing Newtonian mechanics to be false because of being incompatible with discontinuous functions; but maybe those "prominent" philosophers are just ignoring such absurdities while being capable of recognising such propositions to be absurd, since such propositions are for their cause of disproving science and proving theism, I guess. Or not.
But really, who else then, if not a false analytical philosopher, would come up with the idea of Newtonian mechanics to be false because of being incompatible with discontinuous functions? ¯\_(ツ)_/¯

IanS said...

Zsolt Nagy:
I don’t read the post as any sort of argument against Newtonian physics. The last line “… this is all on the false premise of Newtonian mechanics” simply acknowledges that the fundamental physics of this world is not in fact Newtonian.

I read the post as an attempt to frame the 2nd law in such a way that it could plausibly be seen as having causal force, rather than merely being descriptive. Of course, you may think that this is misguided. If so, you would be in good company (Hume and followers). But it is not straightforwardly silly.

Alex:
QM neatly avoids problem you raise, though it raises new ones. The state follows Schrödinger’s equation, which contains only the first time derivative. The catch is that the state is a distribution function (not just a few numbers) and is unobservable. Position and momentum become complementary observables acting on the state.

Zsolt Nagy said...

Again, I know, IanS, that the fundamental physics of this world is not in fact Newtonian. But approximately under certain conditions physics of this world is very much so Newtonian. So it is still very bold to just simply state and claim after such a nonsensical post, that "… this is all on the false premise of Newtonian mechanics".

If Alexander R. Pruss gives an attempt here to frame the 2nd law in such a way, that it could plausibly be seen as having causal force, rather than merely being descriptive, then his attempt is not very productive here to say at least. I almost contend with to not say, that his attempt is rather counterproductive and very missguided.
If he wants to propose and suggest position x of objects to be capable of being a discontinuous function of time t, then he shouldn't analyze such propositions with the current formulation of the second law of motion, since of course with the current formulation the second law of motion is incompatible with his suggestions.
Duh!
I propose and suggest him to first reformulate the second law of motion in such a way, which doesn't require twice or any differentiable and continuous functions and after that give a proper account of the possibility and probability of position x of objects to be capable of being a discontinuous function of time t according to his reformulated version. Before any such reformulations any such proposition and suggestion and analysis with the old and current formulation is really just nonsensical and rather counterproductive than productive.

Note on comment to Alex regarding QM (, before he might give again a nonsensical and irrelevant explanation): Again, QM in the Classical limit also entails Newtonian mechanics under certain conditions and interpretations. Besides that Feyman has already shown in the last century, that the path integral formulation of QM is basically equivalent to the Schrödinger equation/operator formalism of QM and of canonical quantization:
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
Really, this blogpost and discussion here is very much so missguided and beyond any rational belief. It is really nothing but a blasphemy.

Alexander R Pruss said...

Even though a QM-based world is approximately Newtonian in certain domains, that does not mean that the *metaphysics* of a QM-based world is approximately like the *metaphysics* of a classical world even in those domains. All it means is that the two theories yield approximately the same empirical consequences in these domains.

Yes, of course, discontinuous positions are incompatible with Newtonian mechanics. But that's why I imagined a *miracle*, and I posed the thought experiment of what would happen in that miraculous case. Now, there are two things you could think about this thought experiment. If like me you think there is a God, you will probably think such miracles are actually possible--God can do it! In that case, there is an interesting philosophical question of what would happen if such a miracle were to happen in a Newtonian world. Or you might think miracles are impossible, and laws of nature cannot be violated. If so, then the thought experiment is less intesting, but there can still be a value in asking what would happen in an impossible hypothetical case.

Consider Einstein's thought experiment of what would happen if you were to go at the speed of light alongside a beam of light. That's a valuable thought experiment, even though it is a violation of Relativity Theory to suppose an object that has mass (you) moving at the speed of light.

By the way, Mr Nagy's wording at a number of points lacks the courtesy that academic conversation should exhibit, and makes his comments very close to being worthy of deletion. I will allow them, this time (partly because the question of counternomic thought experiments is very philosophically interesting in itself), but in the future I may be stricter.

Zsolt Nagy said...

By the way, Mr Nagy's wording at a number of points lacks the courtesy that academic conversation should exhibit, and makes his comments very close to being worthy of deletion. I will allow them, this time (partly because the question of counternomic thought experiments is very philosophically interesting in itself), but in the future I may be stricter.

Fair enough. I'm just a fellow truth-seeker, who is a bit worried about, how such a supposedly weird truth of Newtonian mechanics simply being false because of being incompatible with discontinuous positions is coming about or has been acquired.
I mean, you appear to assume here, that a discontinuous and *non-differentiable position supposed to be compatible with Newtonian mechanics and hence that supposed to be differentiable and continuous. Since that's not the case, therefore Newtonian mechanics is "of course" false.
It just simply appears to me, that you have more problems with discontinuous and non-differentiable positions being not differentiable and being not continuous than with Newtonian mechanics itself.
Really you might be the first philosopher and mathematician, who basically suggests and proposes an ordinary differential equation to be wrong or false, since it is supposed to have discontinuous solutions - but it hasn't any such solutions, so it is "of course" false. And this is something, I guess - something I have never witnessed, that's for sure. (Is this all right as a form of an evaluation or "constructive" critique?)
But then according to this why is Newtonian mechanics supposed to be false exactly? I have no idea.

*Why are any (arbitrary or to say all) discontinuous functions also non-differentiable?
Since any (arbitrary or to say all) differentiable functions are continuous functions or to say, that any (arbitrary or to say all) not continuous functions are not differentiable functions and since a function is discontinuous, if and only if it is not a continuous function, and since a function is non-differentiable, if and only if it is not a differentiable function, therefore any (arbitrary or to say all) discontinuous functions are not continuous functions and not differentiable functions or to say nondifferentiable functions.
Besides this by the way ordinary differential equations have continuous solutions or to say no discontinuous solutions.
So then why try to dissolve oil in water, when that can not be done?
Hey if you can do it or to say, that you have found a method to dissolve oil in water or to properly differentiate discontinuous functions, then be my guest and show it to me and to the world. But please don’t expect me to simply agree to your suggestion and proposition of Newtonian mechanics having the fault of you being incapable of doing so (to dissolve oil in water or to properly differentiate discontinuous functions), since that’s obviously and trivially not the fault of Newtonian mechanics but the fault of mathematics itself.