If a particle was at position X for zero time, was it there?

[Dale]

You seem to think that as a matter of logic if you posit the existence of something, there is nothing you can further predicate of that something by which you can negate your positing of its existence.

Correct. It is simply not possible for the statement “Particle A was at position X for duration T” to be true and for the statement “Particle A was at position X” to be false, for any A, T, and X. You can suppose that the first statement cannot be true for T=0, but given the first then the second statement is a direct and inevitable consequence.

Defining for a first-order quantificational calculus a universe of discourse that expressly excludes any objects having a zero-value temporo-durational property would not render that system less sound or complete.

Sure, but under such a calculus the first statement could not be true, so such a calculus is irrelevant to the question at hand where the first statement is given as true.

 

[sysprog]

Correct. It is simply not possible for the statement “Particle A was at position X for duration T” to be true and for the statement “Particle A was at position X” to be false, for any A, T, and X.

It's possible for the statement “Particle A was at position X for duration T” to entail the proposition that "Particle A was at position X" only if duration T is non-zero. If duration T is zero, the statement “Particle A was at position X for duration T” is equivalent to the statement “Particle A was at position X for duration 0”, which is equivalent to “Particle A was for duration 0 at position X, which is equivalent to “Particle A was not at position X”.

 

[Dale]

I agree with that, but maintain that it's possible for the statement “Particle A was at position X for duration T” to be true only if duration T is non-zero.

That is your right to maintain, but it is not the topic of this thread where the statement was given to be true for T=0.

You should however note that the statement is a self consistent statement which is theoretically valid for classical point particles. To assert its falsity a priori requires a rejection of classical point particles. (I am ok with such a rejection, but not the arguments you have been using) Without rejecting classical point particles there is no valid objection to the statement, so since the statement is given as true it is reasonable to assume that classical point particles are the intended subject. My comments have been operating under that assumption.

 

[sysprog]

Correct. It is simply not possible for the statement “Particle A was at position X for duration T” to be true and for the statement “Particle A was at position X” to be false, for any A, T, and X. You can suppose that the first statement cannot be true for T=0, but given the first then the second statement is a direct and inevitable consequence.

If the first statement is true, and T=0, the first statement is itself the negation of the second statement.

Sure, but under such a calculus the first statement could not be true, so such a calculus is irrelevant to the question at hand where the first statement is given as true.

If the duration were non-zero the statement could be admissible, and true or false depending on whether the state of affairs was as predicated or not.

 

[sysprog]

It's not entirely uncommon to use zero time as negation:

Sergeant: Private, were you off base today without a pass?
Private: No, Sergeant, at no time today was I off base without a pass.
Sergeant: Were you at some time on some day other than today off base without a pass?
Private: Yes, Sergeant. but never since I first arrived at the base after being assigned here.
Sergeant: Carry on, Private.

That's of course not the same as "for a duration of zero time units", but the negating effect is the same.

 

[sysprog]

I agree with that, but maintain that it's possible for the statement “Particle A was at position X for duration T” to be true only if duration T is non-zero.

That is your right to maintain, but it is not the topic of this thread where the statement was given to be true for T=0.

You quoted (accurately) an earlier version of my post which I subsequently edited. To it I would append "if the truth of that statement is to entail the truth of the statement that "Particle A was at position X", if the truth of that statement is to entail the truth of the statement that "Particle A was at position X", so that the edited statement would read:

I agree with that, but maintain that it's possible for the statement “Particle A was at position X for duration T” to be true only if duration T is non-zero, if the truth of that statement is to entail the truth of the statement that "Particle A was at position X".​

You should however note that the statement is a self consistent statement which is theoretically valid for classical point particles. To assert its falsity a priori requires a rejection of classical point particles. (I am ok with such a rejection, but not the arguments you have been using) Without rejecting classical point particles there is no valid objection to the statement, so since the statement is given as true it is reasonable to assume that classical point particles are the intended subject. My comments have been operating under that assumption.

Let's please for a moment look at a proposed modification of the statement set to make the classic point particle paradigm more explicit:

The path of spatially dimensionless moving Particle A is posited to have intersected with 3 dimensional spatial point Position X, at time T' and for duration T, with duration T being zero and time T' being a specific instantaneous point in time on the timeline of Position X and on the timeline of Particle A. It is asserted that is not self-inconsistent to assert the existence of such a state of affairs, and further, that whether such a description is of a state of affairs that is factually possible in the real world is not decisively determined.​

This scenario being non-static, in that it references a moving particle, requires that the particle have a speed, that is, a distance which the particle traverses during a non-zero amount of time. If the amount of time is zero, the speed of the particle is infinite. This means that over the path the particle traverses at infinite speed, it intersects with all points simultaneously, so that it is being asserted to at time T' be at Position X, and also at time T' to be at other not Position X positions along its travel path. The notion of a point particle being at a specific spatial position at a specific point in time is inconsistent with the notion of it also being at some other specific spatial position, arbitrarily remote therefrom, at the same specific point in time.

It is not, ipso facto, logically inconsistent, unless it is further asserted that being at Position not X is logically equivalent to being not at Position X. It is equally possible to construct a consistent system of symbolization which would allow or disallow such translation. If we construct one which does not allow it, we cannot transform "being at Position X and being at Position not X" to "being at Position X and being not at Position X", because "Position X" and "Position not X" are atomic in this disallowing construction, so that the interior negation is not relocatable to the other side of the "at" in the expression "at Position not X", and so does not allow generation of the "at Position X and not at Position X" contradiction.

Whether the notion of infinite speed is self-consistent or not is a question that is system-dependent -- one could define the division by zero as equal to infinity, or declare it to be not defined, or take some other approach.

I see the original problem statement as systematically misusing language in a manner which produces unsatisfactory results in the attempts at answers. This is a problem with inquiry in general -- when we find that we've formulated a question clearly enough, we'll often have to re-examine foundations that we'd rather leave unperturbed. For how that quandary may be addressed, I see no deficiency to be ascribed in particular to either the person asking the question, or to anyone who tries to answer it.

 

[Herman Trivilino]

Consider particle with trajectory $x(t)=t^3$ at time t=0 it's acceleration and speed are 0, but it does not remain in point x=0.

You are of course correct. The context of my comment was a ball thrown upward, under the influence of gravity only. So the equation of the trajectory might be $x=19.6t-4.9t^2$.

 

[Dale]

If the first statement is true, and T=0, the first statement is itself the negation of the second statement.

No, it isn’t. The statement “Particle A was at location X for duration T” consists of the following three assertions: “there exists a particle A, a location X, and a duration T”, “A was at X”, and “the amount of time A spent at X was T”. So the second statement is indeed implied by the first.

a distance which the particle traverses during a non-zero amount of time. If the amount of time is zero, the speed of the particle is infinite.

It seems like your position is so untenable that you are being forced to ignore calculus to support it. You may want to rethink your position.

If a point particle has a finite velocity v(t) then the distance it traverses over a duration T is $$\int_{T’}^{T’+T}v(t)\;dt$$ which is 0 for T=0 regardless of v. To first order the time required to traverse a small distance $\Delta x$ is $\Delta t=\Delta x/v$ which clearly goes to 0 as $\Delta x$ goes to zero.

 

[hmmm27]

Remind me to never ask you guys to help my Uncle Jack off a horse : regardless of interpretation, both relative and equine would be high and dry. I kid , though I did spend enough time ruminating on dimensional interaction in 3+1 Newtonian space that not only do I not recall whatever TV episode was going on in the background, I'm not even sure what show it was.

 

[jbriggs444]

Remind me to never ask you guys to help my Uncle Jack off a horse : regardless of interpretation, both relative and equine would be high and dry.

If he was only up on the horse for an instant of zero duration then he'll be on the ground soon enough.

 

[hmmm27]

If he was only up on the horse for an instant of zero duration then he'll be on the ground soon enough.

That would be a point(s) contact for "Help my uncle get a leg over the horse".

 

[A.T.]

If the amount of time is zero, the speed of the particle is infinite.

Unless the distance traveled during that zero time is also zero. Then the speed is indeterminate, it can have any finite value. A position is just a point and has no size, so a point-like object travels zero distance, while being at that position.

 

[Vanadium 50]

I can't believe this is still going on.

You have a baseball traveling according to x = 2t + 1. At t = 3 you take a picture and sure enough, the photograph has a baseball at x = 7. Does anyone seriously want to argue that the baseball isn't "really" there? There's a picture and everything.

 

[hmmm27]

What I find interesting is that the OP's actual question - if he meant 2 0dim points - is non sequitur because - outside of a coordinate driven abstract, two 0dim points will never coincide : neither intersection nor duration. You need at least a total of 2 (spatial) dimensions between two objects to get an intersection, and at least 3 to get a duration of the intersection.