WLC is Wrong: The Argument for Actual Infinities

[CliveStaples]

MT, I think I might now understand your location argument.

Suppose you had a countably-infinite list of guests G = {guest 1, guest 2, ... }, and you give each guest either a blue nametag or a red nametag. The guests then fill up the infinite hotel H = {room 1, room 2, ...}; let f:H -> G be the list of which guests are in which rooms (so if guest n is in room k, f(room k) = guest n) You then have all the blue nametag people leave their rooms, and take a headcount of the remaining guests.

But suppose now that the guests decide to rearrange themselves in the hotel by switching rooms, so now there's a new assignment g:H-> G. You then have all the blue nametag people leave their rooms again, and take a headcount again of the remaining guests.

I think your argument can be phrased like this:

(1) Necessarily, the before-rearrangement headcount must agree with the after-rearrangement headcount.
(2) If infinities are actual, then there are rearrangements so that the before-rearrangement headcount won't agree with the after-rearrangement headcount.
(3) Therefore, necessarily infinities aren't actual.

This argument is valid; if (1) and (2) are true, then (3) necessarily follows. However, this argument is not sound, since premise (2) is necessarily false (proof: part i and part ii)

[Squatch347]

Emphasis mine; I think past-Squatch was pretty clear. Which only leads me to conclude that either my reading comprehension is incredibly poor, you phrased your argument in an incoherent manner, or that you were, in fact, genuinely confused over the difference between begging the question and proof by contradiction. I'm pretty sure it's not the former two.

The more coherent explanation is that you took me out of context GP.

I didn’t say that offering the assumption necessarily leads to a begging the question fallacy, I said that it would lead to a begging the question fallacy in this situation, ie that your hypothetical is a response to all of Dr. Craig’s argument.

Follow the train of discussion back an you’ll see that this is part of my initial objection that your argument doesn’t cover all of Craig’s point as your OP suggested, but only a limited section.

If we were to assume that your argument is a coherent and holistic defense of the existence of actual infinities in this universe, than your assumption here would be begging the question since it simply assumes away one of Craig’s objections. It isn’t begging the question if we confine your response to one part of one of Craig’s sub-points.

It was preemptively responded to partially at the end of the OP itself, a different more thorough argument was given at the end of post #5, and in addition to those an additional alternative argument was offered at the end of post #10. I've quoted it once or twice to you already and summarily neglected to read it each time, so you can look up yourself this time and see if you want to respond.

I’m assuming you are referring to your references to QM in both posts. That’s great. I’ve read them three times, and again for this post a fourth time. The problem (as I noted both in post 7 and 11) is that it doesn’t answer the question offered, all of which stems from you inability (or unwillingness) to concede the point that your objection is specifically tailored to a sub-section of Craig’s argument, not to his argument in general.

Your OP is a rebuttal of a specific section of Craig’s objection that certain properties within an actual infinity make an actualized version of that concept untenable. Specifically, the section where he objects to adding a guest to a “full” hotel. I think there are problems with your rebuttal, but we’ll save that for later.

What I found objectionable (and what drove this response) is that your OP makes a somewhat broader conclusion; that actual infinities are feasible within this universe. My point was that this broader conclusion rests on the back of some relatively speculative (or at least unsupported assumptions) some of which you have defended, some of which you’ve simply asserted.

My point in the comment you are replying to above is that if you want to tie that very specific argument to the broader conclusion, we would need to accept your assumptions. This becomes even more problematic if we try to tie your argument back to the original premise of Craig’s (which you very well might not be trying to do here to be fair).

And upon said casual reading of the thread, one can verify that I have not made this claim, and in fact my actual claim was that you have been systematically ignoring huge sections of my posts that are, in fact, my rebuttal to your arguments,

So…you can’t quote a specific section, argument, or question that I haven’t responded to? This would seem to be more grandstanding than actual argument. Given that our responses are generally confined to a single page of debate, one would assume defending your accusation would be easy. But perhaps not.

Addressed in the OP, the end of post #5, and the end of post #10.

Well, no you didn’t.

Please show where you’ve answered these three questions in the posts asserted.

The context for this statement seems to be missing.

That is because you had declined to acknowledge the point when it was initially made. Ctrl+F in post 17 would have given you the appropriate context.

This has already been answered twice, and variations of the question ("What's the momentum?" and "What's the energy?") were directly addressed in the OP, but I'll repeat myself one last time:

After applying the mass regularization scheme proposed in post #10, the mass inside of the interval is 2m at the beginning, and 3m at the end of the time interval. The total mass in the universe at the beginning is 3m, coming from the 1m for the outside particle and 2m for the particles inside the interval. The total mass at the end of the universe is just the 3m lying inside the interval. Thus there is no violation of the conservation of mass, nor as the OP states a violation in the conservation of energy, or momentum.

Actually, this is the first time you’ve responded to that question, and you never brought up any such issue in the OP. What’s more, you clearly didn’t understand what was being objected to, which clearly wasn’t a conservation of mass issue in my post.


Rather, I was asking a question in response to this statement:

The mass starts out infinite and ends up being infinite in this scenario.

Now, you do seem to have altered the scenario by arguing for a non-standard mass amongst the particles, which is fine and why I asked the question I did.

But all that serves to do is dodge the question. I didn’t use mass as an example because of some fundamental property of mass, I used it because it happened to be infinite and could well explain the objection Craig noted.


So we return back to the original question, but replace mass with number of particles. How many particles are there in the set after the addition of the extra particle? The same number as before right?


Ok, here is where you respond with a pithy (you just don’t get infinite values) response a la the responses MT has been getting. But that isn’t an explanation, nor a rebuttal. It is simply a bare assertion. It would seem to warrant some explanation that X+Y=X, X>0.


What’s more, your solution seems to offer a further problem. The mass of each particle simply needs to be a positive, non-zero mass right? Axiom 3 in the OP.


Ok, but given your metric of how mass is related, the mass of the ‘infinitieth’ particle is zero, which would seem to be an issue with your hypothetical. Now I think an objection you might offer would be that “infinity is not a number.” Perhaps, but it doesn’t matter here, we can ask what the mass is of the particle closest to the position “2,” and given the relationship you’ve described, it would be zero since there are in your hypothetical an infinite number of particles.


And all of the above aside, we still need to discuss your assertion that actual particles in our universe do not occupy space, and, more relevantly, that they can be packed infinitely densely within any finitely sized region of space.

I agree that it would be odiously poor form for a debater to do this, yes. I'm pretty happy that neither Clive nor MT has done this.

Totally agreed. Now if we could just get you to stop doing it we would be in great shape.

[GoldPhoenix]

I’m assuming you are referring to your references to QM in both posts. That’s great. I’ve read them three times, and again for this post a fourth time. The problem (as I noted both in post 7 and 11) is that it doesn’t answer the question offered, all of which stems from you inability (or unwillingness) to concede the point that your objection is specifically tailored to a sub-section of Craig’s argument, not to his argument in general.

Your OP is a rebuttal of a specific section of Craig’s objection that certain properties within an actual infinity make an actualized version of that concept untenable. Specifically, the section where he objects to adding a guest to a “full” hotel. I think there are problems with your rebuttal, but we’ll save that for later.

Noted. As previously stated, I will not address any of your arguments or posts any further until you address what was forwarded to you in my posts (the size of particles in #5, #10; the claims regarding successive additions in #12), by either conceding the issues to me or by rebutting directly the arguments given.

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Well, no you didn’t.

I literally discuss this exact issue (particle size) at the beginning and end of post #5, re-quote it the relevant sections in the middle of post #10, and I give a secondary argument against this line of objection at the end of post #10. I have no idea why you've challenging me to defend something so trivially verifiable.

[CliveStaples]

So we return back to the original question, but replace mass with number of particles. How many particles are there in the set after the addition of the extra particle? The same number as before right?

Ok, here is where you respond with a pithy (you just don’t get infinite values) response a la the responses MT has been getting. But that isn’t an explanation, nor a rebuttal. It is simply a bare assertion. It would seem to warrant some explanation that X+Y=X, X>0.

The explanation is that there exists a 1-to-1 correspondence between the two sets of particles.

What’s more, your solution seems to offer a further problem. The mass of each particle simply needs to be a positive, non-zero mass right? Axiom 3 in the OP.





Ok, but given your metric of how mass is related, the mass of the ‘infinitieth’ particle is zero, which would seem to be an issue with your hypothetical. Now I think an objection you might offer would be that “infinity is not a number.”

The objection I have is not infinity is not a number, but rather "the infinitieth particle" fails to refer.

1. Failure to refer

To which particle does "the infinitieth particle" refer? If we're using the same ordering suggested by the OP, where the nth particle is located at 2[(1-(1/2)^n], then it doesn't make sense to talk about an infinitieth particle, since every particle is ordered with a natural number (so "the 2nd particle", "the 16th particle", etc.), and none of these are equal to infinity.

Compare this by analogy to the question, "Who is the 13th juror?" On a jury of 12 people, "the 13th juror" fails to refer, because even though 13 is a number there is no juror whose ordinal is 13th. So a failure to refer doesn't imply that the proposed ordinal (13th in the juror case, infinitieth in your case) is incoherent / not an ordinal.

2. Making Sense of "Infinitieth element"

Now, if you had a different ordering, it might make sense to talk about an infinitieth particle. Say that you had the same particles in the same order, but additionally a particle at 2. This particle might rightly be called the infinitieth particle, in accordance with the following expression:

x₁ < x₂ < x₃ < ... < x_∞

Here it would make sense to speak of x_∞ as the infinitieth particle.

3. Alternative orderings

You could, of course, define a different order on the particles and come up with an infinitieth particle, e.g.:

Original ordering (<) : x₁ < x₂ < x₃ < ...

New ordering (<') : x₂ <' x₃ <' ... <' x₁

In this case, by analogy to the earlier example, x1 could rightly be called the infinitieth particle with respect to <'.






4. Orderings and sequences

The orderings above are equivalent to functions from totally-ordered sets to the set of particles (which I'll denote as P). Since P is countable, there exists a bijection c:N⁺ -> P. Let p_k denote c(k). The original ordering, <, is equivalent to the function f:{2[1-(1/2)ⁿ] | n a positive integer} -> P where f(n) = c(n). Then f is a bijection, and < can be defined as p_i < p_j iff f^-1(p_i) < f^-1(p_j) (as real numbers).

The alternate ordering, <', is equivalent to the function g:{2[1-(1/2)ⁿ] | n a positive integer} U {2} -> P where g(n) = c(n+1) for n > 1 and g(2) = 1. g is also a bijection, and <' can be defined as above where p_i <' p_j iff g^-1(p_i) < g^-1(p_j) (as real numbers).

This process can be reversed to derive the functions from the orderings through transfinite induction, since the sets in question are well-ordered.

Perhaps, but it doesn’t matter here, we can ask what the mass is of the particle closest to the position “2,” and given the relationship you’ve described, it would be zero since there are in your hypothetical an infinite number of particles.

Here again "the particle closest to the position '2'" fails to refer to any particle, since there is no particle that is described by this property; if there were, there would be a particle p_i such that no other particle is closer to 2 than p_i, which is false (proof). Therefore a fortiori there is no particle that is described by this property and has zero mass.

[Squatch347]

Noted. As previously stated, I will not address any of your arguments or posts any further until you address what was forwarded to you in my posts (the size of particles in #5, #10; the claims regarding successive additions in #12), by either conceding the issues to me or by rebutting directly the arguments given.

And as I noted in my last post, this has already been done. You seem unable to connect the fact that your OP is about actual infinities existing within our universe with the clear requirement to defend the latter half of that premise.

It would be like arguing the following:

A married person can be a bachelor.

It is possible to have defined a married person as happy.

Bachelors are happy.

Therefore married people are bachelors.

With no realization that there is a hidden premise to defend. [/i]That married people are actually defined as being happy rather than just possibly[/i].

Your hypothetical doesn’t shift the burden of proof here, you still have to show that it comports with our universe in all the ways that are relevant to the section of Craig’s argument you are objecting too.

That claim is what led to the objections you are talking to (point particle idealizations, formation of the infinite, etc). All of those are relevant objections because they strike at the connection between your hypothetical and the actual state of the universe and the proposed argument it is meant to attack.

I literally discuss this exact issue (particle size) at the beginning and end of post #5, re-quote it the relevant sections in the middle of post #10, and I give a secondary argument against this line of objection at the end of post #10.

Well no, you discussed the use of point particles in idealizations, not in physical interpretations, which is what we are discussing here. Further, you don’t answer the relevant question as to whether their use extends to allowing them to be placed arbitrarily close together.

I’ve already explained how post 5 fails to answer the question earlier in thread, sufficed to say, nothing in that discussion concerns the physical interpretation of point particles or QM’s ability to accept an infinite density of particles.

In post 10 you argue that a point particle could be fundamental objects in nature, but you don’t actually show that in a way consistent with your example. Your description uses the bohemian simplification of a particle to a point particle and a pilot wave, but that wasn’t a physical interpretation, it was a simplification (Bohmian interpretations treat them as point-like particles because the relevant spatial relationship is handled in the pilot wave, and the point is a stand in for location, to my knowledge). Nor does it seem clear that the concept of a pilot wave can be resolved with the idea that we can fundamentally pack particles arbitrarily close.

The explanation is that there exists a 1-to-1 correspondence between the two sets of particles.

I don’t see how the bijection resolves the issue, it appears to be the issue. After all, my point was that the number of particles is identical, regardless of the addition of new particles, which is what the bijection in this case is telling us.

Here again "the particle closest to the position '2'" fails to refer to any particle, since there is no particle that is described by this property

Ok, let’s go with that for a second. Let me ask a question. What is the physical interpretation of that statement?

What does it mean to say that there is no particle described by that property?

For example. Let’s say that there was a particle at point 3 moving towards our set of particles. What does your above conclusion mean in that context? That there is no first particle to impact in the set? So what does it impact when it reaches that region of space?

[CliveStaples]

I don’t see how the bijection resolves the issue, it appears to be the issue. After all, my point was that the number of particles is identical, regardless of the addition of new particles, which is what the bijection in this case is telling us.

What is still in need of resolution? Two sets are equinumerous iff there exists a bijection between them. There exists a bijection between the two sets in question; therefore, the two sets are equinumerous.

Ok, let’s go with that for a second. Let me ask a question. What is the physical interpretation of that statement?

That in this physical model, and for this arrangement of particles, there is always a particle that is closer to the given position.

What does it mean to say that there is no particle described by that property?

For example. Let’s say that there was a particle at point 3 moving towards our set of particles. What does your above conclusion mean in that context? That there is no first particle to impact in the set? So what does it impact when it reaches that region of space?

This is a different question, and one that I actually brought up to GP, so I'll let him respond here.

[GoldPhoenix]

Well, you certainly have not precisely rebutted my arguments. However, you've addressed enough that I can at least continue the conversation. With that said, this conversation does seem to be approaching a stopping point, as in order to convincingly tailor the scenario to your questions, I need more and more math that you are not familiar with, and additionally we can't seem to move past some very basic points regarding mathematical infinity (For instance, you weren't even aware that there is no point nearest to another point in the continuum, you seem to continually be nearly erring on a Zeno's paradox error, and so on). So with some hesitation, I will continue the discussion, but it's unlikely that I will continue to respond to anything more than your technical points. In fact, I'm likely to adopt a policy that unless you respond to me with math, I will simply respond by telling you that you haven't met the burden of proof as laid out in the OP, and summarily disregard that line of argument until math is presented. This means that if you make a claim that "X isn't possible," you will be expected to have a rigorous proof, at least at the level of a 100-level physics or math course. That's not a very high standard (cf. Clive is giving fully rigorous proofs, although the content is regarding what is roughly expected of a third year math student).

And as I noted in my last post, this has already been done.

We've been down this road before in this thread and others, but declaring that you've done something and you actually having done said thing are two different things. I've seen a lot of the former, but very little, if any, of the latter.

You seem unable to connect the fact that your OP is about actual infinities existing within our universe with the clear requirement to defend the latter half of that premise. […] With no realization that there is a hidden premise to defend. That married people are actually defined as being happy rather than just possibly.

Your hypothetical doesn’t shift the burden of proof here, you still have to show that it comports with our universe in all the ways that are relevant to the section of Craig’s argument you are objecting too.

That claim is what led to the objections you are talking to (point particle idealizations, formation of the infinite, etc). All of those are relevant objections because they strike at the connection between your hypothetical and the actual state of the universe and the proposed argument it is meant to attack.

You can bloviate as much as you wish (and throughout this thread, you've certainly taken that liberty) about these non-starters, Squatch, but you've entirely ignored this argument, which explains why you don't want to (and cannot) go there:

There's a reason why you can't object to me using point particles; the primary reason is that, for all we know, classical point particles are the fundamental object in Nature.

Let me take a convoluted example to make this point: You are familiar with the Bohmian interpretation of quantum mechanics, which makes use of a literal point particle and a "pilot wave." Now, is that model likely to be correct? No, not at all. But is it literally impossible? No, not so far as I know. Our universe possibly could be described by this "quantum mechanics" (which isn't quantum mechanics, but simulates the effects of quantum mechanics). But if this model is correct, then everything that I'm describing is literally fine. Yes, there's a pilot wave, which one would need to account for its interactions, but this is essentially no different than setting up the gravity example, because all the wave would do is add forces to the particle. You might even be able to make the pilot wave itself be constructed out of point particles that interact non-locally. But I could deterministically setup the system, and evolve precisely in the manner than I'm describing. The mere fact that this is possible is enough to invalidate everything that WLC is asserting.


It is precisely because of the fact that we don't understand the fundamental nature of reality that WLC cannot appeal to specific sets of laws if his conjecture about actual infinities is correct, and thus what he is saying must apply very generally to any conceivable, sensible, robust version of physical laws. And the mere fact that every currently known and used theory of physics allows or actively involves various kinds of actual infinities inside of them --this should give anyone who hears WLC's conjecture some serious pause.

In other words, you can't simply appeal to the "real" laws of physics, Squatch, because you and WLC are as ignorant about them as everyone else is. Thus it is openly wrong for you to argue that my example proves nothing because it doesn't comport with “Nature as it really is,” because precisely the same argument can be made (and has, by myself in other threads) that this cuts down WLC's argument, as well. We don't know what the “real” laws of physics are, thus we cannot argue that they necessarily are incompatible with actual infinities. Either you can offer a very robust argument --which means that it's very insensitive to the specific set of physical laws that you're saying actual infinities contradict, and then inductively argue that it's reasonable, but not necessarily true, that the still-currently-unknown "real" laws of physics will probably also be broken by them-- or you haven't got anything.

The chief error (either logically, metaphysically, or scientifically) with WLC's argument is that he is actually raising a question, he isn't proving a theorem or giving an argument. WLC is saying "Look, there's a pathological (read: unintuitive) result for systems that obtain actual infinities!" and rather than saying, "Okay, that's an interesting potential bug for infinity, let's explore the ramifications on known physical laws or try to make robust arguments about a wide range of plausible physical or metaphysical laws", he's simply stopping the discussion there because it doesn't make sense to him and thus “must” be wrong. Pathological examples are not proofs by contradiction. Thus this isn't an argument, and the point of this thread is just to try to make the proponents of WLC's position actually think through what it is they are actually arguing --specifically, what is the nature of the claim that they are making, what is the robustness of the conclusion, what are the specific assumptions going into the argument, and so on. You all are definitely presuming something, you just aren't explicating what that is and bringing it into the open for critique and discussion. And whatever it is that you're assuming, it clearly has serious ramifications for abstract mathematics and fundamental physics.

Well no, you discussed the use of point particles in idealizations, not in physical interpretations, which is what we are discussing here. Further, you don’t answer the relevant question as to whether their use extends to allowing them to be placed arbitrarily close together.

A.) Another example of taking a creative license with the history of this conversation. I discussed (for instance, with MT) how point particles have been used literally as the fundamental objects of Newtonian mechanics, for instance the early models of electrons and other fundamental particles. I also said in response to MT that they have been used as an idealization (e.g. in gravitational calculations, for instance, that Newton worked with, the Earth and Sun are modeled as point particles). But you're employing a hasty generalization fallacy if you want to assume that all point particles are necessarily idealizations, and I certainly never said nor implied that they were.

B.) Contrary to what you've implied, Bohmian mechanics as stipulated does contain a literal point particle. We'll get back to this, however. So does Newtonian mechanics, Quantum Mechanics (at least for the configuration of infinite momentum incertainty), Special Relativity, General Relativity, and so on. At least, the point particle is described as something that is physically consistent (in the case of GR, it will be a little black hole, but there's not any real problems with this).

C.) I'll modify the scenario, with no consequences, to make the particles have finite size, since you're very obsessed with this finite size business.

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An Infinite Number of Rods with Finite, Continuous Density

Let's expand each particle to be a size, so now we're dealing with rods. The n^th rod will have an extent of 1/8^n, ending on the point x_n. We could give it a triangle density, an upside-down cup-shaped quartic density or whatever your favorite density is, which is continuous and integrates the mass. Except now to keep the density finite, we can make the masses scale as m_n = m_0 (1/ 8)^n.

You'll notice that since I chose the rods to all lie within a finite distance, and additionally as the particles jump forward one slot, they don't bump into each other. This can be checked by giving each end point of the rods are necessarily outside of each other's intervals by checking that x(start)_n > x(end)_(n+1) when they shift forward.

Thus: finite mass density, finite lengths, finite mass, finite energy, and finite momentum. Now then, what were you saying?

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The problem for you, Squatch, is that it's essentially trivial to modify this scenario to evade your objections. At this point, I can actually modify this to make a fully satisfactory Quantum Mechanics model, by the way. It can't evolve into Hilbert's hotel and the particles would still be free, but otherwise QM is definitely fine with this. (Although the uncertainty in each particle momentum will get larger and larger for each particle closer to 2.)

I’ve already explained how post 5 fails to answer the question earlier in thread, sufficed to say, nothing in that discussion concerns the physical interpretation of point particles or QM’s ability to accept an infinite density of particles.

Firstly, a finite density example is given above. Secondly, the example of Bohmian mechanics was an example of physical interpretation. So that assertion rings very hollow.

In post 10 you argue that a point particle could be fundamental objects in nature, but you don’t actually show that in a way consistent with your example. Your description uses the bohemian simplification of a particle to a point particle and a pilot wave, but that wasn’t a physical interpretation, it was a simplification (Bohmian interpretations treat them as point-like particles because the relevant spatial relationship is handled in the pilot wave, and the point is a stand in for location, to my knowledge). Nor does it seem clear that the concept of a pilot wave can be resolved with the idea that we can fundamentally pack particles arbitrarily close.

1.) Firstly, . Support or retract the claim that you aren't supposed to think of Bohmian mechanics as being about point particles.


2.) My example can trivially be embedded into the de Broglie-Bohm theory by simply making the choice of a constant pilot wave, i.e. \psi = 1 (or your favorite pure number). You can check that everything I've done satisfies the Bohmian equations by noticing that this choice of solution plus what was given in the OP solves the Bohmian mechanics equations. (The first Bohm equation is the equation for a particle, with the force given by a spatial derivative of a the wave, which if the wave is constant then this is zero; additionally, the equation of a wave is satisfied by a constant solution, which is the second equation. Then the linear solution given the OP is a solution to Bohm's equations. A statistically improbable one, but, again, an allowed choice of initial conditions.)

Let me ask a question. What is the physical interpretation of that statement?

What does it mean to say that there is no particle described by that property?

For example. Let’s say that there was a particle at point 3 moving towards our set of particles. What does your above conclusion mean in that context? That there is no first particle to impact in the set? So what does it impact when it reaches that region of space?

So Clive and I have actually already discussed this question. The correct answer is that, “No, actually, it isn't a problem.”

The issue here is that you're taking the scattering picture to literally, in other words thinking about fundamental particles like they are literal billiard balls that have to literally collide into each other before a force can be imparted. If that were true, then yes, that scenario would violate the laws of physics.

However, crucially this isn't how forces work –and this is already true for Newtonian mechanics, and stays true in E&M, SR, GR, QM, and QFT. Particles (classical or quantum, relativistic or no) do not collide with each other. The correct statement is that they are exchanging forces with each other through a commonly shared force field. The prototypical example of this is that an electron interacts with another electron because each creates an electric field, which imparts a force on all other charges particles. When two billiard balls collide, the electrons on the outside of the atoms that make up the billiard balls creates very local, short-ranged forces. Then they come very close (but still finite distance), it creates a strong enough force to cause the balls to move away from each other. Thus there's no point in time when balls literally touch each other (Unless we define “they touch” to mean “they interact strongly through their electric fields”). But crucially particles don't bounce off of each other because of some metaphysical need to repel each other; in fact, around 65 billion neutrinos (and probably dark matter particles) pass freely through every square inch of your body every second, and this is because there's (effectively) no force field to cause an interaction between the electrons in your body and neutrinos (and ditto for dark matter if it exists).

Likewise, for this scenario, we'd need to introduce a force field (or a set of them) for which the particles could interact with each other. Then your question of “Which one interacts first!?” doesn't actually mean anything, because it just creates a field which all of the particles respond to. The only question is if there's a valid solution for the force field.

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For the interested reader:

Now, I could setup an example that uses an infinite number of fields or I could add an (ever decreasing) charge to each of my particles (or rods, if you insist on that), and I could combine this information into something called the Hamiltonian formulation of classical mechanics.

From there, I could choose the parameters of the theory and the initial conditions (velocities) to maintain finite physical quantities. I could probably even sit down and work through the exact initial field profile. The upshot of the Hamiltonian formulation is that it keeps all physical conserved quantities conserved throughout the entire evolution, so the only question is if the solution exists, but there's no obvious physical obstruction to the solution (in particular, if I can show that the initial field profile made sense, then there's definitely no reason why there wouldn't be a solution to that differential equation). You probably need to make the charge density drop to zero, so q_n ~ (1/4)^n would accomplish this. To make the energy manifestly finite, you'd probably need something like (1/16)^n.

I am, however, not interested in doing this. I see no obstructions to doing any of this and it all seems like a rather simple 300-level homework problem in a classical mechanics course, other than that it would be very tedious, and I'm quite certain it would do very little to convince anyone here because probably I'd be the only one who understood why it worked, and additionally I'm quite certain that it'd get brushed aside as "unphysical" for other reasons.

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It's also worth stressing, giving how much you're bloviating about "the actual laws of physics" that you haven't countered a single thing about my examples of actual infinities in Quantum Field Theory combined with semi-classical General Relativity, which so far is the most accurate known theory of physics. Yes, it's true that we've moved away from the specific scenario in the OP, but as mentioned in the OP and subsequent posts, finding allowed (in fact, logically necessary) actual infinities in physics is not challenging:

In Quantum Field Theory, you are forced into accepting the existence of Fock space, which is the space that contains all possible particle configurations (The Hilbert space of all possible particle states; a discussion of this can be found in the first few chapters of any book on QFT, so take your pick, but section 2.4 of Tong's intro QFT book is free and discusses this). The dimension of the Fock space is necessarily infinite (In this case, not being infinite would lead to an actual violation of physical law), and this introduces several very simple "actual infinities." [GP: The Fock space is the (graded) space of states. The first space is the space is single particle states with given momentum and helicity; the second space of two particles states, the third the space of three particle states, and so on. Each is a Hilbert space, and the Fock space is the infinite collection of all of them. In order to work with known laws of physics, you must crucially have all possible collections, including an infinite particle state.]

The first is that I have an actual infinity of different possible particle states. Two examples where this is used, for instance, is in the description of coherent states, for instance in a pulse of light, and the another example is Unruh radiation. Unruh radiation necessarily contains an infinite number of modes in the radiation (or else it violates physical law, see equation 3, which implicitly sums j from minus to plus infinity). Note that even though Unruh radiation hasn't been directly observed a lab, it is a necessary consequence of the current physical laws. So this means if you are accelerating, you see a universe which has a bath containing an infinite number of particles (But all physically finite quantities, like energy, momentum, and temperature), but if you're standing still (or in an inertial frame), you see a universe with zero particles. This highlights the point that trying to demand that infinity is unphysical just doesn't make any sense from a QFT perspective nor does it comport with any known facts of the matter. These theories may break down and render the actual number of particles finite, but as of now, the evidence supporting the assumptions of QFT lead us to these conclusions; and even if so, the only thing that could even conceivably do this is quantum gravity (But quantum gravity also could conceivably lead to past infinities, so this kind of appeal is a double-edged sword for WLC).

Back to the point of Fock space: This means if I want to start my universe out with an infinite number of point particles, there actually cannot be any principle that forbids me from doing so, because that principle would then by direct implications break the previous physical principles (In this case, Lorentz invariance and unitarity). Thus the laws of QFT + WLC's new "No infinities" principle entails a contradiction. [GP: Thus it is "Necessarily false" for the current known laws of physics.]

So going to quantum mechanics doesn't provide a way to dispute my overall point about actual infinities, it simply makes WLC's argument even more obviously wrong from the outset. Again, all I'm trying to do here is give a much simpler, much more intuitive example of an "actual infinity." Beyond which, if we accept that we should take quantum gravity into consideration, then his entire argument fails for other reasons, like our lack of knowledge of quantum gravity means we may very well be capable of living in a past-infinite universe.

You should probably address this at some point in time if you want me to take your argument regarding "the real world" seriously, in addition to responding to the previous points made in this post.

[MindTrap028]

I am really sorry that I just don't have the time to craft the response I would like, but I don't want the conversation to get so far ahead without bringing up this point/question.

So GP, your going to be requiring a mathematical response/proof, or at least prepared to. So my question is more of how to represent mathematically a simple problem.

You have an infinite number of hands, holding an infinite number of apples.
Given the nature of apples and hands, (that they must be held or else they fall). How exactly does adding another apple occur? Suppose that we agree, that there is no free hand for the apple to go to, so the process of "re-assignment" (that is so quickly appealed to). How do you show mathematically that the process is negated? The important aspect here, and the common element in both examples, is the lack of a free hand.

I mean, I'm not even sure how to present that with only 2 hands and 3 apples.
I suppose the inability to execute this re-assignment is shown by the lower number of hands to the higher number of apples.
But with infinities, this is no longer represented in the math (again... as far as I can tell).. yet the reality is every bit as real.

----
As a note, I think in past threads I have identified this as the man in the hallway issue (where the actual process of assigning guests to a room, means that someone is perpetually in the hallway)
My suspicion is that the disconnect between the math and reality is that it is very easy for math to insert a "+" it is another thing for it to occur in reality.
I hope my example brings this issue out more by eliminating the hallway. Apples must be in a hand, or the process fails, and there are no free hands.

I look forward to your explanation of how math represents this reality.

[GoldPhoenix]

So my question is more of how to represent mathematically a simple problem.

You have an infinite number of hands, holding an infinite number of apples.
Given the nature of apples and hands, (that they must be held or else they fall). How exactly does adding another apple occur? Suppose that we agree, that there is no free hand for the apple to go to, so the process of "re-assignment" (that is so quickly appealed to). How do you show mathematically that the process is negated? The important aspect here, and the common element in both examples, is the lack of a free hand.

It doesn't sound like you're asking me to give you a precise mathematical model of an infinite number of hands creating room for another apple --although I can and will do that-- it sounds like you're asking how could that could be physically realized. I'll answer both at the same time. It's contrived, but the math is the math:

---

The Setup:

Suppose we have an infinite number of people, each a meter apart, with in apple in their left hand(they all have their right hand tied behind their backs), and each with a precisely timed watch. At, say, one minute after noon, they all agree to throw the apple in their left hand over over one meter to the next hand. So everyone starts throwing their apples at the same instant, and it lands to the person next to them. So the first person throws their apple at noon, and 5 seconds later the second person catches it. At the same time, the second person throws their apple at noon, and five seconds later the third person catches it. But their hand is free, because at noon (five seconds previously), they threw their apple to the person next to them.

The reason why this works for an infinite number of people is that infinity is never ending. The reason why there's always "room for another apple" is because if there were a finite number of people, then you'd finally reach the end, i.e. the last person, and they would throw their apple to the right, but there wouldn't be anyone there to catch the apple. However, if there's an infinity of people, then there is no last person. Thus, each person throws the apple to the person next to them, and there's never a "last person" from which an apple would fail to find a hand. That's the crucial difference between a finite set and an infinite set.

---

The Mathematical Model:

The model of the n^th apple's trajectory given by x_n(t) for the apple's movement between the hands, and y_n(t) for the apple's height above the hands, all at the time t. We'll define "t=0" to mean "noon", the parameter T is defined as T=5 seconds, g is the gravitational constant (9.81 m/s/s) and we'll define the distance between hands as L = 1 meter. Then we have:

y_n(t) = (1/2gT) t - (1/2g)t²
x_n(t) = n L + (L/T) t

Note that x_n(T) = x_n+1(0), so the n^th apple's new spot, after time T, has moved to the (n+1)^th apple's old spot, where it gets caught by the (n+1)^th hand. So this has the effect of taking apple 0 and adding it to the sequence of apples in each hand.

Now hand 1 is holding nothing. So we can throw a new apple to hand 1, and we've accommodated the new apple, which we'll label as apple 0. Stated mathematically, at time t=0 (apple 1 in hand 1, apple 2 in hand 2, ...) and that evolves to (apple 0 in hand 1, apple 1 in hand 2, apple 2 in hand 3, ...), so the n^th slot of the sequence represents what is physically in the n^th hand. But the total time it takes to accommodate the new apple, apple 0, is T = 5 seconds.

I mean, I'm not even sure how to present that with only 2 hands and 3 apples.

I suppose the inability to execute this re-assignment is shown by the lower number of hands to the higher number of apples.
But with infinities, this is no longer represented in the math (again... as far as I can tell).. yet the reality is every bit as real.

Well, the third apple would hit the ground, but you could take the cases 1-3 of the formulas above, for throwing the apples. This would allow the new apple, let's call it apple zero (plug in n=0, and it'll work for that apple's trajectory).

[MindTrap028]

I'm sorry, I think you missed my meaning a bit.
While your solution is clever, it doesn't really address the heart of my question.

My point is to introduce a physical barrier, and see how math recognizes that barrier or if it can. I don't believe you have done that here, rather you have worked around it. Which is reflective of a poor example on my part. But it should be easy enough to correct. Suppose instead of taking the reasonable first apple, we could choose to use the 1 millionth apple on. Now no hands could throw 1 million meters away... but how does the math of infinities recognize that?

We could also suppose that the people are blind and unable to coordinate, or simply define that the hands must be touching an apple. The idea of course is not to think of some plausible and clever way in which it can be done, but see how math recognizes the clever, and plausibly real impossibilities.

Do you know those puzzles with the empty piece, where you are supposed to move the wooden tiles to form a picture?
https://www.google.com/search?q=slid...odlXAAUA&dpr=1

Here we are supposing an infinity large puzzle, with no empty space, and trying to insert a new one.
As far as I can tell.. the math says it can be done.. but it is a defined impossibility.

----
Conversely, using your solution here...two hands could simply juggle an infinite number of apples. So, this appears to be more of a work around, and I'm not sure how to present possible physical impossibilities.. Unless of course, we are to suppose that infinities negate all possible physical impossibilities.. Now that would be a claim indeed.

[CliveStaples]

Here we are supposing an infinity large puzzle, with no empty space, and trying to insert a new one.
As far as I can tell.. the math says it can be done.. but it is a defined impossibility.

Math doesn't tell you about the real world. Math tells you about mathematical objects; mathematical knowledge is a priori knowledge. If there's a connection between some mathematical object and some real world phenomenon, then this connection is discovered not by doing math but by investigating the real world.

I'm not sure what contradiction you're trying to describe, here. The only argument I can think of goes something like this:

(1) If infinities are actual, then the ways of rearranging apples in hands is identical to the the ways of rearranging guests in Hilbert's hotel.
(2) Every way of rearranging apples is physically possible.
(3) There is a way W of rearranging guests in Hilbert's hotel that has the following properties:
(3a) W is identical to a way W' of rearranging apples in hands
(3b) W' is physically impossible

Thus we derive a contradiction between (3b) and (2). But why should we think that (2) and (3b) are both true?

We could also suppose that the people are blind and unable to coordinate, or simply define that the hands must be touching an apple. The idea of course is not to think of some plausible and clever way in which it can be done, but see how math recognizes the clever, and plausibly real impossibilities.

What does blindness or ability to coordinate have to do with whether this situation is physically possible? This seems like asking whether it's physically possible for a blind person to shoot a bullseye with a rifle; the blindness is just totally irrelevant to the question.

[MindTrap028]

Math doesn't tell you about the real world.

Well, then this whole discussion is not really relevant to proving a real world example of infinite can exist.
Of course, this last exchange is really me questioning how to describe a specific physically bared event (IE assignment re-assignment) mathematically, in response to a possible requirement to formulate all challenges mathematically.

This is my ignorance of mathematical expression showing.
So what I see are a few choices.

1) There is some possible physically bared mathematical assignments.
Thus (a or b)
A) Math has an expression for those blocks, and thus I must use those expressions in my formulated objections.
or
B) Math does not have expressions for those blocks, thus mathematics is not a sufficient language to discusses the actual possibility, as it specifically excludes any possible blocks.

What does blindness or ability to coordinate have to do with whether this situation is physically possible? This seems like asking whether it's physically possible for a blind person to shoot a bullseye with a rifle; the blindness is just totally irrelevant to the question

Seeing the watch. My point was that for every clever work around.. one can imagine a equally clever counter.

---To add an objection, The idea of throwing apples in the air is to suppose more space than is intended by the example. It's like putting people in the hallway.
That is fine for all examples with hallways and air. But in examples where all points are used (such as the op... or at least one reading of it) there is no other points to put the apples.
So I'm trying to get past the clever work arounds.

[GoldPhoenix]

My point is to introduce a physical barrier, and see how math recognizes that barrier or if it can.

Okay, that's a very different question.

1.) Firstly, to re-emphasize a point made by Clive, math is an a priori subject, this means that its truth or falsity does not depend upon facts of the matter (truths about physical reality). Its truth or falsity can be verified purely by thinking through the definitions, primitives, and rules of logical inference. Science is an a posteriori subject, which means that its truth or falsity depends upon facts of the matter (which means you have to explore reality). Mathematical models (and physical interpretations of the variables) are given to describe physical theories. There are two separate questions:

A.) How do we know that the mathematical theory X' comports with reality?

Answer: We perform an experiment many times in a given regime and convince ourselves that X' precisely models the results of our experiments. Thus we say that X' correctly models Nature in those regimes, and we may extrapolate its validity beyond those regimes.

B.) How do we know that the mathematical theory X' is consistent, i.e. contains no contradictions?

Answer: We do calculations (pure math) and check that the results of the theory are self-consistent.

You're asking, I believe, about when we can see that something goes wrong in (B), not in (A). In other words, you're asking about "Given that we've done (A), and our model accurately describes the universe (or some hypothetical universe), and we know that situation Y is unphysical (nomologically impossible), how would we see that situation Y is unphysical in the math?


2.) The way to see contradictions in the mathematical models of nature would be:

1.) If a physical principle is violated, e.g. a physical quantity that should be finite is infinite (Like having infinite force or momentum).
2.) An actual logical contradiction in the equations. (e.g. physical quantity X can be calculated to be X = 1, and also we calculate it to be X = 2).

Let's take an example in Newtonian mechanics.

---

Diagnosing a Violation (in the Mathematical Model) of Physical Law

Suppose that we have two objects, A and B. Let's suppose that Object A (say, a basketball) and Object B (a brick) collide and interact (as explained to Squatch, due to some microscopic, conservative interaction between them, but that's not important here). The momentum, P, of each particle is given by:

Eq 1: P_A = m_A v_A
Eq 2: P_B = m_B v_B ,

where v is the velocity of the object and m is the mass of the object. Now then, there's an important physical law, called the conservation of momentum, which says that when two particles are incoming with velocities v, interact (collide), and then bounce off of each other with outgoing velocity v', the momentum must stay constant:

Eq 3: P_A + P_B = P'_A + P'_B .

Now, suppose I tell you that the mass of particles is simply 1 kg each, which we'll just write as m_A = m_A =1. Next, I tell you that the object B is stationary (v_B=0) and object A travels towards it at 2 m/s, so v_B=2. Then I tell you that object A hits object B, and A bounces off of B, and B gets pushed forward. So A is moving leftwards with a new velocity, v'_A=-2 m/s (the minus sign just indicates that it's moving to the left), and object B moves forward with a velocity of v'_B=2. Using equation (1), equation (2) plugged into the left hand equation (3) for the total momentum before, we see that:

P_A + P_B = (1) (1) + (1)(0) = 1 + 0 = 1 ,

and to get the total momentum afterwards:

P'_A + P'_B = (1) (-2) + (1)(2) = -2 + 2 = 0 .

Thus we can see that the conservation of momentum has given us:

1 = 0 .

This situation has broken the conservation of momentum. Thus, this situation is physically impossible according to the laws of physics. (This scenario has also broken conservation of energy, too, for the record. 1/2 =/= 4)

---

I don't believe you have done that here, rather you have worked around it. Which is reflective of a poor example on my part. But it should be easy enough to correct. Suppose instead of taking the reasonable first apple, we could choose to use the 1 millionth apple on. Now no hands could throw 1 million meters away... but how does the math of infinities recognize that?

That's actually easy to accommodate as well. You just throw the apple to slot M, for an arbitrary distance away (In this case 1 million meters away), and then have apple M thrown to hand M+1, apple (M+1) thrown to hand (M+2), and so on. It requires a minor modification of the equations listed in my previous post.

I think the example above gives a much more precise illustration of what you're looking for.

[MindTrap028]

You're asking, I believe, about when we can see that something goes wrong in (B), not in (A). In other words, you're asking about "Given that we've done (A), and our model accurately describes the universe (or some hypothetical universe), and we know that situation Y is unphysical (nomologically impossible), how would we see that situation Y is unphysical in the math?

Honestly, I'm not sure where it would fall in those two categories.

This situation has broken the conservation of momentum. Thus, this situation is physically impossible according to the laws of physics. (This scenario has also broken conservation of energy, too, for the record. 1/2 =/= 4)

So, has the math shown
1) that Basketballs and bricks can't collide
or
2) They just don't have the result thought.

If it is showing #2, then it isn't the same kind of proof we are talking about here. I mean, given the puzzle I referenced earlier, I'm not certain how math represents no room for a tile.
For another example, of my level of ignorance, I would have never guessed that in the example you gave that the negative results were an indication of left handed movement. (is that inherent to mathematical equations, or just a convention of the model you set up?)

That's actually easy to accommodate as well. You just throw the apple to slot M, for an arbitrary distance away (In this case 1 million meters away), and then have apple M thrown to hand M+1, apple (M+1) thrown to hand (M+2), and so on. It requires a minor modification of the equations listed in my previous post.

... so your supposing that human hands can throw an apple 1 Million Meters, and the mathematical model doesn't show that as false or absurd? I'm guessing this would be due to an over simplified model.

... My problem.
I don't know how to express mathematically the idea that two hands are full and can no longer hold any apples.(an obvious and agreeable truth.. clever juggling aside)
I don't know how to show mathematically that full hands can not accommodate more, or what physical law that violates.

Thus, I can't Answer your challenge as you may desire it to be stated with regard to known, agreeable ends.. So i don't have much confidence in attempting to answer in that manner on ends we disagree on.
Now certainly I don't expect you to make my case for me, but I'm not sure your examples have filled in the blanks for me, though they are appreciated.


--

2.) The way to see contradictions in the mathematical models of nature would be:

1.) If a physical principle is violated, e.g. a physical quantity that should be finite is infinite (Like having infinite force or momentum).
2.) An actual logical contradiction in the equations. (e.g. physical quantity X can be calculated to be X = 1, and also we calculate it to be X = 2).

How have we not shown that #2 has occurred with infinities? We know that subtracting infinite quantities results in contradictory results.
As long as subtraction represents guests leaving a hotel in a model, then any model that disallows a subtraction of a real thing (IE a real infinite) is clearly self contradictory in the manner of #2.
It has been said that subtraction of infinite's are disallowed, thus those models are self contradictory and can not be fully realized (even if they are handy imaginations).

[CliveStaples]

How have we not shown that #2 has occurred with infinities? We know that subtracting infinite quantities results in contradictory results.

This is incorrect; there are no sets A,B such that |A-B| or |B-A| are equal to multiple different cardinals.

As long as subtraction represents guests leaving a hotel in a model, then any model that disallows a subtraction of a real thing (IE a real infinite) is clearly self contradictory in the manner of #2.
It has been said that subtraction of infinite's are disallowed, thus those models are self contradictory and can not be fully realized (even if they are handy imaginations).

You can do set subtraction with any two sets. The result will be a set.

Every set has a unique cardinality, i.e. if |S| = x and |S| = y then x = y.

So you can always do set subtraction, and the resulting set will have a unique cardinality. Where's the contradiction? What statement is both true and false at the same time?

You might be tempted to say something like aleph0 - aleph0 = 1 and aleph0 - aleph0 = 2, but 1 != 2. However, both of these statements are false; it is false that aleph0 - aleph0 = 1, and it is false that aleph0 - aleph0 = 2. Cardinal subtraction is different than set subtraction.

Consider this analogy:

|(-2) - (2)| = |-4| = 4

|(-2)| - |(2)| = 2 - 2 = 0

So even for integers, |x-y| != |x| - |y|. So just because you know |x| and |y| doesn't mean you can find |x-y|. Or, to put it another way, |x-y| isn't uniquely determined by specifying |x| and |y|.

Do you think that subtraction of integers is contradictory? Do you think that the absolute value function is contradictory?

[GoldPhoenix]

So, has the math shown
1) that Basketballs and bricks can't collide

No, clearly not.

or
2) They just don't have the result thought.

Correct. Any real life scenario cannot have the results that were thought up. It is not physically possible for that scenario to occur with those incoming velocities and those outgoing velocities, given the masses of the objects in question.

For another example, of my level of ignorance, I would have never guessed that in the example you gave that the negative results were an indication of left handed movement. (is that inherent to mathematical equations, or just a convention of the model you set up?)

You certainly learned about this in a high school class on algebra, but this is how all Cartesian coordinate systems work: We use a variable, e.g. "x", to signify the position in space (for instance, where the point lies on a line). To accomplish this, we declare one point on the line to be the origin, (0), and the negative numbers are to the left of that point, and positive numbers are to the right of it. It is a convention that we choose negative numbers to the left and positive to the right, but one side must be positive (which is conventionally the right) and one side must be negative (which is conventionally the left). A point that is two units to the left of the origin will be issued the point (-2), and a point that is 7 units to the right away from the origin is issued the point (7). For every dimension, we must include a new number. For a plane (e.g. a piece of paper), we have to have one number to describe left/right and a second number to say how far above/below the origin (now given by (0,0) as a point in the center of the plane).

... so your supposing that human hands can throw an apple 1 Million Meters, and the mathematical model doesn't show that as false or absurd? I'm guessing this would be due to an over simplified model.

1.) The model isn't oversimplified here, it's a question of context, MT. A hand can throw a ball a million meters very easily, for instance, in space. When I throw a ball, it exits my hand with a certain amount of momentum, which is will continue to unless external forces impart changes in momentum to it. Therefore, it may take the ball a very long time to go a million miles (although as we dial down the mass, the force will impart a much larger velocity), but if you're an astronaut in space and you throw a ball of mass M = 1 gram (the mass of a paper clip) with a force of F = 360 newtons (about the force imparted of a 100 apples falling on you, which by comparing to baseball pitcher's strengths is what an exceptional pitcher can do) for t = 1/8 second, and you want it to travel a distance of 10 million meters, this will take T amount of time for the ball to travel a distance L = 10 million meters:

V = L/T = P/M = (Ft)/M
==> T = L M/(Ft) = 10^7 * (.001) / (320 * 1/8) = 10^3 / 4= 2,500 seconds ,

or a little over 40 minutes. Because there's no air friction (which is normally what causes low mass objects to travel slowly), utterly negligible gravity (so it won't just fall to the ground and hit the ground), because the mass of the ball is rather small (e.g. its made of styrofoam), and because humans can throw things reasonably well (again this is estimates for a professional baseball pitcher), it radically changes your expectations for fast a human can throw a ball.


2.) But what you asked for, which is what you got, is an example of what happens when a physical principle is broken. Yes, in the context of being on the surface of the earth, in the context of a human hand doing the throwing, in the context of balls lying in a particular normal range of masses and sizes, then a human hand can't throw a ball one million meters.

---

The problem here has to do with range of forces that a hand can exert on an object (which is due to the materials involved in the human hand and the physical properties they have), which is due to the biochemical/physical properties of your hand. Those rules governing the properties of the materials, of course, themselves are derived from simple principles of physics, like conservation of momentum, energy, the laws governing electrostatics, and so on. So it's much more expedient to ask what goes wrong in an idealized case where we can easily diagnose failures in physical principles. This is how Newton and Galileo made progress where no progress can be made otherwise. If you can't tell me what goes wrong in a simple system, then what hope do you have of telling me what's going to go wrong in an extremely complex system that models the kinematics of an organism?

Edit: Just as a point of clarification, I'm not putting this in large font to indicate yelling or condescension. What you're asking here is an important question, and it's an answered question in physics. You cannot make progress in questions of Nature by demanding that we have exact models that take care of all of the details of, say, a human hand (They couldn't in Newton's day, anyways, but now with much more knowledge, we can). Not at the beginning, anyways, do you try to ask exact answers for really complex systems. And moreover, generally it isn't worth asking those questions, anyways, because all of the physics going on inside of a human hand can be understood individually in much simpler physical systems (pulleys, levers, rotors, etc). So even if you want to understand how the mechanics of a hand works, you won't make progress by asking "How does the whole hand work at once?" Instead, you learn about simple systems (tendons are just pulleys, joints are just rotors, etc) which are totally understood in terms of the physical principles. Now that's crucial, because really what this means is that we should try to understand what "actual infinities" do to much simpler systems, because if actual infinities do not mess up very the very simple systems, it's not really plausible that they will mess up complex systems that are just made up of many of the simple systems (lots of gears, wheels, pulleys, rotors, etc). That's how actual progress in science is made, so if we want to understand what actual infinities imply, we study first how they affect simple systems. Only after we understand simple systems very well can we move onto more complex scenarios. Additionally, if something does go wrong in the more complex system, it's absolutely going to be the case that it's because something went wrong in a subsystem, which means that subsystem went violated a fundamental physical principle.

---

I don't know how to show mathematically that full hands can not accommodate more, or what physical law that violates.

There's an infinite number of different ways to express this system, but a simple model is with sequence that encodes how many items are held in a hand with a principle that a hand can either hold nothing or else hold exactly 1 item.

Let the hands be given by a sequence, X_n that lists each hand (each slot in the sequence signifies a specific hand). Let the number in each slot indicate the number of items the hand is holding. So if the n^th slot of X_n is 0, then the n^th hand isn't holding anything. If the n^th slot is 1, then the hand is holding 1 object. Then what you would want to do is assert a physical principle, like, "No hand can hold more than one object." To implement this mathematically, you would model this with a fancy-sounding name, e.g., "the 1-holding axiom" which states that:

"For all n, either X_n = 0 or X_n =1."

If you could ever prove that this wasn't upheld, then you could claim a violation of this principle.

How have we not shown that #2 has occurred with infinities? We know that subtracting infinite quantities results in contradictory results.
As long as subtraction represents guests leaving a hotel in a model, then any model that disallows a subtraction of a real thing (IE a real infinite) is clearly self contradictory in the manner of #2.
It has been said that subtraction of infinite's are disallowed, thus those models are self contradictory and can not be fully realized (even if they are handy imaginations).

Because to have done so would require giving a precise principle, P, that is at least as mathematically explicit as what I gave above, and then a concrete proof (meaning, going through symbolic manipulations) that my physical scenario, X, entails ~P. Additionally, you would also need to justify why P is something I should actually believe is true.

Without that, you don't have anything. To repeat what I said before, a pathological, counter-intuitive result is not a proof by contradiction (i.e. that the result is false). A contradiction means that you asserted P and ~P, and nothing else. That's the only way that you can determine that the scenario is impossible.

[Squatch347]

What is still in need of resolution? Two sets are equinumerous iff there exists a bijection between them. There exists a bijection between the two sets in question; therefore, the two sets are equinumerous.

That there is a bijection and that the sets are equinumerous is the objection. That a set can undergo addition to, but whose size (in the cardinality sense) is unchanged is the objection being raised given that the definition of cardinality is, essentially, the number of elements in a set.

To write it out essentially we are saying that we can add to the number of objects in a set and leave the number of objects in the set unchanged.

Now so far, the response has been to say, “I don’t see why that would be a problem, we just need a definition of infinite sets that allows for it,” which seems a bit like defining away the objection. We know that X+Y=X in the scenario described above is internally incoherent for finite sets, so to then categorically change that view for infinite sets would seem to beg for a bit more of an explanation as to why infinite sets, by their nature, are capable of displaying a property that makes incoherence, coherent.

That in this physical model, and for this arrangement of particles, there is always a particle that is closer to the given position.

And how then would you ask the very reasonable next question?

What does it mean to say that there is no particle described by that property?

For example. Let’s say that there was a particle at point 3 moving towards our set of particles. What does your above conclusion mean in that context? That there is no first particle to impact in the set? So what does it impact when it reaches that region of space?

For instance, you weren't even aware that there is no point nearest to another point in the continuum, you seem to continually be nearly erring on a Zeno's paradox error, and so on

An interesting rebuttal given that it wasn’t my point at all. You’ll note that I never compared two points along a continuum. Rather, I noted that there is no clear boundary in your continuum (which I clearly was aware of, hence why I raised it as a rebuttal), which causes some very obvious physical problems (see my last response).

I will simply respond by telling you that you haven't met the burden of proof as laid out in the OP, and summarily disregard that line of argument until math is presented.

You are free to take whatever tack you like, but I think it is pretty obvious that the points laid out as objections aren’t dismissed so easily. After all, if they were so easily dismissed, we both know you would have done so by now.

There's a reason why you can't object to me using point particles; the primary reason is that, for all we know, classical point particles are the fundamental object in Nature.

Given that it is your OP, and I think I’ve been relatively patient on you defending your premises so far, Please support or retract that point particles are fully consistent with our understanding of the physical universe. And, please support or retract that point particles can be placed infinitely close together and at precise locations with a known velocity.

An Infinite Number of Rods with Finite, Continuous Density

Let's expand each particle to be a size, so now we're dealing with rods. The n^th rod will have an extent of 1/8^n, ending on the point x_n. We could give it a triangle density, an upside-down cup-shaped quartic density or whatever your favorite density is, which is continuous and integrates the mass. Except now to keep the density finite, we can make the masses scale as m_n = m_0 (1/ 8)^n.

You'll notice that since I chose the rods to all lie within a finite distance, and additionally as the particles jump forward one slot, they don't bump into each other. This can be checked by giving each end point of the rods are necessarily outside of each other's intervals by checking that x(start)_n > x(end)_(n+1) when they shift forward.

Thus: finite mass density, finite lengths, finite mass, finite energy, and finite momentum. [B]Now then, what were you saying?

First, let me offer a side question. In this scenario, the rods can only move forward once, correct? (After the first movement they would seem to occupy a space smaller than their extent. Not that it is a fatal problem at all, since you are only invoking one movement in the OP, I just want to be sure I accurately understand what you are saying here).

Ok, I think we still run into the same problems at some point. Given how you’ve defined both mass and extent, the mass and extent of the “infinitieth” particle is zero. Again we are faced with the distinction between taking a limit that approaches infinity, and an actual infinite set. In this set there are an actual infinite number of particles that exist, which means that some particle has no extent and no mass.

Given that I’m not really sure that giving the particles a small extent escapes the objection as you seem to think.

1.) Firstly, . Support or retract the claim that you aren't supposed to think of Bohmian mechanics as being about point particles.

2.) My example can trivially be embedded into the de Broglie-Bohm theory by simply making the choice of a constant pilot wave, i.e. \psi = 1 (or your favorite pure number). You can check that everything I've done satisfies the Bohmian equations by noticing that this choice of solution plus what was given in the OP solves the Bohmian mechanics equations. (The first Bohm equation is the equation for a particle, with the force given by a spatial derivative of a the wave, which if the wave is constant then this is zero; additionally, the equation of a wave is satisfied by a constant solution, which is the second equation. Then the linear solution given the OP is a solution to Bohm's equations. A statistically improbable one, but, again, an allowed choice of initial conditions.)

I don’t think I said you weren’t supposed to think about them that way, my point was that you weren’t supposed to think about them only as point-like particles. A point particle is only one aspect of the particle’s nature in a Bohmian interpretation. Which is what you refer to in part 2 of your response above.

But 2) doesn’t seem to resolve the question asked. I didn’t ask if the particles and their motion could be described by de Broglie-Bohm, obviously they can, that would be a trivial question as you put it. Rather, what I asked was can you show that a de Broglie-Bohm interpretation of your OP allows for point particles to be placed infinitely close together?

Again, perhaps this is due to the limitations of my knowledge on this subject, but I was under the impression that a particle must occupy a space at least as large as its wavelength (http://www.math.ucr.edu/home/baez/lengths.html) . And while wavelength is decreased by additional energy, in order for it to be infinitely small, we would need infinite amounts of energy (obviously a problem, http://arxiv.org/pdf/gr-qc/9403008v2.pdf).

So Clive and I have actually already discussed this question. The correct answer is that, “No, actually, it isn't a problem.”

The issue here is that you're taking the scattering picture to literally, in other words thinking about fundamental particles like they are literal billiard balls that have to literally collide into each other before a force can be imparted. If that were true, then yes, that scenario would violate the laws of physics.

However, crucially this isn't how forces work –and this is already true for Newtonian mechanics, and stays true in E&M, SR, GR, QM, and QFT. Particles (classical or quantum, relativistic or no) do not collide with each other. The correct statement is that they are exchanging forces with each other through a commonly shared force field.

I think there are two issues with this response. The first is that it seems to miss the main thrust of the objection by invoking a detail set that you had previously said you were going to ignore for simplicity. (Obviously I know that fundamental particles don’t literally touch each other, I’m not even sure what that sentence would mean for a particle with no extent anyway). But the nature of the interaction doesn’t really affect the fundamental question I asked, only its phrasing.

Rather than “which particle does it impact first” we ask essentially the same question, “which particle in the set contributes the most to the net field effect on our particle at point 3?” The answer you seem to gave is, “there isn’t one.” Or perhaps another way to describe that response is “there is always another particle whose influence is greater.” That is problematic given that the set of particles you are describing are precisely ordered and that there is a defined, static field effect on the particle on point 3.


The second issue would seem to be to return to the scenario with the added detail and ask, how does the invoking of a field affect the placing of particles infinitely close together? A classical field to be invoked here is gravity, which, of course results in a singularity and the problems we are currently looking to resolve and to which you allude earlier. So I don’t really see how invoking that complexity makes your system more conforming to our universe rather than less.

How would you go about showing that the field effects don’t become nonsensical at the distances you are invoking here?

[CliveStaples]

That there is a bijection and that the sets are equinumerous is the objection. That a set can undergo addition to, but whose size (in the cardinality sense) is unchanged is the objection being raised given that the definition of cardinality is, essentially, the number of elements in a set.

To write it out essentially we are saying that we can add to the number of objects in a set and leave the number of objects in the set unchanged.

Now so far, the response has been to say, “I don’t see why that would be a problem, we just need a definition of infinite sets that allows for it,” which seems a bit like defining away the objection. We know that X+Y=X in the scenario described above is internally incoherent for finite sets, so to then categorically change that view for infinite sets would seem to beg for a bit more of an explanation as to why infinite sets, by their nature, are capable of displaying a property that makes incoherence, coherent.

I'm not sure I understand what you're asking for. There's two issues that might be the problem:

(1) Why does the definition of equinumerosity rely on bijections?

(2) Why do bijections for infinite sets work differently than for finite sets vis-a-vis proper subsets?

To (1):

Bijections capture the notion of a 1:1 correspondence between two sets. Another characterization is that injections like f:A→B are B-relabelings (or B-embeddings) of A in which each element a of A is assigned just one label f(a) from B, and no two different elements in A are assigned the same label. If f is a bijection, then it is both a B-relabeling (since it's an injection) and the resulting collection of labels is equal to B. So that just by renaming elements of A after elements in B, we end up with all the same elements in B. Another way of saying this is that A, viewed B-wise, is B. This B-wise equivalence to B is the central concept of equinumerosity.


But let's think about equinumerosity on its own, for a moment. Let's list out a few properties that we think the equinumerosity relation should have. For the sake of this discussion, let's say that |A| = |B| if and only if A and B are equinumerous (without requiring the existence of a bijection or the like). I'll formulate the properties in English first, then I'll rewrite them symbolically for easier reference.

(1) In deciding whether two sets contain the same number of elements, we aren't concerned with the names of the elements, so that the number of elements in a set X remains the same under a relabeling. If the labels are unique (no two elements are given the same label), then the number of elements in X is identical to the number of labels used. Let's call this property label invariance. I imagine this will be the most controversial equinumerosity property that I propose.

(2) Every set is equinumerous with itself. Simple enough.

(3) If two sets are equal, then they are also equinumerous. Set equality is defined in terms of mutual containments, so two sets are equal just when they contain precisely the same elements, i.e. A = B precisely when x ∈ A ⇔ x ∈ B. If two sets have precisely the same elements, then they should also be equinumerous with one another.

(4) If X is equinumerous with Y, then Y is equinumerous with X. This is a sort of general equality principle, and equinumerosity seems like a kind of equality relation.

(5) If X is equinumerous with Y, and Y is equinumerous with Z, then X is equinumerous with Z. This is another general equality principle, and (again) equinumerosity seems like an equality relation.

Expressed in set theory (with the caveat that |.| refers to equinumerosity rather than the standard set-theoretic definition of cardinality), these properties are given by:

(1a) For all sets A,B: ∃ f:A→B ⇒ |A| = |{(a,f(a)) | a ∈ A}|
(1b) For all sets A,B: ∃ an injection f:A→B ⇒ |A| = |f(A)|
(2) For all sets X: |X| = |X|
(3) For all sets X,Y: X = Y ⇒ |X| = |Y|
(4) For all sets X,Y: |X| = |Y| ⇒ |Y| = |X|
(5) For all sets X,Y,Z: |X| = |Y| and |Y| = |Z| ⇒ |X| = |Z|

Consider the following argument:

(i) f:A -> B is a 1:1 correspondence. [Premise]
(ii) |A| = |f(A)| [(i),(1b)]
(iii) f(A) = B [(i)]
(iv) |f(A)| = |B| [(iii),(3)]
(v) |A| = |B| [(ii),(iv),(5)]

If we take (1) - (5) as premises along with (i), this argument is valid. To dispute its conclusion, you must either show that it is invalid or that one of its premises is false.

To (2):

If you take an infinite set like N = {0,1,2,3,...} (if you don't like this way of writing down sets, you could also define this as the collection of all constructible non-negative integers) and remove, say, the element 0, you're left with N - {0} = {1,2,3,...}.

There's an obvious 1:1 correspondence between these sets, given by f(n) = n-1 for all n in N-{0}. We know this is a map because of how it's define (it takes as inputs elements of N-{0}, and each of its outputs lie in the set N, and no input is assigned two different outputs). We know this map is a bijection / 1:1 correspondence because we can show that it is both injective (f(n) = f(m) implies n=m) and surjective (for all y in N there exists an x in N-{0} such that f(x) = y).

But say we tried this for a finite section of N, say N_k = {0,1,2,...,k} (if you don't like this way of writing down sets, you could also define this set as the subset of all elements in N less than or equal to k). We remove an element, say, 0, and are left with N_k - {0} = {1,2,...,k}.

If we try to use the same map as before, f(n) = n-1, we still have an injection (f(n) = f(m) implies n=m) but we're no longer surjective, since there's nothing that maps to k (Suppose there were, then there is an x in Nk - {0} such that f(x) = k. Then x-1 = k so x = k+1. But k+1 > k, which contradicts that every element in Nk, and therefore in all of its subsets (including Nk-{0}), is less than or equal to k).

The defect in this particular injection is characteristic of the central issue with trying to find surjections from finite sets to their proper supersets. Finite sets "run out of elements" more easily than infinite sets do, which affects the surjectivity of maps with finite domain sets.

---------- Post added at 02:28 PM ---------- Previous post was at 02:26 PM ----------

What does it mean to say that there is no particle described by that property?

That, rather than there being some particle that is described by that property, none are. Cf. there is no person described by the property "is currently King of the United States".

For example. Let’s say that there was a particle at point 3 moving towards our set of particles. What does your above conclusion mean in that context? That there is no first particle to impact in the set? So what does it impact when it reaches that region of space?

I'm not the physicist, so I'll just direct you to his response.

[GoldPhoenix]

An interesting rebuttal given that it wasn’t my point at all. You’ll note that I never compared two points along a continuum. Rather, I noted that there is no clear boundary in your continuum (which I clearly was aware of, hence why I raised it as a rebuttal), which causes some very obvious physical problems (see my last response).

It's fascinating to see you try to wiggle out of openly specious claims. To wit, you said:

Ok, but given your metric of how mass is related, the mass of the ‘infinitieth’ particle is zero, which would seem to be an issue with your hypothetical. Now I think an objection you might offer would be that “infinity is not a number.” Perhaps, but it doesn’t matter here, we can ask what the mass is of the particle closest to the position “2,” and given the relationship you’ve described, it would be zero since there are in your hypothetical an infinite number of particles.

Now maybe you would argue that you either weren't aware that if such a particle existed, it would have to exist at a specific point on the continuum and thus you couldn't know that you were comparing two points (I don't think you want to argue that you misunderstood this), or perhaps you'd like to argue that you simply were ignorant of the fact that there doesn't exist a natural number, N, such that ([1 - (1/2)^N] == 1 (I don't think you want to admit this either). In fact, I think you might just want to accept the criticism and move on, Squatch, because there's no version of this scenario where you didn't bungle 200-level mathematics.

You are free to take whatever tack you like, but I think it is pretty obvious that the points laid out as objections aren’t dismissed so easily. After all, if they were so easily dismissed, we both know you would have done so by now.

They have been easily dismissed, Squatch. Other than the instances where you naysay me and then ask me fill in the details for you, the only valid line of argumentation you've made was rebutted. But you keep on making the same point that was addressed fully two pages ago, but your response to my line of argumentation has been literal silence. At this point, your argumentation is verging on being little more than:

"NUH UH! CAN'T HEAR YOU! LA LA LA!"


Here's my quote, referenced to you for the eighth time, regarding the nature of infinity in modern physics that you keep on ineptly referencing:

In Quantum Field Theory, you are forced into accepting the existence of Fock space, which is the space that contains all possible particle configurations (The Hilbert space of all possible particle states; a discussion of this can be found in the first few chapters of any book on QFT, so take your pick, but section 2.4 of Tong's intro QFT book is free and discusses this). The dimension of the Fock space is necessarily infinite (In this case, not being infinite would lead to an actual violation of physical law), and this introduces several very simple "actual infinities." [GP: The Fock space is the (graded) space of states. The first space is the space is single particle states with given momentum and helicity; the second space of two particles states, the third the space of three particle states, and so on. Each is a Hilbert space, and the Fock space is the infinite collection of all of them. In order to work with known laws of physics, you must crucially have all possible collections, including an infinite particle state.]

The first is that I have an actual infinity of different possible particle states. Two examples where this is used, for instance, is in the description of coherent states, for instance in a pulse of light, and the another example is Unruh radiation. Unruh radiation necessarily contains an infinite number of modes in the radiation (or else it violates physical law, see equation 3, which implicitly sums j from minus to plus infinity). Note that even though Unruh radiation hasn't been directly observed a lab, it is a necessary consequence of the current physical laws. So this means if you are accelerating, you see a universe which has a bath containing an infinite number of particles (But all physically finite quantities, like energy, momentum, and temperature), but if you're standing still (or in an inertial frame), you see a universe with zero particles. This highlights the point that trying to demand that infinity is unphysical just doesn't make any sense from a QFT perspective nor does it comport with any known facts of the matter. These theories may break down and render the actual number of particles finite, but as of now, the evidence supporting the assumptions of QFT lead us to these conclusions; and even if so, the only thing that could even conceivably do this is quantum gravity (But quantum gravity also could conceivably lead to past infinities, so this kind of appeal is a double-edged sword for WLC).

Back to the point of Fock space: This means if I want to start my universe out with an infinite number of point particles, there actually cannot be any principle that forbids me from doing so, because that principle would then by direct implications break the previous physical principles (In this case, Lorentz invariance and unitarity). Thus the laws of QFT + WLC's new "No infinities" principle entails a contradiction. [GP: Thus it is "Necessarily false" for the current known laws of physics.]

---

Given that it is your OP, and I think I’ve been relatively patient on you defending your premises so far, Please support or retract that point particles are fully consistent with our understanding of the physical universe.

A clever attempt to engage in shifting the burden of proof, but I'm not biting. I am not saying it is necessarily true that point particles are fully consistent with our understanding of the universe. What I am saying is that WLC's argument only passes muster if it can be applied very generally. Additionally, it is not inconceivable (but very implausible) that our universe is described by point particles (as in, to my knowledge it doesn't imply a literal contradiction).

---

Since you ask, however, I'll edify you. As for QFT coming from statistical mechanics, cf. certain QFT's are arbitrarily well-approximated by Ising models (of vast numbers of particles) or read this. Alternatively, there are attempts to work with causet theory (cutting spacetime up into infinitesimal points) or to try to push the de Broglie-Bohm theory of point-particles into a QFT regime (N.B. Here they keep the point-particle interpretation). As stated, neither of these are promising (very, very far from it), but they are not demonstrably wrong because they haven't been completed as programs and thus haven't made predictions yet (and will probably remain that way).

You should be familiar with some of these very non-standards efforts, Squatch. After all, you defended at least two of them (starting with relativistic attempts at de Broglie-Bohm[1] and then additionally defended causet theory (which involves cutting spacetime up into an infinitesimal points, so the degrees of freedom are necessarily point-like) a year ago as plausible and well-founded programs, and you did it for an entire 8 months. (You'll notice that I never said they were wrong, only that they hadn't bore any fruit and that as such they should be taken seriously. But that isn't a proof that they are wrong, as I also said in that thread.)

[1] Where all attempts at this are assuming point-like particles ("Thus, to define particle trajectories, one needs an additional rule that defines which space-time points should be considered instantaneous."), which you ardently defended a year ago.

---

And, please support or retract that point particles can be placed infinitely close together and at precise locations with a known velocity.

I might start with the OP. I would also suggest doing some of the longhand work.

First, let me offer a side question. In this scenario, the rods can only move forward once, correct? (After the first movement they would seem to occupy a space smaller than their extent. Not that it is a fatal problem at all, since you are only invoking one movement in the OP, I just want to be sure I accurately understand what you are saying here).

Ok, I think we still run into the same problems at some point. Given how you’ve defined both mass and extent, the mass and extent of the “infinitieth” particle is zero. Again we are faced with the distinction between taking a limit that approaches infinity, and an actual infinite set. In this set there are an actual infinite number of particles that exist, which means that some particle has no extent and no mass.

Given that I’m not really sure that giving the particles a small extent escapes the objection as you seem to think.

Squatch, it's already been explained to you both simply and in full technical glory why this isn't the case and that there is no "infinitieth" particle. If you don't understand that there isn't an "infinitieth" particle, then that's fine. But you should be aware that if you continue to assert that there's an "infinitieth" particle, you should know I'm just going to repeat that this argument is a strawman because there does not exist an "infinitieth" particle with zero mass in my scenario.

I don’t think I said you weren’t supposed to think about them that way, my point was that you weren’t supposed to think about them only as point-like particles. A point particle is only one aspect of the particle’s nature in a Bohmian interpretation.

Fine, so there is a point-like particle contained in Bohmian mechanics?

I didn’t ask if the particles and their motion could be described by de Broglie-Bohm, obviously they can, that would be a trivial question as you put it. Rather, what I asked was can you show that a de Broglie-Bohm interpretation of your OP allows for point particles to be placed infinitely close together?

Yes, Squatch, the fact that is a solution of the Bohm equations with finite physical quantities is sufficient to say that it is an allowed possibility to the de Broglie-Bohm theory. That's how all of physics work. If there's a solution that doesn't break physical principles, then the physical scenario is not merely allowed by the theory, but that scenario (given the initial conditions of the system) is the prediction made de Broglie-Bohm theory for how that system must be obeyed. That is the de Broglie-Bohm "interpretation" of my OP.

Again, perhaps this is due to the limitations of my knowledge on this subject, but I was under the impression that a particle must occupy a space at least as large as its wavelength (http://www.math.ucr.edu/home/baez/lengths.html) . And while wavelength is decreased by additional energy, in order for it to be infinitely small, we would need infinite amounts of energy (obviously a problem, http://arxiv.org/pdf/gr-qc/9403008v2.pdf).

Yes, you're correct that it is due to the limitations of your knowledge. All of those calculations lie in the regime of quantum mechanics proper. But the wavelength of a particle being the literal the size (uncertainty) of a particle is what de Broglie-Bohm theory is explicitly avoiding in order to remain deterministic. It preserves the uncertainty principle as a statistical form of uncertainty, but it keeps each of the particles deterministic and thus breaks the wavelength-momentum relationship imposed by Fourier analysis. Additionally, the particles are allowed to remain point-like in the description, which is impossible for true quantum particles.

I think there are two issues with this response. The first is that it seems to miss the main thrust of the objection by invoking a detail set that you had previously said you were going to ignore for simplicity.

Sure, and I've said that I do not believe (outside of a conformal field theory) that you can stack interacting particles into an infinitely

Rather than “which particle does it impact first” we ask essentially the same question, “which particle in the set contributes the most to the net field effect on our particle at point 3?”

To keep finite energy, you need the charge to decrease rather rapidly, thus it's unclear that this ethereal "last particle" you insist exists would actually contribute the highest force.

You are, of course, free to do some math yourself to demonstrate that it is true that it would be your hypothetical "last" particle.

The answer you seem to gave is, “there isn’t one.” Or perhaps another way to describe that response is “there is always another particle whose influence is greater.” That is problematic given that the set of particles you are describing are precisely ordered and that there is a defined, static field effect on the particle on point 3.

I haven't answered the question. Right now, I'm saying "I don't know." I'm tired of being asked to full in the details of your vacuous speculations, so if you want to prove that there is a problem, then start do some math and show me that there's a problem. If you think the "last" particle contributes the strongest part of the force, them please demonstrate this.

The second issue would seem to be to return to the scenario with the added detail and ask, how does the invoking of a field affect the placing of particles infinitely close together? A classical field to be invoked here is gravity, which, of course results in a singularity and the problems we are currently looking to resolve and to which you allude earlier. So I don’t really see how invoking that complexity makes your system more conforming to our universe rather than less.

Agreed, there'd be pathological results if we turned on gravity and didn't give the particles an escape velocity (so they never touched each other, resulting in an infinite force) or if you gave them some angular momentum. Otherwise, they'd wouldn't collapse onto each other.

However, in the real world there's a compensating electrostatic force that repels particles, which tend to balance each other (hence why you're not flying up into the air due to electric forces or flying down to the earth due to gravity). So you'd probably want to include a repulsive force, as well, so that might be avoided. Still, it's not obvious to me that you can't use a delicate balancing to charges and masses to compensate each other to yield finite energy, momentum, forces, and so on, to prevent the system from pathologies.

Since you're making the claim that they do and must exist (including in this scenario), I await your mathematical proof that such a pathology exists in this scenario.

How would you go about showing that the field effects don’t become nonsensical at the distances you are invoking here?

I wouldn't (I'm just claiming that the energy and momentum is finite). But you're the one claiming that this system is necessarily impossible/there's something necessarily pathological about this scenario. So would you mind telling me what the pathology is that makes it physically impossible? Again, you'll need to churn through the equations for me, so you can show me what has mathematically gone wrong.

[MindTrap028]

Sorry, once again a kind of drive by post.. as far as I am concerned I have 2 very long posts to respond to, and this is sort of a side road we have gone down.
Thanks again for your patience. Only addressing one point here.

Because to have done so would require giving a precise principle, P, that is at least as mathematically explicit as what I gave above, and then a concrete proof (meaning, going through symbolic manipulations) that my physical scenario, X, entails ~P. Additionally, you would also need to justify why P is something I should actually believe is true.

Without that, you don't have anything. To repeat what I said before, a pathological, counter-intuitive result is not a proof by contradiction (i.e. that the result is false). A contradiction means that you asserted P and ~P, and nothing else. That's the only way that you can determine that the scenario is impossible.

I don't think such a detailed understanding is what you were calling for before. You have provided the principle, namely the method of showing a disproof of mathematical models.

2.) An actual logical contradiction in the equations. (e.g. physical quantity X can be calculated to be X = 1, and also we calculate it to be X = 2).

given your Op
Infinity - infinty = X

It is true that X = 1, and X = 2
(In fact, X can equal any amount ).


So, either the above is not true, or you were incorrect to say that showing X= 1 and X = 2 is a disproof.
of course maybe I have misunderstood the nature of this disproof of mathematical models, and the above doesn't follow the format you were pointing out.
Please, take your pick or explain where I am mistaken(as you guys have been so patient and generous in doing so far).



*Final note*
The general reason for my taking down this side track, is that If in the end you wish for an answer to be given in mathematical terms, then I know I will not be able to do so ,because of my personally lacking proficiency with that kind of expression. I will be limiting(and limited) my answers to a more philosophical expression of it. So if that form is inherently not sufficient(even if it were correct), then I would respectfully have to withdraw from the thread. To the extend that you are willing to bare with that limitation.. I will gladly continue on. I have enjoyed the discussion, but I don't want to waste your time if I have no intention of fulfilling your basic request.

So, the above question (which is mathematical in nature) aside.. just let me know which one it is. Your patient dealings with philosophical answers, or your cold hearted mathematical mind crushing all the hopes and dreams of a poor, white, father of five, dreams of one day addressing the deeper meanings of the infinite world. *J*
No pressure.. totally your choice