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).

Originally Posted by

**Squatch347**
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.

Originally Posted by

**Squatch**
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.

Originally Posted by

**Squatch**
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.

**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?**

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.)

Originally Posted by

**Squatch**
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.

Originally Posted by

**Squatch**
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.)

Originally Posted by

**Squatch**
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.

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.

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.

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