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I've already asked this question on Physics.SE, but it got no response; its not a conventional physics question, but really on how to interpret physical equations and physics.

Newtons law of gravity for two particles is proportional to the product of their masses divided by the square of their displacement.

Supposing that the particles are point particles then gravitional attraction will bring them closer together, and in fact infinitesimally closer together. Now in Newtons time there was no theory, as far as I am aware of inter-atomic forces that would have kept these two particles apart, so the gravitional attraction is asymptotically infinite. This is nonsensical, and either one can say that point particles cannot arbitrarily approach one another, or that particles can never be point particles and must have extension - this in fact includes the previous solution, as the notional point positions of the centre of mass of a particles with extension cannot obviously approach one another.

In Classical Mechanics, in retrospect this could have counted as evidence of either particles cannot be point masses; or of some then unknown repulsive force that acts at very small distances - are there any other alternatives? Why would physicists ignore then such a simple observation? What does this tell us about the process of theory-formation in Physics?

2Classical point particle mechanics and Newtonian gravity do admit one (and only one) possible way to explain the stability of matter without resorting to additional inter-atomic forces: conservation of momentum and mechanical energy. As the gravitational potential energy becomes asymptotically infinite, so do the relative velocities. Since point particles have zero cross sectional area, the probability of collision is zero, and so particles that fall towards one another due to gravity get closer and closer until their distance equals zero, and then proceed to fall

awayfrom each other. – David H – 2014-06-23T16:41:50.347"... and in fact infinitesimally closer together ..." Care to elaborate? If two masses are a distance d > 0 apart, at what time will they be "infinitesimally" close together, and what does that even mean? You know better, right? – user4894 – 2014-06-23T17:45:56.687

Not sure I understand the question or that it has an answer. It could just mean that having d=0 was somehow impossible some other way. – James Kingsbery – 2014-06-23T18:02:06.480

@DavidH: interesting perspective - I hadn't considered that the probability of collision would be zero! – Mozibur Ullah – 2014-06-23T18:13:58.523

@user4894: its just short-hand for talking about a limit; and it is exactly that this question is probing. – Mozibur Ullah – 2014-06-23T18:15:27.777

@Kingsbery:sure, thats a possibility; which is why I suggested this meant particles would have extension; – Mozibur Ullah – 2014-06-23T18:16:42.727

1Before even using the phrase "the centre of mass of a particles" (and talking about mass in general) you need to be precise about what you mean by "mass". This is such a difficult question (though to laymen and physicists it doesn't seem like a question at all) that there's even a book devoted to it:

Concepts of Mass in Classical and Modern Physicsby Max Jammer. I highly reccomend reading it and then coming back :) – user132181 – 2014-06-23T18:20:22.417Also, this is a very cool question :) – user132181 – 2014-06-23T18:26:02.263

@user132181: Thanks for the recommendation - it looks very interesting. – Mozibur Ullah – 2014-06-23T18:27:38.160

It's very ironical that I would also recommend reading

Concepts of Forceby the same author. Force is also a very troublesome concept in physics. Having read these books you will understand that the words "force" and "inertial mass" don't have any real meaning. – user132181 – 2014-06-23T18:30:39.127@user4894: I'm talking physics and not mathematics; in which case the issues that you mention, though important, are at least in the perspective that I'm taking are irrelevant; early modern physicists were using calculus without caring about the notion of a limit; but the focus here is on theory formation, ideas of infinite divisibility and so on. – Mozibur Ullah – 2014-06-23T18:32:59.710

@user132181: sure, but thats 'real' in the sense that physicists use it; and not as how we experience it; and that division is important: The force of a cannon-ball is pretty real, as is the mass & the weight of it. – Mozibur Ullah – 2014-06-23T18:36:19.143

@MoziburUllah It's true that physicists generally think of dx and dy as "infinitesimals" even as the mathematicians next door roll their eyes. But I don't think a physicist would ever refer formally to two distinct objects being an infinitesimal distance apart. Because the word can't be made precise. A nonnegative number is either zero or greater than zero. Those are the only two choices, in both (standard) mathematics and physics. But you know this! So I am asking you to clarify your use of the word. – user4894 – 2014-06-23T18:39:16.080

@user4894 it's not true that "infinitesimal distance apart" can't be made precise - take a look at the nonstandard analysis. Also, thinking about dx and dy as infinitesimals is the right way to think about them. The mathematicians who roll their eyes because of this are too naive and don't know that the way calculus is usually taught is full of problems and it is nonstandard analysis who saves the day :) – user132181 – 2014-06-23T18:44:17.067

@MoziburUllah ps -- If you say "arbitrary" I'll be happy. Do you mean arbitrary? – user4894 – 2014-06-23T18:44:58.500

@user4894:perhaps not; but in Robinsons non-standard analysis they quite casually talk about numbers being at an infinitesimal distance away. There is also a geometrical notion of a rigid line that has infinitesimal length; I don't mean arbitrary - what does that mean :)? – Mozibur Ullah – 2014-06-23T18:45:50.513

@user132181 I did write (standard) so as to

preemptsome clever person saying "Oh yeah what about nonstandard analysis." However, since you did it anyway, I'm aware of NSA. Let's not discuss this here, it's off-topic. In modern math we use the phrase "arbitrarily small" instead of "infinitesimal." And dx and dy are differential forms. It makes a huge difference in our ability to give a logical foundation to the enterprise. My context is standard mathematics and physics. Surely that is the OP's context as well. Peace. – user4894 – 2014-06-23T18:47:36.030@user4894 you cunning bastard... how could you. – user132181 – 2014-06-23T18:52:46.907

@MoziburUllah If the context of your original question is NSA, you should say that up front. "We are working in Robinson's NSA. Consider two point masses an infinitesimal distance from each other." I would have made no objection. Methinks you are suggesting this context after the fact :-) By the way if there are physicists who use NSA in their work ... name one!! And ps -- I never heard of an infinitesimal rigid line. But I don't know much geometry. – user4894 – 2014-06-23T18:53:20.890

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Pleaseconsider taking these sorts of discussions to chat!! I will probably be coming through and cleaning these up... :( – Joseph Weissman – 2014-06-23T22:34:41.930