To be fair though, even Newton's equations have flaws from a pure mathematical physics standpoint [0].
The problem is that certain classes of initial conditions produce infinities. The obvious case is for particle collisions, where gravitational attraction grows without bound in finite time. However, it turns out that similar infinities can arise even if particles don't collide (cf. Xia 1995).
Infinities from colliding particles aren't seen as such a big problem, because they require impossibly precise initial conditions, i.e. the initial conditions sit on a measure-zero set in configuration space.
AFAIK, the jury is out on whether we can say the same about non-colliding singularities. It seems reasonable to susept that's the case, but if not then Newtoian gravity alone would be demonstrably broken even in the math.
The problem with points (which is also present in electrodynamics) is a wonderful example of when the hole in your present theory points the way to the next theory. We already know that Newtonian mechanics does not predict the behavior of the very small, so it is fitting that it doesn't pretend to give an answer. You could almost suggest that it points the way to the diffuseness of quantum mechanics.
It seems all of these "problems" arise because you're taking particles to have no volume (point masses).
This feels like a garbage in garbage out kind of deal. If your starting assumptions are physically impossible, it's not surprising to get strange results.
The blowup shown by Xia doesn't rely on point particles. It works just the same even for "small spheres". I didn't really make that clear.
Your thought is in good company though. Lots of physicists over the years have worked to remove point particles from the classical theories. This issue is that these fixes tend to end up fixing nothing (like with Newtonian gravity), or producing nonsense and contradictions elsewhere. A famous result in electragnetism, if we replace point charges with tiny charged spheres, we get solutions where particles accelerate exponentially without any external forces. Not at all obvious, but what the hell, right?
Anyway, to take a card from your deck and muse haphazardly, my guess is that these problems are extremely hard to avoid when dealing with a continuum. There is some interesting work on discrete theories, but I'm not all that informed about them.
>> It seems all of these "problems" arise because you're taking particles to have no volume (point masses).
No, these problems arise, as the parent had said - because those zero-point particles have a non zero probability of occupying the same coordinates if you're simulating their positions using e.g. floating-point variables.
In your example, floating point numbers are an abstraction for the underlying math which assumes space is continuous. If we're talking about events over a continuous sample space, the probability of two particles being in exactly the same space is identically zero since we can only assign finite probabilities to _intervals_.
The problem is that certain classes of initial conditions produce infinities. The obvious case is for particle collisions, where gravitational attraction grows without bound in finite time. However, it turns out that similar infinities can arise even if particles don't collide (cf. Xia 1995).
Infinities from colliding particles aren't seen as such a big problem, because they require impossibly precise initial conditions, i.e. the initial conditions sit on a measure-zero set in configuration space.
AFAIK, the jury is out on whether we can say the same about non-colliding singularities. It seems reasonable to susept that's the case, but if not then Newtoian gravity alone would be demonstrably broken even in the math.
[0]:https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture