Almost everyone commenting here is worried by zero and infinity, except Tom and Peter who walk where angels fear to tread (see Tom's reply on Oct. 15, 2014 @ 01:50 GMT and Peter's infinite brackets within brackets on 'Alternative models thread'). Eckard is worried and has asked Tom thrice to name examples but I can't see any physically real example. John M has also commented. Jonathan has raised probably the most relevant issues [Jonathan J. Dickau replied on Oct. 14, 2014 @ 16:04 GMT and Jonathan J. Dickau replied on Oct. 14, 2014 @ 15:21 GMT], "If Relativity is correct, the notion of point masses can't be...". I generalize the Relativity here to all (Galilean, Special and General).
The following descriptions and discussions can be found on what Jonathan raised:
"A point particle (ideal particle[1] or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension: being zero-dimensional, it does not take up space".
From Fermilab: "Point particles are much more bizarre and are sometimes said to have zero size. This statement has raised more than one eyebrow. How can something have no size at all? And if it has mass, does the zero size mean it has infinite density? (And by the way, as you read on, you'll see the answer to that last one is no.) You begin to see why some people are skeptical when a scientist says a particle is point-like.....
In summary, extended particles have a fixed size, although they may have a fuzzy edge; point-like particles are mathematical abstractions with zero size. But even zero-size particles have an extended effect, due to the effect of the field surrounding them"
Then, in response here to: Why isn't the electron considered a black hole? It does have mass and its radius is infinitely small, isn't it?
"In actuality, we don't know how big the electron is. All of our measurements point to the electron having no size, but we haven't measured down far enough. The electron, if it were a black hole, would have to be smaller than 1x10-57 meters, quite a bit smaller than we've ever measured! ...
The electron (and all truly 'fundamental' particles) are considered to be true mathematical points in the sense that they have no classical spatial extent. This is known, for example in the case of the electron, by performing scattering experiments: the way particles scatter off one another is quite different if the target is represented as a point as opposed to having some finite size. All of the electron scattering experiments done so far are consistent with the hypothesis that the electron is truly a 'point particle.'
For an electron we would obtain 1.35x10-51m. If the electron were a point particle, it seems it would be within even this fantastically small radius and would indeed be a black hole"!
What inferences can we draw?
-If an electron despite having mass, does not form a black hole, then there must be an end to gravitational contraction. That is a limit to how small a radius can be.
-If all electrons are the same, any lower limit must then be non-random but a prescription of Nature, that is "No matter what mathematicians say, thus far, but no more!!". This is what appears to prevent the electron from reaching 1.35x10-51m. Trying to tease Peter, you can 'bracket' thus far, but no more.
- In view of the foregoing should the concept of point particle not be abolished from our physics? Recall this is also partly what necessitates the use of renormalisation in quantum theory as Tom himself noted.
- Have Aristotle and Proclus not been proved right about what a geometric point can and cannot be as I referenced in my 2013 essay.
- Can there be genuine unification in physics if point particle is still persistently accepted as existing in reality and of zero dimension?
- As I referenced in that essay, the original architects of zero dimension, Plato and latter followers like Leibniz abandoned this route before they passed on. Why their modern day followers continue on this route is an area worthy of study.
Best regards,
Akinbo