Sorry, Bizerl, I was not meaning to get over-technical and leave you behind. There are some very deep philosophical issues involved, and I guess my own interest was in putting some of them before the theoretical physicists who participate in this forum to see how they would react to some of the ideas in my own mind.

You are quite right. The Greek philosophers and mathematicians had a significant problem with the concept of infinity, and lacked the mathematical tools to deal with it. That is another aspect of Zeno's paradoxes. With the mathematical approaches we have been taught, especially the inclusion of differential and integral calculus, we have effective and consistent ways of dealing with the infinite and infinitesimal. That does not address philosophical questions of whether or not we should consider the infinite and the infinitesimal as existing in the real world, but as far as science is concerned it does provide us with reliable terms for observation and description of the real world in a physical model.

From here on, I am getting to a personal, and perhaps slightly tongue-in-cheek viewpoint. There are huge differences between the ways that chemists and physicists look at things. There are also huge differences between the work that theoretical physicists actually do, and the way that they present their findings to a lay audience. When they are actually working, researching, and discovering things, they are manipulating high-powered and completely inaccessible mathematical expressions as they develop and explore mathematical models in an attempt to describe the physical phenomena that they are trying to address. When they try to report their finding to a lay audience, they often use analogies and make philosophical pronouncements that their work really does not warrant, but they find themselves unable to communicate in lay-accessible terms in any other way.

Now physicists like aesthetic simplicity in their models, and often this search for aesthetic simplicity is very productive of new results. For example, chemists recognize and work with three types of sub-atomic particle -- protons, neutrons, and electrons. They are very happy with that picture because it represents a simplification of their previous conception of 80 or 90 individual elements with puzzling patterns and differences in their properties -- a chaotic picture indeed! But theoretical physicists were intrigued by the fact that protons and neutrons had nearly the same mass, but different electrical charge, and that isolated protons were stable whereas isolated neutrons underwent radioactive decay. So they worked hard to produce a mathematical model where protons and neutrons were symmetrically equivalent -- where a proton could emerge as a neutron under some sort of symmetry operation. When this model was developed, it suggested that there should be a number of other sub-atomic particles, some of which were known from previous experiments, and some of which were discovered when they were specifically looked for. These other particles had approximately the right masses and lifetimes that the mathematics had suggested. The theory was very fruitful.

From bizerl:

Also, in terms of real stuff, I was led to believe that when you get down to fundamental particles, they exist in a kind of probability field. Wouldn't this resolve the granular universe by saying that each particle could exist anywhere, in infinite detail?

You certainly have reproduced pretty much what a physicist would probably tell you in the first sentence. The problem is in the last three words of your post. The Copenhagen interpretation of quantum mechanics, which seems to be the only viable one although many scientists are trying to come up with alternatives, says that not only is there an unavoidable minimum uncertainty in any measurement you make on a particle (well, most measurements), but that this uncertainty is a property of the particle, and not just a failure of the measurement method. The problem is that if you try to look at the detail of a particle -- for example to examine its shape or its parts -- then the uncertainty in the measurement of each part is greater than the uncertainty in the measurement of the whole: that is that the parts of a particle are blurred or defocussed over a larger amount of space than the particle itself occupies. That is certainly the sort of thing that the mathematics says; I am not at all sure of my translation of it back into the real world! But those last three words are the essence of the problem.