Is matter discrete? or continuous?

[Daniel]

I was just reading some Augustine, and he seems to prove that matter is continuous. Because every object which has dimension necessarily has at least a right part and a left part, and a top part and a bottom part, and so on. So no matter how small you go you can always continue to divide the thing into those parts, at least conceptually.

But I was under the impression that matter was discrete? You can only divide an object so many times before you end up with some sort of an indivisible piece? Yet I would think that this indivisible piece has dimension, right? So why is it that this piece is immune to further division? Doesn't even this smallest piece still have a top and bottom, left and right, etc. etc.?

I know next to nothing about modern physics, But I was just wondering whether there is any straightforward answer to this paradox, which the non-learned (such as myself) will be able to understand?

[GloriaPatri]

Fundamental particles, like electrons and photons, have no parts that they can be divided into. Furthermore, atoms are mostly empty space. So any extended object from the size of an atom up is not truly continuous. And the fundamental particles that make up those atoms are themselves not continuous. There's no such thing as half an electron or half of a photon.

The more interesting question is whether space-time is continuous or discrete. That is currently an unanswered question in physical cosmology.