If I wanted to disagree with Aristotle, I would do it on the following grounds.
In modern mathematics, we know how to attach a lot more data to individual points; e.g.
- In physics, at any instant, particles have momentum. Physical laws relate momentum to how the position changes
- Similarly, in differential geometry, each point has a cotangent space: data from the cotangent space tells us the rate at which a function is varying
- More generally, each point has a stalk: the germ of a function at a point is enough information to tell us everything about the function in an entire open neighborhood of the point. (although not how big the neighborhood is)
and we've well-studied how to assemble data at individual points into a continuous whole.
We can insist that at each instant - each "indivisible now" - we're not limited to knowing precisely where something is, but instead its entire germ of motion. And consequently, we are able to study motion as a continuum of "indivisible nows".
That said, I think this objection is more of a technicality than disagreeing with the spirit of Archimedes position. My intuition is that the ideas above (and more) are all fleshing out a "fuzzy"* notion of point that somehow has an extension despite** just being a point. And in some sense these fuzzy points still manage to be "divisible"; e.g. by passing from the stalk to the fiber. (i.e. forgetting the "germ" of motion and only remembering the position at that instant).
A more charitable interpretation is that this fuzzy notion of point still satisfies the spirit of Aristotle's position.
*: No relation to fuzzy logic
**: Some even manage it more transparently, such as how nonstandard analysis creates about a standard point an entire halo of "infinitesimally nearby" nonstandard points.
Also, to expand upon something I mentioned in another comment, I would object to the objection above anyways, on the grounds that it still needs more than "points and extra data" -- i.e. there is still additional information (such as a topology) that encodes how the points themselves relate to each other. This additional information is at least as important (and arguably more important) than the points themselves.