Indeed, the history of 20th century physics was in large measure about how to avoid the infinities that crop up in particle theory and cosmology. The idea of point particles is convenient but leads to profound, puzzling troubles.

To us large creatures, space-time is like the sea seen from an ocean liner, smooth and serene. Up close, though, on tiny scales, it's waves and bubbles. At extremely fine scales, pockets and bubbles of space-time can form at random, sputtering into being, then dissolving.

Mathematics cannot handle physical quantities like density that literally go to infinity.

Gregory Benford Quotes