J.J. Thomson's discovery of the electron disproved the idea that atoms are indivisible and fundamental particles.

Here's why:

* Dalton's Atomic Theory: Prior to Thomson's work, John Dalton proposed that atoms were the smallest unit of matter and could not be broken down further.

* Thomson's Experiment: Thomson's cathode ray tube experiments showed that atoms contain negatively charged particles (electrons). This proved that atoms were not indivisible, but rather had internal structure.

Therefore, the statement about the atom that J.J. Thomson proved false is "Atoms are indivisible and fundamental particles."