I'm confused. Protons are "splittable" in the sense that they are not elementary particles, but are instead comprised of multiple quarks. The Large Hadron Collider [0] "splits" protons. In fact this is how they claim to have observed the Higgs boson. See the image caption:

   Simulated Large Hadron Collider CMS particle
   detector data depicting a Higgs boson produced
   by colliding protons decaying into hadron jets
   and electrons

[0] http://en.wikipedia.org/wiki/Large_Hadron_Collider

oops!

Protons are made of Quarks. They are potentially divisible.

[1]http://en.wikipedia.org/wiki/Quark

[2]http://hyperphysics.phy-astr.gsu.edu/hbase/particles/proton....

[3]http://physics.knoji.com/can-protons-or-neutrons-be-split/

This is actually more delicate than that. It is not possible to split a proton into its constituent quarks because quarks can never exist on their own. As far as we know they only appear in pairs or three at a time as in the proton.

There is also the issue of baryon number conservation. This is physical law (as far as we know) that says that the number of baryons must be constant. A proton is a baryon and neuron is a baryon but there are lots more. Anti-particles count negative so we can destroy a proton by combining it with an anti-proton, if we would happen to have one. But we cannot outright split the proton into some smaller set of parts as with an atom.

>It is not possible to split a proton into its constituent quarks because quarks can never exist on their own.

In the same way that the atom was not divisible 100 years ago?

The explanation I was given was that the binding energy of the quarks in a proton is greater than the energy it takes to create a new set of quarks. So before you can pump enough energy into a proton to split it, you've pumped enough energy in for new quarks to appear, and you end up with two particles but no naked quarks.

Back then scientists didn't have a model for what is inside an atom though, while we do have a model for what is inside a proton.

>Back then scientists didn't have a model for what is inside an atom though

Yes, they did:

http://en.wikipedia.org/wiki/Bohr_model

From 1913.

Not a complete model, as we now know, but nonetheless, scientists did have a model of the atom 100 years ago.

Gahh but that's not how quarks work. Paired up, always.

i'm not saying we're able to do it, but it seems doable:

https://en.wikipedia.org/wiki/Quark_Star