I've already mentioned that I think they are reasonable axioms to work with.
Ah, OK, missed that part.
No doubt. But they did not gain the necessary traction in the Islamic world.
Well, they did, but science took a backseat after a century of war with Western armies and then an invasion by the Mongols, the Muslim culture became fractured and began to decline. Combined with the subsequent rise of Western civilization and science, building on what they learned from the Muslims, Western civilazion began to carry the torch.
It wasn't so much that the ideas didn't gain traction, but rather than invasion and war eventually took it's toll.