His views of time and the cyclic universe are well documented and he himself has described them numerous times in interviews found on the web.

I have never heard him describe himself as a platonist and even he was it doesn’t mean he accepts as a religion which demands strict obedience to some written dogma. He obviously believes in abstract mathematics and obviously believes mathematics can accurately reflect and describe reality. Some people may call that Platonism. Having said that I get the feeling platonists want to claim he is because it will afford their theory some prestige to be accepted by a famous scientist.

Penrose is first and foremost a scientist. He has a bedrock belief in empiricism and the importance of strict observation conducted using the scientific method.