At 34:00, he describes mathematical truth.
At 35:00, he describes mathematical Platonism.
At 40:30, he says physical theories are approximations to truth.
I can accept all of that, but then at 40:50 he says:
"The one hypothesis that underlies every viable physical theory is that the Universe is a mathematical object."
Here is where I differ. I don't think that our Universe is a mathematical object, or that any of our theories assume that it is. Our physical theories are mathematical approximations to a non=mathematical object.
This puts me at the opposite extreme from Tegmark's hypothesis. While he says all mathematical objects are universes, I say that none of them are.
My skepticism is based primarily based on:
(1) All of the interesting mathematical statements are about infinities, but there are no infinities in the physical universe.
(2) We don’t really even have any candidate for the Universe as a mathematical object. There are those who talk about having a quantum wave function of the Universe, but they end up talking about many-worlds and other ideas that have never made any sense or had any predictive value.
(3) Attempts to realize the world as a mathematical object are what led Bell to believe in local hidden variables. We now know that local hidden variables are impossible, so maybe the assumptions that led to that belief are also wrong.
Penrose has given lots of interviews, and will probably give some more now that he is a big-shot Nobel prize winner. I don't remember him giving an opinion on this issue, but it is possible. He is as close to being an authority on this as anyone.