Contrary to what some people, especially those who reject philosophy for their pop-scientism, think, you can discuss and test ideas without rejecting or accepting them and while doubting or rejecting the idea that you can know the ultimate status of the idea.
With that last point in mind, I think the argument Penrose made against the idea that mathematics is a human invention and a product of our minds because mathematics would have been valid during the ages of the dinosaurs isn't a very strong argument. The physical nature of the world and universe would have been what it was then too, before any human being was around to articulate or express a perception of it in words but it was still the same universe. However, I don't think Roger Penrose would entertain the idea that words and language are not human inventions and the area of a separate realm of reality which stands in mysterious relation to the physics he has studied and explained in numbers and equations. Since the numbers and equations couldn't explain a single thing about the physical universe without the words that give the equations and numbers any explanatory power at all, and without which it is doubtful that human beings could know any mathematics at all, why doesn't that produce a fourth realm of reality, the kinds of articulations through which the human conceptions of both mathematics and physical reality are known to us, exist?
I doubt that we can ever tease out the basic nature of any of this without reference to our own minds because our minds are always and in the most subtle and profound and determinative way an intrinsic part of our understand of everything. Even if you reject an idealist conception of reality, that mind is the primary causative aspect of reality, for all human purposes of articulating anything about the universe, you can't possibly NOT admit that everything we know of the universe which we talk about or can measure, all of the relationships, both verbally and numerically expressible by us, are created as intellectual "things" by our minds. A more profound question than the status of mathematics while dinosaurs but not people were around is to ask is how an infant without language, without numbers, gains an ability to, as a human being, address the world in an effective way. Penrose sort of dealt with that in terms of his mathematical realm, which he admitted had no causal power in physical reality, could be knowable to a young child. I think that if mathematics is a human construct, the product of human minds and mental capacity, then what he finds so mysterious is far more plausible.
I have no way of knowing which ideas are right in this or if any of them are. If an idealist conception of this makes things seem more "elegant" or an explanation of numbers, mathematical relationships or anything else as a product of human minds is more "parsimonious" doesn't necessarily make it more correct than Penroses "trialist" model. "Elegance" and "parsimony" (as in what has been reduced to a pretty empty slogan "Occam's razor") might be more pleasing to human understanding but that isn't necessarily getting us closer to what we also imagine being "objective reality". Even that last categorization is a product of our mind and might be accurate or it might be as much of a delusion as those things the atheist-materialist loves to classify as illegitimate "subjective illusion".
There is one thing that I think is quite obvious, that anyone who attempts to limit reality to what is contained in materialist, "physicalist" or "naturalist" dogma is willfully ignoring that even their own ideology has to extend past what they claim is contained in it to be an articulable or to gull people into believing in it. The current materialism of popular atheism, a facile, superficial ideology that extends up through academic culture and is quite popular with some very famous and even, at times, accomplished scientists, is based on an ignorance of the complexity of these issues and a refusal to admit that they are basic to any claims of what is real or what exists and aren't left behind at any stage of the discussion and that you don't make any progress at all without accepting the role that our minds, including the ones articulating materialist-scientistic dogma, inevitably play in coloring and shaping and creating the ideas we form and manipulate in the argument. You can't pretend that because you come up with numbers and equations and claims about the nature of subatomic particles and the entire physical universe that you've surpassed the fact that you've done all of that with your mind and based entirely on your own, particular and not objective perceptions, measurements and analyses. Well, you can get away with that among other superficial people but it doesn't impress anyone who refuses to ignore those things or compel assent or agreement because you claim to be doing it with "objective" scientific assertions.
Update: Someone sent me the right link to the part of the interview Robert Kuhn did with Roger Penrose that motivated this.
I would point out that sometimes Penrose seems to contradict himself, such as at c. 10:05 when he claims that mathematics is "controlling the universe". If mathematics has no casual relation to his proposed physical reality, how can he support the contention that it controls the universe? Though as he says a little while later, these matters are mysterious. His point that there are enormous parts of verified, confirmed, tested mathematical truth for which there seems to be no physical connection.
I do think in another part the interview, his discussion of the folly of imagining that our minds are a computable thing, that it could be expressed in mechanical computation might be the most important part of the discussion. I agree with a lot of what he says as part of his "c" alternative, the one he holds that pretty much rejects the computer model, the vulgar conception of Alan Turnings' claims but I suspect that his "d" that says science can't deal with consciousness because it surpasses of the realm of both science and math is very likely true. But, for my purpose, if Roger Penrose dismisses that popular notion and, worse, a widely held computer model of the human mind as one of the "brain only" alternatives popular today, then I don't think there is anything wrong with us rejecting it. I think his observation about the limits of math and science in relation to that, at least today, are politically important. Anything that discredits the claim of physical determinism of minds is vitally important for democracy, anything that places human beings outside of the realm of physical objects is as important for egalitarian legal treatment of people and in asserting metaphysical moral obligations which are essential for everything from egalitarian democracy down to the very basis of civil society, decent behavior.
There is a lot more to be said about that.