Our mathematical models of physical reality are far from complete, but they provide us with schemes that model reality with great precision - a precision enormously exceeding that of any description that is free of mathematics. There seems every reason to believe that these already remarkable schemes will be improved upon and that even more elegant and subtle pieces of mathematics will be found to mirror reality with even greater precision. Might mathematical entities inhabit their own world, the abstract Platonic world of mathematical forms? It is an idea that many mathematicians are comfortable with. In this scheme, the truths that mathematicians seek are, in a clear sense, already "there...." To a mathematical Platonist, it is not so absurd to seek an ultimate home for physical reality within Plato's world.What Penrose doesn't acknowledge is that some convergences are coincidences--and that mathematics can be entirely internally consistent, yet need not map onto any external reality, or are adaptable to entirely variable realities. (The gaping holes in the middle of math brought by Gödel go completely unmentioned.)
This is not acceptable to everyone. Many philosophers, and others, would argue that mathematics consists merely of idealised mental concepts, and, if the world of mathematics is to be regarded as arising ultimately from our minds, then we have reached a circularity: our minds arise from the functioning of our physical brains, and the very precise physical laws that underlie that functioning are grounded in the mathematics that requires our brains for its existence. My own position is to avoid this immediate paradox by allowing the Platonic mathematical world its own timeless and locationless existence, while allowing it to be accessible to us through mental activity. My viewpoint allows for three different kinds of reality: the physical, the mental and the Platonic-mathematical, with something (as yet) profoundly mysterious in the relations between the three.
In other words, math would be an epiphenomenon: predictable, and predictive, but not on its own level of existence--like a literary character in the pages of a novel.
As a bonus, the article offers Nick Bostrom's simulation argument. If we grant that a sufficiently advanced civilization could create a workable simulation of existence, we have every right to suspect we inhabit that simulation.