I am not sure that I get Penrose's argument, but it goes something like this.
If one thinks that the mind IS a kind of algorithm (sounds like functionalism), then one cannot maintain that all algorithms exist only in minds. Otherwise, one would have to agree that any algorithm necessarily exists in an algorithm, which necessarily exists in an algorithm, etc. Something circular here... or maybe an infinite regress. Penrose's solution is to posit Platonic mathematical entities, including algorithms. But to get this right, I suppose I need to read this argument instead of just listening to an audio version of the book.
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