Roger Penrose, in his book Shadows of the Mind, outlines an idea adapted from Karl Popper ? that there are “three worlds.” The physical universe needs no explanation, except perhaps to Bishop Berkeley, while the subjective world of our own conscious perceptions is one we each know well. The third world is the Platonic world of mathematical objects.
Penrose says of the third world: “” (Shadows of the Mind, p. 413)
Edward Everett, whose dedication speech at Gettysburg was so famously upstaged by Abraham Lincoln, put it more poetically: “.” G. H. Hardy was ambivalent about the Divine, but like most mathematicians he believed “.”
This trilogy of worlds raises some questions, of course. The first is what Eugene Wigner called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” As William Newton-Smith asks, “” Why does the…
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