The modern world, from its microchips to its architecture, is built upon the foundations of mathematics. It's easy to take that fact for granted, just as it's easy to take in stride the fact that, in today's universities, "applied math" majors are on the lookout for more ways to connect the mathematical with the practical. And yet the connection between math and reality, is, when you think about, actually quite strange. Writing in the math magazine Plus, mathematician Phil Wilson explores the implications of the "stunning fact" that "the world can be understood mathematically": "Why," he asks, "is applied maths even possible?"
No one's mystified by the way that language describes the world, albeit imperfectly: it's a human invention that's evolved, over time, to be fairly accurate and illuminating. Mathematics, though, seems like something else altogether, because of two apparently contradictory facts. First, there's the fact that math describes the world with breathtaking exactness, to the degree that we say that reality is "governed" by mathematical laws. And second, there's the fact that math also seems like its own autonomous world, full of intricate patterns and relationships that don't (yet) have any real-world analogues. How can math be simultaneously perfectly realistic and totally bizarre?
Wilson explains the different ways that philosophers of math, or "metamathematicians," have thought about these problems through the ages. Today, he argues, most mathematicians have settled on a form of Platonism. They understand themselves as exploring "what feels like a platonic realm -- they don't invent their mathematics, they discover it." Many believe that the physical world "emerges," somehow, out of a mathematical one, and that both worlds are, in a sense, real. Mathematicians keep their platonism under wraps, however, because it feels mystical and weird: "It is not natural," Wilson explains, "for a philosopher or scientist to wholeheartedly embrace such a view (even if they may wish to) since it tends to encourage the preservation of mystery rather than the drawing back of the obscuring veil." And yet there the mystery is, plain as day to anyone who thinks about it, and suggestive of all sorts of metaphysical and theological questions.
Wilson offers, unsurprisingly, no solution to to the mystery -- but he does point to ways to think about it more clearly. One way is illustrated proposed by the metamathematician Roger Penrose in his "three world diagram":
The diagram, Wilson writes, shows how at least some of the platonic, mathematical world is embedded or expressed in the physical world; how at least some of that physical world is embedded or expressed in our mental world; and how at least part of that mental world is embedded or expressed in the platonic world. It seems as though there are three worlds, or three levels of reality; "each world," Wilson concludes, "remains a mystery," especially when it comes to its relation to the others. A fascinating discussion: read the whole thing here.
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Joshua Rothman is a graduate student and Teaching Fellow in the Harvard English department, and an Instructor in Public Policy at the Harvard Kennedy School of Government. He teaches novels and political writing.