I just saw The Guardian’s Science Weekly podcast for November 11, 2013 which included a discussion with mathematician Edward Frenkel about his new book Love & Math: The Heart of Hidden Reality. I then listened to a Huffington Live segment from January 7 where Max Tegmark and Brian Greene talked about the link between mathematics and reality. Tegmark was speaking from the perspective of his new book, Our Mathematical Universe. I’ve only just ordered both books and so I haven’t read them yet. But I would like to say a few things about how each of the authors introduced their ideas. While I find both works encouraging, both bold attempts to reorient the popular view of mathematics, I’m struck by how different they are.
Edward Frenkel began by addressing the need to bring, to a broad audience, some heart-felt appreciation of the beauty in mathematics. The reason no one can see it, he suggests, is the fault of teachers, and not entirely because the ideas are complex. Equally complex ideas like space-time, quantum mechanical behavior,black holes, the Higgs particle, even DNA have found their way into the popular culture. There is no reason that the real subject of mathematics can’t be made similarly accessible for a popular audience. I very much agree.
Frenkel also spent a fair amount of time talking about the Langlands Program, which he referred to as the effort to find a ‘grand unified’ theory of mathematics. In this discussion, he imagined the different branches of mathematics as continents, fully separated land masses. When one finds (as Langland did) a way to translate questions from one area into questions in another, mathematics, Frankel tells us, becomes a teleportation device. While this work has some obvious pragmatic implications, Frenkel uses this idea to point out that the deeper impact of the Langlands Program is in how it reveals the way things (in a more general sense) are connected. Mathematics, he says, tells us about hidden structure that “we still don’t see.” “The more we know about mathematics,” he continues, “the more tools we will have to understand how the world works.” He likens mathematics to an unfinished jigsaw puzzle that’s giving us glimpses of a hidden reality whose final image we don’t know.
Notice he doesn’t say “the more mathematics we know,” but rather, “the more we know about mathematics.” One of the reasons that mathematics seems dry and uninteresting to so many is that it is rarely thought of as an exploration in itself, a search for new meaning and new possibilities or, as Frenkel puts it, for an image you have not seen before.
Frenkel also makes the argument that everything in our world is migrating to the digital. Three-dimensional printers “will be able to print everything out on demand, like a table and spoon, and so on.” The deepest level of physical reality is becoming a digital layer. In this world, he exclaims, “mathematics is going to be king!” And this is because “mathematics’ function is to order information.”
When asked about how the universe could be mathematical, Max Tegmark begins with an appeal to the progress made in modern physics. While the properties of familiar objects were once reduced by science to the properties of atoms, the properties of elementary particles have now been reduced to numbers. We give names to the numbers like spin and charge, but they’re really just numbers. But the way Tegmark speaks about the mathematical universe still feels like he’s giving that mathematical universe the kind of independent, objective reality which I think hampers the effectiveness of seeing things this way. I completely agree that what he proposes leaves the doors to our future understanding wide open. The only limits that exist would be the limits of our creativity and imagination. But because his mathematics still seems to stand outside of us, the question inevitably arises (and did during this interview) about whether consciousness or emotion will ultimately be described mathematically. He and Brian Greene seem optimistic that the answer to that question is yes. Frenkel, on the other hand, has already overlapped mathematics and love in a film he co-created called Rites of Love and Math.
I find Frenkel’s perspective more familiar. I haven’t yet seen the film nor read the book but I will do both shortly. I’m also interested in the conference, organized by the Foundational Questions Institute, and still underway today. It’s a conference on the Physics of Information, and includes a session on Mind, Brain, information and consciousness with neuroscientists Giulio Tononi and Christof Koch.
My views on mathematics share something with both Frenkel and Tegmark, but I’m also influenced by an idea described by Humberto Maturana and Francisco Zarela in their 1987 work, The Tree of Knowledge. A key to their understanding of cognition is their idea of ‘coupling.’ Structural coupling, in its most fundamental sense, is described as “a history of recurrent interactions leading to the structural congruence between two (or more) systems.” It is possible that we are beginning to glimpse something of our own coupling – the coupling of our conscious mind with the world we inhabit (a world that grows with our thoughts). And that may be why we can’t decide which side the math is coming from.
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