New Scientist published an article by Amanda Gefter in their August 15 issue which describes how and why the notion of infinity has come into question again. The distinction between a potential infinity (the process of something happening without end), and an actual infinity (represented, for example, by the set of real numbers) was disputed among mathematicians for a long time until Cantor brought new meaning to the nature of infinities in the set theory he created. Now infinities are part of the living tissue of mathematics. But infinities have thwarted the success of some physical theories and continue to do so. The survey of challenges from mathematicians and physicists alike make for an interesting read. One of the more reasonable claims comes from Max Tegmark of MIT.
When quantum mechanics was discovered, we realised that classical mechanics was just an approximation,” he says. “I think another revolution is going to take place, and we’ll see that continuous quantum mechanics is itself just an approximation to some deeper theory, which is totally finite.
I believe Riemann himself had not decided whether space was continuous or discrete.
One of the links within this piece was to another New Scientist article by Gefter, published in October of last year. The older piece had the title: Reality: Is everything made of numbers? In it, Amanda Gefter surveys some of emerging perspectives in physics that equate, in one way or another, mathematical reality with physical reality. She recalls Einstein’s fix for the equations that described an expanding universe, years before there was clear evidence that the equations were correct.
How did Einstein’s equations “know” that the universe was expanding when he did not? If mathematics is nothing more than a language we use to describe the world, an invention of the human brain, how can it possibly churn out anything beyond what we put in? “It is difficult to avoid the impression that a miracle confronts us here,” wrote physicist Eugene Wigner in his classic 1960 paper “The unreasonable effectiveness of mathematics in the natural sciences”
With respect to current physics investigations she says:
The prescience of mathematics seems no less miraculous today. At the Large Hadron Collider at CERN, near Geneva, Switzerland, physicists recently observed the fingerprints of a particle that was arguably discovered 48 years ago lurking in the equations of particle physics.
Gefter then tells us about how some prominent physicists, like Brian Greene and Max Tegmark, are tackling the riddle. She has more to say about Tegmark’s view than Greene’s, and Tegmark’s view is fairly extreme. Gefter quotes him:
I believe that physical existence and mathematical existence are the same, so any structure that exists mathematically is also real.
I’ve always liked this kind of talk. While the content of a remark like this might be difficult to specify, I suspect that it’s full of insight. Tegmark goes on to suggest that the mathematical structures that have no physical application in this universe correspond to other universes. But his ideas rest, in part, on his judgment that mathematical structures don’t exist in space and time. ” Space and time themselves,” he says, “are contained within larger mathematical structures.”
Now, one might argue that brains and computers exist in space and time, and without them there is no mathematics, or at least not the kind we used to thinking about. But I don’t want to move here into a difficult philosophical debate. Instead, I’d like to suggest that if we consider that everything that exists is known only in relationship, some new light might be shed on the question of how mathematics can do what it does. Physicists like Tegmark can seem to be replacing the physical subjects of their investigations with mathematical ones, putting the mathematics ‘out there.’ Cognitive scientists seem to be finding mathematical notions mirrored in sensory and learning processes, suggesting that mathematics is part of how the body learns about its world. But the body’s perceiving mechanisms are built entirely in relation to its surroundings, or more specifically, to the properties of things like air and light, or even gravity. If we can see mathematics as one of the body’s actions, action that stretches the reach of its perceiving and learning mechanisms, then perhaps we can imagine that it develops not ‘for’ the world we live in, or from it, but ‘with’ it. Like color, perhaps.
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