Time, mathematics and Plato’s cave

Sean Carroll, Theoretical Physicist at the California Institute of Technology has recently published a new book.  Entitled The Particle at the End of the Universe: How the Higgs Boson Leads us to the Edge of a New World it discusses the importance of the Higgs boson as well as the significance of the extraordinary work done at the Large Hadron Collider at CERN where evidence for the Higgs existence was finally found.  The attention brought to Carroll from reviews and comments about the book, and the references to his exploration of time in physics, led me to look at some of his earlier work.  I listened to a TED talk he gave last year on distant time.   The talk was only about 15 minutes long, and one of the things that struck me was how concept-driven physic’s notion of time has become.   It’s not exactly the experience of time that has physicists’ attention, but more whether (or how) it fits as a piece of the universe puzzle we’ve constructed.

Carroll’s description of time begins with the observation that entropy, the amount of disorder in a system, can be quantified.   Entropy is described as “the number of ways we can rearrange the constituents of a system so that you don’t notice, so that macroscopically it looks the same. ”  And so the reason entropy increases is simply because there are many more ways to be high entropy (more disordered) than low entropy (largely ordered).  This is as much an appeal to numbers as it is to a physical system.  With this, Carroll defines time:

Every difference that there is between the past and the future is because entropy is increasing — the fact that you can remember the past, but not the future. The fact that you are born, and then you live, and then you die, always in that order, that’s because entropy is increasing.

He goes on to describe how, in books and lectures, Richard Feynman emphasized that

The arrow of time cannot be completely understood until the mystery of the beginnings of the history of the universe are reduced still further from speculation to understanding.

Whether time is perceived or constructed is open to debate in physics today.  I believe Carroll is of the opinion that it is perceived.

Inevitably I thought about how time has moved in and out of mathematics – how the derivative, for example can be thought of as a rate or as speed or movement along a curve, while at the same time mathematicians required that it be shown to have a purely arithmetic meaning.  It is a conceptual challenge to us to say, outside of these many things it can be, what the derivative actually is.  The structure that our biology gives to the world in experience inspires both concepts and processes in mathematics, but we consistently unravel mathematics from our experience in our search for indisputable meaning.  The sciences, though defined by empiricism, necessarily follow paths that are opened with mathematics.

I also read an article today by Anil Ananthaswamy  in New Scientist called Quantum Shadows The Mystery of Matter Deepens.  I thought the article contained a related observation. In recent experiments that were designed to explore the wave/particle (dual) nature of light, an unexpected thing happened.  The mixture of both aspects could somehow be seen.

… it took only a few months for the experimentalists to catch up with the theorists. But when three independent groups, led by Chuan-Feng Li at the University of Science and Technology of China in Hefei, Jeremy O’Brien at the University of Bristol, UK, and Sébastien Tanzilli at the University of Nice, France, performed different versions of the experiment last year, the results were unnerving – even to those who consider themselves inured to the weirdnesses of quantum physics (Nature Photonics, vol 6, p 600; Science, vol 338, p 634 and p 637).

…”Our experiment defies the conventional boundaries set by the complementarity principle,” says Li. Ionicioiu agrees. “Complementarity shows only the two ends, black and white, of a spectrum between particle and wave,” he says. “This experiment allows us to see the shades of grey in between.”

Near the conclusion of the article the author writes:

So, has Bohr been proved wrong too? Johannes Kofler of the Max Planck Institute of Quantum Optics in Garching, Germany, doesn’t think so. “I’m really very, very sure that he would be perfectly fine with all these experiments,” he says. The complementarity principle is at the heart of the “Copenhagen interpretation” of quantum mechanics, named after Bohr’s home city, which essentially argues that we see a conflict in such results only because our minds, attuned as they are to a macroscopic, classically functioning cosmos, are not equipped to deal with the quantum world. “The Copenhagen interpretation, from the very beginning, didn’t demand any ‘realistic’ world view of the quantum system,” says Kofler.

The outcomes of the latest experiments simply bear that out. “Particle” and “wave” are concepts we latch on to because they seem to correspond to guises of matter in our familiar, classical world. But attempting to describe true quantum reality with these or any other black-or-white concepts is an enterprise doomed to failure.

Paths like this one (where we can see something with certainty but have no way to say what it is we see) are remarkable.  And it is mathematics that takes reveals them.  That we have not yet, in any significant way, looked back at ourselves and wondered at what we are doing is equally remarkable.  I’ll close with this from the article:

It’s a notion that takes us straight back into Plato’s cave, says Ionicioiu. In the ancient Greek philosopher’s allegory, prisoners shackled in a cave see only shadows of objects cast onto a cave wall, never the object itself. A cylinder, for example, might be seen as a rectangle or a circle, or anything in between. Something similar is happening with the basic building blocks of reality. “Sometimes the photon looks like a wave, sometimes like a particle, or like anything in between,” says Ionicioiu. In reality, though, it is none of these things. What it is, though, we do not have the words or the concepts to express.