This question was addressed recently by Max Tegmark in an interview published in Discover Magazine. Peter Woit is not too happy with Tegmark’s thinking on it, but he’s fascinated with the question. The trouble is, Peter feels, Tegmark does not recognize that all mathematics is not equally important. In other words, it’s important to understand that there exists a fundamental mathematics that somehow relates to fundamental physics. Peter writes:
…the evidence is that the mathematical structure we inhabit is a very special one, sharing features of the very special structures that mathematicians have found to be at the core of modern mathematics. Why this is remains a great mystery, one well worth pursuing from both the mathematician’s and physicist’s points of view.
The truth seems clear to me that it is both a mathematical and physical fact. Indeed, one can begin with 0 directions in 0 dimensions, and work up to 8 directions in 3 dimensions, looking at the mathematical and physical relationships that come from that study and find a lifetime of work cut out for oneself.
It turns out that a good example is found in the Discover interview. In it, Tegmark regards abstract mathematical structures, such as the integers, as existing independently of time, statically, all at once, from the outside point of view, so-to-speak, but then they can also be regarded from an inside point of view, within time, he thinks, as when Einstein combines the three dimensions of space and one dimension of time. He explains:
The integers are not a mathematical structure that includes time, but Einstein’s beautiful theory of relativity certainly does have parts that correspond to time. Einstein’s theory has a four-dimensional mathematical structure called space-time, in which there are three dimensions of space and one dimension of time…the important thing to remember is that Einstein’s theory taken as a whole represents the bird’s perspective. In relativity all of time already exists. All events, including your entire life, already exist as the mathematical structure called space-time. In space-time, nothing happens or changes because it contains all time at once. From the frog’s perspective it appears that time is flowing, but that is just an illusion. The frog looks out and sees the moon in space, orbiting around Earth. But from the bird’s perspective, the moon’s orbit is a static spiral in space-time.
For Tegmark, the passing of time is just an illusion. He attributes the fact that the universe is not predictable not withstanding this, except on the basis of statistics, to quantum mechanics:
If the history of our universe were a movie, the mathematical structure would correspond not to a single frame but to the entire DVD. That explains how change can be an illusion…[However,] things are more complicated than just relativity. If Einstein’s theory described all of physics, then all events would be predetermined. But thanks to quantum mechanics, it’s more interesting.
But then the interviewer interjects the natural question arising from our experience with using mathematics to understand physics: “Why do some equations describe our universe so perfectly and others not so much?” To which Tegmark responds:
Stephen Hawking once asked it this way: “What is it that breathes fire into the equations and makes a universe for them to describe?” If I am right and the cosmos is just mathematics, then no fire-breathing is required. A mathematical structure doesn’t describe a universe, it is a universe. The existence of [my] level IV multiverse also answers another question that has bothered people for a long time. John Wheeler put it this way: Even if we found equations that describe our universe perfectly, then why these particular equations and not others? The answer is that the other equations govern other, parallel universes, and that our universe has these particular equations because they are just statistically likely, given the distribution of mathematical structures that can support observers like us.
Of course, this drives people like Woit up the wall, but he and Smolin are in the minority. Woit contends that the only reason Tegmark can get away with this nonsense is due to the fashionable status of speculative science today. Again, I’m on Woit’s side, but for a very specific reason that won’t be found on any of the blogs discussing mathematics and physics, fashionable or not.
Readers will have to read our New Physics and New Math blogs to get the details, but it has to do with a different understanding of numbers, dimensions, time and space. When Hamilton redefined number in his Algebra as the Science of Pure Time, and when Larson redefined space in The Structure of the Physical Universe, it opened the way to understand the four dimensions of space and time, not as one possibility in an endless array of static mathematical structures, but as an inevitable, dynamic, mathematical structure of evolving space|time. The new structure takes us back to the crossroads of math and science history, when the introduction of a symbol for the square root of two and the ad hoc invention of the imaginary number, propelled us into the modern age of perplexing physics and floundering mathematics.
It starts us down a new path, one in which the square root of 2, derived from a triangle of unit sides, does not lead us to the mysteries of infinity and real numbers, but the 1:1 ratio of the same triangle’s sides, derived from a progression of inverse integers, leads us to the enlightenment of integers in four dimensions. Indeed, it leads us to a whole new structure of mathematics and physics, one for which, as we saw in the previous post, Einstein pined. Here we find the fundamental relationship defining both mathematical and physical structure: Each dimension of our reality has two directions. From this fundamental symmetry, all the rest of mathematics and physics follows.