It’s the end of the first week of spring semester 2012, and my classes are already winding up into full gear. This time around, I am pleased to engage in Math 315: Chaos Theory with Dr. Berry Perratt. My professor, who works professionally to solve chaotic problems, was taught by none other than James Yorke, who shares the esteemed Japan Prize for Science and Technology with the late Benoit Mandelbrot, for their contributions to understanding dynamical systems, its controls, and applications. The class is all about dynamical systems, fractal geometry, and self-organized criticality. This first week has been straight philosophy, but from here out it is all numbers and computer programs. Since lecture was mostly just that, a lecture without discussion, I figured I’d use my blog (with inserts from my first assignment) to try to wrap my head around this intriguing emerging field.
The famous ‘Mandelbrot set’, an image collecting all of the Julia sets for quadratic functions. “One of the most intricate and beautiful objects in mathematics”.
So we had to read the first part of our text book, which was a history of the evolution of thought between researchers from a diverse array of fields. Then we read three articles, the last of which I found particularly remarkable. It was called “Faith and Quantum Theory” written by Particle Physicist Stephen Barr for the theology journal First Things. My assignment was to answer the following questions:
1. What is wave-particle duality?
"Once and for all I want to know what I'm paying for. When the electric company tells me whether light is a wave or a particle I'll write my check."
Wave-particle duality is a term that describes the nature of light and electrons. Empirical observations confirm that light and electrons behave both like a continuous wave and a discrete particle simultaneously. The phenomenon is also part of Quantum Theory, and has successfully predicted a vast array of physical systems with stunning accuracy.
2. Describe the contributions of both Max Planck and Louis de Broglie to the wave-particle conundrum.
In 1900, Planck assumed that the energy in light waves came in discrete, indivisible sections (quanta), which was not consistent with earlier assumptions that light energy is like a continuous wave. This perspective allowed him to successfully resolve various theoretical conundrums.
Later, de Broglie assumed that if waves (like light) could behave like particles, then particles (like electrons) could behave like waves. This perspective allowed others to develop a coherent, consistent mathematical description of wave-particle duality.
3. What does the author describe as the main contributions of: (1) Newton’s theories, (2) Maxwell’s equations, (3) Einstein’s relativity, and (4) Quantum Theory.
1) Newton unified celestial and terrestrial phenomenon.
2) Maxwell unified electricity, magnetism, and optics.
3) Einstein unified space and time.
4) Quantum Theory unified particles and forces.
4. Briefly identify one inconsistency to which Quantum Theory gives rise.
“A probability must eventually get resolved into a definite outcome if it is to have any meaning at all, and yet the equations of quantum theory, when applied to any physical system, yield only probabilities and not definite outcomes.”
The author then went on to explain the importance of the observer, a knowledgeable mind, and reality as the collapse of a wave function.
5. The author identifies the three main interpretations of Quantum Theory as: (1) Copenhagen, (2) Many Worlds, and (3) Bohmian. For each one, write a sentence that describes the advantage and disadvantage of the interpretation.
1) The traditional Copenhagen interpretation would believe that since the fundamental equations can only describe probabilistic answers to physical systems, and since the human mind can have certainty through rational thought and empiricism, that therefore the mind cannot be merely physical and deterministic in nature. This would imply that the mind is metaphysical, which is consistent with dualism and Western religious philosophy. This would also imply that reality is created by our mind, and perhaps lies only within our mind. The interpretation also suggests that there is not an objective reality, and that the mind couldn’t be described by physics.
2) The Many Worlds View interprets quantum theory to mean that all probable situations are simultaneously part of reality. If the fundamental equations only give probabilistic answers, then it may be possible that the entire nature of reality is probabilities. This would also imply that there is no objective reality, but that all realities are possible and exist in proportion to their relative likelihoods. The advantage of this perspective is it takes the observer off the pedestal, but the disadvantage is that it would imply infinite alternate realities, all represented as a proportional percent, which seems highly unlikely.
3) The Bohmian Theory interprets quantum data in a manner that distinguishes waves from particles but also explains their entanglement. Waves are the metaphorical track through which particles move through space and time. This would explain why a wave went through two windows at once while each particle just went through one, and also would imply that there is no contradiction in saying a fraction of a wave enters a light collector and simultaneously a whole number of particles enters. The advantage is that it successfully predicts similar results as the traditional interpretation. Although, the view has been said to be ‘artificial’ and deterministic, and essentially just a fresh perspective of Newtonian philosophy.
Here’s a link to one of my all-time favorite humor sites explaining the topic, among others, in “5 Scientific Theories That Will Make Your Head Explode”.
I would like to note that although the author of “Faith and Quantum Theory”, Particle Physicist Stephen Barr, believes in the Copenhagen interpretation, when I think it over for myself, I would put my money on Bohmian Theory. From a scientific perspective, the traditional viewpoint of quantum physics is tremendously unsatisfying. There isn’t an objective reality? We cannot use any known system of analysis for understanding the complexity of the human mind? I mean really? It is a dead end, intellectually speaking.
Although the Copenhagen perspective is harmonious with traditional religious belief systems, it is also the interpretation that has led to the flourishing of new age quantum quackery. To me, it doesn’t make the most sense to hold a scientific opinion because “it seems quite congenial to the worldview of the biblical religions…[and] it seems quite uncongenial to eastern mysticism.” Clearly that remark distinguishes the author as writing from a Western religious bias (to a Western religious audience, I might add), which is a non-secular perspective inappropriate for scientific dialogue. It seems irrational to dismiss Bohm’s clever idea just because “it undoes one of the great theoretical triumphs in the hisory of physics: the unification of particles and forces.” Ok, so what? Maybe we were wrong, and the unification of particles and forces really isn’t a great theoretical triumph, but rather a philosophical thought experiment that proves to be useless. Maybe our first intuitions regarding the experimental results were false, like they quite frequently are in scientific inquiries. It is possible we have been wrong, and if that’s possible, we need to look at other plausible alternative interpretations if we wish to claim scientific understanding. While complete consideration of all possible truths must be a part of the process, it would not be conducive to our future knowledge to conclude, at present and from here forth, we are unable to use the scientific method, mathematics, physics, or technology to advance our knowledge. How absurd! Then we would be left with knowledge based only in faith rather than continuing to hold knowledge based in empirical reason with the highest regard. In my experience, an honest scientific mind would claim to believe in a theory because it provided clarity rather than because to do otherwise would “get rid of the mysteriousness” as Barr puts it. Is a deterministic reality really so philosophically caustic? I think not. Many think not, and many believe moving beyond mystery into a place of clarity is far more remarkable, as poetically expressed by this video.
It seems to me that this research implies something not touched on directly by any of the authors we were assigned to read. It has been said by Jim Yorke, the man who coined the term ‘chaos’, that Chaos Theory, in a nutshell, is that a very small, simple change now can amount to a huge, complex change later. It seems to me that the statement ‘a very small, simple system now can amount to a huge, complex system later’ is also true. The math also appears to back up the idea.
Microscopic image of a fractal cross-section of plant tissue.
The pattern is in our veggies...
I mean, just putz around on google Earth for a little bit and you’ll start to see the pattern everywhere. It’s like once you become aware of it, you can’t help but notice.
To continue the theology theme, it has been humans’ historical intuition that the stunning complexity of life surrounding us is too beautiful, too functional, too adapted to its environment that it just had to be there because of some great cosmic engineer (ie God). Chaos theory has provided us with mathematical evidence that unspeakable beauty with literally infinite complexity is possible with a simple system that starts out with just 0 and 1. (0,1,1,2,3,5,8,13…) The computer is another system that operates on this fundamental truth: a highly functional, complex system from just 0 and 1. Just ONE from nothing, and mathematically, this universe is theoretically possible. Now, if you want to say that whatever that “1” was, is what you’re calling ‘God’, or that the mathematical system is how you’re defining ‘God’, well fine then. Have your God of the Gaps. But it would still imply that it is possible for the universe to begin with nothing, 0. At least, theoretically speaking. I wonder if this has been explored in depth by anyone? A quick google search assures me indeed it has! Now I have a new book to put on the ‘to read’ list – “A Universe from Nothing: Why There is Something Rather than Nothing” by Lawrence Krauss, an actual theoretical physicist.
Who knew math class could be such a doozey!
For some fun videos about math, fibonacci numbers, patterns, and biology, see this blog.
“It is far better to grasp the universe as it really is than to persist in delusion, however satisfying and reassuring.”