Book Notes: Beyond Reason: 8 Great Problems That Reveal The Limits of Science
Written by A.K. Dewdney
I occasionally pick up non-fiction, science books to see if my increasingly fuzzy brain can keep up with the concepts. With Beyond Reason, I could generally follow the logic, but the math formulas proving exactly why — completely went over my head. The book is an examination of the limits of science through eight ideas that are currently considered — for all practical purposes — impossible. Dewdney provides the historical context, a summary of the example, and then digs down into the math.
Impossible Machines: You can’t build a machine that runs forever without an external source of energy, that produces useable energy. Perpetual motion, and why “free energy” will never be practical because energy must be conserved. A frictionless machine in a weightless environment is the closest we can get, but as far that generating energy it’s useless. The history takes us back to the Industrial revolution with pumps and waterwheels and various hucksters who tried to fool people with elaborate machines (one of which revealed a person in an adjacent room turning a crank). It’s also worth noting that this fruitless quest continues today.
The Cosmic Limit: Nothing can travel faster than the speed of light. Light as the absolute speed limit. Einstein’s thinking process as to how he came up with his theory of relativity. I think the interesting thing is how Einstein’s contemporaries laid the groundwork, but seemingly nobody dared make the mental leap that required the speed of light to be the absolute and everything else in the world went along with it.
The Quantum Curtain: It’s impossible to know the detailed behavior of any quantum system — properties of a particle don’t exactly exist until observed. We learn how photons either act as particles or waves, Bell’s Theorem, and eventually the metaphor of a dead cat.
The Edge of Chaos: Some classical systems produce long-term behavior which cannot be predicted. There are many situations that are just too complex to predict the results — even for ever increasingly powerful computers — the situations just get too out of control in order of magnitude. Because of this the author believes that weather prediction — such as will we ever have a computer that can predict the weather a year in advance — is impossible. I found this chapter very interesting. Some systems are naturally chaotic, starting with the weather-prediction computer experiments of Lorenz to come up with “Lorenz attractors”. These patterns are from analysis of seemingly simple systems that produce very complex results merely by changing one input. The charts resemble the “automata” that Wolfram obsesses over in his book A New Kind Of Science.
The Circular Crypt: You can’t construct a square equal in area to a given circle using only a ruler and a compass. This is a geometric discussion of “squaring the circle.” This problem boils down to the impossible (or transcedental) nature of pi. In order to accomplish this task, you have to know the circumference of the circle and translate that into a straight line.
The Chains of Reason: There are some theorems that are impossible to prove. This part had way too much math and I couldn’t get my head around it. It was basically going over a the “incompleteness theorem” (Godel) that proved that unproveable theorems could exist. I guess Godel’s theory doesn’t fall into that category (?).
The Computer Treadmill: There are some yes / no questions that no computer will ever answer. Turing machines suggest that any algorithm can be reduced to simple instructions in the form of long string of “yes, no” (1,0) . Increasingly powerful computers, can be taught to simulate simpler ones. It’s easy to assume that because of this, computers will one day be able to solve the most intractable problems. There are many things that are simply not reduceable to an algorithm and therefore will always be uncomputable — even if you have a faster computer. Then as another does of reality, Church’s thesis states that the mythical Turing machine (simple as it may seem) doesn’t exist in the real world (we have physical limitations of power consumption and finite speed), and for some reason the author tacks on Ackerman’s function as an example of a simple algorithm that would quickly bring a powerful computer to its knees.
The Big-O Bottleneck: Some mathematical problems solvable by computers still need an exponential amount of time. Even if you get a problem that can be defined by an algorithm that a computer can understand, some are found to be so difficult that the only way to deal with these problems is brute force raw computing power, limited by speed and power consumption. There’s a simple chart that orders complexity in order of increasing computing time: logarithmic, linear, quadratic, exponential. The last type really starts to take a toll quickly, as each “next step” is an order of magnitude larger than the one previous. At this point the math took a serious toll on my brain and I couldn’t make heads or tails of Cook’s Theorem.
Anyhow, the end result of reading this book makes me think that I could never have been a mathematician or a physicist, but I can at least comprehend the general ideas, but which I quickly forget and have difficulty summarizing as proved by this blog post. I might call this the Blog Information Entropy theorem.