The Computational Beauty of Nature Quotes

“If we increase r [in a logistic map] even more, we will eventually force the system into a period-8 limit cycle, then a period-16 cycle, and so on. The amount that we have to increase r to get another period doubling gets smaller and smaller for each new bifurcation. This cascade of period doublings is reminiscent of the race between Achilles and the tortoise, in that an infinite number of bifurcations (or time steps in the race) can be confined to a local region of finite size. At a very special critical value, the dynamical system will fall into what is essentially an infinite-period limit cycle. This is chaos.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

“Later Turing proved that Turing machines could compute exactly the same functions as lambda calculus, which proved that all three models of computation are equivalent. This is a truly remarkable result, considering how different the three models of computation are. In Church's 1941 paper he made a statement that is now known as the Church-Turing thesis: Any function that can be called computable can be computed by lambda calculus, a Turing machine, or a general recursive function.

Recall the point that was made about functions describing relationships between numbers and models of computation describing functions. Well, the Church-Turing thesis is yet another level more fundamental than a model of computation. As a statement about models of computation, it is not subject to proof in the usual sense; thus, it is impossible to prove that the thesis is correct. Once could disprove it by coming up with a model of computation over discrete elements that could calculate things that one of the other models could not; however, this has not happened. The fact that every posed model of computation has always been exactly equivalent to (or weaker than) one of the others lends strong support to the Church-Turing thesis.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

“Now, take all of your computer's memory and arrange it as one long line of zeros and ones: 0,1,1,1,0,0,0,1,1,0,1....Take this very long number and put a zero and a decimal point in front of it. We've just translated one huge number into a rational number between 0 and 1. By placing this single point at exactly the right spot on the number line, we can store an unlimited amount of information. Ah, if only it were so simple. In the real world, we simply don't have the precision required to put this method of storing memory into practice. We never will, either, but it's an interesting mental exercise to see that it can be done in theory in an idealized world. The point of this whole mental exercise is that in many ways an irrational number has as much "information" as an infinite number of natural numbers.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

“Any discrete piece of information can be represented by a set of numbers. Systems that compute can represent powerful mappings from one set of numbers to another. Moreover, any program on any computer is equivalent to a number mapping. These mappings can be thought of as statements about the properties of numbers; hence, there is a close connection between computer programs and mathematical proofs. But there are more possible mappings than possible programs; thus, there are some things that simply cannot be computed. The actual process of computing can be defined in terms of a very small number of primitive operations, with recursion and/or iteration comprising the most fundamental pieces of a computing device. Computing devices can also make statements about other computing devices. This leads to a fundamental paradox that ultimately exposes the limitations not just of machine logic, but all of nature as well.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

“The goal of this book is to highlight the computational beauty found in nature's programs.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

“Moreover, multiplicity, iteration, and adaptation are universal concepts in that they are apparently important attributes for agents at all levels-from chemical reactants to biological ecosystems.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

“We have, then, three different ways of looking at how things work. We can take a purely reductionist approach and attempt to understand things through dissection. We also can take a wider view and attempt to understand whole collections at once by observing how many agents, say the neurons in a brain, form a global pattern, such as human intelligence. Or we can take an intermediate view and focus attention on the interactions of agents. Through this middle path, the interactions of agents can be seen to form the glue that binds one level of understanding to the next level.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

“Looking back at the organization of the sciences, we find that at teach level of understanding, traditional scientists study two types of phenomena: agents(molecules, cells, ducks, and species) and interactions of agents (chemical reactions, immune system responses, duck mating, and evolution). Studying agents in isolation is a fruitful way of discovering insights into the form and function of an agent, but doing so has some known limitations. Specifically, reductionism fails when we try to use it in a reverse direction. As we shall see throughout this book, having a complete and perfect understanding of how an agent behaves in no way guarantees that you will be able to predict how this single event will behave for all time or in the context of other agents.”
― Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation