Complexity Quotes

“This statement is not provable.” Think about it for a minute. It’s a strange statement, since it talks about itself—in fact, it asserts that it is not provable. Let’s call this statement “Statement A.” Now, suppose Statement A could indeed be proved. But then it would be false (since it states that it cannot be proved). That would mean a false statement could be proved—arithmetic would be inconsistent. Okay, let’s assume the opposite, that Statement A cannot be proved. That would mean that Statement A is true (because it asserts that it cannot be proved), but then there is a true statement that cannot be proved—arithmetic would be incomplete. Ergo, arithmetic is either inconsistent or incomplete.”
― Melanie Mitchell, Complexity: A Guided Tour

“As the nineteenth-century philosopher Henry David Thoreau put it, “All perception of truth is the detection of an analogy.”
― Melanie Mitchell, Complexity: A Guided Tour

“Whew, this might be getting a bit confusing. I hope you are following me so far. This is the point in every Theory of Computation course at which students either throw up their hands and say "I can't get my mind around this stuff!" or clap their hands and say "I love this stuff!"

Needless to say, I was the second kind of student, even though I shared the confusion of the first.”
― Melanie Mitchell, Complexity: A Guided Tour

“complex system: a system in which large networks of components with no central control and simple rules of operation give rise to complex collective behavior, sophisticated information processing, and adaptation via learning or evolution.”
― Melanie Mitchell, Complexity: A Guided Tour

“We have a poor mathematical, as well as a poor intuitive understanding of the nature of coincidence.”
― Melanie Mitchell, Complexity: A Guided Tour

“Linearity is a reductionist’s dream, and nonlinearity can sometimes be a reductionist’s nightmare. Understanding the distinction between linearity and nonlinearity is very important and worthwhile. To”
― Melanie Mitchell, Complexity: A Guided Tour

“In short, what Brown, Enquist, and West are saying is that evolution structured our circulatory systems as fractal networks to approximate a “fourth dimension” so as to make our metabolisms more efficient. As West, Brown, and Enquist put it, “Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional … Fractal geometry has literally given life an added dimension.”
― Melanie Mitchell, Complexity: A Guided Tour

“How is it that those systems in nature we call complex and adaptive—brains, insect colonies, the immune system, cells, the global economy, biological evolution—produce such complex and adaptive behavior from underlying, simple rules?”
― Melanie Mitchell, Complexity: A Guided Tour

“The defining idea of chaos is that there are some systems—chaotic systems—in which even minuscule uncertainties in measurements of initial position and momentum can result in huge errors in long-term predictions of these quantities. This is known as “sensitive dependence on initial conditions.”
― Melanie Mitchell, Complexity: A Guided Tour

“intellectual”
― Melanie Mitchell, Complexity: A Guided Tour

“forward to the day when we can together tour those new”
― Melanie Mitchell, Complexity: A Guided Tour

“has definitive answers. Although I do not address the first question here, there has been some fascinating research on it in the complex systems community. The second question—what is life, exactly?—has been on the minds of people probably for as”
― Melanie Mitchell, Complexity: A Guided Tour

“You”
― Melanie Mitchell, Complexity: A Guided Tour

“But twentieth-century science was also marked by the demise of the reductionist dream. In spite of its great successes explaining the very large and very small, fundamental physics, and more generally, scientific reductionism, have been notably mute in explaining the complex phenomena closest to our human-scale concerns.”
― Melanie Mitchell, Complexity: A Guided Tour

“As in all adaptive systems, maintaining a correct balance between these two modes of exploring is essential. Indeed, the optimal balance shifts over time. Early explorations, based on little or no information, are largely random and unfocused. As information is obtained and acted on, exploration gradually becomes more deterministic and focused in response to what has been perceived by the system. In short, the system both explores to obtain information and exploits that information to successfully adapt.”
― Melanie Mitchell, Complexity: A Guided Tour

“Information, as narrowly defined by Shannon, concerns the predictability of a message source. In the real world, however, information is something that is analyzed for meaning, that is remembered and combined with other information, and that produces results or actions. In short, information is processed via computation.”
― Melanie Mitchell, Complexity: A Guided Tour

“reached. In dynamical systems theory, each of these abrupt period doublings is called a bifurcation. This succession of bifurcations culminating in chaos has been called the “period doubling route”
― Melanie Mitchell, Complexity: A Guided Tour

“Turing’s first goal was to make very concrete this notion of definite procedure. The idea is that, given a particular problem to solve, you can construct a definite procedure for solving it by designing a Turing machine that solves it. Turing machines were put forth as the definition of “definite procedure,” hitherto a vague and ill-defined notion.”
― Melanie Mitchell, Complexity: A Guided Tour

“Tony Rothman points out, “Why the second law should distinguish between past and future while all the other laws of nature do not is perhaps the greatest mystery in physics.”
― Melanie Mitchell, Complexity: A Guided Tour

“idea models—models that are simple enough to study via mathematics or computers but that nonetheless capture fundamental properties of natural complex systems.”
― Melanie Mitchell, Complexity: A Guided Tour