On Thresholds (-1999)

"We who stand on the threshold of a new century can look back on an era of unparalleled progress. Looking into the future an equally bright prospect greets our eyes; on all sides fruitful fi elds of research invite our labor and promise easy and rich returns. Surely this is the golden age of mathematics!" (Pierpont, James Pierpont, "The History of Mathematics in the Nineteenth Century", Bulletin of the American Mathematical Society, 2nd Series, Vol. 11, 1904–1905) 

"Those terrible logarithms, when I happened to open a table of them, made my head swim, with their columns of figures; actual fright, not unmixed with respect, overwhelmed me on the very threshold of that arithmetical cave." (Jean-Henri Fabre, "The Life of the Fly", 1913)

"When we are thrilled with the wonder of the world, the heights and depths of things, the beauty of it all, we approach the door of natural religion. And when the Nature-feeling is not superfi cial but informed with knowledge, with no gain of the hard-won analysis unused, we may reach the threshold. And when we feel that our scientifi c cosmology leaves Isis still veiled, and when our attempts at philosophical interpretation give us a reasoned conviction of a meaning behind the process, we may perhaps enter in." (J Arthur Thomson, "The System of Animate Nature" Vol. 1, 1920) 

"The scientific spirit brings about a particular attitude towards worldly matters; before religious matters it pauses for a little, hesitates, and fi nally there too crosses the threshold. In this process there is no stopping; the greater the number of men to whom the treasures of knowledge become accessible, the more widespread is the falling-away from religious belief…" (Sigmund Freud, "The Future of an Illusion", 1927) 

"There are scientists who make their chief discovery at the threshold of their scientific career, and spend the rest of their lives substantiating and elaborating it, mapping out the details of their discovery, as it were. There are other scientists who have to tread a long, diffi cult and often tortuous path to its end before they succeed in crowning their efforts with a discovery." (V Safonov, "Courage",  1953) 

"[The] system may evolve through a whole succession of transitions leading to a hierarchy of more and more complex and organized states. Such transitions can arise in nonlinear systems that are maintained far from equilibrium: that is, beyond a certain critical threshold the steady-state regime become unstable and the system evolves into a new configuration." (Ilya Prigogine, Gregoire Micolis & Agnes Babloyantz, "Thermodynamics of Evolution", Physics Today 25 (11), 1972)

"As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics." (Lotfi A Zadeh, 1973)

"Fuzziness, then, is a concomitant of complexity. This implies that as the complexity of a task, or of a system for performing that task, exceeds a certain threshold, the system must necessarily become fuzzy in nature. Thus, with the rapid increase in the complexity of the information processing tasks which the computers are called upon to perform, we are reaching a point where computers will have to be designed for processing of information in fuzzy form. In fact, it is the capability to manipulate fuzzy concepts that distinguishes human intelligence from the machine intelligence of current generation computers. Without such capability we cannot build machines that can summarize written text, translate well from one natural language to another, or perform many other tasks that humans can do with ease because of their ability to manipulate fuzzy concepts." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)

"Threshold functions (are described) which facilitate the careful study of the structure of a graph as it grows and specifically reveal the mysterious circumstances surrounding the abrupt appearance of the Unique Giant Component which systematically absorbs its neighbours, devouring the larger first and ruthlessly continuing until the last Isolated Nodes have been swallowed up, whereupon the Giant is suddenly brought under control by a Spanning Cycle." (Edgar Palmer, "Graphical Evolution", 1985)

"[…] an epidemic does not always percolate through an entire population. There is a percolation threshold below which the epidemic has died out before most of the people have." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"In the realms of nature it is impossible to predict which way a bifurcation will cut. The outcome of a bifurcation is determined neither by the past history of a system nor by its environment, but only by the interplay of more or less random fluctuations in the chaos of critical destabilization. One or another of the fluctuations that rock such a system will suddenly 'nucleate'. The nucleating fluctuation will amplify with great rapidity and spread to the rest of the system. In a surprisingly short time, it dominates the system’s dynamics. The new order that is then born from the womb of chaos reflects the structural and functional characteristics of the nucleated fluctuation. [...] Bifurcations are more visible, more frequent, and more dramatic when the systems that exhibit them are close to their thresholds of stability - when they are all but choked out of existence." (Ervin László, "Vision 2020: Reordering Chaos for Global Survival", 1994)

"When a system is 'stressed' beyond certain threshold limits as, for example, when it is heated up, or its pressure is increased, it shifts from one set of attractors to another and then behaves differently. To use the language of the theory, the system 'settles into a new dynamic regime'. It is at the point of transition that a bifurcation takes place. The system no longer follows the trajectory of its initial attractors, but responds to new attractors that make the system appear to be behaving randomly. It is not behaving randomly, however, and this is the big shift in our understanding caused by dynamical systems theory. It is merely responding to a new set of attractors that give it a more complex trajectory. The term bifurcation, in its most significant sense, refers to the transition of a system from the dynamic regime of one set of attractors, generally more stable and simpler ones, to the dynamic regime of a set of more complex and 'chaotic' attractors." (Ervin László, "Vision 2020: Reordering Chaos for Global Survival", 1994)

"Once we overcome our fear of being tiny, we find ourselves on the threshold of a vast and awesome Universe that utterly dwarfs - in time, in space, and in potential - the tidy anthropocentric proscenium of our ancestors." (Carl Sagan, "Pale Blue Dot: A Vision of the Human Future in Space", 1994)

"The resolution of how to divide the stakes in an uncompleted game marked the beginning of a systematic analysis of probability - the measure of our confidence that something is going to happen. It brings us to the threshold of the quantification of risk." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)