"[…] all roads in mathematics lead to infinity. At any rate, most of the attempts to do the impossible have called upon infinity in one way or another: not necessarily the infinitely large, not necessarily the infinitely small, but certainly the infinitely many." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)
"The mathematician's circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable." (Ian Stewart, "Letters to a Young Mathematician", 2006)
"Within a few decades of its discovery, the fantasy of infinitesimals had completely overpowered the honest method of exhaustion by joining forces with algebra to form an infinitesimal calculus - a symbolism for solving problems about curves by routine calculations, like those already known for the geometry of straight lines. The calculus, as we know it today, is perhaps the most powerful mathematical tool ever invented, yet it originated in the dream world of infinitesimals. We look at these strange origins in the next section." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"[...] a high degree of unpredictability is associated with erratic trajectories. This not only because they look random but mostly because infinitesimally small uncertainties on the initial state of the system grow very quickly - actually exponentially fast. In real world, this error amplification translates into our inability to predict the system behavior from the unavoidable imperfect knowledge of its initial state." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)
"Probability is often expressed using large but finite numbers: ‘one in a thousand’, ‘one in a million’. But perhaps the probability of life, intelligent life, appearing somewhere in our universe is infinitesimal. If so, a universe would need infinitely many planets to produce even a finite number of civilisations (i.e., one)."
"One of the remarkable features of these complex systems created by replicator dynamics is that infinitesimal differences in starting positions create vastly different patterns. This sensitive dependence on initial conditions is often called the butterfly-effect aspect of complex systems - small changes in the replicator dynamics or in the starting point can lead to enormous differences in outcome, and they change one’s view of how robust the current reality is. If it is complex, one small change could have led to a reality that is quite different." (David Colander & Roland Kupers, "Complexity and the art of public policy : solving society’s problems from the bottom up", 2014)
"Calculus is the study of things that are changing. It is difficult to make theories about things that are always changing, and calculus accomplishes it by looking at infinitely small portions, and sticking together infinitely many of these infinitely small portions." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"But in another way infinitesimals were not like imaginaries. While it is true that imaginaries are not part of the ordinary number system, that system can easily be enlarged to accommodate them – for example, by defining imaginaries as ordered pairs of ordinary numbers, as Hamilton did in 1835. It is not nearly as easy, or convenient, to enlarge the number system to include infinitesimals. It was not even known to be possible until the twentieth century, long after mathematicians had decided that it was better to avoid infinitesimals the way Euclid and Archimedes avoided infinity." (John Stillwell, "A Concise History of Mathematics for Philosophers", 2019)
"Infinitesimals were a bit like imaginary numbers. They seemed to contradict accepted principles – just as imaginaries contradicted the principle that squares are positive, infinitesimals contradicted the Archimedean axiom for geometric quantities, [...] yet they enabled calculations that were otherwise difficult or impossible. For mathematicians of the seventeenth and eighteenth centuries this was generally good enough reason to accept them." (John Stillwell, "A Concise History of Mathematics for Philosophers", 2019)
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