"Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity." (Charles Seife, "Zero: The Biography of a Dangerous Idea", 2000)
"While mathematical truth is the aim of inquiry, some falsehoods seem to realize this aim better than others; some truths better realize the aim than other truths and perhaps even some falsehoods realize the aim better than some truths do. The dichotomy of the class of propositions into truths and falsehoods should thus be supplemented with a more fine-grained ordering - one which classifies propositions according to their closeness to the truth, their degree of truth-likeness or verisimilitude. The problem of truth-likeness is to give an adequate account of the concept and to explore its logical properties and its applications to epistemology and methodology." (Graham Oddie, "Truth-likeness", Stanford Encyclopedia of Philosophy, 2001)
"Today, the whole subject of geometry extends way beyond the world of right-angled triangles, circles and so on. There are even branches of the subject in which the ideas of length, angle and area don’t really feature at all. One of these is topology – a sort of rubber-sheet geometry – where a recurring question is whether some geometric object can be deformed ‘smoothly’ into another one." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)
"A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently." (Sir Michael Atiyah, 2004)
"It is clear today that modern science developed when people stopped debating metaphysical questions about the world and instead concerned themselves with the discovery of laws that were primarily mathematical."
"It is also a good idea to not apply any given technique or method blindly, but to think ahead and see where one could hope such a technique to take one; this can allow one to save enormous amounts of time by eliminating unprofitable directions of inquiry before sinking lots of effort into them, and conversely to give the most promising directions priority."
"There are three reasons for the study of inequalities: practical, theoretical and aesthetic. In many practical investigations, it is necessary to bound one quantity by another. The classical inequalities are very useful for this purpose. From the theoretical point of view, very simple questions give rise to entire theories. […] Finally, let us turn to the aesthetic aspects. As has been pointed out, beauty is in the eye of the beholder. However. it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive."
"Contemporary mathematics is often extremely abstract, and the important questions with which mathematicians concern themselves can sometimes be difficult to describe to the interested nonspecialist." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)
"Classification is only one of the mathematical aspects of catastrophe theory. Another is stability. The stable states of natural systems are the ones that we can observe over a longer period of time. But the stable states of a system, which can be described by potential functions and their singularities, can become unstable if the potentials are changed by perturbations. So stability problems in nature lead to mathematical questions concerning the stability of the potential functions." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])
"Despite its deductive nature, mathematics yields its truths much like any other intellectual pursuit: someone asks a question or poses a challenge, others react or propose solutions, and gradually the edges of the debate are framed and a vocabulary is built."
"Economists are all too often preoccupied with petty mathematical problems of interest only to themselves. This obsession with mathematics is an easy way of acquiring the appearance of scientificity without having to answer the far more complex questions posed by the world we live in." (Thomas Piketty, Capital in the Twenty-First Century, 2013)
"Mathematical modeling is the application of mathematics to describe real-world problems and investigating important questions that arise from it." (Sandip Banerjee, "Mathematical Modeling: Models, Analysis and Applications", 2014)
"We tend to think of maths as being an 'exact' discipline, where answers are right or wrong. And it's true that there is a huge part of maths that is about exactness. But in everyday life, numerical answers are sometimes just the start of the debate. If we are trained to believe that every numerical question has a definite, 'right' answer then we miss the fact that numbers in the real world are a lot fuzzier than pure maths might suggest." (Rob Eastaway, "Maths on the Back of an Envelope", 2019)
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