"The mathematics of physics resides in physical phenomena themselves - there are ellipses in the elliptical orbits of the planets, fractals in the fractal shapes of leaves and branches, logarithms in the logarithmic spirals of snails. This means that 'the books of nature is written in mathematics', which implies that the language of mathematics is the language of nature and that only those who lznow mathematics can truly understand nature." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)
"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)
"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"The calculus is a theory of continuous change - processes that move smoothly and that do not stop, jerk, interrupt themselves, or hurtle over gaps in space and time. The supreme example of a continuous process in nature is represented by the motion of the planets in the night sky as without pause they sweep around the sun in elliptical orbits; but human consciousness is also continuous, the division of experience into separate aspects always coordinated by some underlying form of unity, one that we can barely identify and that we can describe only by calling it continuous." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"At all events, our world pictures may have components of all three kinds: perceptual, conceptual, and praxiological (action-theoretical). This is because there are three gates to the outer world: perception, conception, and action. However, ordinarily only one or two of them need be opened: combinations of all three, as in building a house according to a blueprint, are the exception. We may contemplate a landscape without forming either a conceptual model of it or a plan to act upon it. And we may build a theoretical model of an imperceptible thing, such as an invisible extrasolar planet, on which we cannot act." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)
"Physicists use mathematics to study what they amusingly call the real world. It is real, in a sense, but much of physics addresses rather artificial aspects of reality, such as a lone electron or a solar system with only one planet. Physicists are often scathing about proofs, partly out of fear, but also because experiment gives them a very effective way to check their assumptions and calculations." (Ian Stewart, "Letters to a Young Mathematician", 2006)
"Differential equations have found their way into all areas of physics from the motion of planets around the Sun to standing waves on a rope or a drum, to electrical properties of conductors, and the behavior of electromagnetic fields and beyond. As is always the case, no mathematics can draw more attention than that which deals directly with Nature. The urgency of finding solutions to these differential equations prompted many mathematicians of the latter part of the eighteenth and the beginning of the nineteenth centuries to concentrate heavily on certain specific differential equations. It appeared that every differential equation dictated by Nature gave rise to a new function. The most common scheme for solving these differential equations was to assume a power series solution, substitute the assumed solution in the differential equation, and determine the (unknown) coefficients from the resulting equality of power series." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)
"One of the great advantages of vectors is their ability to express results independent of any specific coordinate systems. Physical laws are always coordinate-independent. For example, when we write F = ma both F and a could be expressed in terms of Cartesian, spherical, cylindrical, or any other convenient coordinate system. This independence allows us the freedom to choose the coordinate systems most convenient for the problem at hand. For example, it is extremely difficult to solve the planetary motions in Cartesian coordinates, while the use of spherical coordinates facilitates the solution of the problem tremendously." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)
"Standard economists don't seem to understand exponential growth. Ecological economics recognizes that the economy, like any other subsystem on the planet, cannot grow forever. And if you think of an organism as an analogy, organisms grow for a period and then they stop growing. They can still continue to improve and develop, but without physically growing, because if organisms did that you’d end up with nine-billion-ton hamsters." (Robert Costanza, "What is Ecological economics", 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)." (Daniel Tammet, "Thinking in Numbers" , 2012)
"The goal of physics is to explain the workings of the nonliving world. At first, philosophers described the properties of real objects: the wandering of planets across the night sky, the formation of ice, or the sound of a lyre. When attention turned to things that couldn’t be seen or measured so easily, physicists invented mechanical models to take the place of real things." (Hans C von Baeyer, "QBism: The future of quantum physics", 2016)
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