"The reality is that without mathematics, modern-day cosmologists could not have progressed even one step in attempting to understand the laws of nature. Mathematics provides the solid scaffolding that holds together any theory of the universe. […] Mathematics appears to be almost too effective in describing and explaining not only the cosmos at large, but even some of the most chaotic of human enterprises." (Mario Livio, "Is God a Mathematician?", 2011)
"There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed ‘the unreasonable effectiveness of mathematics’), one more astonishing than the other. First, there is an aspect one might call ‘active’. When physicists wander through nature’s labyrinth, they light their way by mathematics - the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. […] But there is also a ‘passive’ side to the mysterious effectiveness of mathematics, and it is so surprising that the 'active' aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons - with absolutely no application in mind - turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality!" (Mario Livio, "Is God a Mathematician?", 2011)
"I believe that by asking simply whether mathematics is
invented or discovered, we ignore the possibility of a more intricate answer:
both invention and discovery play a crucial role. I posit that together they
account for why math works so well. Although eliminating the dichotomy between
invention and discovery does not fully explain the unreasonable effectiveness
of mathematics, the problem is so profound that even a partial step toward
solving it is progress." (Mario Livio, "Why Math Works", ["The Best Writing
of Mathematics: 2012"] 2012)
"Mathematics is an intricate fusion of inventions and discoveries. Concepts are generally invented, and even though all the correct relations among them existed before their discovery, humans still chose which ones to study." (Mario Livio, "Why Math Works", ["The Best Writing of Mathematics: 2012"] 2012)
"Not only do scientists cherry-pick solutions, they also tend to select problems that are amenable to mathematical treatment. There exists, however, a whole host of phenomena for which no accurate mathematical predictions are possible, sometimes not even in principle." (Mario Livio, "Why Math Works", ["The Best Writing of Mathematics: 2012"] 2012)
"The laws of physics seem to display symmetry with respect to
space and time: They do not depend on where, from which angle, or when we
examine them. They are also identical to all observers, irrespective of whether
these observers are at rest, moving at constant speeds, or accelerating. […] If
the universe did not possess these symmetries, any attempt to decipher nature’s
grand design - any mathematical model built on our observations—would be doomed
because we would have to continuously repeat experiments at every point in space
and time."
"The predictive value of any theory relies on the constancy
of the underlying relations among variables. Our analyses also fail to fully capture
systems that develop chaos, in which the tiniest change in the initial
conditions may produce entirely different end results, prohibiting any long-
term predictions. Mathematicians have developed statistics and probability to
deal with such shortcomings, but mathematics itself is limited […]"
" […] we adopt mathematical tools that apply to our world - a
fact that has undoubtedly contributed to the perceived effectiveness of
mathematics. Scientists do not choose analytical methods arbitrarily but rather
on the basis of how well they predict the results of their experiments."
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