Why the Glove of Mathematics Fits the Hand of the Natural Sciences So Well :How Far Down the (Fibonacci) Rabbit Hole Goes
European Scientific Journal May 2016 edition vol.12
Why the Glove of Mathematics Fits the Hand of the Natural Sciences So Well : How Far Down the (Fibonacci) Rabbit Hole Goes
David F. Haight 0 1
0 Department of History and Philosophy, Plymouth State University Plymouth , New Hampshire , U.S.A
1 Robert Latta, ed., Leibniz's Monadology, Oxford: Oxford University Press , 1965, page 409
Why does the glove of mathematics fit the hand of the natural sciences so well? Is there a good reason for the good fit? Does it have anything to do with the mystery number of physics or the Fibonacci sequence and the golden proportion? Is there a connection between this mystery (golden) number and Leibniz's general question, why is there something (one) rather than nothing (zero)? The acclaimed mathematician G.H. Hardy (1877-1947) once observed: “In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.” Is this also true of great physics? If so, is there a simple “preestablished harmony” or linchpin between their respective ultimate foundations? The philosopher-mathematician, Gottfried Leibniz, who coined this phrase, believed that he had found that common foundation in calculus, a methodology he independently discovered along with Isaac Newton. But what is the source of the harmonic series of the natural log that is the basis of calculus and also Bernhard Riemann's harmonic zeta function for prime numbers? On the occasion of the three-hundredth anniversary of Leibniz's death and the one hundredth-fiftieth anniversary of the death of Bernhard Riemann, this essay is a tribute to Leibniz's quest and questions in view of subsequent discoveries in mathematics and physics. (In the Journal of Interdisciplinary Mathematics, Dec. 2008 and Oct. 2010, I have already sympathetically discussed in detail Riemann's hypothesis and the zeta function in relation to primes and the zeta zeros. Both papers were republished online in 2013 by Taylor and Francis Scientific Publishers Group.)
Fibonacci sequence; the “prime” prime number; the fine structure constant of hydrogen; Planck's constant; other physical constants; the principle of indeterminacy; the pre-established harmony; the harmonic
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series, the natural log, the binary code, and the digital code
Mathematics is the science of the most complete abstractions to
which the human mind can attain. As such, it has provoked and inspired
more philosophical and scientific questions about its foundations and its
relation to the world than any other scientific discipline. The abstractions
known as numbers are the backbone of science. If a subject cannot be
described by numbers, it is not scientific. What, however, are these abstract
entities known as numbers and why do they relate so well to the concrete
physical sciences? Einstein’s felicitous use of Bernhard Riemann’s
fourdimensional curved geometry and metric sphere to mathematically model his
general theory of relativity is a stunning example of this enigmatic
relationship between hand and glove. The glove of abstract mathematics
existed for over half a century before the physical hand of the universe put it
on, so to speak. Was mathematics also ready-made for quantum physics?
Both the fine structure constant of hydrogen (1/137+)—the prime
(first) element—and the 1/137+ probability of the electron jump are derived
from Schrӧdinger’s sovereign wave equation, which is itself derived from
Euler’s and Fourier’s unique exponential growth function that nature had
been wearing long before Euler and Fourier discovered it (and Schrӧdinger
rediscovered it). In recent years, Leibniz’s and Euler’s exponential growth
(catenary) curve has also turned up in foundational cosmology in
mathematically modeling the inflationary growth of the newborn universe.
Even the so-called “imaginary” number, the square root of minus one, has
turned out to be “real.” Vital and fundamental to both relativity theory and
quantum mechanics, this unusual number is also essential to the
dimensionless origin of dimensions and is equal to the speed of light for
reasons that we will mention.
All of these examples, plus many more, strikingly attest to the
prescient nature of mathematics and what the philosopher-mathematician
Gottfried Leibniz, co-founder of calculus, referred to as the “pre-established
harmony” between mind and nature [Robert Latta, 1965]. He based this idea
on the exponential growth function (necklace), y = ex = the harmonic power
series, which is surprisingly unique in that it is the only exponential function
(along with its constant multiples) that is its own derivative. This means that
its state of growth is the same as its rate of growth—they are clasp together
in a unique harmony. As we will see, this function that is unlike any other
gives us a big clue as to the fundamental symmetry or “pre-established
harmony” of mathematics and the physical sciences.
That there is such a fundamental connection between (...truncated)