"Central to the development of the calculus were the concepts of convergence and limit, and with these concepts at hand it became at last possible to resolve the ancient paradoxes of infinity which had so much intrigued Zeno." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)
"Geometry is the study of form and shape. Our first encounter
with it usually involves such figures as triangles, squares, and circles, or
solids such as the cube, the cylinder, and the sphere. These objects all have
finite dimensions of length, area, and volume - as do most of the objects
around us. At first thought, then, the notion of infinity seems quite removed
from ordinary geometry. That this is not so can already be seen from the
simplest of all geometric figures - the straight line. A line stretches to
infinity in both directions, and we may think of it as a means to go 'far out' in a one-dimensional world."
"Indeed, it is hard to see how mathematics could exist
without the notion of infinity, for the very first thing a child learns about
mathematics - how to count - is based on the tacit assumption that every integer
has a successor. The notion of a straight line, so fundamental in geometry, is
based on a similar assumption - that we can, at least in principle, extend a line
indefinitely in both directions. Even in such seemingly 'finite' branches of mathematics as probability, the notion of infinity plays a subtle
role: when we toss a coin ten times, we may get five 'heads' and five 'tails', or we may get six 'heads' and four 'tails', or in fact any other outcome; but when we say that the
probability of getting 'heads' or 'tails' is even, we
tacitly assume that an infinite number of tosses would produce an equal outcome." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)
"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic. Only such a chain of deductions can establish the validity of a mathematical law, a theorem. And unless this process has been satisfactorily carried out, no relation - regardless of how often it may have been confirmed by observation - is allowed to become a law. It may be given the status of a hypothesis or a conjecture, and all kinds of tentative results may be drawn from it, but no mathematician would ever base definitive conclusions on it." (Eli Maor, "e: The Story of a Number", 1994)
"A real number that satisfies (is a solution of) a polynomial
equation with integer coefficients is called algebraic. […] A real number that
is not algebraic is called transcendental. There is nothing mystic about this
word; it merely indicates that these numbers transcend (go beyond) the realm of
algebraic numbers."
"An aura of mysticism still surrounds the concept that has
since been called 'imaginary numbers', and anyone who encounters
these numbers for the first time is intrigued by their strange properties. But 'strange' is relative: with sufficient familiarity, the strange object of yesterday
becomes the common thing of today."
"Because it is not a real number, infinity cannot be dealt with in a purely numerical sense." (Eli Maor, "e: The Story of a Number", 1994)
"Great inventions generally fall into one of two categories:
some are the product of a single person's creative mind, descending on the world
suddenly like a bolt out of the blue; others - by far the larger group - are
the end product of a long evolution of ideas that have fermented in many minds
over decades, if not centuries. The invention of logarithms belongs to the
first group, that of the calculus to the second."
"In contrast to the irrational numbers, whose discovery arose
from a mundane problem in geometry, the first transcendental numbers were
created specifically for the purpose of demonstrating that such numbers exist;
in a sense they were "artificial" numbers. But once this goal was
achieved, attention turned to some more commonplace numbers, specifically π and
e."
"Inherent in the concept of limit is the assumption that the
end result of the limiting process is the same, regardless of how the independent
variable approaches its 'ultimate' value."
"Of course, in mathematics we are free to define a new object
in any way we want, so long as the definition does not contradict any previously
accepted definitions or established facts."
"The acceptance of complex numbers into the realm of algebra
had an impact on analysis as well. The great success of the differential and
integral calculus raised the possibility of extending it to functions of
complex variables. Formally, we can extend Euler's definition of a function to
complex variables without changing a single word; we merely allow the constants
and variables to assume complex values. But from a geometric point of view,
such a function cannot be plotted as a graph in a two-dimensional coordinate
system because each of the variables now requires for its representation a two-dimensional
coordinate system, that is, a plane. To interpret such a function
geometrically, we must think of it as a mapping, or transformation, from one
plane to another."
"The story of π has been extensively told, no doubt because its history goes back to ancient times, but also because much of it can be grasped without a knowledge of advanced mathematics." (Eli Maor, "e: The Story of a Number", 1994)
"Why did he [Euler] choose the letter e? There is no consensus. According to one view, Euler chose it because it is the first letter of the word exponential. More likely, the choice came to him naturally, since the letters a, b, c, and d frequently appeared elsewhere in mathematics. It seems unlikely that Euler chose the letter because it is the initial of his own name, as occasionally has been suggested. He was an extremely modest man and often delayed publication of his own work so that a colleague or student of his would get due credit. In any event, his choice of the symbol e, like so many other symbols of his, became universally accepted." (Eli Maor, "e: The Story of a Number", 1994)
"There is a certain ambiguity in the concept of angle, for it describes both the qualitative idea of 'separation' between two intersecting lines, and the numerical value of this separation-the measure of the angle. (Note that this is not so with the analogous 'separation' between two points, where the phrases line segment and length make the distinction clear.) Fortunately we need not worry about this ambiguity, for trigonometry is concerned only with the quantitative aspects of line segments and angles." (Eli Maor, "Trigonometric Delights", 1998)
"Mathematics has, of course, much to say about the more technical aspects of music, such as the tuning of musical instruments or the design of acoustically satisfying concert halls. But as to its influence on music as an art, it was, with a few notable exceptions, rather limited; the two disciplines simply followed their own separate ways." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)
"[...] the relations between the two [mathematics and music] disciplines were never truly symmetric. Yes, there are many similarities between the two. For example, mathematics and music both depend on an efficient system of notation - a set of written symbols that convey a precise, unambiguous meaning to its practitioners (although in music this is augmented by a large assortment of verbal terms to indicate the more emotional aspects of playing)." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)
"The significance of Fourier’s theorem to music cannot be overstated: since every periodic vibration produces a musical sound (provided, of course, that it lies within the audible frequency range), it can be broken down into its harmonic components, and this decomposition is unique; that is, every tone has one, and only one, acoustic spectrum, its harmonic fingerprint. The overtones comprising a musical tone thus play a role somewhat similar to that of the prime numbers in number theory: they are the elementary building blocks from which all sound is made." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)
"Ultimately, music is meant to move our souls, to stir our emotions, to arouse us to swing by its rhythms, and this cannot be achieved by mathematical principles alone." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)
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