"A definition in mathematics is an exercise in uncovering the
essence of things, one reason that good definitions are so hard to pull off,
since a definition brings the essence to light, and the light brings the
definition to life."
"A five-dimensional space is not a strange deformation of ordinary space, one that only mathematicians can see, but a place where numbers are collected in ordered sets. When string theorists talk of the eleven dimensions required by their latest theory, they are not encouraging one another to search for eight otherwise familiar spatial dimensions that have somehow become lost. They are saying only that for their purposes, eleven numbers are needed to specify points. Where they are is no one’s business." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"A group is a collection of objects, one that is alive in the sense that some underlying principle of productivity is at work engendering new members from old. […] Like many other highly structured objects, groups have parts, and in particular they may well have subgroups as parts, one group nested within a large group, kangarette to kangaroo." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"A proof in mathematics is an argument and so falls under the controlling power of logic itself. […] Within mathematics, a proof is an intellectual structure in which premises are conveyed to their conclusions by specific inferential steps. Assumptions in mathematics are called axioms, and conclusions theorems. This definition may be sharpened a little bit. A proof is a finite series of statements such that every statement is either an axiom or follows directly from an axiom by means of tight, narrowly defined rules. The mathematician’s business is to derive theorems from his axioms; if his system has been carefully constructed, a gross cascade of theorems will flow from a collection of carefully chosen axioms." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"Beyond the theory of complex numbers, there is the much
greater and grander theory of the functions of a complex variable, as when the complex
plane is mapped to the complex plane, complex numbers linking themselves to
other complex numbers. It is here that complex differentiation and integration
are defined. Every mathematician in his education studies this theory and
surrenders to it completely. The experience is like first love."
"But like every profound mathematical idea, the concept of a group reveals something about the nature of the world that lies beyond the mathematician’s symbols. […] There is […] a royal road between group theory and the most fundamental processes in nature. Some groups represent- they are reflections of - continuous rotations, things that whiz around and around smoothly." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"If the method of proof offers the mathematician the prospect of certainty, it is a form of certainty that is itself conditional. A proof, after all, conveys assumptions to conclusions, or axioms to theorems. If the hammer of certainty falls on the theorems, it cannot fall on the axioms with equal force." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"Mathematics is insight and invention and the flash of
something grasped at once, but it is also something salt-cleaned and stout as a
Gothic cathedral." (David Berlinski, "Infinite Ascent: A short history of
mathematics", 2005)
"Practical geometry is an empirical undertaking, living and breathing and sweating in the real world where measurements are always approximate and things are fudged or smeared or jumbled up. Within Euclidean geometry points are concentrated, lines straightened, angles narrowed; idealizations are made, and some parts of experience discarded and other parts embraced." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"Set theory is unusual in that it deals with remarkably simple but apparently ineffable objects. A set is a collection, a class, an ensemble, a batch, a bunch, a lot, a troop, a tribe. To anyone incapable of grasping the concept of a set, these verbal digressions are apt to be of little help. […] A set may contain finitely many or infinitely many members. For that matter, a set such as {} may contain no members whatsoever, its parentheses vibrating around a mathematical black hole. To the empty set is reserved the symbol Ø, the figure now in use in daily life to signify access denied or don’t go, symbolic spillovers, I suppose, from its original suggestion of a canceled eye." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"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)
"There is weirdness in non-Euclidean geometry, but not because
of anything that geometers might say about the ordinary fond familiar world in
which space is flat, angles sharp, and only curves are curved. Non-Euclidean
geometry is an instrument in the enlargement of the mathematician’s
self-consciousness, and so comprises an episode in a long, difficult, and
extended exercise in which the human mind attempts to catch sight of itself
catching sight of itself, and so without end."
"What a wealth of insight Euler’s formula reveals and what
delicacy and precision of reasoning it exhibits. It provides a definition of
complex exponentiation: It is a definition of complex exponentiation, but the definition
proceeds in the most natural way, like a trained singer’s breath. It closes the
complex circle once again by guaranteeing that in taking complex numbers to
complex powers the mathematician always returns with complex numbers. It
justifies the method of infinite series and sums. And it exposes that profound
and unsuspected connection between exponential and trigonometric functions;
with Euler’s formula the very distinction between trigonometric and exponential
functions acquires the shimmer of a desert illusion."
No comments:
Post a Comment