''Mathematics and the Imagination'' is a book published in New York by
Simon & Schuster
Simon & Schuster LLC (, ) is an American publishing house owned by Kohlberg Kravis Roberts since 2023. It was founded in New York City in 1924, by Richard L. Simon and M. Lincoln Schuster. Along with Penguin Random House, Hachette Book Group US ...
in 1940. The authors are
Edward Kasner
Edward Kasner (April 2, 1878 – January 7, 1955) was an American mathematician who was appointed Tutor on Mathematics in the Columbia University Mathematics Department. Kasner was the first Jewish person appointed to a faculty position in ...
and
James R. Newman. The illustrator
Rufus Isaacs provided 169 figures. It rapidly became a best-seller and received several glowing reviews. Special publicity has been awarded it since it introduced the term
googol
A googol is the large number 10100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeros: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, ...
for 10
100, and
googolplex
A googolplex is the large number 10, or equivalently, 10 or . Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes; that is, a 1 followed by a googol of zeroes. Its prime factorization is 2 ×5.
History
In 1920, ...
for 10
googol. The book includes nine chapters, an annotated bibliography of 45 titles, and an index in its 380 pages.
Reviews
According to
I. Bernard Cohen, "it is the best account of modern mathematics that we have", and is "written in a graceful style, combining clarity of exposition with good humor".
According to T. A. Ryan's review, the book "is not as superficial as one might expect a book at the popular level to be. For instance, the description of the invention of the term ''googol'' ... is a very serious attempt to show how misused is the term ''infinite'' when applied to large and finite numbers."
By 1941 G. Waldo Dunnington could note the book had become a
best-seller
A bestseller is a book or other media noted for its top selling status, with bestseller lists published by newspapers, magazines, and book store chains. Some lists are broken down into classifications and specialties (novel, nonfiction book, cookb ...
. "Apparently it has succeeded in communicating to the layman something of the pleasure experienced by the creative mathematician in difficult problem solving."
Contents
The introduction notes (p xiii) "Science, particularly mathematics, ... appears to be building the one permanent and stable edifice in an age where all others are either crumbling or being blown to bits."
The authors affirm (p xiv) "It has been our aim, ... to show by its very diversity something of the character of mathematics, of its bold, untrammelled spirit, of how, both as an art and science, it has continued to lead the creative faculties beyond even imagination and intuition."
In chapter one, "New names for old", they explain why mathematics is ''the science that uses easy words for hard ideas''. They note (p 5) "many amusing ambiguities arise. For instance, the word
function probably expresses the most important idea in the whole
history of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...
. Also, the theory of
rings is much more recent than the theory of
groups. It is found in most of the new books on algebra, and has nothing to do with either matrimony or bells. Page 7 introduces the
Jordan curve theorem
In topology, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region Boundary (topology), bounded by the curve (not to be ...
. In discussing the
Problem of Apollonius, they mention that
Edmond Laguerre's solution considered circles with orientation.(p 13) In presenting ''radicals'', they say "The symbol for radical is not the
hammer and sickle
The hammer and sickle (Unicode: ) is a communist symbol representing proletarian solidarity between industrial and agricultural workers. It was first adopted during the Russian Revolution at the end of World War I, the hammer representing wo ...
, but a sign three or four centuries old, and the idea of a mathematical radical is even older than that." (p 16) "Ruffini and Abel showed that equations of the fifth degree could not be solved by radicals." (p 17) (
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means t ...
)
Chapter 2 "Beyond Googol" treats
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set ...
s. The distinction is made between a
countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
and an
uncountable set
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger t ...
. Further, the characteristic property of infinite sets is given: an infinite class may be in 1:1 correspondence with a proper subset (p 57), so that "an infinite class is no greater than some of its parts" (p 43). In addition to introducing
Aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used t ...
s the authors cite Lewis Carrol's ''
The Hunting of the Snark'', where instructions are given to avoid boojums when
snark hunting. They say "The infinite may be boojum too." (p 61)
Chapter 3 is "Pie (, i, e) Transcendental and Imaginary". To motivate
e (mathematical constant)
The number is a mathematical constant approximately equal to 2.71828 that is the base of a logarithm, base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Eule ...
, they discuss first
compound interest
Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.
Compo ...
and then
continuous compounding. "No other mathematical constant, not even , is more closely connected with human affairs" (p 86).
"
has played an integral part in helping mathematicians describe and predict what is for man the most important of all natural phenomena – that of growth."
The
exponential function, ''y'' = e
''x'' ... "is the only function of ''x'' with the rate of change with respect to ''x'' equal to the function itself." (p 87)
The authors define the
Gauss plane and describe the action of multiplication by i as rotation through 90°. They address
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality
e^ + 1 = 0
where
:e is E (mathematical constant), Euler's number, the base of natural logarithms,
:i is the imaginary unit, which by definit ...
, i.e. the expression ''e''
{{pi i + 1 = 0, indicating that the venerable
Benjamin Peirce
Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philoso ...
called it "absolutely paradoxical".
A note of idealism is then expressed: "When there is so much humility and so much vision everywhere, society will be governed by science and not its clever people." (pp 103,4)
Chapter 4 is "Assorted Geometries, Plane and Fancy". Both
Non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
and
four-dimensional space
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...
are discussed. The authors say (p 112) "Among our most cherished convictions, none is more precious than our beliefs about space and time, yet is more difficult to explain."
In the final pages the authors approach the question, "What is mathematics?" They say it is a "sad fact that it is easier to be clever than clear." The answer is not as easy as defining ''biology''. "
mathematics we have a universal language, valid, useful, intelligible everywhere in place and time ..." Finally, "Austere and imperious as logic, it is still sufficiently sensitive and flexible to meet each new need. Yet this vast edifice rests on the simplest and most primitive foundations, is wrought by imagination and logic out of a handful of childish rules." (p 358)
References
* I. Bernard Cohen (1942)
Review ''
Isis
Isis was a major goddess in ancient Egyptian religion whose worship spread throughout the Greco-Roman world. Isis was first mentioned in the Old Kingdom () as one of the main characters of the Osiris myth, in which she resurrects her sla ...
'' 33(6):723–5.
* G. Waldo Dunnington (1941)
Review ''
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...
'' 15(4):212–3.
* T.A. Ryan (1940)
Review ''
American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
'' 47(10):700–1.
External links
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.
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