''From Here to Infinity: A Guide to Today's Mathematics'', a 1996 book by mathematician and science popularizer Ian Stewart, is a guide to modern

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

for the general reader. It aims to answer questions such as "What is mathematics?", "What is it for " and "What are mathematicians doing nowadays?". Author

Simon Singh Simon Lehna Singh, (born 19 September 1964) is a British popular science author, theoretical and particle physicist. His written works include ''Fermat's Last Theorem'' (in the United States titled ''Fermat's Enigma: The Epic Quest to Solve th ...

describes it as "An interesting and accessible account of current mathematical topics".My Favourite Mathematics Books
{{Webarchive, url=https://web.archive.org/web/20080917053516/http://simonsingh.net/Mathematics_Books.html , date=2008-09-17 , Simon Singh

Summary

After an introductory chapter ''The Nature of Mathematics'', Stewart devotes each of the following 18 chapters to an exposition of a particular problem that has given rise to new mathematics or an area of research in modern mathematics. *Chapter 2 - ''The Price of Primality'' -

primality test A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whet ...

s and integer factorisation *Chapter 3 - ''Marginal Interest'' -

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...

*Chapter 4 - ''Parallel Thinking'' -

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 geo ...

*Chapter 5 - ''The Miraculous Jar'' -

Cantor's theorem In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A, the set of all subsets of A, the power set of A, has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be ...

and

cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...

s *Chapter 6 - '' Ghosts of Departed Quantities'' -

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

and

non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...

*Chapter 7 - ''The Duellist and the Monster'' - the

classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ...

*Chapter 8 - ''The Purple Wallflower'' - the

four colour theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sha ...

*Chapter 9 - ''Much Ado About Knotting'' -

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

and the

Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...

*Chapter 10 - ''More Ado About Knotting'' -

knot polynomial In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. History The first knot polynomial, the Alexander polynomial, was introdu ...

s *Chapter 11 - ''Squarerooting the Unsquarerootable'' -

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s and the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...

*Chapter 12 - ''Squaring the Unsquarable'' - the Banach-Tarski paradox *Chapter 13 - ''Strumpet Fortune'' -

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and

random walks In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

*Chapter 14 - ''The Mathematics of Nature'' - the

stability of the Solar System The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ...

*Chapter 15 - ''The Patterns of Chaos'' -

chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...

and

strange attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...

s *Chapter 16 - ''The Two-and-a-halfth Dimension'' -

fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...

s *Chapter 17 - ''Dixit Algorizmi'' -

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

s and

NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by tryi ...

problems *Chapter 18 - ''The Limits of Computability'' -

Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...

s and

computable number In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive ...

s *Chapter 19 - ''The Ultimate in Technology Transfer'' -

experimental mathematics Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with th ...

and the relationship between mathematics and science

Editions

Important advances in mathematics necessitated revisions of the book. For example, when the 1st edition came out,

was still an open problem. By the 3rd edition, it has been solved by

Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awar ...

. Other revised topics include

Tarski's circle-squaring problem Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Mikl ...

Carmichael numbers In number theory, a Carmichael number is a composite number n, which in modular arithmetic satisfies the congruence relation: :b^n\equiv b\pmod for all integers b. The relation may also be expressed in the form: :b^\equiv 1\pmod. for all integers ...

, and the

Kepler Problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...

. * 1st edition (1987): published under the title ''The Problems of Mathematics'' * 2nd edition (1992) * retitled/revised edition (1996)

References

Books by Ian Stewart (mathematician) Popular mathematics books