''Beyond Infinity : An Expedition to the Outer Limits of Mathematics'' is a

popular mathematics Popular mathematics is the presentation of mathematics to an aimed general audience. The difference between recreational mathematics and popular mathematics is that recreational mathematics intends to be fun for the mathematical community, and p ...

book by

Eugenia Cheng Eugenia Loh-Gene Cheng is a British mathematician and concert pianist. Her mathematical interests include higher category theory, and as a pianist she specialises in lieder and art song. She is also passionate about explaining mathematics to ...

centered on concepts of

infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...

. It was published by

Basic Books Basic Books is a book publisher founded in 1950 and located in New York, now an imprint of Hachette Book Group. It publishes books in the fields of psychology, philosophy, economics, science, politics, sociology, current affairs, and history. H ...

and (with a slightly different title) by

Profile Books Profile Books is a British independent book publishing firm founded in 1996. It publishes non-fiction subjects including history, biography, memoir, politics, current events, current affairs, travel and popular science. Profile Books is distribu ...

in 2017, and in a paperback edition in 2018. It was shortlisted for the 2017 Royal Society Insight Investment Science Book Prize.

Topics

The book is divided into two parts, with the first exploring notions leading to concepts of

actual infinity In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extend ...

, concrete but infinite mathematical values. After an exploration of

number system A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

s, this part discusses

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

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, and

ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...

transfinite arithmetic In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...

, and the existence of different infinite sizes of sets. Topics used to illustrate these concepts include

Hilbert's paradox of the Grand Hotel Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely m ...

Cantor's diagonal argument In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...

, and the unprovability of the

continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

. The second part concerns mathematics related to the idea of

potential infinity In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, exten ...

, the assignment of finite values to the results of infinite processes including growth rates,

limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

, and

infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

. This part also discusses

Zeno's paradoxes Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in pluralit ...

Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rat ...

s, the

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

s of spaces, and the possibility of spaces of infinite dimensions, with a mention of

higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher cate ...

, Cheng's research specialty. The mathematics is frequently lightened and made accessible with personal experiences and stories, involving such subjects as the

Loch Ness Monster The Loch Ness Monster ( gd, Uilebheist Loch Nis), affectionately known as Nessie, is a creature in Scottish folklore that is said to inhabit Loch Ness in the Scottish Highlands. It is often described as large, long-necked, and with one or mor ...

puff pastry Puff pastry, also known as ', is a flaky light pastry made from a laminated dough composed of dough (') and butter or other solid fat ('). The butter is put inside the dough (or vice versa), making a ' that is repeatedly folded and rolled out befo ...

, boating, dance contests, shoes, "Legos, the iPod Shuffle, snorkeling, Battenberg cakes and Winnie-the-Pooh".

Audience and reception

The Royal Society judges called ''Beyond Infinity'' "a very engaging introduction to a forbidding subject". Similarly, reviewer Anne Haworth calls it "engaging and readable", and ''

Wall Street Journal ''The Wall Street Journal'' is an American business-focused, international daily newspaper based in New York City, with international editions also available in Chinese and Japanese. The ''Journal'', along with its Asian editions, is published ...

'' reviewer Sam Kean writes that its "chatty tone keeps things fresh". It is aimed at a popular audience, not assumed to have a significant background in mathematics, including "the young or those brimming with curiosity" as well as college or secondary-school students, although it may be "too elementary for mathematicians or mathematics students". As similar reading material, reviewer Andrew James Simoson suggests placing this book alongside ''The Book of Numbers'' by

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

and

Richard K. Guy Richard Kenneth Guy (30 September 1916 – 9 March 2020) was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathemati ...

(1996), '' One Two Three... Infinity'' by

George Gamow George Gamow (March 4, 1904 – August 19, 1968), born Georgiy Antonovich Gamov ( uk, Георгій Антонович Гамов, russian: Георгий Антонович Гамов), was a Russian-born Soviet and American polymath, theoreti ...

(1947), and ''Really Big Numbers'' by Richard Schwartz (2014).

References

{{reflist, refs= {{citation, title=Review of ''Beyond Infinity'', publisher=European Mathematical Society, work=EMS Reviews, date=April 2017, first=Adhemar, last=Bultheel, author-link=Adhemar Bultheel, url=https://euro-math-soc.eu/review/beyond-infinity-expedition-outer-limits-mathematics {{citation, title=Review of ''Beyond Infinity'', work=MAA Reviews, publisher=Mathematical Association of America, date=April 2019, first=Zdeňka, last=Guadarrama, url=https://www.maa.org/press/maa-reviews/beyond-infinityan-expedition-to-the-outer-limits-of-mathematics {{citation , last = Haworth , first = Anne , date = June 2021 , doi = 10.1017/mag.2021.100 , issue = 563 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, pages = 381–382 , title = Review of ''Beyond Infinity'' , volume = 105 {{citation, last=Kean, first=Sam, newspaper=

The Wall Street Journal ''The Wall Street Journal'' is an American business-focused, international daily newspaper based in New York City, with international editions also available in Chinese and Japanese. The ''Journal'', along with its Asian editions, is published ...

, url=https://www.wsj.com/articles/the-neverending-story-1491433940, date=April 5, 2017, title=The Neverending Story (review of ''Beyond Infinity'') {{citation, url=https://www.publishersweekly.com/978-0-465-09481-3, magazine=Publishers Weekly, title=Review of ''Beyond Infinity'' {{citation, url=https://royalsociety.org/grants-schemes-awards/book-prizes/science-book-prize/2017/beyond-infinity/, title=''Beyond Infinity: An Expedition to the Outer Limits of the Mathematical'' by Eugenia Cheng, work=2017 Royal Society Insight Investment Science Book Prize , publisher=Royal Society, access-date=2021-08-29 {{citation, title=Review of ''Beyond Infinity'', first=Andrew James, last=Simoson, mr=3617029 Popular mathematics books 2017 non-fiction books Infinity Basic Books books Profile Books books