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Very Short Introductions #519
Infinity: A Very Short Introduction
Infinity is an intriguing topic, with connections to religion, philosophy, metaphysics, logic, and physics as well as mathematics. Its history goes back to ancient times, with especially important contributions from Euclid, Aristotle, Eudoxus, and Archimedes. The infinitely large (infinite) is intimately related to the infinitely small (infinitesimal). Cosmologists consider sweeping questions about whether space and time are infinite. Philosophers and mathematicians ranging from Zeno to Russell have posed numerous paradoxes about infinity and infinitesimals. Many vital areas of mathematics rest upon some version of infinity. The most obvious, and the first context in which major new techniques depended on formulating infinite processes, is calculus. But there are many others, for example Fourier analysis and fractals.
In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectual exercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite.
In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectual exercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite.
160 pages, Paperback
First published January 1, 2017
45 people are currently reading
466 people want to read
About the author
Ian Stewart
270 books758 followersIan Nicholas Stewart is an Emeritus Professor and Digital Media Fellow in the Mathematics Department at Warwick University, with special responsibility for public awareness of mathematics and science. He is best known for his popular science writing on mathematical themes.
--from the author's website
Librarian Note: There is more than one author in the GoodReads database with this name. See other authors with similar names.
--from the author's website
Librarian Note: There is more than one author in the GoodReads database with this name. See other authors with similar names.
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Displaying 1 - 13 of 13 reviews
May 10, 2024
The following are some quotes that made me resonate with the author's writing and my thoughts on topics he covered on the philosophical portion of the book while the part dealing with the mathematics is laid out in a pretty straight forward manner.
"In virtually every area of the physical sciences, infinity is an embarrassment. A theory that predicts infinities is wrong. That doesn’t mean it’s useless, but it needs tweaking to get rid of those pesky infinities. However, there’s one area of physics in which an actual infinity—physical, not conceptual—is not just tolerated, but presented as a possible truth: cosmology."
Prof. Penrose's theories on the Universal concept of the Cosmos, Consciousness, etc, and even the controversial Many-worlds quantum school postulate that Consciousness is inherently an integral aspect of reality which is also hinted upon in The Kybalion.
I would recommend this to anyone curious about a short as possible introduction to Infinity covering both the mathematical and philosophical questions as a whole.
"THE ALL is MIND; the Universe is Mental.”The author goes on to say that
—THE KYBALION
This Principle embodies the truth that “All is Mind.” It explains that THE ALL (which is the Substantial Reality underlying all the outward manifestations and appearances which we know under the terms of “The Material Universe”; the “Phenomena of Life”; “Matter”; “Energy”; and, in short, all that is apparent to our material senses) is SPIRIT, which in itself is UNKNOWABLE and UNDEFINABLE, but which may be considered and thought of as A UNIVERSAL, INFINITE, LIVING MIND. It also explains that all the phenomenal world or universe is simply a Mental Creation of THE ALL, subject to the Laws of Created Things, and that the universe, as a whole, and in its parts or units, has its existence in the Mind of THE ALL, in which Mind we “live and move and have our being..."
"In virtually every area of the physical sciences, infinity is an embarrassment. A theory that predicts infinities is wrong. That doesn’t mean it’s useless, but it needs tweaking to get rid of those pesky infinities. However, there’s one area of physics in which an actual infinity—physical, not conceptual—is not just tolerated, but presented as a possible truth: cosmology."
Prof. Penrose's theories on the Universal concept of the Cosmos, Consciousness, etc, and even the controversial Many-worlds quantum school postulate that Consciousness is inherently an integral aspect of reality which is also hinted upon in The Kybalion.
I would recommend this to anyone curious about a short as possible introduction to Infinity covering both the mathematical and philosophical questions as a whole.
May 3, 2022
الكتاب بيعرض مفهوم المالانهاية في الدين والفلسفة والرياضيات، والتركيز الأكبر كان في الرياضيات وتعرض أيضا لمفهوم السببية وأفكار من قبيل هل الزمن له بداية أم هو عبارة عن حلقة دائرية ليس لها بداية أو نهاية. وتغير مفهوم المالانهاية بنشأة وتطوير أفكار وأدوات جديدة في الفلسفة والرياضيات.
الجزء المتعلق بالرياضيات كان الأكثر تسلية وإمتاعا بالنسبة لي، رغم عدم فهمي التام لبعض الفقرات وأنا تخصصي الجامعي رياضيات، مما يجعل بعض أجزاء الكتاب في اعتقادي غير مناسبة للبعض. ومع ذلك أسلوب الكاتب سلس لحد كبير، فهو وضّح مفاهيم وأفكار ليست ببسيطة بطريقة موفقة ورائعة في معظم الأحيان.
الكتاب يطرح تساؤلات جيدة مثيرة للتفكير عن مفهومنا للمالانهاية، فهي درجات.. نعم، هناك مالانهاية أكبر من أخرى، كمثال: الأعداد الحقيقية والأعداد الطبيعية. في الحقيقة توجد درجات من الملانهاية وهو ما أثبته كانتور.
ما هو ناتج جمع مالانهاية وواحد؟ تعلمنا في المدرسة أنا الناتج يساوي مالانهاية. نعم ولكن هذا صحيح في أحد المفاهيم للملانهاية، بينما مالانهاية زائد واحد أكبر من المالانهاية بأحد المفاهيم الأخرى. ذلك يعتمد على أي مفهوم نتعامل معه.
وعالم الرياضيات الألماني چورچ كانتور دخل قائمتي للشخصيات المُحتذى بهم المُتطلع إليهم - فئة العلماء، بعد قراءة ذلك الكتاب مؤكد.
الجزء المتعلق بالرياضيات كان الأكثر تسلية وإمتاعا بالنسبة لي، رغم عدم فهمي التام لبعض الفقرات وأنا تخصصي الجامعي رياضيات، مما يجعل بعض أجزاء الكتاب في اعتقادي غير مناسبة للبعض. ومع ذلك أسلوب الكاتب سلس لحد كبير، فهو وضّح مفاهيم وأفكار ليست ببسيطة بطريقة موفقة ورائعة في معظم الأحيان.
الكتاب يطرح تساؤلات جيدة مثيرة للتفكير عن مفهومنا للمالانهاية، فهي درجات.. نعم، هناك مالانهاية أكبر من أخرى، كمثال: الأعداد الحقيقية والأعداد الطبيعية. في الحقيقة توجد درجات من الملانهاية وهو ما أثبته كانتور.
ما هو ناتج جمع مالانهاية وواحد؟ تعلمنا في المدرسة أنا الناتج يساوي مالانهاية. نعم ولكن هذا صحيح في أحد المفاهيم للملانهاية، بينما مالانهاية زائد واحد أكبر من المالانهاية بأحد المفاهيم الأخرى. ذلك يعتمد على أي مفهوم نتعامل معه.
وعالم الرياضيات الألماني چورچ كانتور دخل قائمتي للشخصيات المُحتذى بهم المُتطلع إليهم - فئة العلماء، بعد قراءة ذلك الكتاب مؤكد.
July 1, 2018
The “Very Short Introduction” series by Oxford University Press attempt to take a moderately deep dive into various subjects in a slimline volume. Professor Stuart addresses the apparent paradox of tackling the subject of the infinite in such a small volume right at the start, along with the observation that the topic of infinity has long provided such paradoxes. This particular VSI aims to tackle infinity as found in numbers, geometry, art, theology, philosophy and more, and so it is a tightly packed volume indeed.
Infinity is a concept that is, at least now, embraced in Mathematics, but also reaches into Physics, Philosophy, Theology and Language in significant measure. In this book, Ian Stewart sets off almost immediately into the mathematical interpretations and concepts of infinity, starting with examples that are likely to be accessible to a wide range of readers, but also touching on some that will cause more mathematically advanced readers to consider them carefully and which may be challenging to less mathematically literate readers.
These examples are of the paradoxical issues surrounding infinity; all but one of these is explicitly mathematical, some geometrical, others more algebraic; by the end of the first short chapter we have visited David Hilbert’s famous hotel and explored some of the implications for the arithmetic of the infinite.
The second chapter then moves into a more detailed exploration of the consequences of the infinite in numbers, and in particular explores the infinite, non-repeating decimal representations of irrational numbers and the continuity of the real numbers.
Stewart then explores the history of the infinite in the third chapter, and how it weaved through early Greek philosophy and the classic paradoxes of Zeno, and how for the Greeks issues of infinity were closely tied to their thoughts and theories about motion, and indeed whether motion was in fact possible or an illusion. Some time is spent with Aristotle, and how he dismissed the idea of an “actual” infinity in favour of a “potential” infinity. We then move through with both Locke and Kant to the beginning of more modern philosophical analyses of the infinite. Some time is taken to explore the philosophy of the infinite in Christian theology, particularly through Thomas Aquinas, a philosopher heavily influenced by Aristotle and how he used the infinite in his “proof” of God.
Stewart also explores how, in the modern era, mathematicians take the infinite very much as a normal and integral part of mathematics, with little concern about the distinction of actual and potential infinities that were the great concern of the philosophy of the ancient world.
We dive then, from the infinitely large to the infinitely small in the fourth chapter where the seeds of calculus and analysis are to be seen, and the philosophical objections from Bishop Berkeley to the use of infinitesimals. It is interesting to note that these and other concerns about the theoretical underpinnings of calculus were largely ignored in the face of its obvious utility, until others tried to explore these foundations more deeply. Stewart takes us through this work through Cauchy and eventually to the work of Bolzano and Weierstrass who finally introduced the e and d notation that has undoubtedly delighted many undergraduates since and ushered in the start of analysis proper. Stewart than dips into an examination of non-standard analysis, a topic that at least I was never knowingly exposed to in my formal studies; it was intriguing to read of these numbers with “standard” and “infinitesimal” parts.
There follows a chapter on the geometrically infinite, which in particular looks at the role of the infinite in art, but which again after an informal discussion dips into the mathematics of what is going on. The chapter after this focuses on infinities that arise in Physics, particularly in optics, Newtonian and Relativistic gravity, moving on then to discuss the size of the known universe and its curvature. These two chapters are both short and may require some unpacking by readers with less background knowledge.
The final chapter is mostly dedicated to work of Cantor and his systemization of modern mathematical thinking around the concept of the infinite. Here we meet the distinctions between the finite, countably infinite and uncountably infinite, transfinite cardinals and transfinite ordinals. But even here we find the objections of some philosophers, in this case Wittgenstein. This is interesting to read in an era where Cantor’s formulations are considered uncontroversial and part and parcel of the “paradise” of Hilbert’s modern mathematics in the same way that the past controversies of complex numbers are of little interest to modern mathematicians.
The approach taken to infinity in the book, is non-apologetically Pure Mathematical in its spirit, and I suppose this may make the work a little less accessible for some readers, particularly those who are not prepared to think through some of the sections, perhaps with a pen and paper. The Very Short Introduction to Mathematics, from the same series, by Timothy Gowers similarly tackles a cross section of challenging examples from the discipline in a relatively small space.
In September 2016, the BBC aired an interesting series on Radio 4: “The History of the Infinite” (this is still happily available online for those interested, at least for those in the UK). In this series, Adrian Moore began discussing the original Greek antipathy to the idea in early philosophy, and then how the idea emerged through Aristotelian Philosophy, Christian theology. It was after this that Moore decided to tackle the more serious implications of the infinitely small and big in mathematics, before emerging back through Physics into more philosophical territory.
I suspect this route, sandwiching the more complicated mathematical treatment between philosophy more related to human experience could be more palatable to a general reader.
The Very Short Introduction to Infinity is nevertheless a fascinating and joyful exploration of the topic, accessible to the committed and careful novice, but with enough detail and asides to delight formally mathematically trained readers.
Infinity is a concept that is, at least now, embraced in Mathematics, but also reaches into Physics, Philosophy, Theology and Language in significant measure. In this book, Ian Stewart sets off almost immediately into the mathematical interpretations and concepts of infinity, starting with examples that are likely to be accessible to a wide range of readers, but also touching on some that will cause more mathematically advanced readers to consider them carefully and which may be challenging to less mathematically literate readers.
These examples are of the paradoxical issues surrounding infinity; all but one of these is explicitly mathematical, some geometrical, others more algebraic; by the end of the first short chapter we have visited David Hilbert’s famous hotel and explored some of the implications for the arithmetic of the infinite.
The second chapter then moves into a more detailed exploration of the consequences of the infinite in numbers, and in particular explores the infinite, non-repeating decimal representations of irrational numbers and the continuity of the real numbers.
Stewart then explores the history of the infinite in the third chapter, and how it weaved through early Greek philosophy and the classic paradoxes of Zeno, and how for the Greeks issues of infinity were closely tied to their thoughts and theories about motion, and indeed whether motion was in fact possible or an illusion. Some time is spent with Aristotle, and how he dismissed the idea of an “actual” infinity in favour of a “potential” infinity. We then move through with both Locke and Kant to the beginning of more modern philosophical analyses of the infinite. Some time is taken to explore the philosophy of the infinite in Christian theology, particularly through Thomas Aquinas, a philosopher heavily influenced by Aristotle and how he used the infinite in his “proof” of God.
Stewart also explores how, in the modern era, mathematicians take the infinite very much as a normal and integral part of mathematics, with little concern about the distinction of actual and potential infinities that were the great concern of the philosophy of the ancient world.
We dive then, from the infinitely large to the infinitely small in the fourth chapter where the seeds of calculus and analysis are to be seen, and the philosophical objections from Bishop Berkeley to the use of infinitesimals. It is interesting to note that these and other concerns about the theoretical underpinnings of calculus were largely ignored in the face of its obvious utility, until others tried to explore these foundations more deeply. Stewart takes us through this work through Cauchy and eventually to the work of Bolzano and Weierstrass who finally introduced the e and d notation that has undoubtedly delighted many undergraduates since and ushered in the start of analysis proper. Stewart than dips into an examination of non-standard analysis, a topic that at least I was never knowingly exposed to in my formal studies; it was intriguing to read of these numbers with “standard” and “infinitesimal” parts.
There follows a chapter on the geometrically infinite, which in particular looks at the role of the infinite in art, but which again after an informal discussion dips into the mathematics of what is going on. The chapter after this focuses on infinities that arise in Physics, particularly in optics, Newtonian and Relativistic gravity, moving on then to discuss the size of the known universe and its curvature. These two chapters are both short and may require some unpacking by readers with less background knowledge.
The final chapter is mostly dedicated to work of Cantor and his systemization of modern mathematical thinking around the concept of the infinite. Here we meet the distinctions between the finite, countably infinite and uncountably infinite, transfinite cardinals and transfinite ordinals. But even here we find the objections of some philosophers, in this case Wittgenstein. This is interesting to read in an era where Cantor’s formulations are considered uncontroversial and part and parcel of the “paradise” of Hilbert’s modern mathematics in the same way that the past controversies of complex numbers are of little interest to modern mathematicians.
The approach taken to infinity in the book, is non-apologetically Pure Mathematical in its spirit, and I suppose this may make the work a little less accessible for some readers, particularly those who are not prepared to think through some of the sections, perhaps with a pen and paper. The Very Short Introduction to Mathematics, from the same series, by Timothy Gowers similarly tackles a cross section of challenging examples from the discipline in a relatively small space.
In September 2016, the BBC aired an interesting series on Radio 4: “The History of the Infinite” (this is still happily available online for those interested, at least for those in the UK). In this series, Adrian Moore began discussing the original Greek antipathy to the idea in early philosophy, and then how the idea emerged through Aristotelian Philosophy, Christian theology. It was after this that Moore decided to tackle the more serious implications of the infinitely small and big in mathematics, before emerging back through Physics into more philosophical territory.
I suspect this route, sandwiching the more complicated mathematical treatment between philosophy more related to human experience could be more palatable to a general reader.
The Very Short Introduction to Infinity is nevertheless a fascinating and joyful exploration of the topic, accessible to the committed and careful novice, but with enough detail and asides to delight formally mathematically trained readers.
February 4, 2021
This is mostly about infinity from the mathematical perspective, though it does touch a bit on how infinity is considered in physics, philosophy, and religion. Sometimes the mathematics is a little rushed: unfamiliar terms are introduced without being defined and things can become a jumble of jargon if you're not already familiar with the terminology.
August 16, 2018
Slightly confusing in certain statements and proofs, though that may just be on my lack of education.
Currently reading
June 29, 2021Contents
list of illustrations xv
Introduction 1
1. Puzzles,proofs, and paradoxes 8
2. Encounters with the infinite 19
3. Historical views of infinity 32
4. The flipside of infinity 54
5. Geometric infinity 70
6. Physical infinity 91
7. Counting infinity 103
References 131
Further reading 133
Publisher's acknowledgements 137
Index 139
list of illustrations xv
Introduction 1
1. Puzzles,proofs, and paradoxes 8
2. Encounters with the infinite 19
3. Historical views of infinity 32
4. The flipside of infinity 54
5. Geometric infinity 70
6. Physical infinity 91
7. Counting infinity 103
References 131
Further reading 133
Publisher's acknowledgements 137
Index 139
July 25, 2017
A great book to come back to whenever I am in danger of thinking I actuallyn know something about anything.
January 6, 2021
Very interesting and a good level of detail :D would reccomend
August 29, 2023
To preface, I am a physics student who could care less about math for math's sake.
Chapters one through four were great, grabbing and kept my attention with lots of facts and cool anecdotes and history, after that I struggled to finish the book. After dragging myself through the last three chapters I found that none of it stuck. Now I love the Very Short Introduction series, and it usually stays true to its word, it introduces you to the topic with little to no prior knowledge required, and I feel that this book didn't hit that mark. If you already know quite a bit about math, be my guest, but otherwise, this book does not feel like a Very Short Introduction.
Chapters one through four were great, grabbing and kept my attention with lots of facts and cool anecdotes and history, after that I struggled to finish the book. After dragging myself through the last three chapters I found that none of it stuck. Now I love the Very Short Introduction series, and it usually stays true to its word, it introduces you to the topic with little to no prior knowledge required, and I feel that this book didn't hit that mark. If you already know quite a bit about math, be my guest, but otherwise, this book does not feel like a Very Short Introduction.
April 28, 2022
My ratings of books on Goodreads are solely a crude ranking of their utility to me, and not an evaluation of literary merit, entertainment value, social importance, humor, insightfulness, scientific accuracy, creative vigor, suspensefulness of plot, depth of characters, vitality of theme, excitement of climax, satisfaction of ending, or any other combination of dimensions of value which we are expected to boil down through some fabulous alchemy into a single digit.
August 22, 2023
A useful introduction to the concept(s) of infinity, made accessible by Ian Stewart's deft writing. At some points, the book becomes unnecessarily challenging when it suddenly leaps from accessible to incredibly technical.
June 30, 2023
They should make the authors of these collab with actual writers
Displaying 1 - 13 of 13 reviews