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Beyond Infinity: An Expedition to the Outer Limits of Mathematics

de Eugenia Cheng

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A mathematician and scientist in residence at the School of the Art Institute of Chicago helps readers explore the concept of infinity through unique concepts including chessboards, a chicken-sandwich sandwich and the creation of infinite cookies from an infinite dough ball.
Adicionado recentemente porca.bookwyrm, Den85, Dgoliber, pinax, Bumb, dubeyak, Henry.Pole-Carew
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I was prepared to be confounded. I came to this book after reading Roger Penrose's book "Fashion, faith and fantasy in the new physics of the universe", where he casually raises concepts like infinity raised to the power of infinity squared. So I was hoping that Eugenia Cheng's book might give me a bit more understanding of concepts about infinity and how they can be manipulated mathematically. Did I come away with this sort of knowledge? No! the sort of calculations that were being thrown around by Penrose were really beyond the scope of this book. But I did come away with a vague grasp of how mathematicians play around with infinity.
There was a throw away line that infinity was something different to a number..and that kind of made sense to me. However when I went back and tried to find that statement I came across things like: "infinity can't be an integer", And "infinity can't be a rational (ratio) number", and "infinity is not a real number (rational and irrational numbers), but Infinity IS a cardinal number and an ordinal number.
And Infinity plus 1 doesn't equal infinity. However 1 plus infinity DOES equal infinity.
I have to hand it to Eugenia, she has a beautiful way of writing (especially for a mathematician) and is able to use some great analogies to help in understanding. OK...I understood most of the stuff as I was reading it....it made sense. But ask me now to explain Hilbert's hotel and I have only the vaguest notion of a hotel with infinite rooms...and all the rooms are full and a busload of additional people show up (or a busload with infinite people show up) and are fitted into Hilbert's hotel by having all the existing residents move up a few rooms ...or an infinite number of rooms. Does this make sense? Not if you have the concept of infinity being something definite. But I like her example of a circle being something that you can run around infinite times and never come to the end.
In talking about time as the fourth dimension she makes the point that if you're meeting up with someone you need to stipulate the place (2 or three dimensions) but also the time....a fourth dimension.
Actually, As I'm writing this review, I realise how little of the book a can really recall in enough detail to be able to explain to somebody else. (I just tried to convince my son about the truth of the non-commutative nature of infinity described above...and his response was ... "Nope....I've just done higher level maths and that doesn't apply". Well so much for the higher level maths he was taught ...and so much for my explaining powers).
I think I need to come back to this book. Somewhere I've read about Cantor's work in more detail than Eugenia goes into and I think I need to study this a bit more.
But overall, a really impressive book......and clearly quite an impressive person. (She is apparently the author of numerous books plus being a concert level musician). Does this mean that she is the product of a "Tiger mum"? I expect so. Anyway, happy to give it five stars and put it in my "re-read pile. (Though my new books are piling up at a much faster rate than I'm reading them...so chances of getting to the re-read pile are slim. ( )
  booktsunami | Jul 23, 2022 |
Mathematics is much more fun without the calculating. Eugenia Cheng understands this; her math class at the School of the Art Institute is titled "The Elegance of Abstraction," which itself is a nice bit of generalization for the math averse. Cheng here plays mind games with infinity, where some things are more infinite than others.

Digressions on food, music and sports help her confront her task with rigor without seeming to go on forever. She fondly recalls the two-line programming loop she wrote on her old Sinclair computer ("I was the kind of child who was not easily bored, so I could do this every day without feeling the urge to write more useful programs.") And she checks in frequently to the infinitely large Hilbert Hotel, named for mathematician David Hilbert, always booked solid but with room enough for one more guest.

I've always been better with logic than long division, so I appreciate that Cheng admits to getting answers wrong working them out by hand. Her rumination on math are like Douglas Hofstadter's 1980s-vintage Scientific American semantic essays, paced so well that it's hard to notice that you've waded into the weeds. A sidebar on how people count reminded me of the eight-fingered aliens who in fourth grade introduced me to set theory (of course they used octal numerals!) and saved math from the ravages of third grade. (I decided Mickey Mouse would make a fine 8-bit programmer.) Cheng's deft use of the physics-for-poets approach should appeal to young adults and the generally curious.
  rynk | Jul 11, 2021 |
Read the Wikipedia article on Hilbert's hotel and Infinite series and you'll learn much more than this book has to offer. The rest is just the author fawning over how quirky and zany she is. That and cutesy stories about children and her childhood. You will be puking rainbows by chapter 3. She also replaced the boring counting of apples with eating cake - truly ground breaking.

The best way by which it conveys infinity is by how many times you get to read about her "10 print 20 goto 10" program. ( )
1 vote Paul_S | Dec 23, 2020 |
There are some big numbers out there, footballers earn a jaw dropping amount per year, for what I am not entirely sure… The global economy is around US$107.5 trillion, there are approximately seven quintillion, five hundred quadrillion grains of sand on the earth and it is thought that there are 10 times as many stars as that. All of these numbers are frankly huge, enormous, gargantuan even, but compared to ∞ they are a mere drop in the ocean. In this book, Eugenia Cheng takes us on a journey to the outer reaches of the mathematical universe to contemplate the slightly abstract concept that is infinity. In it she poses various questions about this number, asking if 1 ∞ is larger than ∞ 1, are some infinities larger than others, can you fit an infinite number of people in Hilbert's Hotel and when does a number start becoming irrational.

Thankfully this book has lots of diagrams as Cheng sets about explaining the concepts of infinity, from the very simplest right up to the most detailed. I found most of it straightforward, but occasionally it was fairly tough going. When trying to get your head around infinity has challenged mathematicians for ages so it is not going to be easy for us mere mortals. Cheng endeavours to keep the prose readable, however, someone who has not picked up a maths book since school might struggle with this, but most of the time she gets the concepts across clearly. Overall a good introduction to infinity. ( )
  PDCRead | Apr 6, 2020 |
Eugenia Cheng ha un modo molto personale di parlare di matematica: nel suo libro precedente paragonava la teoria delle categorie al cucinare dolci, qui il viaggio verso l'infinito è costellato di ricordi di vita personale. Il lettore che sappia già un po' di cose sulla teoria degli infiniti forse non troverà molto di nuovo; il libro in fin dei conti si limita a sfiorare i problemi posti dalla teoria cantoriana. Il neofita invece troverà un grande vantaggio, perché Cheng batte moltissimo sulla constatazione che non possiamo fare matematica dell'infinito con le regole solite, e soprattutto mostra esplicitamente dove e come queste regole saltano. In questo modo si evitano tutti i danni dell'approccio scolastico tipico "si deve fare così", che rende la matematica troppo simile a una collezione di articoli di legge da imparare a memoria. (Sì, anche in diritto le leggi non nascono dal nulla ma hanno un razionale: ma ci siamo capiti). ( )
  .mau. | Nov 6, 2018 |
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A mathematician and scientist in residence at the School of the Art Institute of Chicago helps readers explore the concept of infinity through unique concepts including chessboards, a chicken-sandwich sandwich and the creation of infinite cookies from an infinite dough ball.

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