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Math Without Numbers
An illustrated tour of the structures and patterns we call math
The only numbers in this book are the page numbers.
Math Without Numbers is a vivid, conversational, and wholly original guide to the three main branches of abstract math--topology, analysis, and algebra--which turn out to be surprisingly easy to grasp. This book upends the conventional approach to math, inviting you to think creatively about shape and dimension, the infinite and infinitesimal, symmetries, proofs, and how these concepts all fit together. What awaits readers is a freewheeling tour of the inimitable joys and unsolved mysteries of this curiously powerful subject.
Like the classic math allegory Flatland, first published over a century ago, or Douglas Hofstadter's Godel, Escher, Bach forty years ago, there has never been a math book quite like Math Without Numbers. So many popularizations of math have dwelt on numbers like pi or zero or infinity. This book goes well beyond to questions such as: How many shapes are there? Is anything bigger than infinity? And is math even true? Milo Beckman shows why math is mostly just pattern recognition and how it keeps on surprising us with unexpected, useful connections to the real world.
The ambitions of this book take a special kind of author. An inventive, original thinker pursuing his calling with jubilant passion. A prodigy. Milo Beckman completed the graduate-level course sequence in mathematics at age sixteen, when he was a sophomore at Harvard; while writing this book, he was studying the philosophical foundations of physics at Columbia under Brian Greene, among others.
The only numbers in this book are the page numbers.
Math Without Numbers is a vivid, conversational, and wholly original guide to the three main branches of abstract math--topology, analysis, and algebra--which turn out to be surprisingly easy to grasp. This book upends the conventional approach to math, inviting you to think creatively about shape and dimension, the infinite and infinitesimal, symmetries, proofs, and how these concepts all fit together. What awaits readers is a freewheeling tour of the inimitable joys and unsolved mysteries of this curiously powerful subject.
Like the classic math allegory Flatland, first published over a century ago, or Douglas Hofstadter's Godel, Escher, Bach forty years ago, there has never been a math book quite like Math Without Numbers. So many popularizations of math have dwelt on numbers like pi or zero or infinity. This book goes well beyond to questions such as: How many shapes are there? Is anything bigger than infinity? And is math even true? Milo Beckman shows why math is mostly just pattern recognition and how it keeps on surprising us with unexpected, useful connections to the real world.
The ambitions of this book take a special kind of author. An inventive, original thinker pursuing his calling with jubilant passion. A prodigy. Milo Beckman completed the graduate-level course sequence in mathematics at age sixteen, when he was a sophomore at Harvard; while writing this book, he was studying the philosophical foundations of physics at Columbia under Brian Greene, among others.
224 pages, Hardcover
First published January 5, 2021
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January 9, 2021
In some ways, this is the best book about pure mathematics for the general reader that I've ever seen. At first sight, Milo Beckman's assertion that 'the only numbers in this book are the page numbers' seems like one of those testing limits some authors place on themselves, such as Roberto Trotter's interesting attempt to explain cosmology using only the 1,000 most common words in the English language, The Edge of the Sky. But in practice, Beckman's conceit is truly liberating. Dropping numbers enables him to present maths (I can't help but wince a bit at the 'math' in the title) in a far more comprehensible way. Counting and geometry may have been the historical origin of mathematics, but it has moved on.
The book is divided into three primary sections - topology, analysis and algebra, plus a rather earnest dialogue on foundations of mathematics exploring the implications of Gödel's incompleteness theorems, and a closing section on modelling (including automata and 'science'). What this approach enables Beckman to do brilliantly is to move the image of mathematics away from school maths and onto what professional mathematicians spend their time on. What's more, and perhaps more impressively for a reader who has only ever been interested in applications, it gives the best appreciation I've seen of what the point of pure mathematics is - why some find it so delightful and interesting.
Along the way in those summary headings we come across shapes, manifolds, dimensions, infinity, maps, abstraction, structures and inference. We do eventually meet, for example, sets - though they come surprisingly late when taking a conventional view. Of course not everything can be covered in detail. Groups for example, crop up with brief coverage of both symmetry groups and wallpaper groups - but we are never told what a group is. Of course, most topics have to be handled distinctly briefly. This isn't a long book (I'd say it's just the right length to be enjoyable without being either trivial or getting bogged down), but Beckman fits a lot in.
I do have a couple of small issues. As mentioned, we're told from the start the only numbers in the book are the page numbers. This isn't strictly true - numbers as words crop up reasonably regularly. And though it does provide the freedom I mentioned, in one case - Cantor's diagonal argument for the infinity of the continuum - I found the non-numeric explanation far harder to get your head around than the traditional approach using numbers. It was also, perhaps, a little unfair to include (presumably as a diversion - they aren't given any context) a pair of logic puzzles without providing the solutions: one was straightforward, but the other had some issues. In terms of content, things went ever so slightly astray when Beckman strayed into science, telling us that Newton's gravitational relationship depended on the weights of the two bodies.
No book is perfect, though. The fact remains that Math Without Numbers is a brilliant introduction to pure mathematics and a delight from end to end.
The book is divided into three primary sections - topology, analysis and algebra, plus a rather earnest dialogue on foundations of mathematics exploring the implications of Gödel's incompleteness theorems, and a closing section on modelling (including automata and 'science'). What this approach enables Beckman to do brilliantly is to move the image of mathematics away from school maths and onto what professional mathematicians spend their time on. What's more, and perhaps more impressively for a reader who has only ever been interested in applications, it gives the best appreciation I've seen of what the point of pure mathematics is - why some find it so delightful and interesting.
Along the way in those summary headings we come across shapes, manifolds, dimensions, infinity, maps, abstraction, structures and inference. We do eventually meet, for example, sets - though they come surprisingly late when taking a conventional view. Of course not everything can be covered in detail. Groups for example, crop up with brief coverage of both symmetry groups and wallpaper groups - but we are never told what a group is. Of course, most topics have to be handled distinctly briefly. This isn't a long book (I'd say it's just the right length to be enjoyable without being either trivial or getting bogged down), but Beckman fits a lot in.
I do have a couple of small issues. As mentioned, we're told from the start the only numbers in the book are the page numbers. This isn't strictly true - numbers as words crop up reasonably regularly. And though it does provide the freedom I mentioned, in one case - Cantor's diagonal argument for the infinity of the continuum - I found the non-numeric explanation far harder to get your head around than the traditional approach using numbers. It was also, perhaps, a little unfair to include (presumably as a diversion - they aren't given any context) a pair of logic puzzles without providing the solutions: one was straightforward, but the other had some issues. In terms of content, things went ever so slightly astray when Beckman strayed into science, telling us that Newton's gravitational relationship depended on the weights of the two bodies.
No book is perfect, though. The fact remains that Math Without Numbers is a brilliant introduction to pure mathematics and a delight from end to end.
October 19, 2023
‘—“algorithm,� that’s someone’s name, Muhammad al-Khwarizmi. It’s like “Phillips-head screwdriver”—algorithm just means “This was al-Khwarizmi’s idea.� Algebra too, that’s a mispronunciation of an Arabic word, al-jabr, which there wasn’t a word for in any European language. It was used to describe when you flip a term to the opposite side of an equation. You know, algebra. So that’s all from Africa, yeah? Somewhere around what we now call North Africa�.�
The ‘beauty� of this book lies deeply in how ‘accessible�, and engaging the writing is from start to end. Concise, yet thorough; for all readers � including ‘kids� I’d argue (but don’t argue with me about it; I’m always tired and never in the mood for this) . Don’t even have to be ‘precocious�; just the regular, very average ones will do fine. Aside from the writing itself, the illustrations are brilliant too. Beckman uses ‘quirky� diction like ‘cool�, ‘cute�, ‘fun� � not sure if it’s with the intention to entice, or just because that’s simply how he had wanted to express himself in his writing/work. And that itself adds to why I think any consenting, willing reader would be able to ‘enjoy� the book, if not ‘understand� fully. I don’t see why/how one could not achieve the latter (because there isn’t anything to ‘solve� in the book; you just have to sit back and enjoy whatever Beckman ‘serves�) but leaving room for possibilities anyway.
’To an algebraist, isomorphism—the state of being isomorphic—is the pinnacle of elegance and beauty. Two unrelated situations that secretly have all the same underlying dynamics? Gorgeous. The world has been simplified down one notch. What used to be two different problems or, as the case may be, a hundred or infinity different problems, has been reduced to a single problem.�
Although admittedly, I had wanted more than just an ‘intro�, this felt kind of ‘refreshing� somehow; in any case, I do highly recommend the experience � looking at pure maths in Beckman’s perspective is well ‘fun�. It’s impressive to me how Beckman is able to introduce his ideas of (pure) maths the way he did. I’d even go as far as to say that I hope all maths teachers should have some of his wit and humour; and the patience and creativity to introduce and ‘teach� maths like this.
‘I’ve tried to avoid addressing this question directly because—professional mathematicians really don’t care about real-world applications. That’s the domain of applied math, the opposite of pure math, which should give you a sense for how the word “applied� is meant to sound.
What’s the use of talking about an imaginary fourth dimension when we’re sure we only have three? Why not just classify the manifolds up to three dimensions and call it a day? I can offer two responses: one from a pure mathematician and one from an applied mathematician.
To a pure mathematician, the question is missing the point. We’re not classifying manifolds to be useful. We’re just curious what possible different types of shapes can exist! We don’t have to constrain ourselves to this arbitrary world we happen to live in. Math is general, universal—it’s not made in our image. So we have three dimensions. And? We have ten fingers, and do we stop counting there? That list of sheet-manifolds was out there, somehow, before we ever wrote it down, and it’ll still be the complete list of sheet-manifolds long after our civilizations are lost to history. If that alone doesn’t make you curious about what types of manifolds exist in higher dimensions, just because they aren’t useful, well then you weren’t really in it for the right reasons to begin with.
Then the applied mathematician comes along and ruins everything by making topology useful.�
Of course it’s not ‘flawless�, but it’s Beckman’s first (and I wouldn’t even have thought it if I didn’t look it up), so be decent and cut him some slack? Anyway if you’re not that anal about it, you probably won’t even notice the cracks. But I’d rather this (in terms of ‘style�, ‘tone� and presentation/structure) than a book that reads like a conventional ‘textbook�. This was exactly what I had expected (and then more) and wanted so I can’t complain. It’s one thing to know a lot about a topic/subject, and it’s another to carry that information/knowledge to another well. Beckman can do both! Also, his ‘analogies� are so fucking brilliant.
‘Not all particles are created equal: They have slightly different properties which determine how they’ll move. Whenever you create a particle, you have to give it a “mass� (a positive number) and a “charge� (positive, negative, or zero). And you can’t just pick any mass and charge—there are only seventeen legal combinations of mass and charge to choose from. We call these combinations the seventeen fundamental particles, and we give each one a cute name like “charm quark� or “tau lepton.”�
Would have loved it more if he credited von Neumann (or even just ‘mention� him in his book) a bit rather than directing all his praises towards Gödel, but evidently these are minor/tiny (arguably even irrelevant) issues and I’m just being annoying about it. I blame Benjamin Labatut for this. Why? Read , get fucked (up).
‘There are gaps in our understanding, which you may find suspicious. No one can really say they know exactly how human behaviour arises from electric flashes in neuron circuitry. Artificial intelligence makes the idea plausible, but we haven’t worked out the precise mechanics. You can take this as an opening to argue that there’s something else going on here, some secret sauce that gets added at the level of human brains, which can’t be explained in terms of the interactions of quarks and electrons.
If this is true, if scientific naturalism is correct, then all of reality obeys strict mathematical rules. The entire universe must be identical to some carefully calibrated automaton. Everything going on around you, not to mention inside you, is a direct mathematical consequence of the laws of nature plus the initial configuration of the universe. Which is a pretty trippy thought.
It raises some big philosophical questions, to say the least. If you buy into some version of this naturalist framework—Are these mathematical rules actual, bona fide Laws of Nature, somehow governing the progression of the universe? Or does the universe exist and change in time as a brute fact, and these “rules� are merely patterns we’ve found in it?�
October 20, 2022
I did not understand any of this but I was AMAZED
January 18, 2021
Math Without Numbers is an illustrated tour of the structures and patterns we call "math" and a vivid, conversational, and wholly original guide to the three main branches of abstract math—topology, analysis, and algebra—which turn out to be surprisingly easy to grasp. This book upends the conventional approach to math, inviting you to think creatively about shape and dimension, the infinite and infinitesimal, symmetries, proofs, and how these concepts all fit together. What awaits readers is a freewheeling tour of the inimitable joys and unsolved mysteries of this curiously powerful subject. Like the classic math allegory Flatland, first published over a century ago, or Douglas Hofstadter's Godel, Escher, Bach forty years ago, there has never been a math book quite like Math Without Numbers. So many popularizations of math have dwelt on numbers like pi or zero or infinity. This book goes well beyond to questions such as: How many shapes are there? Is anything bigger than infinity? And is math even true?
Milo Beckman shows why math is mostly just pattern recognition and how it keeps on surprising us with unexpected, useful connections to the real world. The ambitions of this book take a special kind of author. An inventive, original thinker pursuing his calling with jubilant passion. A prodigy. Milo Beckman completed the graduate-level course sequence in mathematics at age sixteen, when he was a sophomore at Harvard; while writing this book, he was studying the philosophical foundations of physics at Columbia under Brian Greene, among others. As someone who dislikes maths quite a bit I found this an intriguing concept for a book and thankfully ended up finding it really accessible, understandable and ultimately fascinating. Written in a down to earth fashion, this helps you to see maths from a wholly different angle and allows even me, someone who despised maths as a kid, to find the concepts discussed interesting. A surprisingly engaging read, I urge anyone prone to avoiding the subject to give this a go. Highly recommended.
Milo Beckman shows why math is mostly just pattern recognition and how it keeps on surprising us with unexpected, useful connections to the real world. The ambitions of this book take a special kind of author. An inventive, original thinker pursuing his calling with jubilant passion. A prodigy. Milo Beckman completed the graduate-level course sequence in mathematics at age sixteen, when he was a sophomore at Harvard; while writing this book, he was studying the philosophical foundations of physics at Columbia under Brian Greene, among others. As someone who dislikes maths quite a bit I found this an intriguing concept for a book and thankfully ended up finding it really accessible, understandable and ultimately fascinating. Written in a down to earth fashion, this helps you to see maths from a wholly different angle and allows even me, someone who despised maths as a kid, to find the concepts discussed interesting. A surprisingly engaging read, I urge anyone prone to avoiding the subject to give this a go. Highly recommended.
April 18, 2021
I cannot believe this dude got me to read a proof and I enjoyed it (?)
June 14, 2021
Milo Beckman set out to show the reading public what real math is, without using numbers. Did he succeed? In some sense, I'm an unreliable reviewer, because I have a university degree in mathematics, so I already have an idea of what real math is. I know very well what Beckman means when he says that at some point, mathematicians stop worrying about or even using "numbers" (that is, quantities represented in Arabic numerals): it's both a joke and also a fact that real mathematicians hate doing arithmetic with numbers greater than, say, 10. And I appreciate the goal of trying to get this across using words and pictures rather than lots of scary notation.
The book is divided into sections for broad areas of the real-mathematics landscape: Topology, Analysis, Algebra, Foundations, and Modeling. Except for Foundations, each section has a few sub-sections. Most of the good stuff is in the first few sections, IMO: Beckman does a nice job of explaining topology's focus on "shape" and leading that into the example of classifying manifolds; he also sketches some good intuition about how the continuum (real line) is used in calculus. The Algebra section starts to suffer from disunity, but the first part about abstraction was pretty good. Modeling has its moments, but no cohesion.
My problems with the book are twofold, possibly describable as "choice of audience" and "choice of mathematics."
First, I started to become unsure who Beckman's target audience was. The selection of mathematics (see below) may or may not be interesting to a high school student, but sufficiently-interested students would probably want a meatier book anyway. He often affects a certain "mystery box" presentation ("There's X, but I can't tell you about X yet") that I found distracting and potentially condescending. It was mildly mysterious.
Second, the choice of what mathematics to present was a bit odd. While the continuum bits of the Analysis section were good, the section overall seemed a bit too trivia-oriented (types of infinities!); similarly with Algebra and Foundations. If the task had fallen to me, I might have gone for more of a themes or tools approach: linearity, invariance, and classification, for example, are three themes that show up across a wide range of mathematical topics. On the tools side, there might be simulation (including probabilistic methods!), induction, and abstraction, these last two being in the book already. These cross-threads could give the book a better sense of unity, and properly conveyed, could give close readers a sense of accomplishment for spotting the connections early.
As a minor point, the Foundations section seemed the most gratuitous, mostly because most "real" mathematicians don't care about foundations, but also because it's structured as a Hofstadterian dialog about the nature of mathematical truth. It spends some pages on a back-and-forth about the history of mathematics as a knowledge enterprise and its association for a few centuries with European imperialism. I'm not going to freak out like some other reviews and say "warrgarble wokeism," but to bring it up, only as shallowly as it was done here, seems more like flag-waving than enlightening or educating. (The other payoff is passing mention of Goedel's theorems, for which GOEDEL, ESCHER, BACH exists but also is something I think is better left to experts.) The Modeling section features other odd digressions, as when Beckman speculates about physics being like an automata (other examples: Conway's "game of life"), which is either naive-realist or Tegmarkian and I don't like either option. Science can use math as a description language but science isn't math!
2.5 stars rounded down. Nice try?
The book is divided into sections for broad areas of the real-mathematics landscape: Topology, Analysis, Algebra, Foundations, and Modeling. Except for Foundations, each section has a few sub-sections. Most of the good stuff is in the first few sections, IMO: Beckman does a nice job of explaining topology's focus on "shape" and leading that into the example of classifying manifolds; he also sketches some good intuition about how the continuum (real line) is used in calculus. The Algebra section starts to suffer from disunity, but the first part about abstraction was pretty good. Modeling has its moments, but no cohesion.
My problems with the book are twofold, possibly describable as "choice of audience" and "choice of mathematics."
First, I started to become unsure who Beckman's target audience was. The selection of mathematics (see below) may or may not be interesting to a high school student, but sufficiently-interested students would probably want a meatier book anyway. He often affects a certain "mystery box" presentation ("There's X, but I can't tell you about X yet") that I found distracting and potentially condescending. It was mildly mysterious.
Second, the choice of what mathematics to present was a bit odd. While the continuum bits of the Analysis section were good, the section overall seemed a bit too trivia-oriented (types of infinities!); similarly with Algebra and Foundations. If the task had fallen to me, I might have gone for more of a themes or tools approach: linearity, invariance, and classification, for example, are three themes that show up across a wide range of mathematical topics. On the tools side, there might be simulation (including probabilistic methods!), induction, and abstraction, these last two being in the book already. These cross-threads could give the book a better sense of unity, and properly conveyed, could give close readers a sense of accomplishment for spotting the connections early.
As a minor point, the Foundations section seemed the most gratuitous, mostly because most "real" mathematicians don't care about foundations, but also because it's structured as a Hofstadterian dialog about the nature of mathematical truth. It spends some pages on a back-and-forth about the history of mathematics as a knowledge enterprise and its association for a few centuries with European imperialism. I'm not going to freak out like some other reviews and say "warrgarble wokeism," but to bring it up, only as shallowly as it was done here, seems more like flag-waving than enlightening or educating. (The other payoff is passing mention of Goedel's theorems, for which GOEDEL, ESCHER, BACH exists but also is something I think is better left to experts.) The Modeling section features other odd digressions, as when Beckman speculates about physics being like an automata (other examples: Conway's "game of life"), which is either naive-realist or Tegmarkian and I don't like either option. Science can use math as a description language but science isn't math!
2.5 stars rounded down. Nice try?
January 11, 2021
Another on a growing list of enjoyable math books appealing to a wider audience. That being said, I do wonder what type of audience the author had in mind, as there are moments where the casualness of tone and hastiness of explanation did not do justice to the complexity of the idea, but maybe that's the point?
The chapter on Dimension was probably my favorite. I will DEFINITELY be using the faucet analogy to introduce polar coordinates from now on. 🚰
The chapter on Dimension was probably my favorite. I will DEFINITELY be using the faucet analogy to introduce polar coordinates from now on. 🚰
February 20, 2021
The first chapter (the sample they use to rope you in) is great, leading you to assume the rest of the book will be as good. Don’t be fooled, it quickly devolves into a mess of halfhearted “proofs� which often use numbers but just spell them out instead of using the number itself. Then the last third of the book stops being about math and becomes a ridiculous woke social commentary.
October 5, 2022
I discovered 'Math without Numbers' by Milo Beckman totally by accident. Sometimes these serendipitous discoveries turn out to be amazing and that is what happened.
At the beginning of the book, Milo Beckman asks the deceptively simple question, 'How many shapes are there?' And before long we are taken into a dizzying tour of different kinds of shapes and how some of them are similar though they look very different and we see the world of shapes in new ways. Then Beckman asks the question, 'Which is the biggest number?' He then proceeds to show us what that is, and while we are reeling from the amazement that we feel with the new things we learnt, Beckman ups the ante and asks us the question, 'Is there a number which is bigger than this biggest number?' The answer to that is even more surprising and amazing! Then Beckman asks the question, 'If someone says something, can we definitely prove that this statement is either true or false?' The obvious common sense answer to this, of course, is that a statement is either true or false. What else could it be? But Beckman shows that there is more to this than meets the eye, and what we discover at the end of this conversation is a revelation.
Beckman's style is conversational and friendly. The whole book is like having a conversation with a friend. It is beautiful. Beckman's breezy style and humour makes us smile. For example, this passage �
"Before you go tell your loved ones that you read a book about math and learned that a square is a circle, keep in mind: Context matters. A square is a circle, in topology. A square is most certainly not a circle in art or architecture, or in everyday conversation, or even in geometry, and if you try to ride a bike with square tires you won’t get very far."
And this one �
"When mathematicians talk about the fourth dimension, we’re not talking about time. We’re talking about a fourth geometric dimension, just like the first three. There’s up-down, left-right, forward-back, and then, let’s say, “flim-flam.� You know, another one."
As promised in the title, there are no numbers in the book. As Beckman is fond of saying, the only numbers which are there in the book are page numbers 😊
When we finish reading the book, we realize that we have been taken on a breezy, fascinating, whirlwind tour of topology, analysis, abstract algebra and Gödel's Incompleteness theorem. All complex parts of mathematics, most of which are taught at the master's level. We don't realize all that, of course, while reading the book, because Beckman makes it all sound simple.
I loved 'Math without Numbers'. It is one of the best books on mathematics and science that I've ever read. It brings out the magical beauty of mathematics to the general reader and it is an absolute pleasure to read. I'll just echo what Ian Stewart has said on the book's cover � "Everyone should read this delightful book."
Have you read Milo Beckman's book? What do you think about it?
At the beginning of the book, Milo Beckman asks the deceptively simple question, 'How many shapes are there?' And before long we are taken into a dizzying tour of different kinds of shapes and how some of them are similar though they look very different and we see the world of shapes in new ways. Then Beckman asks the question, 'Which is the biggest number?' He then proceeds to show us what that is, and while we are reeling from the amazement that we feel with the new things we learnt, Beckman ups the ante and asks us the question, 'Is there a number which is bigger than this biggest number?' The answer to that is even more surprising and amazing! Then Beckman asks the question, 'If someone says something, can we definitely prove that this statement is either true or false?' The obvious common sense answer to this, of course, is that a statement is either true or false. What else could it be? But Beckman shows that there is more to this than meets the eye, and what we discover at the end of this conversation is a revelation.
Beckman's style is conversational and friendly. The whole book is like having a conversation with a friend. It is beautiful. Beckman's breezy style and humour makes us smile. For example, this passage �
"Before you go tell your loved ones that you read a book about math and learned that a square is a circle, keep in mind: Context matters. A square is a circle, in topology. A square is most certainly not a circle in art or architecture, or in everyday conversation, or even in geometry, and if you try to ride a bike with square tires you won’t get very far."
And this one �
"When mathematicians talk about the fourth dimension, we’re not talking about time. We’re talking about a fourth geometric dimension, just like the first three. There’s up-down, left-right, forward-back, and then, let’s say, “flim-flam.� You know, another one."
As promised in the title, there are no numbers in the book. As Beckman is fond of saying, the only numbers which are there in the book are page numbers 😊
When we finish reading the book, we realize that we have been taken on a breezy, fascinating, whirlwind tour of topology, analysis, abstract algebra and Gödel's Incompleteness theorem. All complex parts of mathematics, most of which are taught at the master's level. We don't realize all that, of course, while reading the book, because Beckman makes it all sound simple.
I loved 'Math without Numbers'. It is one of the best books on mathematics and science that I've ever read. It brings out the magical beauty of mathematics to the general reader and it is an absolute pleasure to read. I'll just echo what Ian Stewart has said on the book's cover � "Everyone should read this delightful book."
Have you read Milo Beckman's book? What do you think about it?
January 18, 2024
Hovered a little between 4 and 5 stars, but ultimately had to take a step back and think about how this book made me feel: I loved it.
I gravitate much more naturally to applied math. I think pure math is cool in a sort of wondrous way, if also intimidating, but nothing gives me more pleasure and comfort than being able to explain the world around me with math or make use of it for some greater purpose. I purposely avoided certain pure math courses in college - and for the ones I couldn't avoid, I struggled against some of the very concepts Beckman introduces in his Analysis and Foundations sections. After reading this book I regretted how strongly I pushed against some of these (very challenging!) concepts. Beckman takes ideas that are extremely difficult to explain and conceptualize and he makes them not only accessible, but truly provocative. I felt literal swells of excitement for large portions and a renewed interest in things I haven't thought about in maybe 10 years.
I do agree with another review I saw, which mentioned that the "omission of numbers" or reduction/simplification of ideas is occasionally taken a little too far. The Pythagorean theorem proof, for example, I actually found harder to follow than any other Pythagorean proof I've read, in part because it was so drastically reduced. Maybe it was an issue with the illustrations (which I otherwise found delightful throughout the book).
I gravitate much more naturally to applied math. I think pure math is cool in a sort of wondrous way, if also intimidating, but nothing gives me more pleasure and comfort than being able to explain the world around me with math or make use of it for some greater purpose. I purposely avoided certain pure math courses in college - and for the ones I couldn't avoid, I struggled against some of the very concepts Beckman introduces in his Analysis and Foundations sections. After reading this book I regretted how strongly I pushed against some of these (very challenging!) concepts. Beckman takes ideas that are extremely difficult to explain and conceptualize and he makes them not only accessible, but truly provocative. I felt literal swells of excitement for large portions and a renewed interest in things I haven't thought about in maybe 10 years.
I do agree with another review I saw, which mentioned that the "omission of numbers" or reduction/simplification of ideas is occasionally taken a little too far. The Pythagorean theorem proof, for example, I actually found harder to follow than any other Pythagorean proof I've read, in part because it was so drastically reduced. Maybe it was an issue with the illustrations (which I otherwise found delightful throughout the book).
April 7, 2023
genuinely never gonna forget this book. it writes about the branches of mathematics from a completely accessible and abstract perspective, so you can see the universality and purpose of pure mathematics. as someone who only took up to calculus, this makes me want to learn more about high-level mathematics from a theoretical level.
i have so many extrapolations and notes from the modeling and automaton fundamentals to our psychologically-felt existential consciousness. the idea that math insinuates that reality is a “simulation�, or really a continuously pattern-creating external, and also in another chapter creates the backbone to cognitive psychology (our perceived reality is based on pattern-recognizing/creation- depending on how you imply the patterns origin) is so sick. i always write that we’re “mirrors facing mirrors�, and it’d hold up in a philosophical-math approach to consciousness.
we’re just the universe opening up its eyes to its self - but also awareness of self is the construct of consciousness, so we are of an existential level higher than all previously-felt matter, and are thus both in active and passive roles.
i have so many extrapolations and notes from the modeling and automaton fundamentals to our psychologically-felt existential consciousness. the idea that math insinuates that reality is a “simulation�, or really a continuously pattern-creating external, and also in another chapter creates the backbone to cognitive psychology (our perceived reality is based on pattern-recognizing/creation- depending on how you imply the patterns origin) is so sick. i always write that we’re “mirrors facing mirrors�, and it’d hold up in a philosophical-math approach to consciousness.
we’re just the universe opening up its eyes to its self - but also awareness of self is the construct of consciousness, so we are of an existential level higher than all previously-felt matter, and are thus both in active and passive roles.
July 1, 2024
i actually really do love math
June 12, 2024
لو حابب تقرا و تتعلم عن الرياضيات ك فلسفة تفكير
قبل ما تبقى اثبات رياضية ب رموز و حرفية معينة
ف اليك هذا الكتاب
..و الكتاب نفسه يثير هذه النقطة
ايام ارسطو و سقراط و افلاطون
كانت الاستنتاجات الرياضية بديهية قبل ان تكون حرفية و مكتوبة ب لغة رياضية بحتة
كتاب ممتع جدا يتدرج فى شرح مفاهيم معينة لك
بل و اثباتها بشكل رسومات او حكي و سرد و تدرج منطقي و فكري
دونما اى رموز رياضية
لا اعرف لماذا لا يدرسون مثل تلك الكتب فى المدرسة بدلا من المناهج التلقينية
كيف تقوم بعملية القسمة
ب 3 طرق مختلفة ...!!
مش الاول افهم فلسفة عملية القسمة ؟ يعني ايه عملية قسمة
و بعدها "تقنيا" ابقى اتعلم طرقة و التانية و التالتة
بدل ما اتعلمهم و ابقى بلاعة لطرق الرياضيات دونما فهم لماهيتها و لما هى مجعولة و فلسفتها
كتاب اخر يكمل ما نقص هنا و هى الرموز و الارقام
يعلمك كيف تقرؤها و كيف تكتبها
كعلماء الرياضيات
كتاب يتحدث عن تأريخ الرياضيات
اين و متي تم اكتشاف كل اكتشاف رياضي معين
اقترح عليك هذا الكتاب
للكلام عن المسلمات او البديهات الخمس
و هى لازم لبدء اى اثبات او استنتاج رياضي
مسلمات جوزيبيني
اقترح عليك كتاب عنهم
اما عن الالغاز والفوازير و الارقام ذات الطبيعة و الخصائص الطريفة
اقترح عليك
كتب
أثارت نفس النقطة ... ان يكتشف عربي مسلم ف الدولة العباسية او الاموية
او استاذ صيني من عهد سلالة مينج
اكتشاف "رياضي" ما
و يظل مكتوم او يتم تجاهله لسنين
ثم يجيء اوروبي بارون او دوق او ايرل او كونت يعيد اكتشافه بعد مئات السنين
فيثبت فى الكتب ان النظرية الفلانية او الاستنتاج الفلاني هو للاوروبي
متجاهلين عن عمد دور العربي و الهندي و الصيني فى الاكتشافات الرياضية
الكتاب القادم :
قبل ما تبقى اثبات رياضية ب رموز و حرفية معينة
ف اليك هذا الكتاب
..و الكتاب نفسه يثير هذه النقطة
ايام ارسطو و سقراط و افلاطون
كانت الاستنتاجات الرياضية بديهية قبل ان تكون حرفية و مكتوبة ب لغة رياضية بحتة
كتاب ممتع جدا يتدرج فى شرح مفاهيم معينة لك
بل و اثباتها بشكل رسومات او حكي و سرد و تدرج منطقي و فكري
دونما اى رموز رياضية
لا اعرف لماذا لا يدرسون مثل تلك الكتب فى المدرسة بدلا من المناهج التلقينية
كيف تقوم بعملية القسمة
ب 3 طرق مختلفة ...!!
مش الاول افهم فلسفة عملية القسمة ؟ يعني ايه عملية قسمة
و بعدها "تقنيا" ابقى اتعلم طرقة و التانية و التالتة
بدل ما اتعلمهم و ابقى بلاعة لطرق الرياضيات دونما فهم لماهيتها و لما هى مجعولة و فلسفتها
كتاب اخر يكمل ما نقص هنا و هى الرموز و الارقام
يعلمك كيف تقرؤها و كيف تكتبها
كعلماء الرياضيات
كتاب يتحدث عن تأريخ الرياضيات
اين و متي تم اكتشاف كل اكتشاف رياضي معين
اقترح عليك هذا الكتاب
للكلام عن المسلمات او البديهات الخمس
و هى لازم لبدء اى اثبات او استنتاج رياضي
مسلمات جوزيبيني
اقترح عليك كتاب عنهم
اما عن الالغاز والفوازير و الارقام ذات الطبيعة و الخصائص الطريفة
اقترح عليك
كتب
أثارت نفس النقطة ... ان يكتشف عربي مسلم ف الدولة العباسية او الاموية
او استاذ صيني من عهد سلالة مينج
اكتشاف "رياضي" ما
و يظل مكتوم او يتم تجاهله لسنين
ثم يجيء اوروبي بارون او دوق او ايرل او كونت يعيد اكتشافه بعد مئات السنين
فيثبت فى الكتب ان النظرية الفلانية او الاستنتاج الفلاني هو للاوروبي
متجاهلين عن عمد دور العربي و الهندي و الصيني فى الاكتشافات الرياضية
الكتاب القادم :
March 28, 2025
I’m basically a mathematician now.
March 4, 2022
I heard Milo Beckman interviewed on Tim Harford's podcast and thought the book he was promoting would be interesting. Well, I was wrong.
The book itself is well-written and explains the subject matter reasonably well. It's just that it has no real point. Why is this information being explained? Why should I know this? What should I do with this? It's nice that the information exists, but it lacks a purpose. Worse, the book seems to lack internal connections--at least to the point in the book where I decided to give up. Why do these subjects go together? How do they build on one another? Why would any average person give a damn?
So finally about 110 pages in, I gave up. It was theory without purpose or direction. I don't have the energy for that. This book is the equivalent of a drive without a destination and without the scenery to make the ride worth taking.
The book itself is well-written and explains the subject matter reasonably well. It's just that it has no real point. Why is this information being explained? Why should I know this? What should I do with this? It's nice that the information exists, but it lacks a purpose. Worse, the book seems to lack internal connections--at least to the point in the book where I decided to give up. Why do these subjects go together? How do they build on one another? Why would any average person give a damn?
So finally about 110 pages in, I gave up. It was theory without purpose or direction. I don't have the energy for that. This book is the equivalent of a drive without a destination and without the scenery to make the ride worth taking.
January 18, 2021
Never did I ever think I'd consider the dimensions of personality, or vector maps in my morning coffee, yet here we are. I'm not massively interested in maths (or indeed very good at it) but I wanted to read something different hence I chose "Maths Without Numbers" by Milo Beckman. Some of my favourite parts were the sections about shapes and topology. I enjoyed the analogies throughout (e.g. room allocation at Hotel Infinity). The concept of continuum just about made my head explode. The application of graph theory in social media was particularly informative. Overall an interesting read, accessible to all readers.
June 7, 2021
WOW.
This book is so accessible, taking complicated subjects and translating them lovingly into digestible portions. I have always been enamored by math and numbers, but intimidated to approach the art of M A T H E M A T I C S, seeing it as a whole new language only for those with lots of training and prestige. I will be referring back to this book when I feel overwhelmed in math, especially with modelling. I loved the chapter on algebra, it was really eye-opening.
The illustrations were charming and helped visualize some of the concepts.
I did see one or two numbers in this book though, spelling them out still counts!
This book is so accessible, taking complicated subjects and translating them lovingly into digestible portions. I have always been enamored by math and numbers, but intimidated to approach the art of M A T H E M A T I C S, seeing it as a whole new language only for those with lots of training and prestige. I will be referring back to this book when I feel overwhelmed in math, especially with modelling. I loved the chapter on algebra, it was really eye-opening.
The illustrations were charming and helped visualize some of the concepts.
I did see one or two numbers in this book though, spelling them out still counts!
August 8, 2023
لا أجرؤ على وصفه بإنه كتاب رياضي، لكنه باستخدام أساليب بسيطة يحوم حول مفاهيم رياضية مركبة مثل:
Manifolds
Poincaré's conjecture
Symmetric & Dihedral groups
Gödel's incompleteness theorem
Standard Model
Manifolds
Poincaré's conjecture
Symmetric & Dihedral groups
Gödel's incompleteness theorem
Standard Model
December 1, 2021
Hay quienes dicen que la mejor forma de aprender es cuando no te das cuenta de que estás aprendiendo. Sin embargo, creo que darte cuenta de que estás aprendiendo y que, además, estás disfrutando del proceso, es mucho más valioso, porque conlleva el reconocimiento de que quien te enseña es alguien que ama enseñar y está disfrutando tanto como tú. Y eso es algo que Milo Beckman logra en Math Without Numbers: un libro sobre matemáticas en el que no hay números, a excepción de los números de las páginas. Es en serio.
Beckman es un autor muy joven, un prodigio de las matemáticas que a los 16 años ya estudiaba en Harvard fundamentos de la física con Brian Greene. Ha habido muchos prodigios matemáticos como Beckman, pero pocos han tenido la desbordante creatividad que él ha demostrado en su primer libro.
¿Se puede enseñar matemáticas sin números? Beckman se lanza a probar que sí. El libro está dividido en tres grandes áreas de las matemáticas abstractas: topología (fascinante y delirante), análisis (reflexiva e inquietante), y álgebra (nada que ver con el árido libro de Baldor y su icónica ilustración de Al Juarismi en la portada).
Para mí, #mathwithoutnumbers de Beckman es una propuesta lúdica, filosófica y científica que debería implementarse en todos los colegios y escuelas. Pensar en el infinito, visualizar dimensiones superiores o inferiores a nuestra familiar tercera dimensión, arriesgarnos a resolver acertijos y comprender la lógica de los números, el orden y caos del universo y la belleza de lo simple y lo complejo, es una aventura necesaria y casi irresistible.
Leer este libro es una de las mejores cosas que pueden hacer este año.
Beckman es un autor muy joven, un prodigio de las matemáticas que a los 16 años ya estudiaba en Harvard fundamentos de la física con Brian Greene. Ha habido muchos prodigios matemáticos como Beckman, pero pocos han tenido la desbordante creatividad que él ha demostrado en su primer libro.
¿Se puede enseñar matemáticas sin números? Beckman se lanza a probar que sí. El libro está dividido en tres grandes áreas de las matemáticas abstractas: topología (fascinante y delirante), análisis (reflexiva e inquietante), y álgebra (nada que ver con el árido libro de Baldor y su icónica ilustración de Al Juarismi en la portada).
Para mí, #mathwithoutnumbers de Beckman es una propuesta lúdica, filosófica y científica que debería implementarse en todos los colegios y escuelas. Pensar en el infinito, visualizar dimensiones superiores o inferiores a nuestra familiar tercera dimensión, arriesgarnos a resolver acertijos y comprender la lógica de los números, el orden y caos del universo y la belleza de lo simple y lo complejo, es una aventura necesaria y casi irresistible.
Leer este libro es una de las mejores cosas que pueden hacer este año.
November 26, 2024
I saw this in the Strand and was excited because it looked like a great refresher of some of my favorite classes from college, without all the hard math behind those classes. It was definitely high-level and maybe a bit too general for me to gain much, but to be fair I think that’s the point. It’s a really good overview of topology, analysis, and abstract algebra for the general public, and I loved how the ending even touched a bit on the Eurocentrism of math. Fun logic puzzles thrown in too!
March 3, 2022
Really extraordinary. If only Milo had been my professor, my life would’ve looked so different!
September 24, 2024
Całkiem przyjemna lektura, ale ja zawsze lubiłem matematykę. Wydanie które czytałem ma niestety trochę błędów (głównie niedopasowanie ilustracji do opisów).
December 21, 2021
The book is a nice, non-intimidating , introduction to pure/abstract mathematics with a little divergence to applied mathematics at the end. The book starts by introducing the topic of topology which is the study of the shapes that stay the same under deformations and the shapes of space that they live in. Then it moves to the topic of analysis, where the concept of infinity, the size of infinity, and maps are introduced. In the third part of the book, the author talks about abstract algebra which is the study of abstract objects and the relations and patterns that govern them. Before the author diverges back to applied mathematics and how mathematics are used to model and describe scientific laws, we take a little detour to the problems in the foundations of mathematics, namely the idea of self-referential statements and Godel's incompleteness theorem, which was an interesting read.
I have to say that the author did a really good job in approaching seemingly unapproachable topics in pure mathematics; the language was easy to follow and read and I really loved his emphasis on visualizations and visual proofs.
However, in the foundations section, I didn't care much for the discussion on the sociology of mathematical developments; it felt out of place, unnecessary, and not connected to what follows. Also, in the Science chapter, the author overplayed the reductionism argument. While the author himself admits that he may be overplaying it, I think it was a bit dangerous for someone who's curious about math and science (but not yet experienced) to be subjected to such overplay without recognizing the ongoing philosophical debate around the validity of reductionism.
All in all, I'd recommend this book to any curious mind who wants to take a peek into the abstract world, but with caution. This book sure made me want to study these fields more rigorously! I hope I'm able to do that some day.
I have to say that the author did a really good job in approaching seemingly unapproachable topics in pure mathematics; the language was easy to follow and read and I really loved his emphasis on visualizations and visual proofs.
However, in the foundations section, I didn't care much for the discussion on the sociology of mathematical developments; it felt out of place, unnecessary, and not connected to what follows. Also, in the Science chapter, the author overplayed the reductionism argument. While the author himself admits that he may be overplaying it, I think it was a bit dangerous for someone who's curious about math and science (but not yet experienced) to be subjected to such overplay without recognizing the ongoing philosophical debate around the validity of reductionism.
All in all, I'd recommend this book to any curious mind who wants to take a peek into the abstract world, but with caution. This book sure made me want to study these fields more rigorously! I hope I'm able to do that some day.
May 24, 2020
I thought this book was quite delightful and the main characters were different math terms or different branches of math. This was so well explained and researched it reminded me about some of the math concepts I actually enjoy. I was someone who had severe math anxiety at a young age and through most of high school until I discovered polynomials and algebra; math I was actually good at doing.
I wish math had been introduced like this from the beginning. I like the early chapters on topology and manifolds. This book definitely brings out the inner nerd in everyone. I will definitely be purchasing a hard copy of this when it is available because this makes math fun again.
Thanks to Netgalley, Milo Beckham and Dutton, a division of Penguiin Group publishing for an ARC in exchange for an honest review. I really enjoyed this!
Available:: 10/27/20
I wish math had been introduced like this from the beginning. I like the early chapters on topology and manifolds. This book definitely brings out the inner nerd in everyone. I will definitely be purchasing a hard copy of this when it is available because this makes math fun again.
Thanks to Netgalley, Milo Beckham and Dutton, a division of Penguiin Group publishing for an ARC in exchange for an honest review. I really enjoyed this!
Available:: 10/27/20
October 18, 2023
نحن لا نبتكر الرياضيات لتناسب عالمنا، نحن نكتشف ماهية الرياضيات الموجودة فيه، ثم ندرك لاحقاً أن عالمنا يشبه هذه الرياضيات. "
في هذا الكتاب يشرح لك ميلو بيكمان المفاهيم الاساسية للرياضيات ( أسس الرياضيات والجبر والتحليل والطوبولوجيا والنمذجة ) ، الشيء اللي يميز الكتاب إنه مليئ بالامثلة والرسوم التوضيحية لشرح هذه المفاهيم بعيداً عن ذكر الأرقام ، ومابين الفصول راح تجد عدد من الألغاز الممتعة تسليك في حلها وفهمها .
الشرح في بداية الكتاب بسيط وممتع خصوصاً فكرة الطوبولوجيا ومعلومات الأعداد اللانهائية ، ولكن كل ما تتقدم في الكتاب يصيبك الملل والفتور من أنك تكمله، واعتقد ان السبب مشترك مابين صعوبة الشيء اللي يبي يوصله بيكمان وطريقة المترجم في توصيل الفكرة للقارئ.
في هذا الكتاب يشرح لك ميلو بيكمان المفاهيم الاساسية للرياضيات ( أسس الرياضيات والجبر والتحليل والطوبولوجيا والنمذجة ) ، الشيء اللي يميز الكتاب إنه مليئ بالامثلة والرسوم التوضيحية لشرح هذه المفاهيم بعيداً عن ذكر الأرقام ، ومابين الفصول راح تجد عدد من الألغاز الممتعة تسليك في حلها وفهمها .
الشرح في بداية الكتاب بسيط وممتع خصوصاً فكرة الطوبولوجيا ومعلومات الأعداد اللانهائية ، ولكن كل ما تتقدم في الكتاب يصيبك الملل والفتور من أنك تكمله، واعتقد ان السبب مشترك مابين صعوبة الشيء اللي يبي يوصله بيكمان وطريقة المترجم في توصيل الفكرة للقارئ.
January 21, 2021
Not that I have tried that hard, but when I got this book, I finally was able to enjoy some of the theory of math without the distraction of the process. Milo Beckman brings a fun personality to a serious subject, and he offers basic explanations to complex concepts. This achievement shows his brilliance. These mathematical theory building blocks will stick with me for years to come. I have to say though, as I got toward the middle-to-end of the book, I got pretty bored.
January 22, 2021
This is an engaging and delightful book. Milo Beckman's excitement is contagious - math is a way of seeing and understanding the world, a way of thinking about things. He's humorous & friendly, so if you've previously approached math feeling intimidated, that's over. It's an adventure, and will show you as much about mathematicians as it will about math. Highly recommend!
May 2, 2021
A very clever and concise way of describing the main branches of mathematics.
March 15, 2024
(3.5)
Przyjemna lekturka dla kogoś zaciekawionego tematem, ale nie w temacie bo naprawdę bez liczb, typowo popnaukowo
Przyjemna lekturka dla kogoś zaciekawionego tematem, ale nie w temacie bo naprawdę bez liczb, typowo popnaukowo
March 13, 2024
The title feels a bit misleading - “Math without Numbers� suggests an explanation of basic mathematical concepts without numbers. Instead, this book is more “The Math that People Do That Does Not Focus on Numbers.� Of course, if this book was about the former topic, I wouldn’t have read it; if it had the latter title, no one else would have read it.
This was a very light and easy-to-read exploration of pure mathematics. To be fair, I study pure mathematics, so I can’t speak to the layperson’s experience. But it’s not very long, and the prose reads more like a conversation than a textbook.
I appreciated the author’s focus on why things are the way they are. For example, the Algebra section started with a chapter explaining the purposes and power of abstraction, which set a good groundwork for introducing the abstract work that algebraists do. “Moving symbols around on paper that turn statements into other statements� feels more meaningful once you understand the power of those general statements. The ability to craft a statement that is true about an infinite number of objects is definitely a powerful one!
I did find myself wishing for references or footnotes for the proofs and discussions of math history. Also, I found it strange that the author said pure mathematicians don’t care about application, so neither should we - and then discussed applications in nearly every section. (And I’ve never met a pure mathematician who isn’t extremely excited about real-world applications of their areas of study!) Overall, though, I think the author made these choices to make the book more accessible and engaging, so I can’t really complain.
This was a very light and easy-to-read exploration of pure mathematics. To be fair, I study pure mathematics, so I can’t speak to the layperson’s experience. But it’s not very long, and the prose reads more like a conversation than a textbook.
I appreciated the author’s focus on why things are the way they are. For example, the Algebra section started with a chapter explaining the purposes and power of abstraction, which set a good groundwork for introducing the abstract work that algebraists do. “Moving symbols around on paper that turn statements into other statements� feels more meaningful once you understand the power of those general statements. The ability to craft a statement that is true about an infinite number of objects is definitely a powerful one!
I did find myself wishing for references or footnotes for the proofs and discussions of math history. Also, I found it strange that the author said pure mathematicians don’t care about application, so neither should we - and then discussed applications in nearly every section. (And I’ve never met a pure mathematician who isn’t extremely excited about real-world applications of their areas of study!) Overall, though, I think the author made these choices to make the book more accessible and engaging, so I can’t really complain.
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