Are Numbers Real?: The Uncanny Relationship of Mathematics and the Physical World

by Brian Clegg

Have you ever wondered what humans did before numbers existed? How they organized their lives, traded goods, or kept track of their treasures? What would your life be like without them?Numbers began as simple representations of everyday things, but mathematics rapidly took on a life of its own, occupying a parallel virtual world. In Are Numbers Real?, Brian Clegg explores the way that math has become more and more detached from reality, and yet despite this is driving the development of modern physics. From devising a new counting system based on goats, through the weird and wonderful mathematics of imaginary numbers and infinity, to the debate over whether mathematics has too much influence on the direction of science, this fascinating and accessible book opens the reader’s eyes to the hidden reality of the strange yet familiar entities that are numbers.

Genres: Science, Nonfiction, Mathematics, History, Physics, Philosophy

303 pages, Kindle Edition

Published December 6, 2016

About the author

Brian's latest books, Ten Billion Tomorrows and How Many Moons does the Earth Have are now available to pre-order. He has written a range of other science titles, including the bestselling Inflight Science, The God Effect, Before the Big Bang, A Brief History of Infinity, Build Your Own Time Machine and Dice World.

Along with appearances at the Royal Institution in London he has spoken at venues from Oxford and Cambridge Universities to Cheltenham Festival of Science, has contributed to radio and TV programmes, and is a popular speaker at schools. Brian is also editor of the successful www.popularscience.co.uk book review site and is a Fellow of the Royal Society of Arts.

Brian has Masters degrees from Cambridge University in Natural Sciences and from Lancaster University in Operational Research, a discipline originally developed during the Second World War to apply the power of mathematics to warfare. It has since been widely applied to problem solving and decision making in business.

Brian has also written regular columns, features and reviews for numerous publications, including Nature, The Guardian, PC Week, Computer Weekly, Personal Computer World, The Observer, Innovative Leader, Professional Manager, BBC History, Good Housekeeping and House Beautiful. His books have been translated into many languages, including German, Spanish, Portuguese, Chinese, Japanese, Polish, Turkish, Norwegian, Thai and even Indonesian.

Reviews

[Rossdavidh · Rating 3 out of 5]

The first thing this book has to do, before it tries to answer the question of whether or not numbers are real, is to explain what the question even means. It's sort of obvious, in one sense, that they are not concrete, you can't touch them. I mean, you can touch two of something, but you can't touch the "two" part, only the "something". On the other hand, it seems pretty obvious that the number 2 (or 3 or pi or whatever) are not arbitrary and made-up in the same sense as, say, fictional characters or prices or words for things. Clegg's go-to example of how we use numbers in the ordinary, two-of-something sense is goats. There are a lot of scenarios involving you and your deadbeat neighbor, who keeps borrowing your goats, and who cannot be trusted to bring them all back if you don't count. The early chapters involve a lot of pre-history, some of it necessarily speculative, and much of what he is trying to explain is that humans can, and often have, lived their life without using numbers above the level of 3 or so. There was a point, at which it became important (and therefore common) to know exactly how many of something you had, as opposed to just one, two, or several or many. At that point, numbers were "born". But did they already exist, and we just started noticing them, or did we make them?

There is also, inevitably, some time spent in Plato's cave, with the "real" numbers behind us, and the shadows on the wall our lives, the "two of something" being the shadow, and the real (?) two behind us. But, Clegg would like us to consider that metaphor before we accept it, if we even do.

Then, we move on, through not quite as obviously real numbers like -2, the square root of 2, and the square root of -2. We discuss whether Pythagoras did, or did not, actually have someone killed because they were going to spill the beans that the square root of 2 was irrational. Then, we end up with Newton and Leibniz, wrestling with infinitely small things and finally getting a handle on Zeno's paradox. Eventually, we arrive at modern cosmology, writing of universes of ten (or perhaps eleven) dimensions, all but three of which are curled up very small, and we get the decided impression that somewhere in there, we maybe did leave the land of numbers that were real, and ended up in the land of people just playing with symbols in a way that has nothing to do with reality (Clegg is more diplomatic than I).

How far does a person get to go, away from counting goats, before you are no longer doing something real and are instead just playing games? The question is especially complicated by the fact that on more than one occasion, what appeared to be simply playing with math to the generation that did it, ended up being useful, even necessary, to a later generation of engineers.

So, I have to say that Clegg did most definitely convince me that it is legitimate to question whether or not math (if not numbers exactly) is real. What is less clear, is whether he answered the question, but it may be that truly understanding the nature, and breadth, of the question is more important here than answering it.

[Steve Kimmins · Rating 4 out of 5]

A pleasant and thought provoking popular science read. An author whose popular science reviews on GR I’ve enjoyed for some time but this is the first book by him that I’ve read.

The issue mainly addressed in this book is the apparently uncanny ability of mathematics to represent, model, even predict, physical phenomena. Is this because the universe is in some sense intrinsically underpinned by mathematics or because we’ve created a tool which, when used in a particular manner, can be used accurately in models we construct?
The author very much takes the latter view, of maths being a useful tool of human origin, a particular type of language we’ve created and that we should be cautious in pushing it beyond certain limits in a search for a deeper ‘truth’. I found myself very much on his side, especially as someone who’s spent a long career in engineering, despite having graduated in Physics.
In particular the author seems pretty frustrated that researchers of leading edge physics (anything at the forefront of sub-atomic physics, for example, such as ‘String Theory’) are trying to develop incredibly complex mathematical equations into predictive models without any real grounding from experiments. He makes an excellent case for being very sceptical about this purely theoretical attempt.

The book itself starts by showing the direct links between mathematics and what we experience in life starting from simple counting, as a foundation for numbers, through basic arithmetic. Chapter by chapter we are taken through further mathematical developments (complex numbers, calculus, statistics, etc), often highlighting links with something useful and predictive. Though also often pointing out that the usefulness is only there when we accept the approximate manner in which the maths matches ‘reality’. It’s almost another ‘history of mathematics’ book much of the time. It gets more interesting (for me) when we move onto Set Theory (critical in attempting to provide the foundations of mathematics in the modern era) and the contradictions that can be shown to be present in certain types of Sets. Cantor’s definitions of Infinity are also described and their usefulness assessed. And we conclude by examining the apparent dead-end of mathematical support for modern theoretical physics!

Overall, I think the author does an excellent job in achieving his task, to make us sceptical about the universe being some mathematical construct independent of human thought. To make us realise that a lot of maths, although following logical rules and axioms, doesn’t necessarily relate to anything in our direct experience of the world. That it’s value is when we constrain it by models of reality that we devise and try to assess.
Maybe the writing isn’t as tight as I’d prefer to make this point, as there are a number of diversions in some chapters which although interesting I thought were sometimes unnecessary. As one example, a lot of detail on the medieval scientist, Roger Bacon. And the ‘history of mathematics’ storyline is covered comprehensively in many other books. It is interesting but did we need all that detail to support the point that the book was making?

I think if you accept that you’re going to read a History of Mathematical Concepts in summary form, with interesting diversions, interleaved with a discussion on how useful these concepts are for modelling Reality, then you’ll know what to expect, and enjoy that.

I don’t think this is a book for people with no mathematical grounding, or a dislike of the subject, for whatever reason. My view is that if you got through a high/secondary school grounding in maths, and weren’t alienated by the experience, then you’ll manage fine with the technical content of the book. The technical explanations are commendably simple.
Really, it’s interesting, but I’m a geek!

[Sara · Rating 3 out of 5]

I received an ARC of this book through a Goodreads giveaway.

This one is a bit tough for me to review. I really enjoyed reading Are Numbers Real? for the first third—but as soon as we got to imaginary numbers, I got lost. Just like in high school. I am not the most math-oriented person, but I wanted to give this book a try to get out of my comfort zone a bit.

I found Clegg’s writing to be engaging, and I did learn some things. The idea that numbers don’t actually “exist” is honestly something that had never occurred to me, and I enjoyed reading about the development of mathematical concepts. I was very interested in the history-based exploration of the evolution of mathematical thought, but it was hard to wrap my head around the hardcore mathiness. Still, I’m glad I read it, I did learn some things, and it gave parts of my brain that have been gathering dust and cobwebs since I was 18 some exercise.

[Eustacia Tan · Rating 5 out of 5]

I wouldn’t describe myself as a numbers person, but my brother is and when we saw this book, we decided that it was too interesting to pass up.

Are Numbers Real? is a book that looks into the history behind the math. The first six chapters focus on the history of math – how would numbers come about and how mathematics developed. The history of maths is really interesting, going through the cult of Pythagoras (which reminded me of The Black Tapes podcast), the first proofs, and how the number zero came from the Indians via the Middle East. It’s a very interesting history, and although some of it was familiar to me, I didn’t know a lot and it was good to be able to get the big picture.

Chapter seven bridges the gap between science and maths, and chapter eight onwards is where it gets a little hard to follow (for me, anyway). The book goes into the background behind things like imaginary numbers, statistics, infinity and a few other concepts that quite frankly, I still don’t really understand. It’s not that the writing was very technical – I actually felt it was accessible – but the concepts behind the math (even if it’s something we think it’s familiar, like tending to infinity) is new to me and I think would require a few more rounds of re-reading to fully comprehend.

The main thing I took from this book was the realisation that numbers aren’t concrete. We can see a book, or a cup of rice, or count a hundred sheep, but we cannot see the number 1 (not the words but the actual one-ness of it) or any other number. Which really made me appreciate what it took to move from counting things to thinking of numbers as abstract objects of their own.

Also, I learnt about this really interesting mathematician named Emmy Noether, who is not only one of the few great women in the field but is also the person who is “single-handedly responsible for the way mathematics began to set the agenda in physics” by identifying the importance of symmetry. I have to remember to check if there are biographies written about her because she sounds really fascinating!

If you’re into history and/or maths, I think you would enjoy this! The book doesn’t dumb down the subject, but rather, introduces it in a fairly accessible way. I won’t say that you’d grasp everything on the first reading, but it will definitely whet your appetite for maths.

This review was first posted at Eustea Reads

[Carlos · Rating 3 out of 5]

3.5 stars. This is a great intro to theoretical mathematics, it does takes something away from the narrative if you are not at least heavily interested in mathematics. Not for a amateur mathematics enthusiasts.

[Tom Elliot · Rating 3 out of 5]

Gives an entertaining overview of the history of mathematical physics. However there is little good faith engagement with the titular question. What little there is seems to be a defense of the authors own point of view, with not much in the way of attempts to appraise alternatives.

The author appears to be a sort of instrumentalist, dismissing outright the notion that the Universe is fundamentally mathematical.While this is refreshing to see in a popular science scene that is often dominated by the grandiose language of Super String Theory, Loop Quantum Gravity, and Inflationary Cosmology, he appears, at times, to take it to an almost equal and opposite extreme; a sort of dogmatic scepticism towards the role of mathematical modelling in physics. This leads him on a number of odd occasions to doubt the scientific consensus on the existence of black holes, the Higgs Boson, and even a unifying framework for Quantum Gravity, on the grounds there is no direct evidence for these things. While I expect much of this doubt is not sincere on his part, but rather an attempt to coax his readership in practising scepticism, it often left me with the sense the issue was being dealt with rather superficially; there are good, often quite complicated reasons why working physicists accept these paradigms beyond blind faith that I do not feel are adequately represented here. Similarly, the author commits a lot of ink to emphasising that mathematics is not physics and that not every mathematical idea need be incorporated into our understanding of physics (I have never come across anyone seriously arguing to the contrary). What I feel is lost in this very one-note account is a sense of the uncanniness with which the high-level mathematics that does appear in schemes such as General Relativity and Quantum Field Theory seems to reflect the world around us. Almost all of what is understood about fundamental physics is understood mathematically. The power of mathematical reasoning in gaining insight on physical law is almost absurd, and it is not difficult to see why many see their models as perhaps deeper than mere analogy. In fact, as the author himself points, even when discussing something as apparently basic and "known" as an atom or a photon, we are invoking a mathematical model underpinned by empirical data. I think few would doubt that model describes some underlying reality, and this is a reality on which mathematics seems to have traction where all other intuitions fall short. I think it is not unreasonable to entertain the notion that this tractability speaks to more than a sophisticated allegory, even if such thinking seems ascientific at our present stage in understanding. It's certainly less ascientific than asserting the unreality of the black holes we now have pictures of. It's a shame that no real effort is made to address such ideas in a more nuanced way.

An engaging if somewhat myopic popular introduction to the topic. Well worth a read.

[Todd Stockslager · Rating 4 out of 5]

Review title: Approximating reality

The question at the title of this book is deceptively simple, much like the numbers we use every day to count time, balance our accounts, and measure the miles to the store and our weight on the scales from eating too much of what we bought at the store too soon after buying it.

Clegg progresses chronologically through the history of mathematics, from the use of fingers to count specific objects, to the generalization of the counters ("hands" and "fingers" in his example) to digits that can count any objects, to the use of the digits to add and subtract the counters, and then to the visualization of numbers in geometry. Here, the classical Greeks excelled in drawing and theorizing about lines and angles on two-dimensional flat surfaces--and as Clegg explains, made the first move away from reality: the world isn't flat, even though the mathematical models of geometry were.

Yet these numbers and models have worked for reflecting and counting reality for thousands of years. What Clegg is getting us to think about is the reverse: does reality in fact work the way our numbers and mathematical theories prove?

From my hotel room on the 18th floor, looking out at a landscape of other high rise buildings, an airport and railroad tracks, a network of roads and bridges, and a municipal water supply building below, the answer for these engineering projects is obviously yes. The mathematical equations that underpin these man-made objects clearly define how the world works exactly enough for people to trust their lives with them and for businesses and governments to base their budgets and expected revenues and outcomes on them. This is truly mathematically defined science that is so well proven that is called engineering, a branch of applied sciences so well established that educationally and occupationally science and engineering are separate entities. Much of the mathematical progression Clegg traces reflects this match of numbers to reality.

But as mathematicians progressed from the basics of counting and measuring, they also realized that they need not be bound by the reality of the real world, or of science and engineering models that reflect and build on it. They could create axioms and hypotheses that don't exist in the relationship world, and create mathematical formulas to resolve them. So those mathematical objects (Clegg gives as examples imaginary numbers and esoteric theories of infinite sets) aren't real--but even more bizarrely, mathematicians and scientists found that they could be useful in scientific theories that do seem to reflect the real world! In the end though, Clegg warns us that even when useful, such mathematical theories do not reflect reality and may be driving some sciences, particularly physics and cosmology, down complex trails where their models may no longer work. As Clegg digs deeper, into the basement where math can be conceptualized into esoteric formulas and theories that make your brain hurt and extend for page after page of arcane unprintable symbols and indecipherable text, it turns out that the foundations of all this "real" stuff is based in axioms and constants that don't seem to match reality. Even when models seem to reflect reality, explains Clegg, they are not reality.

This is the basement where relativity and quantum theory have opened up trenches in the 20th and 21st century science, giving us string theory, experiments suggesting multiple parallel universes, and potentially serious discussions about time machines (see my previous review of the book Time Travel by James Glieck) and warp drives through worm holes. Clegg, while not dismissing these scientific applications, warns of assuming that they are real. Numbers are real, and mathematical formulas are really powerful, but scientific theories based on them are not real. Concludes Clegg:

I would suggest that this shows us why those who believe that science is mathematics, or that the universe is mathematical in nature are wrong. . . . Because the two disciplines are inherently different in this way. One (math) is a collection of facts, which we are able to establish because we fix the rules, and the other (science) is a collection of models and theories, which we can test against data, but can never call the actual truth. (p. 260)

When I read books like this I am amazed that all scientists do not believe in God. A universe with such anomalies and oddities in the basement that on the surface works so consistently that we sentient beings on that surface can define formulas to build airplanes, skyscrapers, and solar panels must have a creative force behind it that is itself intelligent enough (omniscient) and unbound by the rules (eternal, infinite, and omnipresent) so that it could create and sustain it. We humans have made up the facts of math that seem work because we "fix the rules", but God fixed the rules that make reality really work and that science can only model imperfectly.

Clegg's book is simple enough to establish the foundations for even those who aren't mathematicians and scientists to think through deep thoughts like these.

[ara ara]

DNF didn’t like the tone of the author

[Amelie el Mahmoud · Rating 1 out of 5]

I have been holding off on giving this book a full review, but here I go. This book made me genuinely angry. The author tip toes around simple seventh grade algebra, basically babying the reader through fractions. He complains about math of a level higher than eighth grade yet attempts to explain concepts in quantum mechanics (make that make sense). The chapter on infinities does not even make an attempt to explain the difference between countable and uncountable infinities, which would have been a good conceptual addition to the book. The book begins with a very lengthy few chapters of imagining how numbers were thought of by humans. Clegg continues by trying to tie in how math has been used to describe the physical world. He begins to hint at how "detached" mathematics has become from reality, and at the end of the book, he laments about the hyper-specific and advanced fields of mathematics that he cannot see as useful. He even poses the question: has mathematics gone too far? Well I am here to say it has not. Mathematics should not be "kept in its place", but maybe this book should.

[Ed · Rating 3 out of 5]

I was doing fine following along with the math and theory until I got to chapter 12 on Infinity and Beyond, and then it was downhill after that!

[Erik · Rating 2 out of 5]

I've been delving into physics and mathematics and have been struggling to make sense of this very question. This book did not help me. It's a solid "popular mathematics" work, and probably a really interesting read if you are looking for a light introduction to why math and physics might be interesting to learn about as a lay person. If you've learned much about the topics already, you wont learn anything new, and you will find some things that will bother you as the author doesn't get them quite right. Alternatives are:

"Mathematics and the Physical World" by Morris Kline, although this one is an overwritten condescending slog.
"The Great Unknown" by Marcus du Sautoy, with better writing and research.

[Dick Hamilton · Rating 4 out of 5]

I enjoyed this book. Each chapter covers a different point of mathematics, however, I would say that the last three to four chapters were not as easy to follow as the first ten to eleven. Granted, the author has much more difficult material to address in the later chapters, however, at times the going is tough. I am not sure if more graphic/visual presentation in these chapters would have helped but this might have been beneficial. On the whole though, a good read.

[Beau · Rating 2 out of 5]

A sketchy look into the history of numbers, math, and physics. The book covers almost no new ground and has some mathematical and historical errors. But it's always fun to read about math and science, and the last chapter somewhat redeemed the book with an insightful look into the role of mathematics in modern physics.

[Rory Fox · Rating 4 out of 5]

A history of mathematics and its relationship to science.

The first half of the book provided a fairly standard account of the history of mathematics. Focusing on the Middle East, Greece, Egypt and India we hear how numbers and mathematical procedures emerged. The story takes us from geometry to algebra, through the development of notation to innovative insights into concepts like infinity and calculus.

The later part of the book moved the issues more towards the relationship of science and mathematics. Beginning with Galileo we saw the way that scientists drew more and more upon mathematics as both a source of ideas and also as a language to describe what was being observed.

That does raise a question, of course: why does mathematics seem to map onto reality? That abstract question was less well handled than the history. In the wider literature there are a range of ways of unpacking the question. Some (evolutionist) thinking even looks at the relationship between the physical brains of humans and the physical universe they have emerged from as an explanation for why there might be a relationship between thoughts and reality. Whether readers agree with that kind of thinking or not, it would have been helpful to include more of a breadth of ways of looking at the problem, otherwise the issues end up being raised and then left hanging.

In places the book did offer some occasional insights of its own. At the end it made the point that the teaching of modern science seems to generally view relativity as a twentieth century discovery. Yet, says the author, there are examples from Galileo which show the fundamental themes of relative frames of reference. He gives an example of Galileo in a boat throwing a key into the air, confident that it will fall into his lap, despite the fact that the boat is moving and despite his fellow passengers thinking that that horizontal motion of the boat will affect where the key falls. Yes, examples like that are indeed nibbling away at the core issues which would emerge in later relativity theory.

One of the strengths of the book is the author’s solid grasp of the history behind the mathematics. This means that he generally tells a good story, with appropriate anecdotes appearing throughout the text. However, sometimes this strength became a weakness, with too much irrelevant background detail imposing upon the text. For example, chapter 7 focused on Roger Bacon. But in doing so we had a ‘long’ account of life in thirteenth century Oxford. Similarly, chapter 9 told us about Isaac Newton. But did we really need to know peripheral details like the size of Newton’s library (2100 volumes) and that it was half the size of the entire library of Trinity College? Those kinds of factoids are interesting in isolation, but cumulatively they become distracting.

Textually around 7% of the book was notes and follow-up materials.

Overall, the book was enjoyable and well worth delving into for anyone interested in the development of mathematics. However, I think the title is a little misleading. It sounds like the book is focusing on philosophical issues (ie the reality of numbers), but those issues ended up being a relatively minor background consideration, to what seemed of far more interest to the author, the relationship of mathematics to science.

[Zachary Matheson · Rating 4 out of 5]

I'll divide this review into three parts:

Part 1, roughly the first 75% of the book: Mostly a history of mathematics, similar to what you'll find in other popular math books like Fermat's Enigma or A History of Pi. I would say I gained a newfound appreciation for many mathematical innovations, such as imaginary numbers or the concept of zero, but I wouldn't say these chapters did much to address the titular question "Are numbers real?" ***

Part 2, starting in the last few pages of the chapter on infinity until somewhere in the chapter about symmetry (~75-93%): Just a few pages, but this is where I had my faith shaken and I said to myself "Okay yeah, I get what the problem is. Why does math work so well for physicists?" In some sense we (as a collective body of mathematicians and physicists) started at the middle: we used math to describe what we could observe, without thinking too hard about why it works (the "beginning" aka axioms) and without a clear understanding of where it might lead us (which may not be consistent with reality when you push far enough). Now people are going back and trying to update the axioms such that the results are consistent with observation no matter how far you push, but it's not clear when - or even if - they will uncover THE axioms (if such a thing even exists). *****

Part 3, the rest of the book: Here I felt like the whole book was just a lead-up for the author to borrow Sabine Hossenfelder's soapbox for a few pages. Except unlike Hossenfelder, this author was heavy on complaints but light on actionable feedback. Basically, "We should do more experiments instead of just doing math, but also we should do math too." I don't know anyone who would disagree with that. Perhaps the author doesn't like that we use mathematical models to decide what experiments are worth doing, but I don't really know how else to do it (and if this book is any indication, he doesn't either). **

[Joseph Sobanski · Rating 3 out of 5]

I picked up Brian Clegg's Are Numbers Real?: The Uncanny Relationship of Mathematics and the Physical World after getting into an argument with my grandfather about if imaginary numbers actually existed (they do grandpa). Are numbers and mathematics representations of natural reality or are they abstract concepts created to create order out of chaos? This is the question being asked in Clegg's book, and he does arrive at a generally satisfactory answer in the conclusion of the text. Structurally this book is a history of mathematics, mostly focusing on the Western world, and it is this history which I found the most interesting aspect of this book. Exploring how numbers and mathematical concepts were developed helps explain how these things can be rooted in the natural world.

The biggest issue I had with Are Numbers Real? is Clegg's style of writing. First, I found it quite annoying how he would reference future and past pages of his book. This perhaps would work in a text book, where one would only read the chapter which interests them, but not in a short book like this. I don't need to see page ## on Isaac Newton when that was just the previous chapter. Second, Clegg seems infatuated with his own knowledge of mathematics, which leads to some tangential paragraphs exploring interesting tidbits about the so and so mathematician. I wish he would stay more focused on the central question of this text, but it seems at times he is more interested in cluing the read into some neat fact about the history of mathematics.

So to conclude, I found Are Numbers Real? to be a satisfactory introduction to the history of mathematics, and it did provide some clarity to the question of if math is a representation of the natural world or an abstraction. But it was also a frustrating read at parts, particularly in the second half, where I felt the author lost focus to impress the reader with unnecessary trivia.

[Theo · Rating 3 out of 5]

This book wasn't really what I was expecting. I was expecting more of a pop philosophy book that would act as an introduction to the philosophy of mathematics, but the book is more of a pop science book about how applied mathematics is used and its relationship to science. For a book ostensibly about mathematics, I found it focused very heavily on science, often conflating science with reality and spending more time explaining scientific models over mathematics. Several chapters are pretty much exclusively dedicated to general and special relativity, quantum mechanics, and other physics concepts with very little exploration into the actual math behind them (aside from an assurance that it's complicated). The first few chapters and the last chapter of the book are pretty good, but the rest of the book seemed rather aimless with only a few cursory sentences connecting the chapters to the supposed theme of the book. The book starts off with a strong focus on mathematics but starts to drift further and further towards physics before ending in a plea to ground physics more in reality and less in mathematical models. For what is ostensibly a philosophical question, there is also very little engagement with philosophy and, when it is mentioned, it is often token and weak. There is mention of Plato's cave, which is philosophy 101 stuff, and a butchering of Popper's philosophy of science but little more than that. As such, while I did enjoy the book and I think the explanations are well-done, I'm disappointed with it as it didn't really provide any interesting or unique thoughts about mathematics and seemed more interested in relating math to physics than it did the math itself.

[Patrick Cumby · Rating 4 out of 5]

The title is a bit misleading, as the book is more concerned about answering the question "Are *mathematics* real?" Clegg does a great job of explaining how the use of numbers (as in counting the number of goats in your neighbor's pasture) led to the development of arithmetic, geometry, statistics, calculus, etc, and how all of these branches of math were developed in order to solve real-world problems. He shows how some of even the most abstract concepts in theoretical math have come to be valuable in the modern age of computers and cell phones. He demonstrates how math has become an indispensable tool for the modern world. He also cautions us that math is a tool to understand reality, and that many complex modern theories may be based too much on the math and not enough on actual empirical evidence.

In presenting a topic as difficult as the whole of mathematical endeavor, Clegg does a good job of keeping the language down to Earth. This is definitely a book that can be read and enjoyed by almost anyone interested in the history of science--no college math or science degree required.

In the end, I would recommend this book to anyone interested in understanding how we search for truth in the physical world around us.

[Makayla Lawrence · Rating 4 out of 5]

“Is there a point when mathematically derived theory becomes a fantasy?” Is the question that Brian Clegg sets us up for over the course of this book.
While thoughts that the book starts with, like “Are numbers like 1 and 2 real, or are they just useful tools humans use to keep track of reality” are fun to ponder, but not very consequential in most people’s lives. However, as the book progresses it becomes clear that much of modern science and mathematics is built on some belief that math is ‘real’ - mathematical theories and ways of thinking can and should be used to understand reality. Should physicists chase a theory just because it is mathematically beautiful? Do waves time travel because a particularly useful equation implies that they do? How much does our number system shape the way we think about the world?

Clegg doesn’t exactly answer any of these questions, but the book is nonetheless an interesting read. I think it’s accessible to people with a wide range of backgrounds. An understanding of proofs, calculus and/or physics may be helpful, it’s not necessarily.

[Ashley Lambert-Maberly · Rating 4 out of 5]

I don't think I'll ever become a math or physics whiz from reading these books, but I did feel like I (mostly) understood the concepts being explained, which is good enough for me (and the intention of the author). This one erred a bit more to the physics than I was expecting (note: no use of the actual word "physics" in the title), and I would have preferred a bit more of a mathematical read, but that's nitpicking.

Spoiler alert: it seems like numbers might indeed be real, in some cases surprisingly real. Pi popped up in one place that we would never expect it, for one instance.

(Note: 5 stars = amazing, wonderful, 4 = very good book, 3 = decent read, 2 = disappointing, 1 = awful, just awful. I'm fairly good at picking for myself so end up with a lot of 4s). I feel a lot of readers automatically render any book they enjoy 5, but I grade on a curve!

[Rayfes Mondal · Rating 4 out of 5]

An interesting topic and it's important to note that while math may model observations accurately that there's no guarantee that it's a reflection of underlying reality. This is even more important for things like finding the Higgs boson where all we have is a lot of indirect evidence and statistics since we can't observe things directly in the quantum realm.

A lot of the book covers the history of how math concepts like imaginary numbers came into being and how they are useful. Later on author's opinions come out which is a nice way to end the book. I was thinking I would be interested in reading more of this author's book but I'm not.

[Becca · Rating 4 out of 5]

There were many chapters where Clegg threw out so many examples, or seemingly went off on multiple tangents, that it was sometimes hard to follow his line of reasoning. However, having read the book in its entirety and understanding the key points, I think a second read would prove that Clegg is not as haphazard as the first read would suggest.
Overall, I think Clegg did a remarkable job of taking numbers from their primitive uses clear through to applications in modern science. All the while highlighting every instance where math takes a step away from reality. An easy read for someone well-versed in math or science, and a moderate but accessible read for someone less than well-versed.

[Teena · Rating 4 out of 5]

This book validates nearly everything I've every believed about mathematics: Beyond arithmetic, it's merely a theory that has no correlation to the physical world.

While I believe that the study of mathematics is important and shouldn't be discounted, I will argue its value as solid proof in the realm of physics or other scientific fields. As described in this book, physics is no longer a hard science, provable by calculation. It's merely the belief system of theoretical physicists, similar to religious doctrine.

[Pamela · Rating 4 out of 5]

I benefited and enjoyed reading it even though much was beyond my knowledge base. I know that is a refrain for me with science books but it is true . Even so I get the opportunity to suggest that exploring can be gratifying.
And it is encouraging that some physicists are open about their own limitations with this challenging area of study including quantum theory, black holes.gravity, etc. Also it is interesting to learn that many scientists including Clegg are increasingly skeptical about string theory.

[E W · Rating 3 out of 5]

Liked the first part, got to 60% and then struggled to finish it. Clegg does a decent job of introducing basic concepts and I got a kick out of trying to figure out other base numbering systems but it's as if the subject itself runs out of gas. The book is written for a general audience so everything is well explained with no knowledge of math needed. I just lost interest and had to force myself to finish.

[Gregory Eakins · Rating 4 out of 5]

While I feel that Clegg failed to make a compelling discussion of question in the title of the book, he does present a fascinating discussion of the history and development of mathematics as we know them today and the driving forces that necessitated their development. As an engineer who has taken far more mathematics than I'll ever need, I've always wondered just how our modern approach to math took form.

The short answer to the title is: all models are wrong, but some are useful.

[Bill Yates · Rating 4 out of 5]

I get it that simple numbers correspond to reality, but that doesn't make numbers real. The author never defines what he means by "real". Certainly it is true that sets with cardinality exist, but it would seem that the numbers themselves are mental constructs. Those abstractions are useful for comparing the size of sets, for example. If numbers are only abstractions, then the they exist only in the mind, and therefore cease to exist at death.

[Gary Schroeder · Rating 4 out of 5]

SPOILER ALERT: Clegg thinks numbers are not inherently baked into the universe, but that math is a very useful tool created by clever humans. He only spends one chapter talking about this. Odd, given the title of the book...which would have been more appropriately titled "A History of Mathematics." Diverting and brief. Very low on the technical content, which was perfectly fine with me.

[Joe Fernandez · Rating 4 out of 5]

I really liked this book more than I thought I would. It gives a great history of how mathematics has evolved from simple arithmetic to a complex, intricate, yet broadly applied field of study. The last two chapters did drag on a bit though, but overall a good read, particularly for those who may not feel an aptitude towards mathematics but still find the subject itself generally interesting.

[Arron Fitzgerald · Rating 3 out of 5]

An interesting exploration of the (pre)history of maths and its relationship to physics and objective reality, however anyone expecting a more rigorous examination of the philosophy of mathematics will be disappointed.