The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg

by Robert P. Crease

Philosopher and science historian Robert P. Crease tells the stories behind ten of the greatest equations in human history. Was Nobel laureate Richard Feynman really joking when he called Maxwell's electromagnetic equations the most significant event of the nineteenth century? How did Newton's law of gravitation influence young revolutionaries? Why has Euler's formula been called "God's equation," and why did a mysterious ecoterrorist make it his calling card? What role do betrayal, insanity, and suicide play in the second law of thermodynamics?


The Great Equations tells the stories of how these equations were discovered, revealing the personal struggles of their ingenious originators. From "1 + 1 = 2" to Heisenberg's uncertainty principle, Crease locates these equations in the panoramic sweep of Western history, showing how they are as integral to their time and place of creation as are great works of art.

Genres: Science, Nonfiction, History, Mathematics, Physics, Popular Science, History Of Science

224 pages, Hardcover

First published January 1, 2008

About the author

Professor Robert P. Crease is Chairman of the Department of Philosophy at Stony Brook University, New York.

Reviews

[Benjaminxjackson · Rating 4 out of 5]

The Great Equations is a science and mathematics book for popular readers, which is its strength and its weakness. The strength is that the text is accessible to just about any reader with some smattering of science and mathematics in their schooling. The weakness is that sometimes I found it a little too basic and too quick to gloss over details and proofs for equations.

One of the interesting things is how the book demonstrates the way science progresses in fits and starts. It also shows how important collaboration and communication are to scientific discoveries.

The other thread of the book that I enjoyed is its commitment that science is something that everyone should know something about -- the chapter on the Second Law of Thermodynamics, the law of entropy, is entitled "The Scientific equivalent of Shakespeare." He thinks we would all have better lives if we treated some basic scientific knowledge like we would knowing about Shakespeare. Crease is not shy to deride things like astrology.

Crease also acknowledges his debt to the "Two Cultures" concept first proposed by C.P. Snow that says knowledge is divided into technical and cultural knowledge essentially. One of the most interesting chapters is the interlude that discusses the humanities and the science. "For only when the humanities couple their inquiries into human dimensions and possibilities with an awareness of what science has disclosed of the dimensions and possibilities of the world will the humanities most effectively be able to provide answers to the questions of what we know, should do, and can hope for."

I enjoyed the chapters on quantum mechanics because they explained things in a way that helped me make sense of things friends of mine who are scientists and mathematicians have explained to me in other ways.

One of the things missing from this book, as another reviewer pointed out, is the quadratic equation. I also would have liked to see mote mathematical proofs in the book. I think Crease missed a chance to make popular readers more comfortable with equations beyond just holding them up as something to be admired.

I also would have liked to see a proof of the Pythagorean Theorem associated with the diagram associated with the U.S. President James A. Garfield used to invent a proof. I don't think enough information is given in the book to recreate the proof myself, though I may need to try again.

I would recommend this book to anyone interested in the history of science, humanities and social science people, and educators who might draw some ideas on how to bring science and math to a wider audience.

[David Raz · Rating 3 out of 5]

Though none of these equations was totally new to me, the history behind these ten important scientific equations was interesting. The decision to avoid mathematics at all cost means the discussion is sometimes too shallow. Even the exact definition of each of the components is not made clear. I also found the philosophy parts even more lacking, when even the basic concepts were left unexplained. Still, a good read for the history itself, three stars out of five.

[Robu-sensei · Rating 3 out of 5]

Excellent presentation of the history behind ten of the most famous equations in science. I would have liked more technical detail. I can see glossing over the derivation of a physical law involving tensor calculus, but Pythagoras's own proof of his famous theorem should be within the grasp of this book's audience.

[Sarah · Rating 3 out of 5]

Fun reading about the history leading up to the equations and why they're significant to mankind. However, the explanation of the math was hard to follow (maybe needed more pictures and diagrams). It was hard to read a lot of the book in one sitting since it felt "dense."

[Jun Wen · Rating 3 out of 5]

Reading this book as a chemical engineer, I probably had an easier time than most, being already familiar with the laws of motion, gravity and thermodynamics.

Even then, it was not a book that could be breezed through easily, despite its short length. This was especially so towards the end of the book as it moved away from classical physics, and in the final two chapters I felt severely out of my depth.

Its difficulty notwithstanding, this is a well written book that summarizes the progress of humankind's understanding of the world through the stories of prominent equations. One can gloss over the technical aspects and still be able to walk away with a good understanding of the journey that was undertaken to get to where we are today. It provides a fresh perspective where progress is not measured in personal terms as we are used to, but in much more meaningful terms as an entire species, with each individual's contribution adding to the entire species' understanding. This alone makes the book worth reading.

Alternatively one can pay attention to the scientific and mathematical aspects of the book, and attempt to follow the train of thought of our eminent scientists, so as to come to their conclusions. This will require a lot of hard work, but the book has made it easier by not omitting the important technical terms and derivation, as well as providing landmark concepts which can be easily Google-d for should one desire more information. At the end of such a reading one will be enriched for having thought through all the concepts himself instead of simply having accepted something handed down to him.

Whichever way the reader decides to explore this book, he will need determination to brave on in the face of unfamiliar symbols and equations, but in the end he will walk away with valuable knowledge.

[TheIron Paw · Rating 3 out of 5]

An enlightening read for those of interested in mathematics but with limited knowledge. This book is mostly history and philosophy but with a modest amount of math that you can either work through or half bleep over without affecting the author's message too much. More difficult to deal with are some of the concepts that don't fit my day to day understanding of reality. One of the interesting concepts brought forth in the discussion of quantum mechanics is the "anschaulicht" - the property of a "thing" to be visualizable by human mind. I'm also reading a book "Inside of a Dog" that talks of the "umvwelt" of dogs - their subjective view of reality that differs from ours largely because of their different sensation of reality (i.e. the colorful world of smells). The difficulties of understanding the quantum universe and of understanding dogs are quite parallel. Now if only my dog would give me the equation...

Overall an enjoyable book, but not one to be read quickly (or when your wife has a list of chores.)

[Augusto · Rating 2 out of 5]

Livro fraco escrito num tom pra lá de "deslumbrado". Um interlúdio cientificista que meio que espezinha as ciências sociais, em especial a história, beira o mau gosto e só piorou as coisas.

[Alastair · Rating 3 out of 5]

The Great Equations is an awkward marriage of slightly-too-basic primers on famous equations (at least for the specialist); and far-too-nuanced philosophical asides on the nature of scientific progress and its communication to the public. As a result I honestly can't say I know who this book is for.

As a mathematical physics graduate, the book really delivered in some chapters, particularly with regards the history of certain equations. I knew very little about the historical context for Newton's second law of motion: F = ma. This book firmly remedied that and would have been extremely useful to have read during my degree. The same is true during the discussion of the second law of thermodynamics (which I won't attempt to render in Goodreads' quasi-html formatting).

Yet other chapters dragged and dragged. I honestly never need to hear another word about why Pythagoras' theorem is important or how it was 'actually' discovered by the Sumerians or whoever. It may be interesting to the totally unaware but I've just heard it one time too many because every book of this kind always covers Pythagoras.

In other places the book - for a reader with my sort of background - is let down by lack of technical detail. Schrödinger's Equation is pretty hard to discuss without a good grasp of mathematics and while author Robert P. Crease attempts to do so it doesn't really come off. The same is true of the Heisenberg chapter. I don't think anyone without a background in this area would get much out of a chapter with subheadings like "Matrix vs. Wave Mechanics". But at the same time, I didn't get much out of it because it didn't delve deeply enough into the really interesting historical debate between competing quantum viewpoints in the 1920s.

Exempting these too-complicated chapters, does this book work for the non-specialist reader? Yes, to some degree, but the interludes between each of the main equation chapters pose something of a problem. The author has a little aside after each Great Equation. These reflections cover things like the humanities inability to engage with mathematics or science; or the fetishisation of equations like E = mc-squared. It is not particularly clear who these somewhat hectoring diatribes are aimed at: specialists may respond to them but are unlikely to feel much beyond 'well, of course this or that is a problem'. And the general audience is unlikely to get why the author has such a strength of feeling. The only audience I can really think of is the rather narrow readership of fellow journalists who steer much of the public's interaction with science. Influencing this group is laudable, but doesn't help this book in its quest for a coherent identity.

In the end, if you want to brush up on a bit of your maths and science history, this book is a decent read. For everyone else, better popularisations of maths and science are out there.

[Martin · Rating 5 out of 5]

This book helped me appreciate a lot of the real life behind the big equations that he talks about. It was good in telling parts of the biographies of the people involved in them and this was enriching to read.

His principle is that just having an objective understanding of the equations misses out a lot of important lessons, and things to reflect on, we can learn from the development of those equations. He does this well.

Generally the equations are described appropriately, since the book is not teaching how to use the equations

I think there were quite a few small errors in the English of the copy I read which made it difficult to read sometimes because you have to keep jumping back to make sense of what you're reading - and this ruins your flow. Another unpleasant part of the book is the way he tries to cram so much into many sentences, meaning he has these huge sentences that are difficult to follow until you get into his rhythm. The book often seemed verbose. This strongly put me off the book in the first couple of chapters, but I am very happy I persevered because some of his musings are well organised and very thoughtful.

Throwing science out to the general public could be seen as a process of simplification, but this book really focusses on how the implications of things in scientific development are complicated and there is a lot we can learn philosophically.

I loved that he thought about these things, that I got a glimpse into the world where these equations were being developed and that the author respected the view that there was a lot that non-experts could gain from reading about this esoteric knowledge.

[John Boyer · Rating 1 out of 5]

Easy to read, but you will be disappointed if you are expecting to learn much about what the great equations equate. The first few chapters are better, reaching an acme in chapter 4, where you can experience exactly how to equate e^{i*pi} - 1 with 0. But the book then descends quickly to a nadir in chapter 6, wherein Maxwell’s great equations are only window dressing at the start of the chapter— there’s discussion of almost none of the variables appearing in the expressions Maxwell equated.

Even the few remaining times when the variables are discussed, you won’t get a good sense of the great equation. For a case in point, consider the chapter on Einstein’s most famous great equation, E=mc^2. The chapter explains that the equation holds because photons emitted forwards and backwards from a moving reference frame have different amounts of kinetic energy, which can only happen if the emitting object _loses_ mass, as measured from the rest frame relative to which it is moving. But this claim doesn’t help one to know _why_ mass is convertible to energy because there is no explanation _why_ the kinetic energies differ between the photons.

From an epistemological standpoint, the explanation has only moved the location in knowledge where the reader must replace reason with _faith_ or trust in authority that matter can be equated with some amount of energy. And even if we take it on faith that the photon energies differ, we only get that E is equal to some amount of mass, not how E is equal to c^2 units of the mass. This is what failing to explain a great equation looks like.

[hkbbgs · Rating 5 out of 5]

Honestly, the best book that I have read so far. When I first picked up the book, never in my life would I have imagined that it would be so funny, insightful, and beautiful. Of course I already knew that it would contain a lot of knowledge, though. The book reads like poetry to me. I have come to appreciate science and the struggles and experiences people have went through for these scientific discoveries.

This book is a mixture of physics, math, history, but most importantly— heart. The author’s passion shines through each chapter and each interlude, and even the acknowledgments. I even read the references. I was struggling with physics, and although I still am, it has made me feel more passionate towards it.

I’ve read many books before but would never be able to answer the question “What’s your favorite book?” and would have to reply with a mediocre “There’s so many that I can’t choose!” when that was a lie. I just couldn’t think of any that would be called a “favorite.” But now I know for sure that this I can call this my favorite book, until I read something better.

Unfortunately my friends don’t feel the same sentiment, and they call me crazy and boring for reading a book about equations. They’re missing out, because it was way funnier than I thought it would be. Crazy!

[Robert B · Rating 3 out of 5]

Explores 10 famous equations: the Pythagorean theorem; Newton’s second law of motion; Newton’s law of universal gravitation; Euler’s equation; the second law of thermodynamics; Maxwell’s equations; E = mc2; Einstein’s equation for general relativity; Schrödinger’s equation; and the Heisenberg uncertainty principle. Crease wants to show how each equation arose from dissatisfactions with the way the world was explained and to capture the wonder associated with their discoveries. In one or two cases – Newton’s second law of motion, in particular – he does a good job and the contrast between our previous understanding and that uncovered by the new equation is made exciting and clear. But in other cases, particularly the equations that arose in the 20th century, Crease does less well in explaining the equations in a clear but not overly simplistic manner. Unfortunately, each equation deserves a book – or two – of its own, and it may be too much to expect any author to boil down these equations into short chapters.

[George Eraclides · Rating 4 out of 5]

A difficult read for the non-mathematically inclined or those with poor scientific knowledge which means they will probably ignore this book, which is a pity. If you are comfortable with (or not frightened by) the use of mathematical symbols, it is worth persevering. The author (or more likely, the publisher) could have made it a little easier by including more diagrams, word-pictures and a glossary for scientific terms used in the text. We have the beauty of equations from Pythagoras to Newton's Laws, Einstein and ending with the uncertainty equations for quantum mechanics. The author enlivens the discussion with some personal narratives of the mathematician-scientists and why the equations matter (besides being beautiful). He adds descriptions of the scientific process (not always logical) and the historical context. He ends the book with a fine essay on the scientific method and the sense of wonder you experience in gaining an insight into nature - the wow factor, a peak experience, not unlike a religious epiphany.

[Kara Babcock · Rating 4 out of 5]

To paraphrase Mr T, I pity the fool who doesn’t see the beauty of mathematics inherent in the world around us. As a teacher, I feel rather complicit at times in robbing children of the joy of mathematics. The systemic, industrial tone of education does not often lend itself well to the investigation and discovery that should be the cornerstone of maths; I find this particularly true in the UK, where standardized tests and levels are the order of the day. There are times when I am conflicted about how to cover subject matter. I have to find a balance between a breakneck schedule and a desire to achieve the comprehension that only comes with time and careful practice, strive to find the equilibrium between exploring interesting lines of inquiry and curtailing those lines in order to teach what’s on the test. I hope that as I become more experienced finding this balance becomes smoother. For now, though, it’s a struggle.

Because the secret that everyone learns as a child and then has beaten out of them by the endless grind of daily mathematics lessons is this: mathematics is not numbers. It is not arithmetic.

There, I said it.

I gave my students a test today on our statistics unit, which involved data collection: designing surveys, selecting sampling methods and sample sizes, etc. As they worked through the test, a few questioned its connection to mathematics. "This is words!" they protested, as if I were somehow an imposter trying to sneak extra English content into their day. Somewhere along the line—I don’t know precisely where—they developed this notion that mathematics is solely about manipulating numbers.

Really, though, mathematics is about relationships between things. Mathematics is a process for understanding the world, as well as understanding theoretical constructs that, while not directly observable in the real world, can still have useful and fascinating properties. Math can be numbers, but it’s also truth, in one of the most fundamental ways possible.

This is what Robert P. Crease attempts to communicate in A Brief Guide to the Great Equations. He foregrounds each equation and carefully explains how it became a part of the great canon of mathematics. He also explains why the result is so exciting, not just to mathematicians but to the population at large. I’m pretty enthusiastic about all this crazy math stuff, but Crease manages to stoke even my considerable flames of fanaticism and set my heart racing. The way he breathlessly extols the beauty and utility of Maxwell’s equations or Einstein’s relativity … it’s like a BBC Four documentary in paper form.

When it comes to books on popular mathematics, I always try to anticipate how a layperson would receive the book. As a mathematician, I don’t have a problem following the equations and explanations; it comes naturally. It still staggers me how some people are able to understand the intense nuances of some of the higher-level mathematics involved in quantum mechanics and relativity; I’m somewhat reassured by Crease’s claims that physicists often rejected new developments that required them to learn a lot of complicated new math. Yet I still know what Crease means when he carelessly bandies around certain terminology, expecting his reader to keep up to speed based on a high school education alone.

As far as pop math goes, A Brief Guide to the Great Equations is not the most friendly book. I’d probably hesitate to recommend it to casual readers, preferring maybe Zero: The Biography of a Dangerous Idea . For someone very interested in the history and philosophy of science, however, this book would appeal even if one’s math knowledge isn’t quite up to snuff. Crease recounts without fail some of the more interesting scuffles and disagreements among famous mathematicians and scientists; he also carefully lays out his own views on what constitutes a scientific revolution, and the role that developments of equations can have in revolutions.

It’s easy enough to follow the history and soak up the spectacle without following the math. I don’t mean to say that you shouldn’t read this unless you’ve studied math in university. If anything, Crease hopefully sheds light on how and why people can find math such an interesting occupation. By reading these stories of how Maxwell and Einstein and Schrödinger dedicated years of their lives to these problems, one gets the sense that the problems are more interesting and worthwhile than the equations themselves indicate. Crease explains how the problems consumed and intrigued these brilliant minds in such a way that, even if one doesn’t understand the nature of the problem—or its resolution—itself, one can still appreciate the passion and dedication involved.

Such passion and dedication are more universal than even the mathematics that unites the great thinkers featured in this book. One need not like math to be good at it or to succeed at it in school or in life. One need only appreciate its versatility, utility, and beauty. Crease tries and succeeds admirably in showcasing such attributes through the equations and history that he includes here.

Math is beautiful. You just need to open your mind, cast aside the "but I just don’t have the brain for it", and embrace the wonderful freedom of being able to figure out how the world works.

[AD LAND · Rating 4 out of 5]

This book started out at a painfully slow pace and I almost chucked it but, luckily, persevered. It got better with Sir Isaac and the sections on relativity and quantum physics were excellent. The author showed the struggles involved in arriving at the equations, the competing schools of thought, the personalities and egos. I liked his extensive use of direct quotes and explanatory footnotes; yes, the footnotes are good reading, too. Would have liked to see some derivations; that would've earned the fifth star.

[Azri Razak · Rating 3 out of 5]

Not a really page turner this one. But ideally good for knowledge.
Though I'd do some further read up about the equations mentioned in this book to get a better understanding of it. Nevertheless, it's an interesting view on how these equations were made and not everything looked so easy and unsophisticated. Made me appreciate my maths, physics and science lessons I went through as a student, as I realized we used these equations for many many applicable things right now. And without them who knows what kind of world would we live in.

[EDpon3 · Rating 5 out of 5]

Does an excellent job at explaining the origins of some of humanity's greatest equations from humanity's greatest minds.

This isn't so much a MATH book as it's a HISTORY book; going into the stories and origins of the greatest equations we learn in school, and one in particular that has become a pop culture icon.

It does an excellent job at explaining WHY these equations are so important to learn, and what they tell us about our place in the world and the universe.

This is the kind of book that should be REQUIRED reading in junior and high schools across the board.

[Andre Henrique de Siqueira · Rating 5 out of 5]

A busca do sentido das equações trás contribuições efetivas para o entendimento da forma como representamos a realidade. Robert Crease descreve com entusiasmo a história e o significado de algumas das mais importantes equações da moderna ciência. Uma leitura imperdível para quem se interessa pelo tema.

[Ogden Jones · Rating 4 out of 5]

Another book I got when moving Mom to a smaller apartment. This was clearly a book Mike gave her, and had probably never been cracked open. It is a heckuva book. I finally learned how to prove the Pythogorean theorem. I was familiar with all of the equations, but still got something out of the history of how they were arrived at.

[Scott Froggers · Rating 3 out of 5]

I learnt a good amount of tasty science, although I found myself skimming particularly near the end of each chapter when the details were less exciting, and more about the scientists being puzzled or something similar. If it were fiction it would be rated 2 star, but I learnt some exciting stuff and I'm glad I read it.

[JMH · Rating 5 out of 5]

One of my favorite books, especially the interlude chapters that place scientific endeavors in historical context -- e.g., the story of Newton and the apple is obviously fiction, but Newton himself wrote it. Why?

[Noah Thomas · Rating 4 out of 5]

This is a great pop-science book on the history of mathematics and scientific thought. Explains complicated concepts clearly for a lay audience and is a great short read for someone interested in the topic.

[Let's talk Novels · Rating 5 out of 5]

this book makes me want to write more about science in a more lucid format.

[Sheila Jungco · Rating 4 out of 5]

Not so highly mathematical and easy to read.

[Howard Thompson · Rating 4 out of 5]

Accessible but technically sound, Crease does a good job explaining 10 mathematical results that shape human thought and civilization.

[Jake Losh · Rating 2 out of 5]

It was ok. It’s a historiography of science as told through some important equations. Not as interesting as I’d hoped. Suitable for a bright high school student interested in math and physics. I enjoyed the vignettes on the Pythagorean theorem and Euler’s Equation (e^i*pi + 1 = 0) but the rest was too physics heavy for my taste).

[Karen Carlson · Rating 4 out of 5]

The focus is on how certain equations – the ones that had great impact, that changed paradigms – were discovered and developed. This means, often, stories of people trying to figure things out via mathematics. The physics gets pretty complicated in later chapters, but the point isn’t to understand the math; it’s to understand the process by which the math emerged.
FMI see my blog post at A Just Recompense.

[Upom · Rating 4 out of 5]

So often our culture forgets science and in our march to progress. Crease even noted that Howard Zinn's A People's History of the United States: 1492 to Present barely mentions science, although mentioning almost every other aspect of a forgotten American history. Sadly, Crease also mentioned a simple rule of thumb publishers use for math in a book: for every math equation added to a popular book for science and readers, the sales of books drop by half. It's for sad reasons like these that I feel Crease is doing a great service in trying to make math mathematical equations as compulsory to know as Shakespeare. The book's explanation of equations is pretty standard for a popular math and science book, though his history seemed to be a bit more comprehensive and insightful than most books I've read. Where this book really shines are the interludes that conclude every chapter. Crease's training as a philosopher shines here as he reflects on the nature of science both as a field of endeavor and its part in the greater society. They are extremely well thought out, and fairly original in their considerations. These interludes alone would have made for a compelling book. The only complaint might have been was the mathematics in the book might have been a little rushed in explanation for non-technical readers. Still, if you have a light background in mathematics, this is a great book for learning and better understanding some of civilization's most quintessential equations.

[Jerad Acosta · Rating 5 out of 5]

Fun, quick, interesting read that gives you an appreciation for the history of a subject so many people fear - and gives you some great cocktail party stories to share about equations everyone knows about.
If you enjoy historical anecdotes of why things are the way they are, or how something was discovered then you will devour this book.
If you respect math and history, you will appreciate these stories - however no talent in either field is required.
This is a great read. The chapters are self contained so you can jump around or just dip in for a quick 30 min chapter without concern for what happened last - though I really enjoyed reading it from front to back which gave me a compounded appreciation for the evolution of equations and questions over time.
The author does a great job of making these accomplishments seem more attainable and the achievers seem more practical - while of course still maintaining the brilliance of each character, which should be implicit by the nature of the matter.

[Max · Rating 3 out of 5]

The Great Equations is a great book that explains everything you will ever need to know about maths history and the way it has become the way it is today.

The Great Equations starts from the first equation being 1+1=2 and ends explaining The Heisenberg Uncertainty Principle. Along the way we see Menos' paradox and how Pythagoras had nothing to do with the creation of the Pythagorean theorem (it was invented possibly a thousand years before Pythagoras was born in India). The book explains how equations came about and why there were people looking for that equation at the time, and how it was then put into practical use in a very simple and interesting way.

I give The Great Equations three stars because though it was a good book, it was only a good book compared to other math related books. But if you're curious about the world i feel its safe to say this book explains it pretty well.