The Road to Reality by Roger Penrose is a comprehensive guide to the laws of the universe.

Published in 2004, this 1099-page book presents the math of modern physics to nonmathematicians. However, it has its drawbacks.

Penrose spells out the simplest concepts but expects readers to know the most complex. As a result, much of the new material can't be followed in detail.

He takes readers into thickets of esoteric algebras with minimal motivation or explanation. The tone, after walking through simple arithmetic, is rather demanding.

The book gives a hazy idea to those without a Ph.D. in math about what physicists have been doing since Planck's discovery in 1900.

Penrose shows his sources by citing hundreds of texts. But he only provides abstract mathematical formalism, not quantum mechanics for chemists.

There are some simple ideas presented, such as the concept of a black hole and its entropy formula.

The book also discusses usable energy, classical mechanics, quantum mechanics, and black-body radiation.

However, there are areas of confusion, like the degrees of freedom of a rigid body.

Hyperbolic geometry and musical harmony are also covered.

Fermat's Last Theorem is mentioned, and solutions are available online.

There are some errata in the book, such as a misprinted Planck blackbody spectrum formula.

Overall, The Road to Reality is a challenging but valuable resource for those interested in the laws of the universe.

I WASNT READY. BREATH BREATH. SO MUCH MATH.

Oh my goodness, I really wasn't prepared for this. I'm taking deep breaths trying to calm my nerves. There is just so much math in this!

ok. I'm gonna have to revisit this one after I practice my math skills. Way too much went over my head.

Well, it's clear to me that I need to go back and practice my math skills before I can truly understand this. So much of it just went right over my head and I feel a bit lost.

If you're gonna pick up just know, penrose does not half ass his mathematical expositions.

If you decide to pick up this work, just be aware that Penrose doesn't do things halfway when it comes to his mathematical explanations. He really goes in-depth and expects a certain level of understanding from the reader.

I recommend this highly for those who want to know how mathematical physicists see the world, and who can breath in the face of high level mathematics.

I would highly recommend this to anyone who is interested in learning how mathematical physicists view the world and who has the ability to handle high-level mathematics without getting too intimidated. It's a fascinating read for those who are up for the challenge.

I give it five stars due to how comprehensive it is, and for how much I love penrose's prose and mind

I initially gave this five stars because it is so comprehensive and I really love Penrose's writing style and the way his mind works. However,

Edit:

I have since then deleted one of my stars given how I still don’t know what reality is. And no, it’s not because of my lacking ability in the “maths.”

After further reflection, I have decided to take away one star because I still don't feel like I have a clear understanding of what reality is. And it's not because I'm not good at math, but rather because the concepts presented in this work are so complex and profound that I'm still grappling with them.

This is indeed a quite challenging read, but it is definitely worth the effort put into it.

The first half of the book is mainly focused on "math", while the second half delves into "physics". Penrose has done an amazing job in writing a guidebook that涵盖了 these two fields to a great extent.

However, I felt that there were some uneven sections. I recall a significant amount of discussion on Fourier transforms and thought that if such a whole section is needed for that, how will the reader manage to keep up with differential topology, twistor theory, diagrammatic and group theory notation, and so on.

Nonetheless, overall, it was great. The writing was almost playful at times, especially as the book progressed. Penrose did a nice job of self-identifying his biases. I really enjoyed the discussions on the relationship between math and reality, as seen on page 631 where the question is raised: "What is the physical justification in allowing oneself to be carried along by the elegance of some mathematical description and then trying to regard that description as describing a'reality'?"

The last two chapters of the book further explored this idea. The discussion of string theory was especially understandable (and self-admittedly biased).

It struck me as a bit funny that, especially later in the book, there was more and more reference to the "magic of complex numbers", and yet in one of the final sections, there was an argument that there is nothing more or less confusing between real numbers and complex numbers.

This book was extremely "hyperlinked" or self-referential. At first, all the links were "forward links" (necessarily), and mostly I ignored them. As I got further into the book, I started following the "links" (both forward and backward) much more. I found myself using the index and really engaging with the text. As I started this review, I can say that it takes some work, but it is definitely worth it!

I plan to reread this book in a few years.

Wow... I actually managed to read it, 1050 pages, every single one of them.

But can I really say that I'm done with this book? I don't think so. Although it took me a year and a half to read it, I didn't even understand a significant part of it. Since I'm a physics student, I understood most of it on some very basic level. However, I'm pretty sure I'll have to open this book again and again to take a peek at some of the awesome ideas put here by Penrose.

Did I say awesome? That's a huge understatement. I meant incredibly brilliant, original, profound and refreshingly sober!

Sir Roger Penrose is the reason why I came to love physics as much as I do. He's probably the main reason why I chose theoretical physics and why I'd like to be a mathematical physicist.

You, sir, are a huge inspiration. Thank you!

I don't want to write anything else because I don't think I could give this book a truly proper review. You should simply try to read it. And if you manage to do it, I hope you'll understand what I mean.

A book with such an ambitious goal as describing the state of scientific knowledge we have about the deep reality of nature and the paths to advance in what remains to be discovered is truly difficult to evaluate.

This is the only scientific book that appears among the top 100 books on Goodreads' list of "Books You'd Take to a Deserted Island". I believe it is there with every deserving reason. It is a book that allows for many rereadings. During the almost two months it has taken me to read it, the things I was reading continuously came to my mind. Everything it presents is interesting, stimulating, and expands your mind.

The only problem with this book is that on a deserted island there is no Internet. Let me explain: despite its more than 1,400 pages, the book is almost impossible to follow only with the information that appears in it. I have had to resort to the web on many occasions to search for the necessary information to "fill in the gaps" of what I was reading. The author leads you by the hand during the exposition but frequently lets go and forces you to make an act of faith in the argumentation or takes for granted the evidence of a mathematical reasoning. I have greatly missed links to Internet resources where the details of the ideas presented are explained or the exercises it proposes are solved. In my case, only when I could follow the argumentative line in detail did I have a clear image of what I was reading.

In any case, the effort of confronting the reading of this book is amply rewarded by the expansion of your mental horizons.

It is also fascinating to glimpse how the mind of a scientist like Roger Penrose works, for me, one of the greatest living thinkers.

The story goes that Stephen Hawking was once told by a book publisher that every formula in his book would halve the amount of readers (or sales, almost the same thing, but not to a publisher). He was, of course, totally and utterly wrong.

The Road to Reality contains about 1100 pages and, on average, there's about 9 formulas per page. That makes roughly 10,000 formulas. According to this publisher's law, that would halve the sales ten thousand times (2^(-10000)), so a rough estimate of the amount of readers left is around 5x10^(-3001), which is a number so absurdly and unimaginably small, you have to be a Belgian surrealist painter to be able to imagine it. Roger Penrose would be left with much, much less than a single quark of a reader. Judging by the amount of readers here, this is obviously not the case.

What this publisher was pointing out, however, was that, in order to attract a lay public, you have to shun formulas, because they might chase them away and you wouldn't be able to teach anybody anything because there wouldn't be anyone left. Hawking followed the advice, wrote a book for the layman and, in my humble opinion, failed miserably. The layman didn't understand anything from his cryptic text, because they were missing several dozens of books of background information needed to understand most of the concepts. The more informed readers were chased away because of the lack of depth, the haphazard and disconnected way the concepts were presented and the absence of the mathematical beauty. It did sell well, so from this perspective, the publisher was undoubtedly right (and this was probably the only perspective he was interested in).

This fiasco teaches us something very important: writing and reading about present-day physics is no small feat. Lots of background is needed (yes, mathematical as well). However, unfortunately, most writers listen to publishers and omit the formulas and lots of the difficult bits as well, so the task for the physics reader is greatly hindered.

Luckily, Penrose is a kind old British gentleman, not afraid to take up the glove and rise to the challenge. The Road to Reality is his magnum opus, introducing the reader to modern physics (mostly cosmology and to a lesser degree quantum mechanics), without oversimplification or the omission of essential formulas. He doesn't muck about either. If mathematics is what you need to understand it, mathematics is what you'll get. No less than 16 of the 33 chapters are devoted solely to mathematical foundations. After that, Penrose gives an overview of the established physics (mostly special and general relativity, quantum mechanics and some thermodynamics. The last few chapters are concerned with the more controversial topics, like string theory.

The mathematical part is, to the best of my knowledge, quite complete. The established physics is concise and omits most things that aren't related to the aim of the book: present-day speculative cosmology. For the last part, Penrose focuses on string theory (currently still the most popular group of theories), loop quantum gravity (the most interesting contender) and twistor theory (Penrose's own field of research). His account of these theories is very honest, making no secret of his critiques on certain fields of research and warning the reader to take his interpretations with a pinch of salt and often pointing the reader towards books containing views conflicting with his own.

Does this make The Road to Reality the ultimate physics book for the dauntless layman? Of course not. The task Penrose set himself is neigh impossible. One cannot replace several dozen textbooks by one, albeit bulky, volume, even if it is written by someone as erudite as him. A lot of his explanations are just too brief. He would need another two to three thousand pages to be able to explain everything thoroughly. Furthermore, the splitting of the maths and physics in different chapters is probably necessary, but it means Penrose refers a lot to previous chapters. Since I have a hard time exactly remembering some subtle mathematical notion from 400 pages ago, I needed to skip backwards and forwards a lot, making for quite a laborious read (not something to do while commuting by bus, for example). The subject itself is very slippery too. Current theories cannot be verified (yet) by experiments, so theoreticians are only guided by very abstract notions, such as mathematical beauty, or instincts honed by years of hard work. Such things are often very hard to convey to anyone not a specialist in that particular field, although Penrose does a good job, nonetheless.

It is probably the closest you'll get to a textbook without actually reading a textbook. The Road to Reality trumps A Brief History of Time in many ways, but ultimately it cannot be avoided that one needs to read a lot more to truly understand physics, but that's not necessarily a bad thing (publishers will agree).

For the past year and a half, I have delved deeply into popular works on physics and astronomy. The range of these works varies from the rather simplistic and awe-inspiring (like Michel Kaku's) to the total beginner level, and then to the somewhat more advanced (such as those by Brian Greene, Kip Thorne, and Lee Smolin). However, almost always, I have been frustrated in my understanding due to the lack of mathematical support for the often metaphorical discussions.

At the same time, I am well aware that the real mathematics of relativity, not to mention quantum and string theory, would be far beyond my comprehension. This was evident when I borrowed and promptly returned one or two books by Penrose himself, as well as those by Wald, Susskind, and others. So, when I saw the description of this book, stating that Penrose intended to provide the necessary mathematics to understand the physics as he progresses, I thought it might be exactly what I wanted - if it were possible. I was also rather skeptical about whether it was truly possible to achieve this in under 1100 pages, and as it turned out, my skepticism was justified.

Penrose fails to make the math understandable. He explains certain concepts at a very basic level and then assumes that the reader understands far more than he has actually explained. When he reaches something especially difficult to comprehend, that's when he provides a footnote to show why it works. At the beginning, I read with a pencil in hand, but it soon became clear that I was unable to work out the problems.

I am not completely devoid of mathematical ability. I minored in math in college, and in addition to the usual high school background in algebra, geometry, and physics (before the dumbing down of high school math and science around 1970), I have taken two years of college calculus, a semester of calculus-based probability and statistics, several "fundamentals" courses, and a calculus-based (but entirely "classical") first-year physics course. Admittedly, it's not a vast amount, but it's probably more than most non-math/science majors. I have also tutored high school and college students in math up to early calculus for over ten years.

So, with this background, what did I understand of this book? The first six and a half chapters. Chapter six explains calculus in nineteen pages, and I'm fairly certain that most of what he discusses was never covered in my two years of calculus (perhaps it would be in a real analysis course?). Chapter seven was on complex analysis, and I understood about the first half. This pattern continued throughout the rest of the book - I understood about the first half of each chapter, where he explains the basic concepts, and then I got lost when he tried to "clarify" things through "the magic of complex numbers" (his favorite expression). The same was true when he moved on to the physics of relativity, quantum theory, and the modern speculations about strings, twistors, and so on - I understood about half of each chapter. (Actually, the chapters on loop quantum gravity and twistors were completely beyond my reach from the beginning.) I would say that the minimum requirement for truly following him would be to have already taken at least a complex analysis course.

Was the book a total loss, then? No. Although I didn't learn the various fields of math he covers (vector and tensor calculus, some projective geometry and topology, etc.), I did learn what those fields deal with. If I didn't learn what I needed to understand the physics, I did learn what I would need to study to learn it, and perhaps most importantly for an autodidact who can never afford more formal education, I learned the order in which I would need to study the various fields. In short, for me (and probably for most readers without a strong math/science background), this was not the "Road" but more of a roadmap. I didn't make the journey, but I got a sense of where the route passes through. At the very least, I was inspired to review my high school and college math and perhaps try to go a little further than where I stopped. At my age (nearing retirement), I will never fully understand the physics, but maybe reading this has kept my mind working a little longer.

I should briefly summarize what he is saying about the questions I've been reading about in other books. He is very skeptical of string theory, mainly because of the higher dimensions (especially the problem of degrees of freedom). He is somewhat more favorable towards quantum loops and other alternatives. Naturally, he is most interested in his own theory of twistors, but he admits that at present, it doesn't have the solution either. He argues that to be consistent with general relativity, there must be significant modifications to standard quantum theory. Obviously, I'm not well-informed enough to evaluate any of this.

I would recommend the book for what it is, but not for what it claims to be, unless the reader has a genuine background in mathematics and physics.

Penrose visited GT and delivered an open lecture on cosmic parameters and cosmological arguments derived from the 2nd Law of Thermodynamics. This topic is covered in chapter 27 of a certain book, which is one of the most ambitious and impressive catechisms I've ever read. Although it may be incomplete, a bit uneven, and as taxing as one has heard. Additionally, he gave a closed lecture on twistor theory, which is presented in chapter 33. What's more exciting is that he signed my copy! w00t! I had the opportunity to shake Sir Roger's hand. As this happened, trillions of neutrinos passed through both of us, completely undetected. Our entangled R-type state evolution left an indelible imprint on all our lightcones forevermore. This occurred at the cost of a little more entropy, with order being traded for disorder in the guise of order, yet still maintaining an orderly appearance.

The image provided shows the signed page 730 (27.13). It gives a visual representation of this memorable event. The photo adds to the overall experience and serves as a keepsake of Penrose's visit and his autograph.

Overall, Penrose's lectures and the opportunity to have my copy signed were truly remarkable and will be remembered for a long time.

I have this distinct feeling that I really need to commence a brand new Goodreads shelf that is aptly titled "What was Roger Penrose thinking?".

It is truly beyond my imagination to fathom for whom exactly he was penning this work. However, it is blatantly obvious that Penrose's intended audience is way above the level of a graduate student in chemical engineering.

It seems as if this book initially aspired to be a textbook. But then, it appears that the author didn't have the inclination to invest that significant amount of effort into it. So, in the middle of the process, he made a sudden switch and resolved to market it towards a general audience.

The end result? A massive failure. It just doesn't seem to hit the mark for either the academic crowd who might have expected a more comprehensive textbook or the general readers who might have been put off by the somewhat disjointed and perhaps overly complex nature of the content.

It's a real pity that a work with such potential ended up in this state of disarray.

This book RULES!

It serves as a primer on the mathematics essential for a true understanding of quantum physics. Of course, the relevant mathematical knowledge is a vast amount, and this book is indeed damn huge. Moreover, it progresses at a rapid pace: the entire theoretical foundation of single-variable calculus is covered in just one chapter. The reader is swiftly led through intense cram sessions in multivariable calculus, algebraic topology, real analysis, and more - everything one needs!

Yet, it doesn't feel dense at all. This is because Roger Penrose is one of the great living stylists in mathematical writing. Even a significant number of equations can't impede the fluidity of his prose, the lucidity of his explanations, or the enthusiasm of his presentation. I had taken classes on some of these topics, and I felt that Penrose's single chapters were at least as beneficial to me. For other topics with which I was only superficially familiar, this book felt like a high-powered introduction. There are even some really excellent exercises, mainly in the form of proofs (the enjoyable kind of exercise).

Exhaustive theoretical math books are rare but do exist. What makes this one truly effective (in contrast to some dogmatic giants like the classic Whitehead & Russell) is that Penrose is actually aiming at a somewhat tangible goal: educating a casually geeky, intellectually curious layperson about quantum physics without leaving anything out. For whatever reason, it works.

I brought this book along as my only reading material on an 8-week tour, and it was perfect. I could get a quick dose of clearly and beautifully written, mind-bending math a couple of times a day. YES!

UPDATE 2022:

Several years later, I discovered that this work, which includes Feynman's lectures, was a far better use of my time. Thanks to it (and COVID), I came to understand fundamental physics. These are the "years' worth of secondary sources" mentioned in the review: I could now comprehend the book. https://www.susanrigetti.com/physics

Let me begin by stating (its relevance will soon become clear) that I hold a bachelor of science in applied mathematics and a PhD-ABD in another highly quantitative discipline* (both from top-50 schools, and I wasn't in the bottom 50% of the class). After reading the first 300 or so pages (out of 1200), I found that the math in this book (which is at least 40% or more pure math, not text, and often without text explaining the math) was far beyond my comprehension and was often left undefined in the text. The author doesn't even have the courtesy to direct the reader to textbooks where these concepts, such as pseudo-Riemannian geometry, anti-de Sitter spaces, and Seiberg-Witten manifolds, are defined and can be learned.

The book fails to fulfill its promise and purpose of being a self-contained guide to the current mathematical or theoretical-physical understanding of the universe. It is far from accessible to the layman (I have postgraduate training in math and was a good student, yet it is inaccessible to me). To grasp the concepts in this book, I would probably have to spend a year of my free time and a thousand or more dollars on secondary sources (if I bought them used and cheap). I purchased this book hoping for a $20 overview (like Collier's 'A Most Incomprehensible Thing' for the theories of relativity [I prefer the original 'invariance'], which was technical but self-contained and comprehensible; reading that was the only thing that gave me any knowledge at all of tensors, which this book is full of). What I got was essentially a 1200-page bibliography without the authors being noted and without the important works being starred.

This is a very ambitious book that fails completely in its execution.

The author starts by explaining what complex and irrational numbers are and why they are useful (this is freshman high school math) in the introduction, accompanied by an apology for the necessity of using a difficult concept like logarithms. Then, 200 pages later, he jumps to pseudo-Riemannian geometry (this is postgraduate pure math). No kidding. Penrose spends about five pages defining all of classical statics and dynamics and then assumes that you understand classical mechanics. This breakneck pace is maintained throughout, which is how he manages to cover everything from logarithms and complex numbers to doctoral-level mathematics in 500 or 600 pages. Once he moves out of pure math and back to applied math (i.e., physics proper), it gets a little easier, but I would still not recommend attempting to tackle this book unless you are a graduate in math or a self-taught prodigy in pure math.

The book promises to be a self-contained guide to the best mathematical understanding of the universe we have, but it ends up more like the author simply inserted the important theorems with a minimum of explanation (he does cover almost all of them: one thing that struck me as unnecessarily erudite - showing off - and odd was the statement of Maxwell's field equations, which are mathematically simple and elegant, in terms of tensors, which are very, very difficult). So, it's a complete guide if you already know all of the math (in which case you don't need the book); it's much more of a refresher and quick reference for people who are already familiar with and understand (or at one time understood) the concepts the author presents.

Required prerequisites: understanding of linear algebra (Lie, Poisson, Frobenius), TENSORS (and more tensors), several varieties of non-Euclidean geometry (Minkowski, de Sitter, Riemann), scalars, topology and n-manifolds, group theory (Lie groups), gauge theory, etc., or the willingness to learn these from expensive secondary sources, because Penrose will not teach you them here and the arguments of the book are incomprehensible without them. Without them, one would be reduced to skimming the 20% of the book that is text (especially the final chapter, which is comprehensible to any semi-educated layman) and taking the author's word for the rest of it. Just about the only thing he explains in full is twistor theory (his own invention).

I still have to award two stars for the obvious intensity and depth of erudition that Penrose poured into this work, but only two because it doesn't even partially fulfill its stated purpose or self-description.

*Redacted to protect the privacy of a member of Vagabond of Letters.

I have a strong suspicion that Penrose hasn't interacted with an undergraduate student in the past 30 years. His concept of "introductory material" is not only incorrect but also extremely strange.

The renowned mathematician dedicates several pages to the discussion of fraction addition and then swiftly skims through holomorphic functions and Riemann spheres.

It seems that his approach might be a bit too advanced for the intended audience. I plan to revisit this book in a year or two when I have accumulated sufficient mathematical background to be considered a "non-mathematician" with a better understanding.

Perhaps by that time, I will be able to make more sense of the complex topics he presents and gain a deeper appreciation for his work.