What We Cannot Know: Explorations at the Edge of Knowledge

by Marcus du Sautoy

Britain’s most famous mathematician takes us to the edge of knowledge to show us what we cannot know.

Is the universe infinite?

Do we know what happened before the Big Bang?

Where is human consciousness located in the brain?

And are there more undiscovered particles out there, beyond the Higgs boson?

In the modern world, science is king: weekly headlines proclaim the latest scientific breakthroughs and numerous mathematical problems, once indecipherable, have now been solved. But are there limits to what we can discover about our physical universe?

In this very personal journey to the edges of knowledge, Marcus du Sautoy investigates how leading experts in fields from quantum physics and cosmology, to sensory perception and neuroscience, have articulated the current lie of the land. In doing so, he travels to the very boundaries of understanding, questioning contradictory stories and consulting cutting edge data.

Is it possible that we will one day know everything? Or are there fields of research that will always lie beyond the bounds of human comprehension? And if so, how do we cope with living in a universe where there are things that will forever transcend our understanding?

In What We Cannot Know, Marcus du Sautoy leads us on a thought-provoking expedition to the furthest reaches of modern science. Prepare to be taken to the edge of knowledge to find out if there’s anything we truly cannot know.

Genres: Science, Nonfiction, Philosophy, Physics, Mathematics, Popular Science, Psychology

450 pages, Kindle Edition

First published May 5, 2016

About the author

Marcus Peter Francis du Sautoy, OBE is the Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford.

Reviews

[BlackOxford · Rating 4 out of 5]

Stories For the Widening Gaps

I was educated in the school of anti-reductionism. That is, I was taught that knowledge of how the fundamental particles of the universe work would not help me understand what I might choose for dinner. I did take university courses in physics and chemistry, but the deterministic implications of science, those which suggested the fictional character of things like purpose and choice and free will, were never allowed to surface fully. It turns out that my views may simply be the result of never having encountered a literate reductionist. Marcus du Sautoy is such a literate reductionist. And I am tempted to rebel at an advanced age against my upbringing.

The classic objection to scientific reductionism - the idea that the world works strictly according to fixed laws of cause and effect - is that it can’t explain science and other apparently purposeful activities. But it turns out that there are good scientific reasons for this. The complexities of scientific thought, actually any thought, are so great that although its physics are deterministic - in the manner of billiard balls on green felt - its trajectory cannot be predicted. We cannot know either the starting point of the billiard balls or their random quantum connections.

It is this complexity, not any extraneous properties like spirit, or intelligence, or God, that leads, deterministically, to our stories about some substances other than energy and matter as important to human life. Properties that appear ‘emergent’ are simply those whose complexity we cannot (yet) penetrate. Our narratives of human motivation, character, and responsibility merely fill in the explanatory and predictive gaps created by this emergent complexity.

Interestingly, these gaps increase as science progresses. More knowledge about the world increases our understanding of its complexity even as we simplify our theoretical descriptions. Every scientific advance multiplies the number of unsolved scientific problems. What du Sautoy (and Donald Rumsfeld) calls the known unknowns blossom exponentially. But rather than project philosophical or theological narratives into these expanding gaps, we can only throw more science. The narratives that science supplies may be less literate but they are certainly more compelling. Human thought, it appears, becomes simultaneously more inclusive and less certain but yet more confident as it develops.

I admire this implication. It is a humbling observation that we are not exempt from the materiality of the rest of the universe. The exceptionalism of humanity in its own mind has been the cause of much misery. Everything we know is uncertain. Kant, it appears has been vindicated - we can never even hope to know the universe as it is. Yet we can have confidence in our scientific narratives because... well because they provide something for which to hope. This hope is not one of personal salvation, or the coming of some other universe but one of realizing human harmony with the universe as it exists. This, I think, is the principle ethical content of modern physics: that there is an inescapable necessity to conform, even when we believe we are rebelling.

Traditionally, ethics have been grounded in one’s view of anthropology. Some believe that appropriate human behaviour can be deduced from the necessities for survival; others that there are divinely mandated rules for conduct. What du Sautoy’s scientific narrative suggests is that this anthropological view is arbitrary in terms of both its level of analysis as well as its presumptions about what it means to be human. Why not a ‘fundamental particulate’ ethics? And why not a complete abrogation of any fixed, final definition of what these fundamental particles might be? This approach would suggest that the search for what it means to be human is the real ethical task, a task which should never result in dogma that is purported to be certain.

It could be that my sympathy for du Sautoy’s account is prompted by his own sympathy for Plato over Aristotle - an attitude roughly equivalent to a psycho-analyst preferring Jung to Freud. Du Sautoy quotes Werner Heisenberg approvingly: ‘Modern physics has definitely decided in favour of Plato. In fact, the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.’ So then, science, art and philosophy appear as random variants of the same deterministic trajectory. We can’t seem to stop ourselves from doing the same thing - telling stories - in different ways.

All sorts of criteria for verifying science - or for preferring Plato over Aristotle - have been used historically: simplicity, elegance, statistical correlation, logical implication, and tradition, among many others. The criteria are as unstable as the results of science itself, and they vary among disciplines and are differentially favoured by individual scientists within disciplines. It doesn’t take much insight to recognize what du Sautoy admits: the criteria of scientific analysis are unavoidably aesthetic; they correspond to what we find beautiful. Beauty may be correlated with things like usefulness and consistency in evolutionary terms but it nevertheless stands alone as the proximate cause of scientific action.

So the stories we tell ourselves are better or worse not by any objective standard but by the inherent arbitrariness and subjectivity of our appreciation of what constitutes the beautiful. Our nervous systems are even apparently constructed so as to provide illusions when confronted with incoherence. Having said that, it is clear that beauty is not entirely arbitrary or subjective. Beauty is something we are taught, like language. It may appear that certain forms or ideas are ‘obviously’ more beautiful than others but only because there is no challenge to them within our social group.

Science, it seems, is the evolved social institution in which just this debate about aesthetic criteria must take place. And the historical narrative of science is one of continuous instability, a story of changing fashion and taste. This, not some fixed standard, is what we mean by ‘reason’ - intense but non-violent argument about how to argue. The content of reason, its meaning, varies more or less continuously. Any attempt to halt the evolution of reason - by religious authority or governmental dictate - will therefore ultimately fail as unreasonable.

Could it be that the narratives of continuity that we tell ourselves - of personality, of memory, of culture, of history - are mere gap-filling? Having disposed of God as the explanation for what we find inexplicable, we seem to rely even more heavily on stories about ourselves - psychological, behavioural, even spiritual stories that have the same function as the old fashioned religious narratives: providing comforting continuity. These stories may be put forward as scientific but are they anything but pseudo-science in their insistence as being ‘truthful’ given that they cannot account for the fundamental discontinuities of our existence? Scientific determinism, therefore, has an unexpected effect - it opens the world to change, innovation, surprise, even opportunity, particularly the opportunity to create new stories.

I feel a certain satisfaction in the uncertainty created by this conclusion. And it gives me some comfort as well. In a way, it seems to me, recent science frees story telling, and therefore all of literature, from the stigma of ‘fiction’. What is the distinguishing mark of science other than its unconstrained swapping of stories? The fact that we can’t know anything definitively, neither the length of the hypotenuse of a right triangle of unit size, nor the character of a photon, means, perhaps, that we have evolved randomly precisely in order to produce and exchange fictions. Jorge Luis Borges would probably be pleased. And Adam Levin in his novel of spiritual determinism, The Instructions, provides a rather apt summary of the situation: “It is good to do justice because God will kill you and your family whether you do justice or not.”

[David Rubenstein · Rating 5 out of 5]

This is a fascinating book about the theoretical limits of our knowledge, our ability to make predictions, and to understand the universe. Marcus du Sautoy is a professor of mathematics at Oxford University. A central theme of the book, is to show how mathematics works as a powerful tool in helping us understand nature, and the limits to our understanding of how it works.

The book begins with a discussion of throwing a 6-sided die. With perfect knowledge of the position, velocity, and rotation of the die, is the outcome predictable? Actually, the answer depends on the friction of the table onto which the die is thrown. For certain ranges of friction, chaos prevents predictability.c

So the book relates the story of King Oscar II of Norway and Sweden, who celebrated his 60th birthday by offering a gold medal prize to whomever solved a math puzzle; "Is the solar system stable?" Poincare worked on this puzzle, and submitted a paper analyzing the orbits of two planets and a speck of dust. He showed that they followed periodic paths. The editor of the journal showed a gap in the proof, so Poincare tried to stop publication, but it was too late. Poincare had discovered that small perturbations to a stable orbit could make the system fly apart. This led to the discovery of chaos. In 2009, French astronomers modeled the solar system's stability many times, adding slight perturbations. The orbits of Jupiter and Saturn were very stable, but in 1% of the simulations, Mercury's orbit had a resonance with Jupiter, allowing Mercury to collide with Venus, which then collides with the Earth.

It is well known that the Earth's atmosphere is a chaotic system. A small perturbation leads to unpredictable outcomes. This idea feeds the claims of climate-change deniers, that we cannot predict a change in climate. Sautoy brings up the quote,

Not believing in climate change because you can't trust weather reports is a bit like saying you can't tell when the next wave is going to break on Bondi Beach because you don't believe in tides.

Sautoy gives the clearest explanation of wave-particle duality and some of the paradoxes of quantum mechanics, that I have ever read. He discusses elementary particle physics. He brings up the paradox, that electrons behave as if they have no volume, yet they have mass. But then, what is their density? Is it infinite? Is each electron a tiny black hole?

Sautoy brings up a very interesting aspect of Heisenberg's uncertainty principle. Heisenberg's original paper on the subject is often misunderstood. It is not that the act of observation affects a system. Heisenberg was forced to put that example into his paper in order to get it approved by skeptical editors. Even without a direct interaction with a particle, the product of uncertainties in its position and momentum is limited. It's not that we cannot simultaneously precisely measure position and momentum. It is that they do not exist until we measure them.

Many scientists equate God with "things we cannot know." Heisenberg's uncertainty principle also holds for time and energy. When you look at empty space for a short period of time, there is uncertainty in the energy content; space can never be truly empty. Since energy can change into mass, particles can spontaneously appear in a vacuum. So, there is no need for a creator. Some people give the definition of God as the solution to a question, such as "why is there something rather than nothing?" The trouble with most religions, is that God is then given so many properties that god ends up having nothing to do with the definition. It is like working backwards, conjuring up properties without understanding the original definition. One could define God as the existence of things we cannot know. To declare oneself as an atheist is to mean there is nothing we cannot know. But to say that there are things we cannot know, then, is a proof that God exists.

Cosmology is a science where not all questions are, in principle, answerable. This is a "Copernican" aspect of things. The universe is expanding and in the past seven billion years, the expansion rate has been accelerating. Dark energy is hypothesized as the cause. In the distant future, we won't be able to see other galaxies, as space will have disappeared from view, beyond the sphere of our visible universe. But we will still see the stars in our Milky Way galaxy. So, we are lucky to know that other galaxies even exist. Some day in the far future, this knowledge may be lost.

Sautoy describes an interesting set of experiments designed to fool the brain into thinking that your seat of consciousness is located elsewhere--in another person or even in a Barbie doll! Then there is a fascinating discussion of experiments dealing with consciousness. A group of philosophers believes that science can never solve the problem of consciousness. But, we are reminded of the prediction of the philosopher Comte, who foretold that we would never know what is at the heart of a star.

This book is written in a very accessible style. Sautoy gives crystal-clear explanations. I enjoyed every page of this book, as it gave me different ways to think about the universe, and the role of mathematics in unraveling its mysteries.

[Marc · Rating 3 out of 5]

Maybe you thought this was a book about science? Well, actually, this is a love story, really! At least every 10 pages du Sautoy confesses his unbridled love to ... mathematics, the perfect universal language of our cosmos. That is not surprising, because Marcus du Sautoy is a professor of mathematics at Oxford, and obsessed by the maths virus since he was a kid.

He uses that ‘weapon’ of mathematics to look at the sciences, starting from the question of whether there is anything that we will never know. Indeed quite a challenge, such a difficult question. And the whole, rather predictable, list passes: the question whether the laws of nature are truly universally valid, the search for the smallest building block of matter, the question of whether our universe is infinite and whether there was anything before the Big Bang, the question what time really is and finally the riddle of human consciousness. Du Sautoy sketches the scientific progress in these domains with a lot of verve, to arrive at very annoying stalemates, as a result of chaos theory, the uncertainty principle of Heisenberg, quantum theory, etc., and that is also a well-known list.

It turns out that indeed there are questions that are (provisionally) unanswerable. And although Du Sautoy believes with heart and soul in the progress of science (especially by using creativity, not so much by observing and experimenting), there are apparently issues that we can never know the ‘truth’ about. Even his beloved mathematics does not escape that fate; perhaps mathematics itself is 'only' a language that is limited by the human brain that constantly works in and through language and that is determined by the physicality of men. He formulates this as a possible explanation, more like a question even, without affirming it (it must have hurt him a lot).

It adorns Du Sautoy that he does not shy away from the thorny question of God in these observations, on the contrary. After all, he is the successor to none other than the polemic "religion-fighter" Richard Dawkins at the professoral chair for the Public Understanding of Science in Oxford. Just as Dawkins, du Sautoy openly proclaims himself as an atheist (with the exception of his football club Arsenal, that is) and in this book he studies various images of God claiming to fill the open spaces in science (that which we cannot know). They are viewed with respect by him, but they are rejected as insufficient or problematic.

In the end, Sautoy argues for a form of scientific opportunism: let us embrace the uncertainties we now identify in the sciences, but then as a challenge to continue our scientific work, as creative as possible, and perhaps some day we will find answers to some seemingly unknowable issues; and if not, we simply continue to work within what we can know, in those areas science has already made huge strides and will continue to be successful.

I certainly recommend the reading of this book, but for the layman this work still is quite difficult. du Sautoy does his uttermost to be as clear as possible, but I had a hard time especially when he brings up mathematical examples or explanations (but, of course, that might be due to my mathematical insufficiency). So, however appealing, it’s a bit of a hard nut to crack. Some disappointment also because about 4/5 of the issues that the author touches upon comes from the world of physics; only the chapter on consciousness is an exception. But for example the whole interesting domain of biology remains completely out of the picture, and then of course we don’t even mention the human sciences.

I read this book to see to what extent the entire movement of Systems Thinking, Cybernetics and Complexity Theory are discussed. To my surprise, they aren't! Yet there are indirect references to these currents: the author occasionally uses the term "emergent property", in the end he admits that the core problem of science is that we belong to the reality that we're investigating, and in the chapter on the mathematics he uses the term 'system' several times. That is strange, because Systems Thinking and its relatives fundamentally embrace the uncertainties that are hidden in our reality, and they offer a method to deal in a pragmatic and scientifically sound way with these uncertainties. However interesting this book may be, this certainly is a gap.

[WarpDrive · Rating 5 out of 5]

Brilliant, highly accessible, extremely readable and riveting exploration of the limits of human knowledge.

The author is a Professor of Mathematics at the University of Oxford, and he is highly renown for his excellent, contagious, enthusiastic divulgation to the wider public of the beauty of the mathematical world.

His exploration of the boundaries of human knowledge is multi-faceted, encompassing a variety of disparate disciplines such as quantum physics, cosmology and relativity, probability, chaos theory, computation theory, neurosciences and, of course, mathematical logic, number theory and the foundations and nature of mathematics. Even some important aspects of philosophy of science and mathematics are briefly touched by the author.

Very easy reading, accessible to everyone but at the same time conceptually accurate and quite informative, this book will probably provide not that much new information to more sophisticated users who are already knowledgeable in these areas of expertise (the book does not really provide much depth on any of the topics it examines), but it is a really pleasant read for everybody - in fact, while addressing with competence and clarity many important and fascinating issues in contemporary science and mathematics, the author manages to accomplish it in such a concise and brilliant way as to require only a minimum pre-requisite knowledge in order to be fully savored.

I also particularly enjoyed his contagious love for the beautiful world of mathematics - in his infectious, almost Pythagorean enthusiasm for this discipline, he can be put on a par with giants such as Penrose.

Highly recommended.

Some of my fav. quotes from the book:
-"mathematics is infinite. It goes on forever. Unlike chess, which has an estimated 10^10^50 different games that are possible, the provable statements of mathematics are infinite in number. Mathematics does not have an endgame."
-"that is the power of mathematics: to use our finite brain to know the infinite"
-"one of the reasons for believing that science is producing true knowledge is its success rate. Science is so successful in its description and prediction of the way things appear to be that we feel like it must be getting close to a reality that most of us believe does exist"
-"if the map that you are using consistently gets you to your destination, that's a good sign that the map is a pretty accurate representation of reality. Science may not "really" represent reality, but there isn't anything that comes even close as an alternative"

[Ettore1207 · Rating 4 out of 5]

Scrivere un libro su "ciò che non possiamo sapere", se ci pensate bene, è un ossimoro e un compito arduo. Non c'è dubbio che l'autore abbia una gran testa, il libro è interessante, e sarà ancor più apprezzato da chi possiede conoscenze di matematica e fisica più solide delle mie traballanti. Alcune parti le ho capite poco, ma non certo per colpa dell'autore.
Non mi preoccupo più di tanto, e invoco a mia parziale discolpa quanto segue:

Devo ammettere che la natura controintuitiva del mondo quantistico mi sta dando grossi problemi. A quanto pare, comunque, è un buon segno: come ebbe a dire il fisico Niels Bohr, «se la fisica quantistica non vi ha profondamente sconvolti, significa che non l’avete ancora capita».
Richard Feynman si spinse addirittura oltre, dichiarando che «non c’è nessuno che abbia capito la fisica quantistica». Nel corso di un discorso d’apertura tenuto quando aveva più di sessant’anni, ammise: «Potrei confessare subito che abbiamo sempre incontrato (segreto, segreto, chiudete le porte!), che abbiamo sempre incontrato, dicevo, un sacco di difficoltà nel comprendere la visione del mondo rappresentata dalla fisica quantistica. Mi mette in ansia ancora oggi».

[Christopher · Rating 2 out of 5]

Modern physics and mathematics have brought some major new discoveries in recent years, but there is a general feeling that we are approaching the limits of knowledge, and soon further discoveries will be much more difficult or outright impossible -- at least according to the current paradigms. In What We Cannot Know Marcus du Sautoy, an English mathematician who was elected Simonyi Professor for the Public Understanding of Science at Oxford in 2008, tries to explain these impasses to a general audience. The book is essentially structured as a series of places where human inquiry is hitting a wall: 1) quantum physics, 2) elementary particles, 4) cosmology, 5) artificial intelligence or consciousness, and 6) abstract mathematics.

There have been a large number of popular science books in recent years trying to explain the insights of quantum physics or cosmology to a general audience with no particular ground in the sciences. Unfortunately, du Sautoy's book is not particularly competitive among them. Du Sautoy is longwinded and repetitive, and often the metaphors he constructs to make all this accessible for a layman audience, become so convoluted that the thread of the narrative is lost.

But perhaps the biggest flaw is the inconsistency in what kind of audience he is writing for. At one moment, du Sautoy is patiently explaining what a "light year" is, so he appears to assume an audience with the least background possible. But then later in the book he starts dropping one mathematical equation after another, and a better than ordinary familiarity with maths would be necessary to understand his point. As I read this, I kept thinking that if du Sautoy is employed as one of the foremost popularizers of science in the UK and the world, he's not doing a very good job.

[Sara · Rating 4 out of 5]

There are so darn many pop science overview books out there. This is a particularly good one. The author has made a career of promoting science to the general public, he's done TV, radio, and previous books, and he's got it down. He expects his readers to be intelligent and to think but he doesn't throw a lot of equations at them. He comes up with clever and accurate metaphors that have not been done to death, and the hand-drawn illustrations are attractive and genuinely helpful.

[Cher 'N Books · Rating 2 out of 5]

2 stars = Meh. Just ok.

“What I want to try to explore is whether there are problems that we can prove will forever remain beyond our knowledge.”

It is always frustrating when a synopsis is misleading, giving the reader erroneous expectations for what they are about to read. This one promises a captivating (nah, bro) read about history, science and philosophy. Those three subjects are present but they are merely the side characters. The summary does not even hint towards what this book actually is - a romance with mathematics, which is the main character. When I say this author loves math, it is not hyperbole. It is practically a religion for him.

“Perhaps we should reverse the old adage “God is a mathematician.” Perhaps mathematics is the god everyone is chasing.”

The first 40% is painfully slow and boring (chaos, matter, quantum physics), but the back half was filled with more compelling topics (the universe, time, consciousness and infinity.) The author does an excellent job of breaking down mathematical equations and scientific theories into simplistic terms that are easy to comprehend. But unlike Neil deGrasse Tyson and other authors, he is not able to do so in a manner that engrosses the reader. It was a struggle to maintain focus and keep turning the pages as the writing was so dull. Most of what I highlighted were quotes from others that are included in the text. Below is an example of the delivery so you can judge for yourself:

“If the curvature continues across the whole surface, then it must join up to form a finite surface. Curvature like that of a sphere is called positive curvature. If a surface is flat, then it could be infinite, extending forever, but it could also be like the game of Asteroids, in which the universe on the screen is flat but finite. Flat surfaces are said to have zero curvature. There is another sort of curvature, which is like the saddle of a horse or a Pringle crisp. The curvature seems to dip down in one direction and up in the other direction. This is called negative curvature, in contrast to the positive curvature of the surface of a sphere. It gives rise to infinite surfaces, rather than finite surfaces like that of the sphere.”

This was educational and easy to understand, but not exactly what one would call riveting, despite the topics themselves being fascinating. I did gain a better understanding of cosmic concepts and appreciated some of the philosophical discourse which is what kept me reading to the end. The most important takeaway, is if you are ever able to find someone that loves and adores you even an infinitesimal fraction of how much this author loves mathematics, then you are immeasurably fortunate and have truly achieved the apex of success in both romance and life.

“When we die, time stops for us, but we know that time will continue on unawares. Just as I cannot experience my own death, I wouldn’t experience arriving at a singularity in space-time.”
-----
First Sentence: Every week, headlines announce new breakthroughs in our understanding of the universe, new technologies that will transform our environment, new medical advances that will extend our lives.

Favorite Quote: Death-awareness is the price we pay for self-awareness.

[ReadsandThings · Rating 5 out of 5]

I find this book really hard to rate. Despite always being told by her teachers and parents that science was not for girls and that I would always be bad at it because I was a girl (which turned out to be a self-fulfilling prophesy), I have always had a deep-seated fascination with science in general and astrophysics in particular. Ever since realising as a child that the universe was either infinite (and how could that be!?) or finite (and then, what was beyond it?!) I have had this aching desire to KNOW, and to know the limits of what we CAN know.
Clearly, I have no background in science at all, and 98 % of the book were completely over my head. That is why I find it hard to rate - I have no shred of context to estimate if the science is sound or if what he says makes sense.
On the other hand, I had so much fun with this book! I listened to the audiobook read by the author himself, who does a great job as a narrator. His own fascination with his subject and his joy in talking about it come across very clearly and are quite infectious (however, as a German I do have to say that his pronunciation of German terms and names is endearingly atrocious).
I kept finding myself wanting to ask the author questions, to clarify things I hadn't quite grasped or to expand on some part I found particularly interesting. I think that is the highest praise I can give to a book aiming to teach the reader something.
So I am giving this book 5 stars for the sheer pleasure I had listening to it, even though I have no way of rating the actual content.

[Jose Moa · Rating 5 out of 5]

A very good,interesting and deep in concepts and reflections book.
In words of the author : "What I want to try to explore is wether there are problems that we can prove will remain beyond knowledge despite any new insights".
In the knowledge subject are famous the words of Donal Rumsfeld :
There are known knowns.
There are known unknowns.
There are unknown unknowns.
The author first explore what is known and then especulate about the unknowns.
The book is divided in seven edges of known and unknown.

The first edge is about probability and chaos.
The second edge is about quantum mechanics and particle physics.
The third edge is about duality particle wave in quantun mechanics,the role of the observer,the incertidumbre principle and related subjects.
The fourth edge is about the shape of the universe,its finitude or infinitude,the Fermi paradox and the posibility of the multiverses.
The fifth edge is about time,the special snd general relativity,the posibles shapes of time,the causality,the black holes and also touch the cicles of time cosmology by Roger Penrose.
The sixth edge is about consciousness ,he says that explaning consciousness or the working of our brain using our own brain is a hard problem because we enter in a autoreference problem ; he also enters in the Integrated Information Theory related with the structure of connectivity and feedbacks in a network and also for example that the question of distinguishing zombies from conscious beings could remain as one of the unanswerable questions of science.
The seventh edge (the author is a mathematician that has written popular books about group theory ant the Riemann hypothesis ) is mostly abot the limits of knowledge in mathematics caused by the first and second Godels theorems that could make that the Goldbach conjecture or the Riemann hypothesis could remain forever out of reach for the human minds.
In the end he compares his list of problems in the edges with the seven riddles posed by de Bois-Reimond :
1.The ultimate nature of matter and force
2.The origin of motion
3.The origin of life
4.The apparent teleological arrangements of nature
5.The origins of simple sensations
6.The origins of inteligent thougth and languaje
7.The question of the free will.
A interesting feature of this book is that at least for me is the first time I have seen a good clear attempt to popularice a explanation of the difficult demostration of the two Godels theorems.

Recomended for everibody that would like a more deep insight of the nature of reality.

[Arturo Herrero · Rating 5 out of 5]

¡Increíble! ¿Tiene límites el conocimiento humano? Sautoy trata de responder a la pregunta explorando las fronteras del conocimiento en distintos ámbitos: ciencia, filosofía, física cuántica, neurociencia, matemáticas, etc. Uno de los libros más complejos y fascinantes que he leído en mi vida.

[Robert Narojek · Rating 5 out of 5]

It is fantastic book - it reads like a crime novel. Of course if one loves science and its "unsolvable mysteries" :)

[Paromita · Rating 3 out of 5]

A bit haphazard especially the final sections on consciousness and mathematical proofs but lots of food for thought.

[Emre Sevinç · Rating 4 out of 5]

If you've missed your share of popular science books on physics, cosmology, mathematics, and neuroscience during the last two decades, then this book can be a not-so-bad starting point. The author has a very down to earth style, and manages to be engaging at the same time. His objective is clear: are there aspects of live, universe, and everything that are in principle unknowable? Are there hard limits to science? The question is simple to ask, yet finding out definitive answers is not so easy. But by setting such a clear motivation, the reader is gently led towards the state-of-the-art in scientific knowledge. Along the way, the author shares nice conversations with scientists who not only know about their respective fields, but had been pioneers who had also spent time thinking about deeper questions. Some of them, such as John Polkinghorne, might surprise you about their perspective on physics and religion.

Unfortunately, no matter how engaging it generally is, the book fails to provide much depth on any of the topics it examines. In other words, if you already know about that topic then you'll be bored, and if the topic is totally new for you, then you'll be left scratching your head because of so many missing details. But if, in that case, the book manages to motivate you to dig a little deeper, your reward can be satisfying, but only after further reading. And that reading does not have to heavily technical, for example, whereas this book talks about dark matter and dark energy only superficially, "Dark Matter and the Dinosaurs: The Astounding Interconnectedness of the Universe" by Lisa Randall will be able to give you a much better picture (while still avoiding 'scary' mathematical formulae).

You might also like seeing a mathematics professor from Oxford struggling with interpretations of quantum physics, and trying to deal with the fuzzy nature of neuroscience research. The physical world is indeed messy, only approximately modeled, always waiting to surprise us, and still leaves a lot of things in the dark. But maybe, just like the author, you will get a deep satisfaction from our unstoppable appetite for scientific search and discovery.

Last but not the least, even if you had your fair share of popular science books, you might still learn a thing or two from this book regarding the edges and limits of mathematics, physics, and consciousness.

[Ian · Rating 3 out of 5]

A 2-star rating would better reflect my own personal enjoyment of this book. I’ve rated it 3 stars partly because there were sections that I enjoyed, and partly in an attempt to include an element of objectivity to the review. Normally if I give a non-fiction book 2 stars, it’s because of easily identifiable errors or because the author has been overly tendentious. I can’t say either of those things in this case. However, my own reaction can be summed up in the old relationship breakup cliché:

“It’s not you, it’s me.”

Professor du Sautoy is a mathematician, and he really, really, loves his subject. Unfortunately for me, maths was the subject I struggled with most at school. My eyes still tend to glaze over when I see an algebraic equation, and there are certainly plenty of those here. Pretty much every discussion in the book turns back to maths. For example, in discussing whether the Universe is either finite or infinite, the author comments that, whilst the finite is knowable, at first glance it would seem impossible to prove the existence of an infinite Universe. He then goes on to hold out hope that mathematics might one day be used to prove infinity by means of proof by contradiction (i.e. “a mathematical contradiction identified under the assumption of a finite Universe”).

I did enjoy the section on consciousness, and the discussion on whether humans have free will. Maybe that’s a sign that I prefer biology to physics. I also liked the discussion of Chaos Theory, and I quite enjoyed the section on quantum physics, if only because it was completely mind-boggling and constantly left me thinking “Whaaaaat?” However, if I’m being honest there were chunks of this book that I struggled to follow. Personally I wouldn’t recommend it unless you have an above average understanding of maths.

[Marián Tabakovič · Rating 5 out of 5]

Keď chcete porozumieť tomu, ako funguje svet - a ja chcem - nemôžete odignorovať tie najvyššie úrovne a úplne fundamentálne otázky. Prečo niečo vôbec existuje a prečo neexistuje nič? Čo bolo na začiatku reťazca, v ktorom sa niečo vytvára z niečoho iného? Prečo si uvedomujeme samých seba? Problémom je, že na to, aby človek dostal odpovede, ktoré nie sú banálne, absurdne zjednodušené, alebo nábožensky podfarbené, sa treba zahĺbiť do viacerých vedeckých disciplín. Vyštudoval som humanitné vedy a hoci mám rád populárnu vedu, nezvládol by som viac ako najzákladnejšie matematické a fyzikálne vzorce a rovnice. Knižka ma preto dosť potrápila, ale prehrýzol som sa ňou. Chcel som si otestovať svoj svetonázor, ktorý som si vytvoril ešte kdesi na strednej škole a rád som ním ohuroval pri krčmových debatách. Inak chýba mi to. V podnikoch, keď sú občas otvorené, sa o tom už nikto nebaví a na Facebooku sa mi to robiť nechce.

Takže, považujem sa za deistu (verím v akéhosi Boha filozofov, prvohýbateľa, ktorý vytvoril dokonalý vesmír a nezasahuje doňho), nevylučujem, že univerzum samotné môže mať vedomie (panteizmus), nevylučujem ďalšie vesmíry (teória multiverza) a nevylučujem reainkarnáciu vedomia. Vo vesmíre sa všetko recykluje, my sami sme vyskladaní z odpadu po výbuchu supernov, prečo by sa teda nerecyklovalo aj vedomie?

Kniha prekvapivo vrátila do hry aj teistický pohľad na Boha. Autor, hoci sa sám považuje skôr za ateistu, vážne rozoberal možnosť, že najvyššia bytosť môže do chodu sveta zasahovať bez toho, aby porušovala vlastné fyzikálne zákony. Môže to robiť cez teóriu chaosu (chovanie systémov je od istej zložitosti nepredvídateľné, hoci sú tam ostrovčeky stability), alebo cez kvantovú neurčitosť. No lenže prečo by dokonalá bytosť vytvorila nedokonalý vesmír? O tomto sa hádali už Newton s Leibnitzom.

Kvantový svet je ešte viac mind-blowing, ako som si myslel. Fascinovala ma informácia, že častica na kvantovej úrovni ako keby vedela, že je pozorovaná a správa sa vtedy inak, ako keď pozorovaná nie je. Neustálym pozorovaním ju vieme udržať v jednom stave, v ktorom by inak nevydržala. Tento bizár nikto nevie vysvetliť a vedci sa v Kodanskej interpretácii viacmenej len zhodli, že to proste tak je a že to nechajú tak. Celá realita je podľa autora akási nejasná a od určitej hĺbky prírodu nemôžeme pozorovať, keďže má v sebe zabudovaný limit.

Je možné, že okrem mnohých budúcností existuje aj mnoho minulostí. Minulosť je možno len akousi superpozíciou možností, ktoré vykryštalizujú až vo chvíli, keď sú pozorované. Dekel v tejto chvíli ulietava. :) Tieto veci vracajú do hry aj radikálny solipsizmus (som len ja, resp. sme len my, ostatné je v našej mysli) a teóriu, že žijeme v simulácii. Ak by všetko okrem prítomnosti a pozorovaných vecí existovalo len v nejakých protostavoch, bolo by to energeticky veľmi šetrné a pripomínalo by to napríklad počítačovú hru, kde sa lokácia vygeneruje až vo chvíli, keď do nej s avatarom vstúpite. Nemám na to jednoznačný názor.

Inak elektrón je po grécky jantár. To je krásne.

[Asim Bakhshi · Rating 3 out of 5]

A popular and extremely readable, at times funny version of complex ideas on the cutting edge of science. I don't think it deserves three stars for a popular science lay-reader; however, Sautoy failed to strike a personal cord inside me, perhaps because most of the material is well-known in the scientific tradition. He is a great teacher for making science publicly accessible but there are very little original insights here; however, as a collective, unified exercise, this whole concept of exploring 'edges' of the known/ unknown divide is very useful and original. Recommended for disciples of Dawkins and other radical newbies.

[Ken Parker · Rating 4 out of 5]

An awe-inspiring journey through some of the boundaries of our knowledge today. While a self-described atheist, du Sautoy seems to endorse the idea/potential for God as a transcendent force outside our reality. Visited seven edges:
Read 06/16/2017 while on Oregon vacation

1. Edge 1. Chaos. Small changes in starting conditions can have enormous changes in results. We cannot have precise knowledge of everything (down to electron positions throughout the universe.) Everything is connected.
=============
2. Edge 2. Matter. Is it continuous or discrete? Are quarks the smallest or is matter further divisible? Unsure. Amazing journey of exploration and discovery. Great symmetry in nature. Allows strong predictions. Current quark model seems to be complete but unknown below that.
=============
3. Edge 3. Quantum Physics. It is completely random (e.g. Radioactive decay). No determinism. Observation can inhibit the progress of a quantum system. The past is also a superposition of possibilities that only crystallize once they are observed. The act of measuring seems to determine the properties of a particle. Einstein called Quantum entanglement "spooky action at a distance".
=============
4. Edge 4. The Universe. Can we use light to determine the curvature of the universe and whether it is finite or infinite. The universe is expanding. (Due to dark energy). One implication, galaxies we could see are going to be pushed beyond the the edge of what we can see. Eventually, only our galaxy will be visible. Currently, we believe 18b light years is the cosmic horizon limiting our observations.
1. Don't currently have a good way to explain the ~20 constants of nature that allow life to exist. Multiverse is one potential explanation- there are a bunch and ours happened to be right.
2. John Barrow, Cambridge professor - won the Templeton Prize (connecting science and religion). Studies limits of our knowledge. Check out his books. Can never prove a theory is correct, only that it is incorrect.
3. Key question- is the universe finite or infinite?
=============
5. Edge 5. Time.
1. Two bodies moving a different speeds relative to one another will both sense that the other's clock is moving slowly. Must think about an integrated concept, space-time. Special relativity.
2. Impossible to tell the difference between gravity and acceleration. Same effects. Big implication: high gravity slows time, just like acceleration does. General relativity: gravity and acceleration are woven into space-time. Basically, things with MASS.
3. Heisenberg stated that both position and momentum cannot be precisely determined - one or the other. Same with time and energy - choose one.
4. Roger Penrose- imagined model of Bib Bang being a continuous connection of universe expanding, followed by contracting. Each cycle called an eon
5. KP: what if black holes are the seeds of a new universe that will be launched (and continue to expand)?
6. Edge 6. Consciousness. Have been trying to understand for centuries. Only with recent tools has our understanding started to grow quickly. In past, structural damage allowed people to see effects. Two hemispheres seem to be somewhat independent maybe redundant. fMRI and EEG are the tools used but they assume homogeneity among brains.
1. EEG measures electrical activity in the brain by recording voltage fluctuations from electrical currents among regions of brain. There are 5 speeds of waves that are associated with different brain activity.
1. Delta waves. 1-4 Hz. Deep, dreamless sleep
2. Theta waves. 4-8 Hz. Light sleep or meditation
3. Alpha waves. 8-12 Hz. Relaxed wakefulness
4. Beta waves. 13-30 Hz. Wide awake
5. Gamma waves. 30-70 Hz. Formation of ideas, language and memory processing and various types of learning.
2. fMRI detects changes in blood oxygenation in brain areas. Oxygenated blood is more magnetic.
3. Fun little experiment: look through paper tube with one eye and look at far off object with both eyes. Will see a hole in your hand.
4. Look up McGurk effect on YouTube. See face making sounds but you won't hear them because they switched the soundtrack.
5. Look up Ehrsson experiment with a fake hand and real hand behind a screen. Stroke fake and real hands simultaneously and people will start to believe it's their own hand.
6. Mind-body problem. Is consciousness separate from our physical body?
7. Networks that are tightly connected but differentiated. High symmetry seems to not be associated with consciousness. Lots of feedback and connectivity is good. Key structural elements: connectivity, feedback, differentiation.
8. Consciousness seems to be nonlinear, like a phase transition- all or none, not gradual. Does this mean that our attempts to create a computer consciousness will not be rewarded with partial success (an idiot but conscious)?
=============
7. Edge 7. Infinity.
1. Gödel proved that mathematics is not free of contradictions. Mathematics is open to falsification. In any system, there must be true formulas that are neither provable or disprovable. Corollary: the consistency of a logical system cannot be proved within that system.
2. Georg Cantor - 1890's, did work with infinity to show that different infinities are equal. E.g. Whole numbers and fractions. Did it by matching things up and showing you didn't run out. There are infinite infinities. The mathematical establishment of the day shunned him.
3. Does Gödel's incompleteness theorem mean we cannot know some of these things since we are inside the system? You cannot step outside the system to prove it because the theorem is an infinite regress...
4. Does a transcendent God stand outside the system? By making God flesh, are we trying to make the unknowable knowable? That seems counter to the concept of God.
9. Summary.
1. It is possible that we cannot know because we are stuck within the system we seek to understand.
2. Our physical limitations will place boundaries on our knowledge.
3. "The trouble with most religions is that the God that is served has so many properties that have nothing to do with the definition."
4. Frederic Fitch. Paradox of unknowability: "Unless you know it all, there will always be truths that are by their nature unknowable."
5. Stephen Hawking: "The greatest enemy of knowledge is not ignorance but the illusion of knowledge."

[Pieter · Rating 4 out of 5]

Du Sautoy, a mathematics professor at Oxford, explores the boundaries of human knowledge, questioning whether there are things that humanity, as a collective or as individuals, will never fully comprehend. He identifies seven key areas of science to examine these mysteries, referring to them as "Edges." Alongside these inquiries, he delves into humanity's tendency to fill gaps in knowledge with concepts of God or a deity, raising profound questions about the existence of a divine being.

The author writes in an engaging and accessible style, guiding readers through his quest that spans topics like chaos theory, quarks, time, the Big Bang, black holes, consciousness, infinity, and Gödel's incompleteness theorem. While certain sections touch on mathematical concepts, Du Sautoy intentionally minimizes the technical demands to ensure the book is approachable. You don’t need to be a science or math expert to appreciate this work—a high school-level understanding of basic science and mathematics is sufficient.

What stands out is how Du Sautoy masterfully covers such a wide-ranging spectrum of science and philosophy within a relatively concise format. It is thought-provoking and captivating, providing a comprehensive update on the major scientific questions of both the past and the present in just 350 pages.

I highly recommend this book for its ability to inspire curiosity and expand perspectives.

4.5 stars

[Chuck Noren · Rating 3 out of 5]

Excellent book. Intended audience, layperson with an interest in mathematics and science. Marcus du Sautoy is a renowned mathematician who looks at 7 areas of knowledge that mathematicians and scientists don't know. I liked how hear clearly explained topics in mathematics and physics.

Wherever Dr. du Sautoy dealt with topics close to mathematics, he was excellent. Other areas, in partucilar, consciousness, was too hazy, almost too pop-science. If the reader listens to a few Brain Science podcasts by Virginia Campbell, they'll get better, more solid information about this topic.

While Dr. du Sautoy is an atheist, he exhibits a gentlemanly respect for others who hold various religious views. I find that refreshing (now I am biased because I am a confessional Reformed Evangelical Christian). In philosophy, I think Dr. du Sautoy did a reasonable job. But he takes a narrow slice among his Oxford friends. There are others to explore, such as Alvin Plantinga.

On the topic of the multiverse, there are some issues dealing with infinities that, in my opinion (not that it's worth much), should be examined. One is the issue of what would be a typical observer in an infinite multiverse across all time. Ludwig Boltzmann raised the issue that we would be far from typical observers that we'd expect in a multiverse.

Overall, the book was excellent. I enjoyed it very much.

[Dan Graser · Rating 5 out of 5]

This is a hugely absorbing and eminently readable account of just where the limits of scientific knowledge currently are in relation to seven areas of inquiry: Chaos, Matter, Quantum Physics, The Universe, Time Consciousness, and Infinity. The current Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford, Marcus Du Sautoy, presents some of these well-known challenges in fresh new light, coming from a highly respected mathematician instead of a physicist or biologist as is frequently the case. The result is a current appraisal of our known-knowns, known-unknowns, and suggestions as to where our unknown-unknowns might be lurking. His knowledge across several fields is vast but there is obviously in a work of this magnitude some unevenness, not surprisingly given his mathematical research, the chapter on Infinity is his finest. This is great fun and hugely education reading for our current state of scientific epistemology as well as some meanderings into information science ontology with respect to our relationship with artificial intelligence. Great stuff!

[Mr John N Pedley · Rating 5 out of 5]

A Mind Blowing Book.

The scope of this book brings together the current thinking in many disciplines from physics, philosophy, mathematics, cosmology, neurobiology etc, in a structured, well researched and authoritative narrative which seeks to explain the boundaries of human knowledge.

[amberle · Rating 4 out of 5]

interessante, non scontato, scritto in un linguaggio abbastanza semplice ma certo non semplicistico, porta anche in luce le opinioni degli "addetti ai lavori".
consigliatissimo ai grandi curiosi.

[Baby Adam · Rating 5 out of 5]

Marcus du Sautoy takes the reader on a journey to the edges of current human knowledge. Most of the chapters are dedicated to physics, and provide a refreshing account of many different topics, with excellent thought-provoking explanations. I particularly enjoyed his explanation of the "many aeons" interpretation of the universe, in which the big bang is "glued" to the heat-death of the universe at which time all matter becomes energy and the notion of distance becomes meaningless.

The section on neuroscience was fascinating. He drew me in with some fun experiments and weird wonders of the brain, and provided some really cool case studies of people whose brain had effectively been split in half. I would definitely like to read more popular neuroscience, having had my appetite whet. I managed not to embarrass myself in talking about the book with my neuroscientist husband either, which is a big positive. #aliegroupisacircle

The final chapter was, inevitably, dedicated to mathematics. After a fun account of problems such as Gödel's incompleteness theorems, which roughly say that within any system of axioms, there are always going to be things you can't prove, and moreover, you can't know that you can't prove them. There is some good philosophical discussion at the end, and a real sense of having been on quite an extensive journey.

This book was a gift from my darling husband, and then I received a second copy from Baby Hair's mum! It shows how much I love Marcus du Sautoy. I look forward to when my husband reads this, and we can discuss all the fun details!

[Paul · Rating 4 out of 5]

Marcus du Sautoy is the new Simonyi Professor for the Public Understanding of Science, at Oxford University, taking over from Richard Dawkins. Overall I enjoyed it. Although i thought he spent quite a lot of time going over the general history of science, or at least some of the major events. He does eventually get to the hardcore questions of 'what is knowable' mathematically and logically, which is what I was really interested in, but only in the last few chapters.

[Edward B. · Rating 3 out of 5]

Pretty good.
Quite readable - *if* you already have some knowledge of things like Relativity and the Planck Length and mathematical proofs, etc.. I'm not sure how much you would get out of it without any of that.
It ends up being vague and wishy-washy on its stated goal of determining "What We Cannot Know" (which is the book's title in Britain).

[Nicholas · Rating 5 out of 5]

He's certainly a gifted communicator. And although the path he treads here is not exactly unbeaten, he manages to find novel ways to enlighten, clarify and entertain. His explanation of special relativity using Pythagoras is brilliant and just one example of many.

[Lance · Rating 2 out of 5]

"I have always been extremely unsettled by things I cannot know. Things I cannot work out. I don't mind not knowing provided there is some way to ultimately work it out."

This is a very important book. Uniting the disciplines within science and seeing all the sciences as a coherent formation against the boundaries of knowledge is an important viewpoint missing from much of popular literature. Chemistry flows seamlessly into physics, scientists and mathematicians collaborate, and big data and mathematical modelling have become staples of all the sciences. I think that having books like these will help break down popularly held stereotypes about individual sciences and increase the breadth of knowledge that every person of scientific mind should strive for.
The book is split into seven sections, each tackling a set of questions which faces science today. The first is the world of probability, which has been developed to help us understand the outcomes of chaotic systems. This is a cool bit of science which spans from meteorology to evolution and benefits from a mathematician's perspective with the awesome up to date study from Kapitaniak et al. He then moves on to particle theory and the awesome chemists Mendelev, JJ Thomson and Rutherford who between ordered the substances of the natural world, cracked the weirdness of electricity, and revealed the internal structure of the atom. I love to see chemists getting their credit! From here he flows into the glitchy random world of quantum dymanics (a field on the boundary of both physics and chemistry). "Quantum physics isn't about knowing answers to old questions, it's about challenging the questions we are allowed to ask." I enjoyed reading about the discovery of muons and quarks, the mind-blowing wave-particle duality was well captured by du Sautoy, and I particularly enjoyed Melissa Franklin who embodies the scientist's passionate fascination with further learning almost regardless of legacy, purpose, or fruitfulness. She was cool. After a diversion from Heisenberg's mind-bending work, the narrative engaged in the large-scale wonders of the universe. I was interested to read that the accelerating expansion of the universe has now been resolved, this may be the first book published since Carl Sagan's Cosmos which prioritised this awesome question. The Einsteinian special relativity was well-written and important, but fell a bit flat as I was already very familiar with the concepts. More interesting was the perspective of eccentric genius Roger Penrose, I was fascinated by his hypothesis of aeons based on the idea that the concept of space and therefore time has no meaning when the universe eventually loses all of its matter and decays into energy only. The final chapter, which prioritised the mathematics of infinity was slightly disappointing as the most recent mathematics discussed, that of Cantor, is still around a century old and I was hoping for something fresh and juicy from the journals to take advantage of du Sautoy's insider knowledge of maths.
The sixth section, that which attempts to chart consciousness, was unfortunately overambitious. Consciousness is a multi-disciplinary research questions, which embraces neuroscience, psychology, evolutionary biology, computer science, philosophy, and contributions from many other fields each of which could fill a whole volume with its own discreet conclusions. So, two chapters was always going to be a push. I was gratified to see that du Sautoy oriented his attack on consciousness from a neuroscience perspective, as the life sciences are rarely represented alongside physics in the same popular book despite the great overlap between skill-sets. And yet, it was immediately apparent that du Sautoy's research for this chapter was limited. He begins with an account of brain structure and approximate function of gross anatomical areas. This is very confused. I don't know where he got the information for this section, but it was not laid out with the clarity or authority that Susan Greenfield or Cordelia Fine bring to their popular neuroscience works. He talks about the left and right hemisphere, when he means the left and right hemisphere of neocortex only (the relations of deeper brain structures to hemisphericity are more complex), and cites the corpus callosum as a "bottleneck" of communication between the two hemispheres of the brain. As a processional developmental neuroscientist, this tells me that he has hasn't done his basic homework. His reading list at the back of the book includes no basic neuroscience texts which would have cleared up these misunderstandings, a quick browse of Eric Kandel's seminar work Principles in Neural Science would have been well advised. For example, such a book would have been able to explain that the hemisphere structure of the brain is almost certainly an evolutionary continuation of the bilaterally symmetrical body plan and that the cortical specialisation arose much later in mammalian evolution. Check it out. Symmetry is extremely efficient for all life forms. Perhaps his time would have been better spent here than pursuing fMRI scans and EEGs. Although these methods still retain a certain space-age charm to the technology, these are established clinical tools for probing gross anatomy and brains functions. I would have been more impressed if du Sautoy took the time to explain electrophysiology, a set of techniques from the 1800s which formed the foundations of our understanding of the neurone, or ventured into the cutting edge of optogenetics, the technique developed by Gero Meisenboeck which uses light to activate or silence neurone individually or in populations. Following EEG to Harvard, du Sautoy interviews sleep researcher Guillio Tononi. I found his description of phi as a measure of consciousness through complexity of the connections within a system highly relevant and impressively clear for a non-expert. However, he then goes on to make a schoolboy error! When providing experimental evidence to back this up, he describes sleep as an "unconscious" state, which is not true according to all scientific definitions of consciousness. Sleeping people can still respond to their environments is stimuli are strong enough, and show the gamma waves (neural activity that is the signature of alertness) through the thalamus and much of the brain. So sleeping humans do not lose their consciousness. Also, Vyasovskiy and Harris' 2013 review described low-phi sleep waves present in local areas of the brain throughout wakefulness, and, amazingly, less complex electrophysiological activity occurs when a mouse undertakes a routine motor activity such as running in a wheel than during low-phi sleep. So is the mouse less conscious while running than it is when it is asleep? It is a great shame that du Sautoy did not take the opportunity to interview Vlad Vyasovskiy for this book, given that Vlad is currently working in Oxford across the road from the Maths Institute ... Even better, he could have picked the miraculous brain of Ray Guillery, whose inspirational book Exploring the Thalamus contains predictions of several amazing consequences of complex neural networks which mathematical biologists and mathematical philosophers such as Karl Friston are only now beginning to validate and fully understand. The amazing Ray Guillery would also have been an excellent proof-reader for this chapter and prevented the small misunderstanding from going to print. du Sautoy salvages the chapter with a couple of excellent interviews with Christof Koch (not he of the fractals) who introduces some clarity to the story of phi, the human connectome, and the ability to download a brain using current computer technology. It's a shame that du Sautoy couldn't get in touch with a research computer scientist who could have spoken to him about the potential for quantum computing to overcome the determinism of contemporary computing technology, as well as the transistor connections problem. A really good effort, and I really appreciate the representation, I do, but I feel that time Marcus borders on giving misinformation.
I was mightily unimpressed with the spirituality sections which occupied the end of every section and dominated the conclusions. I felt that his definition of God as "what we cannot know" lacked meaning. Why equate these two concepts at all? I might as well chose to define God as the total information content of the universe, or inside the event horizon of a black hole, but that doesn't mean that I have somehow accessed meaning through this definition. du Sautoy rejects God as a figure of love and forgiveness, but make no comment on other more objective connotations of throwing the word "God" around, such as God's role as creator or God's sentience. I find assertions about God which have no conclusion about his/her/their/its role in the creation of the universe fairly flaccid. Also, the interviews with religious scientists seemed purely apologistic for Richard Dawkin's strong atheist views. Polkinghorn, the quantum reverend, spouted some stuff about chaos theory as the site of God's action, a field he does not even work in. And then there is Barrow, who says nothing that I disagree with but merely shrouds his belief behind uncertainty. I am interested in religion, and I know and respect many religious scientists, but I do not think du Sautoy was well-versed in the philosophy and I feel that the inclusion of religion was ultimately disjoined within this particular book. So that's there the other star went, the one that wasn't lost in the neuroscience section.
As usual, Marcus du Sautoy's greatest strength is his elegant and playful prose. His facility with language makes him an ideal communicator of complex scientific ideas, and his colourful interdisciplinary metaphors are an exciting medium through which to tell scientific tales. Anyone who uses the verb "ossify" as a metaphor for stagnation of social progress and includes quotations from Tom Stoppard's Arcadia, one of my all time favourite plays, as well as Jorg Luis Borges The Library of Babel is creating some seriously classy popular science. Despite this, imagine my chagrin when du Sautoy began to use the word "schizophrenic" to describe, what? Mixed motives? Ambiguity? I found this metaphor quite inappropriate. However, it's not only scholarly prose that grips in this book. There is also du Sautoy's sense of humour which includes cultural references from Reservoir Dogs to lame Christmas cracker jokes. An exemplary writer, and I hope that his retirement from the Simonyi professorship does not stop him writing.
"Science is king. Science is giving us unprecedented insights into some of the big questions that have challenged humanity ever since we have been able to formulate them."

[Stone · Rating 4 out of 5]

Known known. Known unknown. Unknown unknown. Unknown knowns.

- Chaos theory
- Observation
- Uncertainty
- Nonlocality
- Equivalence (me moving or them?)
- Doppler Effect (siren fading away)
- Gravity
- Singularity
- Entropy
- God
- Brain
- Language (Mathematics)
- Fractal

Notes
I am more interested not in the existence of a God to fill the gap, but in equating God with the abstract idea of the things we cannot know. Not in the things we currently don’t know, but the things that by their nature we can never know. The things that will always remain transcendent.

The unpredictable and the predetermined unfold together to make everything the way it is. It’s how nature creates itself, on every scale, the snowflake and the snowstorm. It makes me so happy. To be at the beginning again, knowing almost nothing. Tom Stoppard, Arcadia

it doesn’t matter how many times in a row you get a 6: this has no influence on what the dice is going to do on the next throw.

Until we discover other inhabited planets and can study their trajectories, it’s difficult to assess whether evolution inevitably leads to mismanaged ecosystems based on just one dataset called Earth.

‘The bad news is that cosmology has unknowable unknowns: things that we are never going to know even though we suspect we might know of their existence.’
(Barrow. Main.)

with an optical microscope I can magnify the dice by a factor of 1500, which would scale my dice up to the size of a large building. Peering inside this huge dice still won’t reveal much about the secrets of how it is built. Everything still looks pretty smooth and continuous.

Everything we call real is made of things that cannot be regarded as real. Niels Bohr

‘Young man, if I could remember the name of these particles, I would have been a botanist.’

The mathematics of symmetry suggests that we’ve reached rock bottom.

It was the code-cracking mathematician Alan Turing who first realized that continually observing an unstable particle could somehow freeze it and stop it evolving. The phenomenon became known as the quantum Zeno effect, after the Greek philosopher who believed that because instantaneous snapshots of an arrow in flight show no movement, the arrow cannot be moving at all.

What is science? How do we attempt to navigate our interaction with the universe? It is only by measuring and observing that we know anything. Mathematical equations can tell us what to expect, but it’s just a story until we measure. So it’s curious that our only way to ‘know’ anything about the universe is to observe and make particles and light make up their mind about where and what they are doing. Before that, is it all just fantasy? I can’t measure an entire wave function, I can only know it mathematically. Is a quantum wave function part of the universe that we can never know because how can we truly know without measuring? At which point it collapses. Perhaps it’s just greedy to believe that I can know more than I can measure. Hawking has certainly expressed such a view: I don’t demand that a theory correspond to reality because I don’t know what it is. Reality is not a quality you can test with litmus paper. All I’m concerned with is that the theory should predict the results of measurements.

My real problem with the current mainstream interpretation of quantum physics is that if you run the double-slit experiment twice, set up with exactly the same conditions, the outcome can be different each time. This goes against everything I believe in. It is why I was drawn to mathematics: the certainty of a proof that there are infinitely many prime numbers means I’m not suddenly going to get finitely many prime numbers next time I check.

Clifford Shull’s experiment confirmed that as the size of the slit is decreased, the statistical spread of the locations of the neutrons increases.
(double-slit experiment)

quantum physics currently posits a limit to how far we will be able to penetrate in our investigation of what’s inside my dice: that beyond the Planck length, it’s simply no-go. There is an edge beyond which we cannot know.

I believe that one can formulate the emergence of the classical ‘path’ of a particle pregnantly as follows: the ‘path’ comes into being only because we observe it.
(Heisenberg)

Quantum physics, however, is trying to tell me that this belief in the existence of a precise value before measurement is a mistake. It is my interaction with the system that creates the properties of the particle. Can it really be my act of measurement that produces the reality of this particle?

According to Heisenberg’s uncertainty principle, it follows from the fact that space exists that you will get particles appearing from nothing. You don’t have any need for a creator. Quantum fluctuations mean that we are seeing something appearing from nothing all the time.

Some commentators have argued that if there is genuine randomness in quantum physics, and the present state of things is not predetermined by past events, then it is evidence of free will at work in the universe.

principle of equivalence. The strange consequences of special relativity are teased out of the principle of relativity, which states that it is impossible to tell whether it is me who is moving or my environment that is moving past me.

When an ambulance drives by with its siren blaring, your ears experience the squashing up of sound waves as the ambulance approaches, which causes the wavelength to shorten, resulting in the siren sounding higher-pitched than when the ambulance passes you. As it pulls away, the sound waves are stretched out, resulting in a longer wavelength and lower pitch. This phenomenon is known as the Doppler effect.

The astronaut at the bottom of the spaceship is going to send a pulse of light to the astronaut at the top each time his clock ticks. The astronaut at the top can then compare the arrival of these pulses with the ticking of her clock. Without acceleration or gravity, the arrival of the pulses and the ticking of the clocks will be in synch. However, let me now accelerate the spaceship in the direction of the top of the spaceship. A pulse is emitted at the bottom of the spaceship, and, as the spaceship accelerates away, the light has further to travel each time, so that it takes longer and longer for each pulse to reach the top of the spaceship, and the astronaut at the top will receive the pulses at a slower rate.
(Wow what an explanation!)

What if I reverse the experiment and get the astronaut at the top of the spaceship to send pulses down to the bottom of the spaceship? Because the astronaut at the bottom is accelerating towards the pulses, he is going to receive them at a faster rate than the pulses of light he is sending. So he will confirm that his clock is running slower than the clock at the top of the spaceship.

According to Einstein’s principle of equivalence, whatever effect acceleration had on the clocks in the spaceship, the effect of gravity must be the same. So when I place a large planet at the foot of our Shard-sized spaceship, the impact is the same as if the spaceship was accelerating through space: clocks run slower at the foot of the Shard than they do at the top.
(Very well put!)

Ina returns younger because she has to accelerate to get to her constant speed. Similarly, when she turns around she needs to decelerate and then accelerate in the opposite direction. This causes her clock to slow down relative to that of her twin on Earth, who doesn’t accelerate. This asymmetry results in Ina heading into Magaly’s future. If I sent both twins off in spaceships in opposite directions and brought them back together then they’d be the same age – and everyone on Earth would have aged quicker.

The ball pulls the surface down, creating a well. Gravity can be thought of as the way things get pulled down into this well.

The speed at which you need to throw the ball is called its escape velocity. But imagine increasing the mass of the Earth more and more. The speed required to escape the gravitational pull also increases. But there will be a point at which the mass of the Earth is so big that the ball would need to be launched at a speed faster than the speed of light in order to escape. At this point the ball is trapped. It cannot go beyond a certain point before it is pulled back to Earth.

the population of the planet would become infinite on 13 November 2026.

The interesting thing is that someone outside the event horizon will never know what is going on inside. From their perspective, time does appear to stop as soon as I cross the event horizon, and for them there is no telling what comes after, although there is an after for me as I carry on towards spaghettification. So asking what happens next appears to be a sensible question. It has an answer. The only trouble is that if you are outside the event horizon, you are denied access to the answer by the laws of physics.

But provided I have complete information about all the atoms and photons involved in my bonfire, it is theoretically possible to rewind the process and recover the information contained inside the magazines. It would of course be extremely difficult in practice, but the science asserts that there is nothing about this process that is irreversible. The laws of physics work both ways. The existence of black holes offers a challenge to this idea. If I throw one magazine in one black hole and the other magazine in another black hole, it is now impossible to find out which magazine went into which black hole. Black holes seem to be the ultimate paper shredders, in which information appears to be genuinely lost.

Just as God is not located at one point in space, there is no need to locate God at one point in time.

If you opt for theism rather than simple deism, then God needs to have a temporal quality to intervene in the world. If God is outside, looking at the whole of space-time, the future is already there in the landscape. Interestingly, although people may argue over the order of events, they won’t argue about their order if they are causally related. This demands a God who steps in and out of time, moulding the geometry of space-time. But a God who acts in the world is a God who acts in time. So it is very difficult to square a timeless God with a God who acts in the universe.
(very well put!)

The question remains: what is this thing called God that is meant to be outside of time? Can anything be outside of time? Actually, there is something that I would regard as timeless: mathematics. And as a timeless thing it is well suited to the job of sparking the creation of what we observe around us. It is the creator responsible for the equations of quantum physics that give us space-time and something from nothing. Mathematics has an attractive quality: you don’t need to ask who created the mathematics. It is something that exists outside of time and doesn’t need a moment when it was created. It just is. Perhaps we should reverse the old adage ‘God is a mathematician’. Perhaps mathematics is the god everyone is chasing. If I replace the word ‘God’ with the word ‘mathematics’ in Aquinas’s attempt at a definition, I think it works quite well: ‘Mathematics is to be thought of as existing outside the domain of existents, as a cause from which comes everything that exists in different forms.’

quidditism: the idea that there is more to the universe than just the relationship between objects – what they are (the quid is Latin for ‘what’) provides another level of distinction.

If mathematics is eternal and outside of time, you don’t need a creator to begin things. The equations of mathematics are truly outside the universe, so they could play the role of something supernatural and godlike. It is not, however, a God that acts in the world, because this is a deistic vision.

I wonder, though, whether, as I come to the end of my exploration at the limits of knowledge, I have changed my mind about declaring myself an atheist. With my definition of a God as the existence of things we cannot know, to declare myself an atheist would mean that I believe there is nothing we cannot know. I don’t believe that anymore. In some sense I think I have proved that this God does exist. It’s now about exploring what quality this God has.

Wetness is the classic example of an emergent phenomenon: one molecule of H2O is not wet; wetness requires that you put many molecules together before it appears.

Haynes can see which button I will choose because there is a region in the brain that lights up six seconds before I press, preparing the motor activity. A different region of the brain lights up according to whether the brain is readying the left finger to press the button or the right finger. Haynes is not able to predict with 100% certainty yet, but the predictions he is making are clearly more accurate than if you were simply guessing. And Haynes believes that with more refined imaging, it may be possible to get close to 100% accuracy.

Koch has a tough battle on his hands convincing others. There is a hardcore group of philosophers and thinkers who don’t believe that science can ever answer this problem. They are members of a school of thought that goes by the name of mysterianism, which asserts that there are mysteries that will always be beyond the limits of the human mind. Top of the mysterians’ hit list is the hard problem of consciousness.

‘If you pursue that to its end you just end up with solipsism.’

Suppose a scientist cooks up a perfectly consistent logical theory of how the universe might work, but then discovers that it doesn’t match the experimental evidence of our own universe. The hypothetical theory is booted out and is no longer of interest to the world of science. If a biologist starts writing papers about a hypothetical beast that potentially could exist, like a unicorn, but doesn’t actually walk this Earth, no one is going to be interested unless it casts some light on the animals that do exist. By contrast, in mathematics such new worlds and beasts are celebrated and embraced. They become part of a richer tapestry of mathematical possibilities. Science does the actual, mathematics does the possible. Science charts a single pathway through a tree of possible universes, mathematics maps every possible journey.

The limitations of language are at the heart of many of the limits of knowledge, and these could possibly evolve and change. Certainly, many philosophers identify language as a problem when it comes to the question of consciousness. Understanding quantum physics is such a problem because the only language that helps us navigate the ideas is mathematics. Try to translate the mathematics into the language of everyday experience and we create the absurdities that make quantum physics so challenging. So the unknowability of position and momentum isn’t really a genuine unknowable. Rather, it is a failure of translation from mathematics to natural language.

Karl Popper proposed: a theory can only be falsified, never proved.

The size of the universe: ‘There is no doubt a lot of universe beyond our horizon, but we haven’t seen it and almost all of it we will never see. So when people ask whether the universe had a beginning, or is it finite or infinite, we can never answer questions like those about the entire universe.’

(time is reference point)

My initial instinct is to think of time as a line, which makes it very difficult for me to think of anything other than a line that is finite and therefore has a beginning, or a line that is infinite. But attempts have been made to show that the question ‘What happened before the Big Bang?’ isn’t one whose answer we cannot know. Rather, the question does not really make sense. To talk about ‘before’ is to suppose the existence of time, but what if time existed only after the Big Bang?

Two thousand years ago Socrates declared: ‘True knowledge exists in knowing you know nothing.’

Let p be the true statement among the two options: ‘there are an even number of dice in my house’ and ‘there are an odd number of dice in my house’. I don’t know which one is true, but one of them must be. The existence of an unknowable truth is squeezed out of the existence of this unknown truth. The following statement is an unknowable truth: ‘p is true but unknown’. It is certainly true. Why is this unknowable? Because to know this means I know that p is true and unknown, but that’s a contradiction because p can’t be unknown and known simultaneously.
(So true)

Wittgenstein summed this up colourfully: ‘You cannot shit higher than your arse.’

Kant believed that the ways things really are will always remain hidden from our view. All we can ever know is the appearance of things.
I think most scientists spend some time reading about this debate concerning ontology and epistemology, and listening to philosophers who question whether science is really telling us how it is. And then they get back to the science, telling themselves that if we can never know what reality is really like, then let us at least try to say what the reality that we apprehend through our senses is like. After all, that is the one that impacts on us.

As Niels Bohr said: ‘It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’

[Sujith Ravindran · Rating 5 out of 5]

What we cannot know is an incredibly fascinating book authored by Marcus du Sautoy. Sautoy is a Mathematics Professor at Oxford University. The central theme of this book is to explore the limits of our knowledge in understanding our nature and true self.

Sautoy has tried to explore this theme by choosing several edges like: Is it possible to predict the future of casino dice? Whether we can know the smallest particle? Is there a beginning to time? What constitutes our consciousness? and so on...

Sautoy shows us that mathematics can be used a powerful tool for finding answers. But, there are limitations or edges beyond which knowledge seems to be inaccessible. Perhaps, our languages are not capable of precisely articulating mathematical proofs.

Our major constraint is we are stuck within the system and trying to understand the system by staying inside. Quantum experiments have proven that we cannot repeat the same experiment exactly because we are unable to isolate the experiment from the universe. Further how can we not be sure that we are fooled by our senses.

"It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature". Niels Bohr

So I think we may live in a world of realities (could be even fantasies) created by our limited knowledge. True knowledge will remain hidden and perhaps unknown.

"The greatest enemy of knowledge is not ignorance, but the illusion of knowledge" Stephen Hawking.