I found the book a bit difficult, in the sense of boring, if you have already read many science non-fiction texts, or have a degree in science, particularly if you are the curious scientist. All the common bits from physics, neuroscience, mathematics, are gathered here and presented in summary form.

There are, however, some magic nuggets that emerge both in terms of interesting interviews, or historical details.

Overall I really wished this book was more systematic and complete. It is a good overview but could have been a text to shape the big picture of what we can’t know, rather than a list of pieces. ( )

Marcus du Santoy explora siete confines del saber para dar respuestas a la preguntas: ¿lograremos algún día determinar qué es la conciencia, develar la naturaleza del tiempo, comprender las paradojas de la mecánica cuántica, vislumbrar el porvenir del universo o entender si el infinito es una entidad real o tan solo un concepto abstracto? ( )

Eerst het slechte nieuws. Dit boek gaat eerder over wat we wel kunnen weten, dan over wat we niet kunnen weten. Daar zijn twee redenen voor. Enerzijds zou Marcus du Sautoy zelf liefst alles weten. Hij is dus enigszins bevooroordeeld. Anderzijds heeft het ook te maken met zijn aanpak. Als wiskundige gaat hij in eerste instantie te rade bij de wetenschappen. En wetenschap is bij uitstek een poging om een antwoord te vinden op open vragen, om het domein van wat we niet weten zo klein mogelijk te maken. Hij kiest zijn wetenschappen ook met zorg: fysica, kosmologie, wiskunde. Een verkenning langs geschiedenis, paleontologie en sociologie had allicht een ander plaatje opgeleverd.

Maar er is ook goed nieuws. Uitstekend nieuws zelfs. du Sautoy is een boeiend verteller die bovendien weet hoe hij moeilijke onderwerpen op een toegankelijke manier moet presenteren. Zeven thema's behandelt hij in zeven 'randen': de chaostheorie, de zoektocht naar het kleinste deeltje, de quantummechanica, de oneindigheid van het universum, het begin van de tijd, het bewustzijn en de grenzen van de wiskunde. Zijn uitleg is bijzonder helder, bijvoorbeeld die over de flexibiliteit van de tijd of over de zoektocht naar het kleinste deeltje. Alleen als het over zijn eigen vakgebied gaat, de wiskunde, heeft hij wat moeite om het essentiële van het bijkomstige te scheiden en de focus van zijn verhaal helder te houden. Het is hem niet kwalijk te nemen. We onthouden in elk geval dat de enige oneindigheid de andere niet is.

De naam van Kant valt pas op de laatste pagina's van het boek. Voor iemand die op zoek is naar wat we (niet) kunnen weten, vat hij diens filosofie wel zéér beknopt samen. Nochtans haalt du Sautoy in het hoofdstuk over de wiskunde zelf aan dat het soms nodig is om van buiten af naar een systeem te kijken om te achterhalen wat waar is en wat niet. En dat is juist wat de filosofie nastreeft. Zij gaat buiten de wetenschap staan om te achterhalen hoe deze functioneert, probeert de grenzen aan te geven van wat we kunnen weten. Volgens Kant kunnen we het ding zoals het werkelijk is, het 'ding an sich' nooit kennen. We kunnen het immers alleen maar kennen via onze zintuigen en via enkele fundamentele categorieën van ons denken, zoals causaliteit. Je zou kunnen stellen dat onze lichamelijkheid - onze zintuigen, de werking van ons brein - beperkingen opleggen aan wat we kunnen weten. Dat is een piste die du Sautoy niet onderzoekt.

Hij wijst er op het einde van zijn boek wel op dat we onze zintuigen aanzienlijk hebben kunnen uitbreiden. We hebben ontdekt dat licht bestaat uit elektromagnetische golven. We hebben ook ontdekt dat er ook elektromagnetische golven bestaan die we niet rechtstreeks kunnen waarnemen en we hebben geleerd om die te analyseren. Toch is dat niet meer dan het oprekken van de grenzen. Hier en daar in het boek van du Sautoy lijken we een glimp op te vangen van een realiteit die ons ontsnapt: snaren die trillen in elf dimensies, donkere materie en donkere energie die we niet kunnen waarnemen maar die er wel moeten zijn. Het zijn niet meer dan uitvloeisels van wetenschappelijke modellen, maar dat was ook zo met Neptunus en met zwaartekrachtgolven. De vraag is of ze wijzen op een dieperliggende werkelijkheid, of op de onvolkomenheid van die modellen.

Stof om over na te denken. ( )

Some interesting areas. Very focused on quantum and astrophysics, but does eventually get around to other topics. Well written. ( )

“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.”

In “What We Cannot Know - Explorations at the Edge of Knowledge” by Marcus du Sautoy

“Until it [the photon of light] reaches the detector plate, like the electron, it is seemingly passing through both slits simultaneously , making its mind up about its location in space only once it is observed. It’s this act of observation that is such a strange feature of quantum physics. Until I ask the detector to pick up where the electron is, the particle should be though of as probabilistically distributed over space, with the probability described by a mathematical function that has wavelike characteristics. And it is the effect of the two slits on this mathematical wave-function that alters it in such a way that the electron is forbidden from being located at some points on the detector plate. But when the particle is observed, the dice is rolled and the probability wave has to choose the location of the particle. [...] The interesting point for me - one that is often missed - is that up to the point of observation, quantum physics is totally deterministic. There is no question of what the nature of the wave equation is that describes the electron as it passes through the slits. [...] The probabilistic character and uncertainty occurs when I observe the particle and try to extract classical information. The highly non-classical and bizarre new feature is this discontinuous shift that seems to happen when the wave is ‘observed’. Suddenly the determinism seems to vanish and I am left with an electron randomly located at some point in space. [...] We should be careful not to over-egg the role of humans here. Worms too can presumably collapse the wave-function. But it is not just living creatures that are doing the measuring. There are particles on the other side of a potentially lifeless universe that are interacting with inanimate objects, causing the wave-function to collapse into making a decision about the properties of the particle. [...] I am having real trouble getting my head around this idea of observation marking a divide between a deterministic electron described by a wave-function and an electron that suddenly has a location determined purely by chance. The whole thing seems crazy. Nonetheless, there is no denying that it works as computational tool. [...] Where’s the dividing line between the quantum world of probability and the classical world of certainties? This dualistic vision of a microscopic quantum world and a macroscopic classical world all seems a bit suspicious. Surely the whole shebang should be described by a wave equation.[...] Can it really be my act of measurement that produces the reality if this particle? ”

In “What We Cannot Know - Explorations at the Edge of Knowledge” by Marcus du Sautoy

Some great scientists over the centuries were convinced that they had discovered the ultimate basis of physical reality - that there was nothing more to be said. A few became possessive and even spiteful to those who questioned them. But the great majority of physicists share their discoveries with their fellows and continue to use scientific methodology to go as far as possible always knowing that final answers or a theory that explains everything is unavailable and beyond our human intellectual capacity.

Now turn to politics and the journalists who become part of it making neutral and "objective" reporting impossible. When the the established order controls the media; when newspapers and TV are owned by powerful financial billionaires any semblance of balance is removed. Now and then an independent voice is allowed but is "balanced" by ten on the establishment side. "Truth in the news" is a hurtful joke.

One of the problems is the concept of the laws of physics or of nature. This implies that there are principles at work that do not actually exist underwriting an ultimately correct scientific explanation. I do not see, though, how you can ever know for sure that any explanation is the right one. It is proper for scientists to try to find out and explain why apples tend to fall downwards from trees rather than upwards, but it is fruitless to try to explain why nature is such that there is that tendency. Nature is just as it is. There are no non-physical laws making it as it is. Science therefore should see itself as having the task, the very important one, of finding out what nature consists of and how its constituents change from being in one state to being in another, rather than attaching itself to a belief in the reality of laws. The so-called professed laws of physics should be considered as temporary shorthand explanations for the general case lacking any great predictive force. There are no general formulae that will tell you the exact outcome of accidentally dropping a plate on the tiled floor of the kitchen. The great physicist Feynman said ‘People say to me ‘Are you looking for the ultimate laws of physics?’ No, I am not. I’m just looking to find out more about the world.’

Observation is prone to experimental error and confirmation bias. Observation can also only tell you about the observed and you can never know if you have observed everything of relevance. Reliance on observation is why geometry fell into a dark age after Euclid's parallel axiom was shown to only be true for some geometries, not all. Theorizing via the Scientific Method is immune to both these forms of error, but is subject to the problem that all theories are limited to modelling observations and cannot extrapolated or interpolated beyond the limits for which the model is defined. Axiomatic reasoning is also immune to both these forms of error, and is also immune to scale and bounds issues in the conventional sense, but attempts to convert all of mathematics to set theory were eventually shown to be doomed. It turns out that all axioms, in general, have bounds issues of sorts.
All approaches are imperfect and have natural domains where each is master. A craftsman relies on many tools of each type and many types of tool, to do the perfect job. The error made by many is to assume any type is disposable. You can always do a hack job, even with just a hammer, but elegance requires something better.

I have to say that I can find no evidence time is emergent. Indeed, we have quaternions today because time cannot be emergent. 3+1 space simply doesn't work. No particles are immune to time. Not even entangled particles, for reasons Kip Thorne can elucidate far better than I. On the other hand, there is a minimum scale. We can't achieve it yet, but we know it's there. Physical particles can be decomposed into pure information as particles. Mathematics becomes the ultimate framework. You cannot go smaller than that.

We can know perfectly well how neurons produce consciousness. John Conway's Free Will theorem proves beyond doubt that if the brain has free will, so do all particles. This implies that if particles lack free will, the brain must be computable. So if Many Worlds is true (which means particles can't choose) then we know for a fact that we can reduce the brain to an algorithm. If Many Worlds is wrong, then particles do choose, then the brain goes beyond chaos into a new realm of living mathematics. We may not understand this realm, but that wasn't the question. The question was how neurons produce consciousness, not why. As long as I can describe living mathematics in formal terms, the rest is an exercise for the enthused.

Kurt Gödel's incompleteness proofs define limits but not serious limits. You cannot define a system that is provably complete and provably correct. Nothing stops you from producing a system that is provably complete, but it must contain paradoxes and inconsistencies. Nothing stops you from producing a system that is provably correct, but it must contain gaps and omissions.

As someone who once completed Edward de Bono's extended course on lateral thinking, I don't see the problem. Lateral thinking contains quite a bit on how to gain full coverage through patchwork methods. Gödel's argument doesn't apply to anything outside of purely logical frameworks, which is likely why he began to suspect that the incompleteness theorems had problems. We can't know now, unfortunately, we only know he saw the potential for exceptions and there really aren't any other avenues he could have considered.

A patchwork approach would consist of N provably correct systems with no overlap where all N use incompatible axioms in order to formulate their area. You can't combine them without paradoxes, so you can't have any single thing that is complete and correct, so it doesn't violate the rules. The edges can be fractal, if you like, eliminating meta-rules on switching rules, so eliminating pseudo-violations. You nonetheless end up with coverage of everything where everything covered is correct. This moves the problem from there not being such a construct to not being able to know what the construct is. It is now the understanding, and not the maths, that cannot be both complete and correct.

Du Sautoy limits his discussion to things that can be considered without human existence. Beauty, etc., have no meaning without human existence. In contrast we can consider things that existed before (or after maybe) humans. Dinosaurs for instance. For me, that is the difference between subjectivity and objectivity. So God equated with Platonism, according to Du Sautoy, but if consciousness itself is one of those things that will never be solved how about equating more to God than just something abstract, i.e. more substance? Otherwise, this belittles the concept of God straight away. Once you do this you then link abstract mathematical Platonism with consciousness itself, as you cannot see and discover the former without the latter.

Finally, the issue of experience itself comes in, anomalous, mystical, religious, creative/intuitive, something Du Sautoy surely must have thought of. ( )