European Mathematical Society - 83f05
https://euro-math-soc.eu/msc-full/83f05
enThe Cosmic Web
https://euro-math-soc.eu/review/cosmic-web
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The original text is from 2016, but this review is written on the occasion of the unaltered paperback version in 2018. Gott sketches the successive cosmological models that were designed to understand why the universe is behaving the way it does. The description is intended to be understood by anyone interested but what is so nice about reading this ongoing evolution of our insights is that he has known as a young researcher the people who were working on earlier models, and he has contributed himself in his later career. This makes his account very personal. His main interest involves the structure of the universe at very large scale. The galaxies are not uniformly distributed, but they cluster on filaments and form a sponge-like web, The term "web" was first introduced in 1983 in a paper by Klypin and Shandarin. This is the story of the universe told at the mind-blowing scale of billions and trillions of light-years.</p>
<p>
The story starts with the models of the 20th century. Using red-shifts, Hubble showed that the universe is currently expanding. Another debate was about the cosmological constant and its value which defined the shape (flat, hyperbolic or elliptic) of our universe. Only near the end of the century it was observed that the universe is not only expanding, but that the expansion was accelerating.</p>
<p>
Fritz Zwicky contributed to the larger structure studying clusters of galaxies, using gravitational lensing. In the course of this research, some effects could only be explained using a smooth massive presence which is now called dark matter.</p>
<p>
Gott himself enters the scene with his research about the possible ways that galaxies are formed and to what they evolve. Two competing possible structures are considered: either the galaxies cluster together and float like isolated meatballs in a mostly empty cosmic soup or they fill up the universe, but leaving large, almost vacuum holes like a Swiss cheese. The clustering depends on perturbations in the initial conditions of the universe and which of both models will result (the soup or the cheese) depends on whether the density is considered low or high. Zeldovich in the USSR considered a Swiss cheese model where galaxies cluster on relatively thin boundaries of some vacuum 3D Voronoi cells</p>
<p>
The observed uniformity in the microwave background (how we observe the early universe) remained an enigma for some time. Uniformity can only be explained if the whole cosmos had been in a contact close enough to exchange photons, but what we observe is that they are too far apart to have ever met. This can only be solved by accepting an inflationary phase right after the Big Bang when space expanded faster than light, doubling in size every $10^{−38}$ seconds. Thus parts escaped outside our causality horizon and only now, because the expansion has slowed down to below light-speed, they re-enter our causality horizon. So they look like being too far apart, but they were actually very close and in causal contact before the inflation. This explanation for the horizon problem can be understood by realizing that one second after the Big Bang we can only see what is less than one light-second away, but as time continues, we can see farther and also see objects that are much farther apart.</p>
<p>
The expansion during the high energy inflation period gave rise to bubbles of lower energy. Such a bubble creates a universe on its own, and since there are more bubbles, this assumption becomes a possible multiverse model, our universe being in one of these bubbles. The formation of galaxies could be explained by assuming cold dark matter that can clump together spontaneously by gravitation. The next question is how these galaxies are distributed and how they will evolve. What follows is a remarkable story of Gott's high school project on regular space filling polyhedral structures. He detected that by removing some faces to connect all the polyhedral interiors but at the same time leaving all the remaining faces connected too, he got some sponge-like space filling surface. For example a truncated octahedron consists of 8 hexagonal faces and 6 square faces (where the vertices of the octahedron are truncated). These polyhedra can fill 3-space. Now remove all the square faces from the structure, and one gets a sponge-like surface that is neither a soup-with-meatballs models (where the meatballs are disconnected) or the cheese-with-holes model (where the holes are disconnected). All the remaining faces and all the empty space of the octahedral tessellation are connected. Thus 3D space is partitioned in two disjunct yet fully connected subsets like in a sponge. Gott defined the genus of such a surface as the number of holes minus the number of isolated regions of matter. It is equal to the integral of the curvature and equals minus half the Euler characteristic (<em>g=-(V-E+F)/2</em>). The genus is negative for the Swiss cheese model and positive for the meatball model and anything in between is sponge-like.</p>
<p>
But the sponge-like distribution of galaxies is only temporally. Galaxies may be attracted to each other too. One may in fact construct fluid flow lines showing how galaxies and clusters of galaxies are attracted to so called super clusters. We are with our solar system in the Milky Way, part of a Local Group, belonging to the Virgo Supercluster, which is a branch of the Laniakea Supercluster. Like the water in the watershed of a river we are attracted to the center of the Laniakea. Laniakea means immeasurable heaven in Hawaiian.</p>
<p>
Simulations were run and observations were made detecting clusters of galaxies arranged in filaments. So the sponge became a web. The term "cosmic web" was used first in the title of a paper by Bond, Kofman, and Pogosyan "How filaments are woven into the cosmic web" that they posted in 1995 on arXiv.</p>
<p>
The iconic elliptic map of the cosmic background radiation (CBR) is a projection of the celestial sphere showing the radio spectrum observed by the WMAP satellite. Investigation of the curvature of the isothermal contour lines can be used to define the genus of the topology. It showed that this matched the sponge-like structure that was also predicted in the simulations when starting from random quantum fluctuations during the inflation.</p>
<p>
Still one element is missing to explain the acceleration of the expansion of the universe detected in 1995. This is explained by the repulsive effect of a negative pressure from some energy density. However since the gravitational effect of matter density is much higher than the effect of energy density, one needs an enormous amount of energy to explain the acceleration. To match all the observations, one came to a consensus in 2015 that it is required that about 70% of the universe should consist of dark energy and only 30% of matter, most of which is dark matter. As the universe increases, the density of matter will decrease, and the energy density will increase. Current estimates are that the size is currently doubling every 12.2 billion years. Depending on the ratio of the pressures caused by dark energy and dark matter, different scenarios for the ultimate future of the universe in a googol or a googolplex years are proposed.</p>
<p>
The particularly nice thing about the way Gott tells this story is that he can tell it because he personally contributed to it and met or collaborated with many of the other people who have shaped our current knowledge of the cosmos. Moreover he not only describes the models but he also explains why new observations made it necessary to modify a previous model. So he explains not not only the "what" but also the "why". Of course he is not showing the field equations, that would be too technical beyond the understanding of a general reader, but he explains what the different interpretation is of a constant placed on the left-hand side or the right-hand side of the equations. He also gives physical and mathematical information about the phenomena discussed. His topological science project is explained in some detail with classic polyhedra, and we can follow his derivation of the genus of a sponge-like structure and we learn the meaning of curvature. He mentions the use of the Mollweide equal area projection to picture the CBR. There are Gaussian and other curves (like many other graphics) throughout the book. Sixteen colour plates are bundled at the end, like some extra notes to the different chapters and references to the literature. Also the subject and name index is quite effective. This is a very readable account of how our understanding about the cosmos has evolved with some interesting mathematical excursions. In particular the web-like structure at an incredible large scale is very well explained, which fully justifies the title of the book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The book describes how our understanding of the cosmos has evolved since the early 20th century. In particular the study of the cosmos at a very large scale with billions of galaxies organized in clusters and superclusters that form a cosmic web is a key topic of this account since the author was an essential contributor to this concept.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/j-richard-gott" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">J. Richard Gott</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-15726-9 (hbk), 978-0-691-18117-2 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">29.95 USD (hbk), 19.95 USD (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">272</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/10539.html" title="Link to web page">https://press.princeton.edu/titles/10539.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/83-relativity-and-gravitational-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83 Relativity and gravitational theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/83f05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83f05</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/85a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85A40</a></li></ul></span>Tue, 26 Feb 2019 10:36:34 +0000Adhemar Bultheel49152 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/cosmic-web#commentsThe “Golden” Non-Euclidean Geometry
https://euro-math-soc.eu/review/%E2%80%9Cgolden%E2%80%9D-non-euclidean-geometry
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Fibonacci numbers and their relation to the golden ratio are among the few mathematical items that gained some publicity among non-mathematicians. The golden ratio ($\phi=1.68033...$) is well known since antiquity and it played an important role in Euclid's <em>Elements</em> and in the work of many other mathematicians. It also shows up in phylotaxis and spirals that appear in nature. And it relates to harmony, another term that has been studied in a mathematical sense since the Greek. The golden ratio has therefore gained some mythical and even mystical status, the latter often has to be understood in a (pejorative) numerological sense.</p>
<p>
Fibonacci numbers (denoted $F_n$) is a term coined by Édouard Lucas in the 19th century, who also introduced the sequence of Lucas numbers (denoted $L_n$). Both sequences are solutions of the difference equation $x_{n}=x_{n-1}+x_{n-2}$. The initial conditions for the Fibonacci sequence are $(x_1,x_2)=(1,1)$, while for the Lucas numbers it is $(x_1,x_2)=(1,3)$. The limit of ${x_n}/{x_{n-1}}$ equals $\phi=({1+\sqrt{5}})/{2}$ in both cases. These numbers are defined for all integer indices by $x_{-n}=(-1)^nx_n$.</p>
<p>
In a first chapter, the authors give a brief historical survey, summarize some properties of the Fibonacci and Lucas numbers and they introduce hyperbolic functions: the symmetric Fibonacci hyperbolic sine $sFs(x)=({\phi^x-\phi^{-x}})/{\sqrt{5}}$ and cosine $cFs(x)=({\phi^x+\phi^{-x}})/{\sqrt{5}}$. Similarly for the Lucas versions $sLs(x)$ and $cLs(x)$ but these do not have the denominator $\sqrt{5}$. Their graphs look very much like the graphs of the standard hyperbolic functions.</p>
<p>
The second chapter is about harmony. The old Greek <em>Music of the spheres</em> was picked up by Pacioli and Kepler. But soon the text comes down to one of Stakhov's pet horses, namely that harmony is a forgotten pillar in mathematics. Counting and classical measure theory are the two other pillars that have resulted in conventional mathematics. However by rejecting Cantor's axiom (a 1-to-1 correspondence between the reals and the points on a line) and a consistent application of the golden ratio and its generalizations, a different measure theory, number system, and geometry can be developed. This is what he calls harmonic mathematics. He considers a delayed version of the above difference equation which leads to the introduction of a new representation of number systems and his $p$-Proportion Codes. However this is soon replaced by another generalized Fibonacci sequence, defined for any real $\lambda>0$ by $F_\lambda(n+1)=\lambda F_\lambda(n)+F_\lambda(n-1)$, with $F_\lambda(0)=0, F_\lambda(1)=1$ and the limiting ratio $\phi_\lambda=(\lambda+\sqrt{4+\lambda^2})/2$ which is a root of the characteristic equation $x^2-\lambda x-1=0$. The above Fibonacci hyperbolic functions can be generalized by replacing $\phi$ by $\phi_\lambda$ and the $\sqrt{5}$ by $\sqrt{4+\lambda^2}$. They are denoted as $sF_\lambda$ and $cF_\lambda$. Note that (up to a factor 2) the classical hyperbolic functions are obtained as a special case of the $\lambda$-Lucas numbers by choosing $\lambda=e-1/e$. For $\lambda=1,2,3,4$ we get the golden, silver, bronze, and copper relations, referred to as the metallic relations.</p>
<p>
The third chapter is about Hilbert's fourth problem, in which it is asked to design new forms of non-Euclidean geometry. The formulation was however rather vague and different proposals were made but it remained unclear whether the problem was (completely) solved or not. So the authors have their own interpretation and solve their form of the fourth problem using the hyperbolic functions introduced above. Lobachevsky's hyperbolic geometry is based on classical hyperbolic functions. Replacing the classical ones by the hyperbolic $\lambda$-Fibonacci functions they get different hyperbolic geometries. To obtain a similar generalization for spherical geometry, yet another type of Fibonacci functions are needed. There are of the form $SF_\lambda(x)=c_\lambda\sin(x\ln\phi_\lambda)$ and $CF_\lambda(x)=c_\lambda\cos(x\ln\phi_\lambda)$ with $c_\lambda=2/\sqrt{4+\lambda^2}$. The $\ln\phi_\lambda$ factor appears here for the sake of harmony. A similar form can be obtained in the hyperbolic case giving a true hyperbolic geometry in harmony mathematics. They consider many more relations and formulas in this context and claim that the Clay Mathematics Institute made a mistake by not putting Hilbert's fourth problem on their list of millennium problems. So the authors claim to have actually solved a self declared millennium problem.</p>
<p>
The next chapter 4 is about the qualitative theory of dynamical systems based on harmony mathematics. Hence the `golden' and also the other metallic proportions show up again. It is a simple observation that a metallic ratio $\phi_\lambda$ (which is an irrational number) can be approximated from above and below by ratios of successive $\lambda$-Fibonacci numbers. This simple fact is exploited in a complicated framework of foliations of a 2D manifold. First foliations of such a manifold are introduced, which is then specialized to the 2D torus $T^2$. These foliations are characterized by a Poincaré rotation number $\omega$. In the particular case that it happens to equal a metallic proportion, then it can be approximated by ratios of Fibonacci numbers and hence the irrational foliation is approximated by rational ones. Since integral curves for flows of a dynamical system are foliations, this may also be applied in a context of dynamical systems. This chapter is much more mathematical with long mathematical proofs which do not seem to be easily accessible for a general public.</p>
<p>
A last chapter is about the fine structure constant in physics. Like the mathematical millennium problems, there is a list of physical millennium problems. The first of these problems is asking whether all dimensionless parameters of the physical universe are calculable. Here the fine structure constant $\alpha$ is declared to be fundamental and hence is the constant to be discussed. The approach taken here is by looking at the Lorentz transform in special relativity theory. It is a transformation of the space-time vector whose matrix can be written as a direct sum of the identity and a hyperbolic rotation over an angle $\theta\in(-\infty,\infty)$. In view of the preceding items it is again a natural thing to replace the classical hyperbolic sine and cosine functions of the rotation angle by the hyperbolic Fibonacci sine and cosine ($\lambda=1$) of an appropriate angle $\psi$ and so obtain a Fibonacci special relativity theory. Here however $\psi\in(-\infty,0)\cup(2,\infty)$ because singularities appear at 0 and 2. Moreover, the speed of light in vacuum has to be made variable. It decreases with the age of the universe. It will be $c^*$ (the classical value) for $\psi\to-\infty$ and it is $c^*/\phi^2$ for $\psi\to-\infty$. The physical meaning is that the Big Bang corresponds to $\psi=0$, the interval (0,2) is the dark age before galaxies were formed (the speed of light is imaginary), and for values larger than 2 this corresponds to the light age, when the stars were formed that created light in the universe. To the left of the origin is the black hole situation with the arrow of time reversed.<br />
In 2000, N.V. Kosinov proposed a formula $\alpha=10^{-43/20}\times\pi^{1/260}\times \phi^{7/130}$. Inspired by this formula, the authors propose to let $\alpha$ depend on $\psi$ by replacing the $\phi$ in this formula by their $\psi$-depending speed of light. The result is an $\alpha(\psi)$ with $\psi=\lambda_0 T$ where $T$ is the age of the universe (in billions of years) and $\lambda_0$ a constant. This $\alpha$ is decreasing with $\psi$ in the black hole range until it becomes 0 at the Big Bang. In the same range the speed of light drops from $1/\phi$ to 0. In the dark age, the derivative is positive and goes from 0 to $\infty$ just like the modulus of the speed of light does, and in the light age it drops from infinity to a little bit below its current value of about $7.29\times 10^{−3}$. Of course as a consequence of the varying $\alpha$, also other values that depend on it will change with the age of the universe. In an appendix allusion to multiverses is made when the $\phi$ in the previous setting is replaced by $\phi_\lambda$ with $\lambda\ne 1$.</p>
<p>
The first author Alexey stakhov is a Ukrainian mathematician with a PhD in computer science, who lives in Canada since 2004. He has published many papers and books in which he has proposed many of his original, sometimes controversial, ideas. Chapter 2 clearly summarizes some of his previous work. The second author is Samuil Aranson who is a Russian mathematician, now living in the USA whose domain is differential equations, geometry and topology. It is therefore clear that he must be the main author for chapters 3 and 4, which also explains the somewhat different and more mathematical style. Scott Olson is a professor of philosophy and religion in the USA, who wrote a book on the golden section and who seems to be helping with the English editing of this book.</p>
<p>
The first two chapters are elementary with a lot of history and simple mathematical relations. Who wants to read more on Fibonacci and Lucas numbers and generalizations can read <a href="http://www.euro-math-soc.eu/review/pell-and-pell%E2%80%93lucas-numbers-applications">Pell and Pell-Lucas Numbers with Applications</a> for a good mathematical treatment and there are of course many popular books on the golden ratio. If you are interested in the golden ratio and harmony, you would certainly want to read <a href="http://www.euro-math-soc.eu/review/fibonacci-resonance-and-other-new-golden-ratio-discoveries">The Fibonacci Resonance and other new Golden Ratio discoveries</a>. However chapters 3 and 4 of this book are much more mathematical and create a complicated mathematical framework of foliations, not suitable for a general public anymore, while it only illustrates and applies the fact that the ratio of two successive Fibonacci numbers tend to the golden ratio and hence that this irrational number can be approximated by rationals. The fifth chapter is devoted to physics. The core idea is to replace a classic hyperbolic rotation by a more general one. The physical interpretation is certainly not mainstream and is probably susceptible to critique by theoretical physicists, if they do not consider it to be just numerological mysticism. However, since there is no experimental proof of what is exactly happening at this cosmological scale, it may be another explanation that is as good as many other fantasies. It is clear that the book is mainly collecting results that the authors have published as papers and that are here somewhat streamlined into a more consistent survey. Long lists of references are added after each chapter with many papers of the authors but several are only available in Russian. That this harmony mathematics and Fibonacci numbers and generalizations can solve all these problems clearly adds to the myth of the golden ratio. The typesetting in LaTeX is nicely done. I could spot a few typos but not that serious. For example page 121, a $(dv)^2$ is missing in the equation and on page 232 the Black Hole should correspond to $-\infty<\psi<0$ and not $0<\psi<2$. Also the graphics of chapter 5 are a bit rough and not always very precise. Anyway there are some original ideas to be found in this book. Whether the reader will agree with them or not will depend on who's reading it.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a book in which the authors give a summary of some of their work. They study Fibonacci and Lucas numbers and show how these give rise to a new kind of mathematics: the mathematics of harmony. Generalizations of these number sequences and their limits the golden and other metallic ratios are applied to derive a new kind of non-Euclidean geometry, to study foliations and dynamical systems and even a golden Fibonacci version of the special relativity theory in which the fine structure constant from cosmology is analyzed.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alexey-stakhov" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alexey Stakhov</a></li><li class="vocabulary-links field-item odd"><a href="/author/samuil-aranson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Samuil Aranson</a></li><li class="vocabulary-links field-item even"><a href="/author/scott-olsen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Scott Olsen</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4678-29-2 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£98.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">308</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/9603" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9603</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11b39" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11B39</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/53a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53A35</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/37d40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37D40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/83a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83A05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83f05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83f05</a></li></ul></span>Fri, 23 Sep 2016 08:30:58 +0000Adhemar Bultheel47182 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/%E2%80%9Cgolden%E2%80%9D-non-euclidean-geometry#commentsThe Singular Universe and the Reality of Time. A Proposal in Natural Philosophy
https://euro-math-soc.eu/review/singular-universe-and-reality-time-proposal-natural-philosophy
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In this book, a philosopher and a theoretical physicist discuss natural philosophy based on three ideas:</p>
<ol>
<li>
the singularity of the universe (there is only one universe that contains everything, although not everything in it is observable)</li>
<li>
the reality of time (everything has a history and hence can vary in time and hence has a causal relation with its past although this causality does not obey timeless laws)</li>
<li>
the selective realism of mathematics (as opposed to Platonism, mathematics wants to formulate the laws of nature and thus it has to play a reasonable, yet limited role in physics).</li>
</ol>
<p>
The first two thirds of the book are written by the philosopher Unger, the rest by the physicist Smolin, followed by a short discussion on topics they do not fully agree. On the fundamentals of the three previous axioms, they however generally agree. Both of them have a reputation of being iconoclasts in a their respective scientific communities. Hence this book, as are their previous publications, is provocative and will be the subject of much debate. With their arguments they want to criticize and redirect science. They conclude that the present, generally accepted views on cosmology are wrong and should be redirected using instruments already at hand.</p>
<p>
Why is our universe as it is? Why precisely these physical constants? Why are we here? All questions current cosmology will not be able to answer. Even though cosmology currently knows its greatest successes, it is in crisis. The standard model of physics is only valid in a middle range of the scale. It fails at a subatomic and at a cosmological scale. The authors suggest a different, historical approach to cosmology following from their three postulates. The role of mathematics is over-emphasized. It cannot be an abstract timeless mirror of the physical reality.</p>
<p>
Unger formulates fallacies of current cosmology: laws and observations applying to a subset of the universe cannot be applied to the whole universe. This subset refers to a limited scale as well as to a limited time window. The standard model is not applicable at a subatomic scale and not at the scale of the universe and we do not have access to the moment of the big bang nor do we know the future of the universe. The one universe may however go through different stages. There may be cyclic phases, or drastic changes with transitions at moments such as the big bang, or it may just evolve in a linearly changing phase. Each of these phases may have different physical laws or these may change during the evolution. This poses the problem of a meta-law. Which laws govern these transitions? Smolin gives several singularity theorems that speculate about the very beginning or the ultimate evolution of the universe, but we have no evidence for any of them. He also presents a theory of his own that the meta-law results from some nonlinear meta-dynamical system with many, yet a finite number of degrees of freedom which will define the physical laws of our universe. But whatever the answer is, these meta-laws are not fixed either and are varying as well. Change changes, which is intrinsic to the reality of time. Such vision is accepted in evolution theory, but it never got accepted in physics.</p>
<p>
Unger continues by placing his arguments against recent developments in physics, cosmology and science in general. He then elaborates extensively on the three main topics and gives arguments why they should be accepted followed by recommendations for the further development of cosmology. Unger does not believe in several universes or stages of one universe, cyclic or not. This reduces the problem of answering the cosmological questions to deciding on the initial conditions. Smolin is more relaxed on the evolution of the universe, and does not want to speculate about what is beyond our horizon of observation in the cosmological past as well as in the future. Unger does not believe in infinity either. Smolin argues that this would also exclude continuity, and accepts it as a useful tool in mathematics.</p>
<p>
Concerning time, Unger rejects a preferred cosmic time. Since this would contradict general relativity, confirmed by experiments, general relativity should be reformulated independent of the Riemannian spacetime concept which has been enforced on physics by mathematics. The only reality is now. The past is not real, but it has been real, so that we can acquire information about it through its consequences in the present time. The future is not real though. Although Smolin argues that we can make predictions of the future, these are always approximate and we can never be sure what the future will really bring.</p>
<p>
While Unger's discussion of the reality of time and the mutability of physical laws is rather extensive, his discussion of the role of mathematics is shorter. The question whether mathematics is invented or discovered is a false one. It is not discovered because that would mean that it exists independent of time. This is impossible because in a causal universe, everything is the consequence of its history. If it is invented, it only follows its own rules and the choice of the inventor without being bound or corrected by the physical reality, and hence would be useless for physics. Smolin says it is 'evoked', meaning that it did not exist before, but it is bound by specific properties. A clear distinction must be made between a mathematical concept and the corresponding non-mathematical reality. Mathematics should never dictate physics, it can only be partially of service to physics. While Smolin accepts the possibility that mathematics evokes structures that are helpful in physics. It is impossible though that mathematics acts as an oracle for the future. Even mathematics itself is subject to evolution and its own future is unpredictable. Unger is less permissive and reproaches mathematics that it is completely timeless, hence violating the reality of time and that it is evolving on its own away from reality.</p>
<p>
In his essay, Smolin goes, with some variations, more or less through the same arguments as Unger but he is at some points a bit more technical. I mentioned some of the differences already. His agenda for science that is the consequence of his views is as follows. The evolution of science should mainly influence cosmology, quantum gravity and the foundations of quantum theory. Multiverses are excluded since they can never be observed and hence are not real, but the universe passing through successive stages needs to be investigated. Passing from one stage to another in the cosmic evolution of the universe will also influence the way the laws of physics will evolve, but the main problem is that we do not know whether such changes can ever be observed. A decisive interpretation of quantum mechanics needs to be developed. As pressing is the investigation of the existence of global time and of the arrow of time. Other open problems are the nature of quantum gravity, and clearly the solution of the meta-laws and the cosmological dilemmas. He gives guidelines and constraints within which all this research should be performed.</p>
<p>
This is not an easy read. It is not a pure philosophical book, and neither is it a book about physics or cosmology. Neither is it a confrontation of the philosopher versus the theoretical physicist. Both do a bit of each, and I can imagine that they will get opponents from all directions, be it philosophy of science, or the mainstream cosmologists or theoretical physicists. In my view mathematics plays an essential role here, but it is somewhat pushed into a corner leaving it not much room to move, being a humble, though 'reasonably effective' servant to physics. Anyway, if you are concerned with your reason of existence as a mathematician or of mathematics in general, this book is something that will give you material to ponder. You may need a bit of a background on philosophy and on cosmology and theoretical physics, but the book is not over-technical in either direction. It may need checking some of the terms or references though. Speaking of references, the numbers in the text of Smolin (Unger gives no references) do not correspond to the numbers in the bibliography at the end of the book, which is very annoying.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
A philosopher and a theoretical physicist give their provoking views of the foundation of physics as a consequence of their three principles: the uniqueness of the universe, the reality of time, and the role played (or to be played) by mathematics.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/roberto-mangabeira-unger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Roberto Mangabeira Unger</a></li><li class="vocabulary-links field-item odd"><a href="/author/lee-smolin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Lee Smolin</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781107074064 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£19.99</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">566</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107074064" title="Link to web page">http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107074064</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/83-relativity-and-gravitational-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83 Relativity and gravitational theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/83f05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83f05</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03A10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/81p99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81P99</a></li></ul></span>Mon, 20 Jul 2015 16:13:34 +0000Adhemar Bultheel46315 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/singular-universe-and-reality-time-proposal-natural-philosophy#commentsDiscrete or Continuous? The Quest for Fundamental Length in Modern Physics
https://euro-math-soc.eu/review/discrete-or-continuous-quest-fundamental-length-modern-physics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Ever since Zeno has formulated his well known paradoxes, there has been a tug of war between mathematicians and philosophers alike about whether continuous mathematics is an approximation of a discrete reality or whether reality is continuous, hence properly described by continuous mathematics, but which can in practice only be computationally approached by discretized approximations. Does not quantum physics suggest that spacetime consists of quanta and hence is not continuous? Is it possible, or will it ever be possible to prove or disprove the discrete or continuous nature of reality? So far we have only been able to verify experimentally up to an order $10^{−16}$ that we can still subdivide, but that is still infinitely far away from true continuity. We have built the Large Hadron Collider to measure the smallest particles, but it needs many orders of magnitudes more to arrive at the Planck length $1.6\times10^{−33}$ cm. And that is beyond reach unless we can capture all the energy of a few galaxies. The only possible alternative is that we may hope to extract information from cosmological observations echoing states of the universe at a very early stage, when those amounts of energy were still available in a condensed form.</p>
<p>
So Amit Hagar sets on a quest for the fundamental length in physics if ever there is one. Because the description of spacetime so far worked perfectly well with the continuous model, there is quite obviously, still great resistance against a discrete alternative. There are only few exceptions. See for example Max Tegmark's plea for a mathematical universe in <a href="http://www.euro-math-soc.eu/review/our-mathematical-universe-my-quest-ultimate-nature-reality">Our mathematical universe</a>. Also Hagar is convinced that there is some fundamental length which has for example the great advantage that a discrete model avoids the current singularities of a continuous worldview.</p>
<p>
The book starts with a historical survey of the mathematical arguments used in favor or against spatial discreteness. The arguments are that the two visions are mutually exclusive and only one can lead to consistency of geometry. An analysis of Zeno's paradoxes by Adolf Grünbaum who criticized the views of Whitehead and Russel, leads to the conclusion that if we want to keep countable additivity, then a line segment must have $\aleph_1$ elements, or if we go for the discrete model, i.e. allowing line segments with $\aleph_0$ elements, then we have to give up on countable additivity. However Hagar objects that such arguments deal with the mathematics itself, and not with the applicability of mathematics to the real world. Other attempts have been made to construct a geometry on a discrete line segment. For example Weyl argued that if the shortest length is the distance between neighboring points, then in a 4 by 4 square, the diagonal would have the same length as the side since both cross 4 squares, and that violates Euclidean geometry. Again, the objection against this argument is that this only says that a discrete geometry should not be Euclidean. In the information age, the Church-Turing test was relying on a discrete Turing machine which could approximate a continuum to any desired accuracy, so it is generally accepted that a physical Turing machine would be able to describe the real world with any desired accuracy. In fact that is what applied mathematics do: they compute solutions good enough for any practical situation even if pure mathematics predict only a continuous solution.</p>
<p>
In this way Hagar continues defusing all the mathematical arguments used against a discrete universe and he goes on to strip the arguments used on a more general philosophical level. A finite viewpoint would for example downgrade metaphysics to epistemology, i.e., what is depends on what we know. That would simplify many discussions. However, this viewpoint brings about a problem if one wants to distinguish classical from quantum probabilities. This can however be resolved by defining an appropriate measure which brings a new, more natural, interpretation to the thought experiment that motivated quantum physics.</p>
<p>
From here Hagar turns to (quantum) physics. Almost by definition, quantum physics is discrete and the renormalization program is a collection of techniques to deal with infinity and singularities that inevitably arise when discrete quantum physics describe dynamics assumed to be continuous, i.e., dynamics in a continuous spacetime. Originally renormalization was used in quantum electrodynamics (QED) in which relativity theory and quantum mechanics were integrated. In this context we also see the notion of fundamental length appear for the first time. It was assumed to be of the order of the electron radius. Also Heisenberg's uncertainty relation imposed a finite resolution on the simultaneous measurements of position and momentum. This led Heisenberg even to speculate about a lattice world. The more general quantum field theory (QFT) considers particles as exited states of an underlying field. QED for example considers only one electron field and one photon field. However, it is not a candidate to describe full reality and shed some light on the problem at hand since it does not include gravity. Thus one should move a further step up and we should put our hope in quantum gravity (QG) theory, which is still under construction.</p>
<p>
So Hagar continues by sketching the history of quantum gravity in which mathematicians and physicists join forces to incorporate gravity into quantum field theory. Discretization of gravity may lead to a fundamental length. Completing a theory of quantum gravity is still a matter of searching in the dark by absence of experimental data. However, accepting that there is some fundamental length, hence a discrete reality, may help the development of the theory. This can be explained as follows. Hagar's historical description of quantum gravity includes a correspondence between Einstein and W.F.G. Swann from which Hagar gets the inspiration for his "thesis L". By this thesis he means that there is the possibility that the dynamics of some (discrete) postulated building blocks are consistent with observable spacetime. In fact this thesis can serve to design these building blocks their symmetry groups and the proper metric. Hagar continues by proving his thesis assuming some discrete system with a fundamental length (or area or volume) exists. However, the problems caused by conflicting with relativistic causality, locality, unitarity, and Lorentz invariance are not yet resolved. So this is the current challenge for quantum gravity theorists: to construct a model that solves these problems on the small scale but still being consistent with what the classical approach can do at other scales. In a final chapter, a summary is given of the whole argumentation in the form of questions and answers.</p>
<p>
This book is rather philosophical and definitely concerns a metaphysical problem, but the discussion is more at a metascience level. How can science, and in particular quantum gravity with all the mathematics that it involves, be advanced by his discrete worldview, and how this theory can eventually lead to an invitation for experimental verification of the existence of a fundamental length. However, it should be clear that the book is not about mathematics or about quantum physics. The text is only occasionally interrupted by a formula. It is certainly of interest for mathematicians and physicists working on quantum gravity theory, but it is not the place to learn about this topic. Although of general interest, it is no easy reading at all. It's a philosophical discussion of the foundations of modern physics placed in a historical context. Preliminary training in advanced mathematics and certainly in quantum physics is required.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Hagar's purpose in this book is to convince the reader that there is some fundamental length in physics, which implies a discrete reality, as opposed to the currently generally accepted belief that we live in a world that is continuous in se. He goes through the historical pros and cons raised in mathematics, philosophy and physics and pleads for his viewpoint. His belief is that in the end quantum gravity, when it will be finally resolved, will allow for a consistent theory for a discrete reality, along which possible experimental verification can be sought.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/amit-hagar" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Amit Hagar</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781107062801 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£60.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">267</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107062801" title="Link to web page">http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107062801</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/83-relativity-and-gravitational-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83 Relativity and gravitational theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/83-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-02</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/03f50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03F50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/70a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70A05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/81q60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81Q60</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83b05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83B05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/83f05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83f05</a></li></ul></span>Thu, 17 Jul 2014 07:51:51 +0000Adhemar Bultheel45639 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/discrete-or-continuous-quest-fundamental-length-modern-physics#commentsTime in powers of ten
https://euro-math-soc.eu/review/time-powers-ten
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Nowadays, one may find several sites where the visitor is taken on a tour, with images or video, zooming in and/or out from an ordinary human size view to cosmological distances on one side and to subnuclear scale in the other direction. There is also a similar, printed, "powers of ten" tradition from the pre-Internet era, that was picked up by Nobel Laureate Gerard 't Hooft and Theoretical Physicist Stefan Vandoren who published a Dutch version of the present book in 2011 (currently sold out). Its success may have triggered the publication of the current updated (yes, the 2013 confirmation of the Higgs boson is there) translation. The original format is kept. It's almost square (267 x 253 mm), has two columns per page with wide margins and is abundantly illustrated. A classic example of a coffee table book.</p>
<p>
The powers of 10 range from minus 44 to plus 1000. The unit of time is the second. The outer ends of this time scale are hard to imagine, things are beyond our current knowledge and one enters speculative ideas. At minus 44, we are dealing with Planck time (the time needed for light to travel the Planck length which is about $1.6\times 10^{-35}$ meter) and beyond plus 100 we enter the so called "Dark Era" when everything has disintegrated into dark matter. In the first part the authors guide us from the familiar 1 second over cosmological time scales up to the vague upper regions with some speculations of what could happen then. Not for our lifetime though because all life will have ended long before. In the second part, we are not taken on a zoom-in journey but instead the authors have chosen to be constructive and build up time and matter from the tiniest time scales and subnuclear particles to end where their time traveling started: at the one second scale.</p>
<p>
Given the background of the authors, one may expect a lot of (theoretical) physics and indeed there are recurrent physical topics that are discussed at almost every time scale, namely phenomena that are astronomical (like orbits of planets), periodic or vibrating (e.g. biological cycles or electromagnetic issues), decaying (like half-lives of radioactive elements), cosmological (the evolution of the universe from just after the Big Bang and onwards) and phenomena at speed-of-light scales. Each of them get their particular color in the margin so that they are easily recognizable. These recurring themes form the bulk of the content, but there are other entertaining stories and interesting facts as well. For example the naming of the numbers or the etymology of words like second, hour, and the names of days and months. Some other bread crumbs to give an idea: the longest war ever was between the Netherlands and the Isles of Scilly (335 years) that was officially ended in 1985, the 13 days of the Cuban crisis (1962), 36 days for Christopher Columbus to cross the ocean (1492), and the 30.07 days half-life of a cesium isotope is flanked by the 30 years war (1618-1648) during the Holy Roman Empire and the 32 years pontificate of Pope Pius IX (1846-1878).</p>
<p>
All powers of 10 get their stories, until we arrive at powers 14-16 (a few billion years). At this scale we are dealing with evolution theory, plate tectonics, the origin of life on Earth, and the lifetime of the Earth itself. Beyond that scale there are some gaps in the time line and we enter the dark eternities with lifetime of huge black holes at exponent 100. Then we are on the verge of current knowledge and the intensely investigated unifying theory of everything. Therefore jumping from that scale of time at the end of the first part to the smallest possible scales in the second part is a dazzling mind-blowing step, but scientifically, seen in the light of the Grand Unification Theory, they are not really that far apart.</p>
<p>
What happened between $10^{-44}$ and $10^{-38}$ seconds after the Big Bang is speculative, but scientists are convinced about the exponential inflation of the universe to a few centimeters about $10^{-36}$ seconds after the Big Bang. With $10^{-25}$ we enter the world of bosons, fermions, and particle physics, moving up to electromagnetic waves, which are the main players of the second part. The time spanned at the lower side of the second in this 2nd part is shorter. Also there are less facts-of-general-interest kind of elements. Observations at these very small time scales are not possible without instruments. Common people are familiar with the hundredths of a second in certain sports timings but they are not so familiar with finer time scales. What could have been a topic is nanotechnology in material science and and electronics and of course microelectronics too. These applications were however not discussed. We do find somewhat familiar items like radio waves, lightning (average duration of 30 microseconds), the correction of GPS satellites because of relativity theory (39 microseconds per day), the reaction of a nerve cell (2 milliseconds), the wing beat of a hummingbird (66 milliseconds), etc. The authors keep amazing us with numbers: the size of the universe 0.32 seconds after the Big Bang was about 4 light-years with a density of 100 tons per cubic centimeter. Hard to imagine for us creatures living in the same universe some 13.7 billion years later.</p>
<p>
All of this illustrates that we actually experience our daily lives and loves in a very narrow time scale. The scales of the larger or smaller numbers are so difficult to imagine once they are outside our usual scope. Many people do not realize how fast exponential growth or decay really takes place. The book makes it so easy to flit back and forth over these scales that the enormity of the numbers is somewhat lost. Sometimes one should take a pause and allow to let it sink in. Fortunately, the format of the book administers small sups at a time. But the somewhat facetious narrating style and the abundance of illustrations are so inviting and rather addictive once you picked up the book. If the book is indeed placed on a coffee table, the coffee break may last somewhat longer than usual.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even"> </div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is an updated translation of the Dutch version that appeared in 2011. It is a coffee table book, richly illustrated, and with an unusual format (267 x 253 mm). Time scales range over 10 to the power k seconds with k ranging from 0 to 1000 and in a second part from minus 44 back to 0. Each time scale gets 2 to 4 pages discussing various phenomena from different scientific disciplines.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/gerard-t-hooft" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Gerard 't Hooft</a></li><li class="vocabulary-links field-item odd"><a href="/author/stefan-vandoren" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stefan Vandoren</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4489-81-2 (pbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 16.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">232</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/8786" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/8786</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/83f05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83f05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/81-xx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-XX</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/92e99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92E99</a></li></ul></span>Mon, 30 Jun 2014 06:30:12 +0000Adhemar Bultheel45590 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/time-powers-ten#commentsOur mathematical universe. My quest for the ultimate nature of reality
https://euro-math-soc.eu/review/our-mathematical-universe-my-quest-ultimate-nature-reality
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In this book Max Tegmark defends his speculative and controversial theory of everything. His claim is the mathematical universe hypothesis (MUH) which he sees as a direct consequence of the external reality hypothesis (ERH). The latter is a kind of Platonic vision of mathematics: there is some external reality independent of human intervention and that we can only observe filtered by the limitations of our senses with subjective interpretations of our minds. In the MUH our universe and everything it contains is just a mathematical structure. Since any mathematical structure is a universe, there is a whole zoo of universes outside ours, and any self aware creature in such a universe will experience its physical environment as real as we do.</p>
<p>
But the book starts in a much less provocative way. Part one (zooming out) begins explaining how people managed to measure the size of the moon, then the sun, planets, stars, galaxies etc. The farther we can see in distance the farther we see in the past, given the time needed by the light to travel through space. So we end up seeing young galaxies forming shortly after the Big Bang. Beyond that: only darkness. And yet there is some microwave background radiation. That is where Tegmark got involved in visualizing the picture of our baby universe in the WMAP project. It shows a bright plasma of a very hot free electron soup, cooling down and transforming hydrogen into helium in our infantile universe of only 400,000 years `young'. His story becomes very lively at this point him being a first hand witness. But if this was the state after the exponential inflation, where did mass and gravitation come from, that will finally form the galaxies? Tegmark gives clear answers to such fundamental questions and many others. If you define our universe as the sphere from where light can reach us since its origin some 14 billion year ago, then one might expect there is more beyond what we can observe, i.e., beyond the boundary of that sphere. So there may be more universes `out there'. This is what Tegmark calls the Level I multiverse. Our Big Bang is not the very beginning, but basically the end of the stage of exponential inflation. The Big Bang is caused by the inflation and not its origin. But the creation of a Big Bang is a very local phenomenon. In chaotic inflation theory there is a multitude of Big Bangs that will form bubbles in this for ever inflating multiverse and in each of these bubbles another universe will exist, some with different fundamental physical laws. This is the Level II multiverse.</p>
<p>
Part two (zooming in) goes in the opposite direction and deals with particle physics and quantum mechanics. Much less details are given about the theory here, but it mainly serves to place the Copenhagen interpretation of a Schrödinger wave function collapse against the many worlds interpretation of Hugh Everett. All possible outcomes of the observation are possible, but they are alive in parallel worlds. Schrödinger's cat will be dead in one of the worlds, but it will be alive in a parallel one. This creates a Level III multiverse. This time the universes are not at a distance out of reach for us, but many versions of you will exist in as many parallel worlds that are separated in the Hilbert space in which the wave functions live. Since any possible outcome will be realized in one of these worlds, the Level III multiverse will include the Level I and Level II multiverses.</p>
<p>
While all the multiverses defined so far have been considered also by others, the Level IV multiverse is Tegmark's idea. It is explained in the third part (stepping back) which fills almost completely the second half the book. Here he builds up his theory of the MUH and all the consequences that implies. For example he needs to explain how inhabitants of such a mathematical structure can be self-aware and how they experience the external reality. Since any mathematical structure is a universe, we are dealing with yet another kind of multiverse. This is what he calls the Level IV multiverse. However mathematics should rule. Thus Gödel's incompleteness or the Church-Turing undecidability should be avoided to form a consistent system. This rules out infinity. The `infinitely small' is related to continuity, but that can be removed because continuity is nothing but an approximation of reality that is only observed at a much higher scale. Zooming in at the details, everything there is just discrete particles, strings or branes or whatever, but always discrete. Real numbers are out too because they contain infinite information and thus are not computable in finite time. And so on and so further. Tegmark tackles one by one all possible objections and possible inconsistencies that may be raised by a critical opponent. Whether the reader will agree with all his arguments or not is of course up to the reader.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Two books for the price of one. On one hand an instructive account of cosmology (Tegmark was involved in generating baby pictures of our universe) and a less extensive part on quantum mechanics. In these parts some fundamental questions about our universe are answered in an elucidating and lively way. These parts are written by the conventional scientist Mark Tegmark. His alter ego is a defender and a source of speculative and controversial ideas including a zoo of parallel universes, claiming that our universe and everything in it, including ourselves, and in fact any other universe or multiverse is nothing but a mathematical structure. Mathematics is all there is, that was, and that will ever be.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/max-tegmark" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">max tegmark</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/penguin-books-allen-lane" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Penguin Books / Allen Lane</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781846144769 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£25.00 (net)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">432</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.penguin.co.uk/nf/Book/BookDisplay/0,,9781846144769,00.html" title="Link to web page">http://www.penguin.co.uk/nf/Book/BookDisplay/0,,9781846144769,00.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/83-relativity-and-gravitational-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83 Relativity and gravitational theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/83f05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83f05</a></li></ul></span>Tue, 11 Mar 2014 17:56:14 +0000Adhemar Bultheel45555 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/our-mathematical-universe-my-quest-ultimate-nature-reality#comments