European Mathematical Society - basic books
https://euro-math-soc.eu/publisher/basic-books
enThe Universe Speaks in Numbers
https://euro-math-soc.eu/review/universe-speaks-numbers
<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 title of the book may suggest that this is about numbers, but there are nu numbers in the book, at all. So the subtitle: "How modern mathematics reveals nature's secrets", is a better description of the content. Because the book is about "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" as Eugene Wigner formulated it back in 1960. Especially with physics, there has been a close and successful interaction with mathematics. But Farmelo explains that this is not so unreasonable and in fact it goes also the other way around since there is also "A Reasonable Influence from Theoretical Physics on Mathematics". Perhaps, there can even be a superstructure that has mathematics and nature as two of its realizations that we humans, with our limited intellectual capabilities, are able to experience, without yet understanding the superstructure, an idea that has been proposed before by Max Tegmark in <a target="_blank" href="/review/our-mathematical-universe-my-quest-ultimate-nature-reality">Our Mathematical Universe</a>.</p>
<p>Clearly both communities, physicists and mathematicians, have their own culture. Mathematics expands in the minds of mathematicians that are driven by abstraction and a mathematical result is true and remains true beyond discussion after it has been proved once and for all. The (traditional) physicists are driven in their urge to explain nature. They care somewhat less about rigour and their models are inspired by observation, and perhaps less by strict abstract deduction. Their models are accepted if they are confirmed by nature itself after appropriate experiments, but acceptance of the model is only guaranteed until it is contradicted by new or better experiments. The latter situation seems to have changed for current theoretical physics that has moved to the mathematical approach.</p>
<p>Since antiquity mathematics and physics have developed in parallel. Even mathematics developed and progressed often driven by practical physical problems. Yet, Greek philosophers already discussed whether mathematics was created by humans or provided by nature and left for humans to be discovered. Farmelo sketched in the first half of the book this on-off relationship between mathematics and physics up till about the 1970's. He tells the history by staging the people who have contributed to the major steps in the evolution of both mathematics as well as physics.</p>
<p>Of course Newton and the scientists of the Enlightenment who envisioned a mechanical clockwork world. It was promoted by Laplace that if all the details of the current state age given, then this would allow to predict the future perfectly. Much of this resided on an atomic idea about the world consisting of particles subject to forces.</p>
<p>The electromagnetic theory of Maxwell introduced the concept of a field. His theory is condensed in the Maxwell equations named after him, but written down by Heaviside. The "beauty" and symmetry in the equations were a source of inspiration for later developments.</p>
<p>Gravity, the source of inspiration for Newton, coupled to a geometric vision, and thought experiments, were the instruments used by Einstein to develop his special relativity theory, only to be confirmed afterwards by practical experiments. At first he was not convinced that physics required advanced mathematics but he had to abandon this idea when he got in trouble for the development of his general relativity theory. Early twentieth century were turbulent times, shaking the foundations of both physics and mathematics. This was promptly followed by a steep increase in our understanding of the world.</p>
<p>Heisenberg and Schrödinger developed quantum mechanics, but it was Dirac who could link this new model, which inherently involved uncertainty, with classical world view of Laplace. He endorsed Einstein's revised view that mathematics is unavoidable for the development of physics. But then, during the war and in the post war period came, what Farmelo calls, "The Long Divorce" where mathematics and theoretical physics each went their own way for about two decades. However, physicists in search for their unifying theory got stuck and were confronted with a zoo of subatomic particles. Feynman made some progress, and Yang and Mills tried to generalize the symmetry of the Maxwell equations, but Freeman Dyson in his 1972 talk to the AMS pointed to the missed opportunities because the physicists were not aware of the most recent developments in mathematics and mathematicians were not interested in physics. Until in November 1974 (hence called the November Revolution) the psion (J/ψ meson) was discovered, a particle that lived much longer than other elementary particles, and gauge theory became the common interest of physicists and mathematicians, also because of the Atiyah-Singer theorem which had shown the power of differential geometry in explaining subatomic quantum mechanics. The Standard Model was realized in the 1970's.</p>
<p>This is where the first part of the book ends, surveying about 3 centuries from Newton till the Standard Model. The second half deals with the 4 decades that follow. Veneziano had written down the formula forming the model for string theory on a napkin already in 1968. It was however abandoned because quantum chromodynamics and quantum gravity (the quantum mechanical approach to study gravity near black holes) had stolen the hearts and minds of physicists. But strings were later reinstalled as the road to take for a Theory of Everything. String theory introduced extra dimensions because supersymmetry is the only way to extend the symmetry between space and time as in Einstein's special relativity theory. Farmelo continues to illustrate the intense interaction between theoretical physicists and mathematics. They mutually helped each other to make progress, with Witten, Deligne, Seilberg, Penrose, Arkani-Hamed and many others as main contributors. However the Large Hydron Collider (LHC) of CERN didn't provide the many particles that were predicted by the theory, the detection of the Higgs boson in 2012 being the last success.</p>
<p>However the state of affairs have brought mathematicians and theoretical physicists closer together than ever before. They are collaborating in the new emerging field of mathematical physics. In the last chapter Farmelo concludes his arguments defending what he has illustrated in this book: it is predestined and the fate of mathematics and physics to work together. He even makes some predictions about ideas that will stand the test of time like space and time are not fundamental but are aspects of a more fundamental concept. He also believes that supersymmetry will be verified experimentally thereby affirming the beauty of mathematics to be basic. And he has a few more like those. So with this book he contradicts Sabine Hossenfelder who in her book <a target="_blank" href="/review/lost-math-how-beauty-leads-physics-astray">Lost in Math</a> complains about the state of affairs that physics is at a dead end caught up in theoretical imaginations remote from reality and just because this vision happens to be mainstream, it absorbs all the research money. Who is right? Time should bring the answer, but in my opinion it doesn't look like it will come in the very near future, despite what mathematicians and physicists may think or hope for.</p>
<p>It is remarkable that this book, that is from the first till the last page about mathematics and mathematical physics, has no formulas (well almost none, I counted five very simple ones and that includes E = mc²). Farmelo does not go into technical details in the sense that he avoids confusing the reader with technicalities. If that reader does not know the exact meaning of the terms (gauge theory, quark, gluon,...) then it does not really harm the basic story that he wants to tell and it does not hinder reading on. He keeps the reader hooked. This alone is a tour de force. Farmelo's style is very entertaining, describing the moments and the circumstances when it was realized that some breakthrough had been found. This is only possible because he interviewed the people involved or he himself was a witness of the events in the second half of his book. He also frames the time and the setting by referring for example to the fact that some physics event took place "in the year that the Beatles produced their first LP" or "when nearby a large group of music lovers flocked together" (referring to Woodstock), or "a few weeks after Obama was inaugurated". He discusses throughout the book the, sometimes subtle, interplay between the mathematics and the physics, and he is as generous about mathematics as he is about physics. I also appreciated how he analyses the important lectures of Dyson, Witten, and others where they made some important statements about the state of affairs. He claims that it is the book he has been writing since his childhood, and I can believe that. A recommended read, and Hossenfelder can be a comparable complementary read to keep the balance.</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 without being technical, the close collaboration between mathematics and physics in the course of history. In particular the mutual influence of mathematics and theoretical physics since the 1970's till now.</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/graham-farmelo" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Graham Farmelo</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/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic books</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">2019</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">978-0465056651 (hbk), 9781541673922 (ebk) </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">£ 24.00 (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">336</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="https://www.basicbooks.com/titles/graham-farmelo/the-universe-speaks-in-numbers/9781541673922/" title="Link to web page">https://www.basicbooks.com/titles/graham-farmelo/the-universe-speaks-in-numbers/9781541673922/</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/81-quantum-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81 Quantum 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/81-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-03</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/81-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-03</a></li></ul></span>Mon, 03 Jun 2019 08:43:09 +0000Adhemar Bultheel49420 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/universe-speaks-numbers#commentsLost in Math: How Beauty Leads Physics Astray
https://euro-math-soc.eu/review/lost-math-how-beauty-leads-physics-astray
<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 title of the book says it all, but it may need some explanation. Physicists have learned by experience that mathematics is very effective in describing the physical laws. It has also turned out to be effective when a simpler explanation is preferred over a more complicated one. Simpler means fewer equations, fewer assumptions, and fewer parameters. These simplifications are often the result of symmetry. So physicists came to accept symmetry as a synonym for beauty. This was a success when in astronomy the paradigm shifted from a geocentric to a heliocentric system, and it was useful in particle physics to unify the theory of different forces as in the standard model. But physicists should be warned by the disaster of the models designed and loved by economists for their mathematical beauty but that dramatically crashed in the 1980s with many years of global recession as a consequence.</p>
<p>
To arrive at a Grand Unified Theory (GUT) and design the mathematics for quantum gravity, physicists, intoxicated by this concept of symmetry, could only think of going one rung further up the ladder, and blindly proposed supersymmetry (susy for the intimi) as the logical next step. And mathematically, it is indeed a beautiful theory: a supergroup that incorporates all the symmetries of the underlying laws as subgroups. However mathematics is not physics, and physics requires experimental confirmation. Unfortunately the physicists are now erring in the dark since the Large Hadron Collator (LHC) did not reveal all the expected particles predicted by susy. The Higgs boson observed in 2012 was the last success. But the Higgs is at a scale that is so far off the scale of the other particles that it conflicts with another paradigm of physics: <em>naturalness</em>. This means that a dimensionless constant should at approximately the same scale as the other parameters. If it is not, then it might be a statistically irrelevant outlier. However if statistics shows that it is an important parameter of the theory, then it needs to be shifted or renormalized, which is called <em>fine tuning</em>. The cosmological constant is an example of fine tuning and it is ugly and still a source of much debate. Einstein was the first to introduce the cosmological constant Λ but he did not like it. It was reintroduced to explain the expansion of the universe and dark energy in the ΛCDM (Cold Dark Matter) model, but dark energy is poorly understood.</p>
<p>
So the question that Hossenfelder as a theoretical physicist asks herself is whether physicists are blinded by this mathematical symmetry principle of beauty and naturalness and do not realize that they are barking up a dead end. Mathematics should not lead the way for physics. It should be the other way around: the appropriate mathematics should be derived from the physics. In the past this ideal of symmetry and naturalness were not the lead. Symmetry was observed a posteriori when the paradigm shift was made. Also naturalness has not always been avoided. The parallax of the stars that could explain the heliocentric system, was way off the scale of distances in our solar system, and so are the distances between stars and between galaxies as compared to our solar system.</p>
<p>
She sets on a mission to ask many of the specialists in the field about their opinion. Skyping and travelling all over the world to interview the leading scientists and everyone who might have an opinion on this matter, trying to convince them (and herself) that physicists have driven research into a cul de sac. This book is a report of her crusade searching for answers. It is clearly her conviction that there is something rotten in the state of physics.</p>
<p>
To explain the problem and to understand the arguments given, she also has to discuss the terms and claims of particle physics, cosmology, etc. So at the same time, this book is a very readable popular science book on the subject. Of course there are no technical details, but for the layman, just enough insight into the concepts and problems are given to know what the discussion is about. All this information is nicely interwoven with her travelling experiences, and the interviews. These interviews are related in a very lively and personal way, often in the form of dialogues that she reconstructed from her recordings. So the reader is painlessly introduced to a broad spectrum of concepts. Of course supersymmetry, but also the standard model in particle physics and the concordance model in cosmology, multiverses, strings and branes, dark matter and WIMP, vortex theory, QBism, symmetry and Lie groups, cosmic background radiation, a simplified particle zoo, and many more. But only just enough info to understand the context.</p>
<p>
The interviews are also very personal. These "big shots", sometimes very busy running a research group, sometimes old retired emeriti, they are only humans, yet convinced of their opinion, defending the foundations on which they built their career. The place and circumstances in which the interview takes place, the hesitation or silence in the discussion, that sometimes drifts off to a philosophical discourse, it all contributes to an entertaining story. And there are many she has interviewed. From Nobel Prize winners Frank Wilczek, Murray Gell-Mann, and Steven Weinberg, to a wind-surfing non-academic Garrett Lisi, enjoying life and research on Maui, Hawaii, who proposed a so far not very successful Theory of Everything based on the exceptional E8 Lie group, and there are about a dozen more people that are staged.</p>
<p>
The frustration of Hossenfelder is that theoretical physicists just go on chasing after mathematical results following these assumptions of beauty and naturalness, and they do not even care about experimental verification, which technically means that they leave science behind. Moreover, they live in a closed, isolated environment, publishing in journals refereed by peers that have the same opinions. The pressure of publishing, getting finances, being accepted by peers, all means that young researchers have to conform themselves to these mainstream ideas. In 2016 the LHC detected something that could not be explained by the standard model. A few months later hundreds of papers had appeared in refereed journals about this so-called diphoton anomaly, when it was announced that the observed bump should be disregarded since it could be explained as noise. This illustrates that theoretical physics can invent explanations for whatever data are presented but so far, no experimental data occurred to confirm susy. Susy is called beautiful, but the concept of beauty can change. Perhaps there are physical laws that are beautiful in an unfamiliar way. Clearly Hossenfelder is throwing a bat in the henhouse. Her message is not very welcome in the community. Will she remain a voice calling in the desert? As a mathematician, (not involved in theoretical physics), I would like the math to triumph, but I think Hossenfelder has some good arguments, and in the past paradigm shifts away from what was considered to be perfect and beautiful have been rewarding. For example it was difficult to abandon the perfectly beautiful circular motion of the planets on epicircles in a geocentric system. However, even though the circles had to be replaced by ellipses in a heliocentric system, it only resulted in an even more general and beautiful mathematical theory. Thus there is no doubt that mathematics will prevail, but nature should be the guide and not the other way around.</p>
<p>
I loved reading this bird's-eye vision of the state of confusion and hope against all odds that theoretical physics is in today. It is written in an entertaining and convincing, yet very human way, showing that science is only produced by people. Perhaps it is worthwhile that scientists take a step back from the rat race of producing papers and that they reflect on what they are doing and recall what the ultimate goal of their science is. Reading this book, can be a good start.</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>
In this book Hossenfelder complains about the current state of theoretical physics. She shows that currently the mathematical beauty of supersymmetry is the guideline for researchers while so far none of it has been experimentally verified. Her plea is to invert the engine and let nature drive theoretical research instead of unverifiable mathematical assumptions. She tells her story by reporting on interviews that she had with peers all over the world.</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/sabine-hossenfelder" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Sabine Hossenfelder</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/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic books</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">2018</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">978-0-4650-9425-7 (hbk), 978-0-4650-9426-4 (ebk)</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">USD 30.00 (hbk); USD 17.99 (ebk)</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">304</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://www.basicbooks.com/titles/sabine-hossenfelder/lost-in-math/9780465094257/" title="Link to web page">https://www.basicbooks.com/titles/sabine-hossenfelder/lost-in-math/9780465094257/</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/81-quantum-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81 Quantum 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/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li></ul></span>Tue, 05 Feb 2019 08:26:43 +0000Adhemar Bultheel49081 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/lost-math-how-beauty-leads-physics-astray#commentsWeird maths. At the edge of infinity and beyond
https://euro-math-soc.eu/review/weird-maths-edge-infinity-and-beyond
<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>
This book is a joint venture of an experienced science writer (Darling) and an exceptionally bright young math student (Banerjee). The result is this book popularising mathematics by presenting a set of curious/interesting/surprising (I would not call them weird) mathematical facts in such a way that they are easily accessible for the layman. The topics they cover are close to the topics that are also discussed in other books written with the same objective. There is of course always a new approach and there is always something to learn from a surprising fact or an unexpected link that is made. The authors have adopted the rule that also Hawking used: avoiding mathematical formulas in a popular science book. Each formula would presumably halve the number of readers. Besides symbols like ℵ and ω (when discussing infinities) and notations like 3↑↑3↑↑3 (when discussing large numbers) there are only very few equations or formulas. The authors state in the preface: If we can not explain it in plain language, then we don't properly understand it. This does not mean that hard topics are avoided since there is quantum theory, cosmology, and physics, as well as the foundations of mathematics with for example Gödel's theorems. According to the preface, Darling is responsible for the philosophical and anecdotal aspects, the relation to music, and he polished everything into the final text. Banerjee was more involved with the technical aspects including large numbers, computation, and prime numbers. The result is a pleasant read that anyone with only a remote interested in mathematics will enjoy.</p>
<p>
The breadth of the topics covered is too wide to enumerate them all, but to give a rough idea, what follows is a fistful of topics that are discussed.</p>
<p>
Historically mathematics is of course inspired by the necessity to count and by our surrounding physical world and the stars up above. But how would a four-dimensional being see our three-dimensional world? For us hard to imagine, but mathematics has no problem to function in higher dimensions.</p>
<p>
With probability one can simply explain the birthday paradox, but it is also essential in quantum theory which is hard to understand, certainly when it eventually leads to vibrating strings in an attempt to construct a theory of everything. In chaotic systems such as the weather, the smallest perturbations, in spite of all the laws of probability, may prevent any valid prediction. On the other hand probabilistic systems can obey simple rules like in Brownian motion, or it may generate complex structures such as fractals. Think of automata like the Turing machine or Conway's Game of Life. If we can model a system, this does not mean that it is practically computable because one can hit the boundaries of complexity like problems of class NP.</p>
<p>
Music and prime numbers are classical topics for books like this one. With the title "music of the spheres", there is a reference to Kepler of course but the story of that chapter also meanders by mentioning the music disk sent into space in the SEFI project as well as singing whales. Obviously there is an obligatory extensive discussion of the mathematics of music. The next chapter is discussing the unavoidable prime numbers. We meet for example the 17 year cycle of cicadas, the Ulam spiral, the Riemann hypothesis, and the twin prime gap.</p>
<p>
Two more classical recreational topics are game theory and logical paradoxes. Game theory is discussed in connection with computers playing chess against humans and later also the more complex game go, but game theory can of course be applied to other games as well. One may for example investigate whether winning strategies exist when the human player can start? Game theory may have been developed to help people win an entertaining game, but when it was applied to very real economics, politics and other modelling and optimization problems it became a serious mathematical subject and John Nash won even the Nobel Prize in economics and the Abel Prize with his results. Paradoxes, the foundations of mathematics, and logic get their separate chapter. They may be applied to entertaining logical puzzles, but when digging a bit deeper, one bumps into much harder problems and the chapter ends by mentioning surprising mathematical results such as for example the Banach-Tarski theorem (a solid ball can be cut up into 5 pieces such that these can be reassembled into two solid balls of the same size as the original).</p>
<p>
With large numbers and transfinite numbers we are back in the realm of numbers and mathematics. Infinity and the orders of infinity are discussed including the continuity hypothesis and the existence or not of the absolute infinity Ω. Big numbers (the really really big ones) like googol, googolplex, power towers (as introduced by Knuth), Graham's number, TREE(3), and several other numbers are featuring in a chapter that is less common in popular math books. Yet there is a subculture of googologists challenging each other in competitions to define ever larger (finite) numbers.</p>
<p>
The last two chapters dive somewhat deeper into the less elementary mathematical topics. Of a more geometric nature is a chapter on topology with objects like the Moebius band, the Klein bottle, and different kinds of geometry and how this is applied to our universe. Finally we arrive at fundamentals discussing the completeness of the mathematical system, Gödel's theorems, proof theory, the axiom of choice, the Peano calculus, ZFC axioms, etc. These are clearly among the more advanced topics discussed in this last chapter.</p>
<p>
As can be seen from this (largely incomplete) enumeration of topics, the discussion is rather broad and sometimes also touching upon deeper theoretical problems. The text remains however very readable, even when more advanced problems are carefully dissected. This is a very nice addition to the popular math literature deserving a warm recommendation.</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 to popularize mathematics written by a science writer (Darling) and a bright young math student (Banerjee). The authors cover many of the topics that are traditionally covered by this kind of books, but some surprising connections are made. For example the topic of transfinite numbers is a classic but the googology chapter on large (but finite) numbers is not so common.</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/david-darling" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Darling</a></li><li class="vocabulary-links field-item odd"><a href="/author/agnijo-banerjee" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Agnijo Banerjee</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/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic books</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">2018</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">9781541644786 (hbk), 9781541644793 (ebk)</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">USD 27.00 (hbk), USD 16.99 (ebk)</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">320</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/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</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://www.basicbooks.com/titles/david-darling/weird-math/9781541644786/" title="Link to web page">https://www.basicbooks.com/titles/david-darling/weird-math/9781541644786/</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</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/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span>Tue, 05 Feb 2019 08:21:18 +0000Adhemar Bultheel49080 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/weird-maths-edge-infinity-and-beyond#commentsFluke. The Math and Myth of Coincidence
https://euro-math-soc.eu/review/fluke-math-and-myth-coincidence
<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>
Coincidence, fluke, and serendipity are closely related concepts that refer to something unexpected. Coincidence is surprising because there is no obvious apparent cause. Fluke means something accidentally happening with an unintended consequence and serendipity is used for a fortunate result that is obtained while striving for another objective.</p>
<p>
Anyway, such events always amaze and astound people. Like you meet a neighbor when on holiday at the other side of the world, or you think of an old friend you haven't seen in many years and he calls in the next hour. But are such experiences really unexpected? Are they exceptional? Is there perhaps a hidden cause? Can we compute the probability of a coincidence? Such are the questions that are discussed in this book. The myth and math of coincidence. Although there is a very light introduction to probability, and some probabilistic analysis of a few cases, the book is perhaps more a collection of some recognizable or legendary examples of coincidences with reflections on what is actually so coincidental about them.</p>
<p>
The book opens with ten examples of such stories, like for example you find a book in a second hand book store that you were looking for and it happens to be the copy you had when you were young, or when you take a taxi in two different cities that happens to have the same driver, and it is known that Abraham Lincoln dreamt about being assassinated in the weeks before he was actually killed. However, although it may be the synchronicity of time and place that amazes us, there can be other connections or perhaps unconscious associations that can explain the coincidence and considerably reduce its spooky character.</p>
<p>
Time to introduce what probability of some event means and what the odds are of something happening. Some historical notes on the origin of probability are given starting with Cardano's <em>Liber ludo aleae</em>, and later contributions by Pascal and Fermat. The Galton board is used to explain the bell shape of the Gaussian distribution. It was Jacob Bernoulli's book <em>Ars conjectandi</em> that can be considered the first book on probability and in which the famous law of large numbers was described for the first time. A not-very-mathematical introduction is then given to explain expected value, mean, distribution, standard deviation, etc. Some examples are given about the probability of winning with poker, or the chance that a monkey arbitrarily hitting the keyboard will produce a Shakespearean sentence, or (this is a classic) how large the group should be to make the odds larger than than even that two people in the group have the same birthday. It is also made clear that it is important to realize what the sample space is. The 'enormity of the world' should illustrate how many times the same event happens simultaneously world wide so that the probability of a coincidence becomes much more probable than originally thought. Nonlinear dynamics and the second law of thermodynamics illustrate the seemingly unrelated and unexpected effects a simple modification can produce while the unknown relation between cause and effect could be relatively simple. One should also take some hidden variables into account, i.e., parameters that can couple the two coinciding events but that are not always obvious. If two people incidentally meet at some place and time, this will depend on all the systems these people are part of: social, familial, biological, environmental, political, etc. and all of these can have highly nonlinear chaotic influence on their meeting. Anyway, Mazur applies and simplifies some of this knowledge needed to analyze how probable the 10 cases, that were introduced at the beginning of the book, can actually be. Or rather Mazur makes several assumptions that allow him to justify whether such an event is 'rather probable' and others are 'extremely improbable'.</p>
<p>
The previously described contents is making up the first three parts of the book: the ten cases, the mathematics needed, and the analysis of the cases. A rather extensive fourth part is called 'the head-scratchers'. Here probability and analysis are not the main topic anymore. Diverse topics are discussed. It is first explained what a DNA analysis in a criminal investigation really means and it turns out that DNA identification is not as irrefutable as the media want us to believe and innocent people may have been condemned or imprisoned based on disputable evidence. I think Mazur makes a strong point here.<br />
Some scientific discoveries were accidental. Famous are the discovery of penicillin, of X-ray fluorescence, and quinine as an antimalarial, but also Turing and his team decrypting the German Enigma code did have some luck. We learn along the way the origin of the famous dictum, usually attributed to Newton: `If I have seen further it is by standing on ye shoulders of Giants'. It has been used by him, but he was not the first.<br />
Then there is the factor risk that has to be dealt with in many situations. Certainly in gambling but also on the financial markets (which is about the same as gambling), but there are also risks of disasters like earthquakes that are still highly unpredictable.<br />
Under the title 'Psychic power' a discussion is given about ESP. There are obviously many frauds, but Mazur seems not to completely exclude the possibility of some 'action at a distance' because of some electromagnetic or other waves we may perhaps not fully understand yet.<br />
Another chapter discusses some coincidents in the literature. Two examples are discussed in detail: There is the Middle English poem of <em>Sir Gawain and the green knight</em>, one of the best known stories of the Arthurian legends, and the story of <em>The three princes of Serendipity</em>. The latter Indian story also dates from a 14th century and is brought to Italy in the 16th century via the Persians. It is actually the origin of the word serendipity in English. Fictional coincidences may be the result of subconscious associations made by the author while writing.</p>
<p>
To summarize, the mathematics of this book are low level, so the 'math' in the title should not frighten any potential reader. On the other hand there is not a strict systematic analysis that can tell you that such and such surprising event could happen with such exact probability. There are too many factors and unknowns, perhaps hidden relations, that make an event happen. There are of course all the laws of physics that guarantee that in principle every event is completely defined by its past so one could even bring a discussion about free will into the argumentation. Fact is that we do not completely understand some coincidences, or certainly we do not know enough to analyze them with certainty via a mathematical methodology. So, if you are a mathematically interested reader, attracted by the 'math' in the title, the content may be a bit disappointing. Strict analysis is not that simple, except for constructed examples. But isn't that exactly why we find these coincidences so fascinating and that is thus probably why one will want to read this book. Mazur just provides elements that will start you thinking about there phenomena. Some may become less freaky, others we may perhaps never understand completely. Nevertheless it will help you distinguish bogus science like spiritism and the likes of it from what is scientifically possible.</p>
<p>
Joseph Mazur is an emeritus professor of mathematics who has written several books on popular mathematics in the last decade. See for example the review of <a href="/review/enlightening-symbols-short-history-mathematical-notation-and-its-hidden-powers"><em>Enlightening symbols</em></a> in this database. He should not be confused with Barry Mazur, another prolific writer of popular math books.</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>
Jozeph Mazur discusses several forms of coincidences and the possibility to analyze the probability that they happen. For some it is possible with a set of assumptions to give at least a qualitative idea, for other 'head-scratcher' cases it is impossible for several reasons. Some light notions of probability are introduced so that no preliminary mathematical knowledge is required. In this book it may not be the coincidences that surprise you but perhaps the results of Mazur's analysis of the events. <br />
</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/joseph-mazur" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Joseph Mazur</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/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic books</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">978-0-465-06095-5 (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">28.88 USD</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">288</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/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</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.perseusacademic.com/book/hardcover/fluke/9780465060955" title="Link to web page">http://www.perseusacademic.com/book/hardcover/fluke/9780465060955</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97k50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97K50</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Sat, 30 Jul 2016 05:25:06 +0000Adhemar Bultheel47085 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/fluke-math-and-myth-coincidence#commentsThe Magic of Math. Solving for x and Figuring Out Why
https://euro-math-soc.eu/review/magic-math-solving-x-and-figuring-out-why
<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>
Arthur Benjamin is a mathematics professor at Harvey Mudd College in Claremont, CA. He has made it part of his mission to bring mathematics to the general public. He performs before audiences and gave TED talks. He also wrote a book on how to perform better in mental calculation, and he recorded the video course <em>The Joy of Mathematics</em> produced by <em>The Great Courses</em>. Elements of this course are used to write this book. Like so many others, he is an admirer of Martin Gardner, and likes puzzles, magic, and mathemagic. Hence probably the title of this book.</p>
<p>
The title <em>The Magic of Math</em> and the cover picture referring to a magician will make you expect magic tricks with numbers and cards, but do not be mistaken. The message of the book is obviously 'mathematics is fun' with fun-elements omnipresent and there are indeed some suggestions to use mathematical properties (like the casting-out nines check) to surprise your audience, but it becomes in the second half of the book also very course-like in the sense that there are theorems and proofs, all at an elementary level, but still.</p>
<p>
Because everything is kept at this introductory mathematical level, there is much in the realm of numbers (meaning natural numbers) to start with, but we also get some algebra, and later geometry and calculus. To some extent, Benjamin follows the historical evolution of mathematics. He starts with numbers and geometry and deals with the properties of numbers much like the ancient Greek did using essentially geometric elements to 'prove' these properties.</p>
<p>
The first four chapters are generally dealing with numbers and a bit of algebra. There are number patterns (e.g., triangular and rectangular numbers, i.e., numbers that can be arranged in this geometric form). But there are also many patterns to be discovered in Pascal's triangle. Furthermore the reader is instructed about modulo calculus, Fibonacci numbers, and combinatorics to do all the counting. The algebra is essentially restricted to first and second order equations. There are almost no formal proofs here, but evidence is sometimes given on a geometric basis with graphical arguments.</p>
<p>
The sixth chapter introduces 'the magic of proofs' with some elementary examples like a formal proof of the property that "the product of two integers is odd if and only if both numbers are odd". The proof that the square root of 2 is irrational, and the proof that there are infinitely many prime numbers are classics.</p>
<p>
Once the reader is familiar with the rigor of a formal proof, it is time to switch to a more axiomatic environment. The most classic rigorous system is provided by Euclid's <em>The Elements</em>. So the next chapter introduces some elements of geometry and it has more formal proofs, ending in several variants for the proof of the Pythagoras theorem.</p>
<p>
Chapter eight is semi-geometry semi-calculus. It is a discussion of the number pi and how it relates to circular area and circumference but also with mnemonics to memorize its digits and a mock tribute to pi, in the form of a parody of a popular song. The number pi is the best known number that everybody knows about, so it deserves a separate chapter in a popular book on mathematics.</p>
<p>
Somewhat less popular, but mathematically equally important are the numbers e and the imaginary unit i. These are however more 'mathematical', meaning that they are further away from people's daily common experience. Therefore the subsequent chapters are more serious lecture-like dealing with trigonometry, the numbers e and i with logarithms and complex numbers, some elements of calculus such as differentiation, and finally some infinite series. The 'fun-element' returns at the end with the proof that the sum of all natural numbers equals −1/12 (an amazing paradox that you may find on the Web in several versions) and magic squares.</p>
<p>
The book is richly illustrated and it has many grey boxes, called 'asides', that give some more information, or a proof, or something extra that will appeal to the more advanced readers. These can be skipped without any harm.</p>
<p>
From the previous, it will be clear that this is a minimal introduction to the mathematics that one would get at a secondary school level. Is it the mathematics book that I would have loved to I have had then? I doubt it. I did not need that much of show element to be interested. But it might help for others who find more traditional textbooks terribly boring. If, as a teacher, you need to 'force' the math upon some unwilling student, this might be a very helpful alternative.</p>
<p>
On the other hand the market for popular science books, and that includes popular mathematics, has never been as big as it is today. So there is great interest for this kind of books. I doubt that the buyers of this kind of books are the secondary school pupils. Perhaps the main target readers are the adults who lost interest during adolescence and regret that later. So they want to catch up, but in a less scholarly way. For this kind of readers this is a marvelous read.</p>
<p>
And then there are the ones who already became mathematicians or math teachers. They will not find new mathematical elements and they do not need the motivation anymore, but they can always pick up some of the fun elements. I'm sure some of these will be new even for them. So also for them, there is a reason to enjoy 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>
This is a popularized introduction to elementary secondary school mathematics. There are many fun-elements, but also theorems and proofs. The text is readable for anyone after primary school.</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/arthur-benjamin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Arthur Benjamin</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/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic books</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">978-0-465-05472-5</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">US$26.99</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">336</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/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</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.perseusacademic.com/book/hardcover/the-magic-of-math/9780465054725" title="Link to web page">http://www.perseusacademic.com/book/hardcover/the-magic-of-math/9780465054725</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Fri, 02 Oct 2015 13:55:27 +0000Adhemar Bultheel46425 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/magic-math-solving-x-and-figuring-out-why#commentsQuantum Mechanics
https://euro-math-soc.eu/review/quantum-mechanics
<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>
<a href="http://theoreticalminimum.com/"><em>The theoretical minimum</em></a> is an online lecture series given by Leonard Susskind, professor of theoretical physics at Stanford University. The first volume in the book series with the same name treated <em>Classical Mechanics</em>, a course given in the <a href="http://theoreticalminimum.com/courses/classical-mechanics/2011/fall">fall 2011</a>, and the current volume on Quantum Mechanics resulted from the <a href="http://theoreticalminimum.com/courses/quantum-mechanics/2012/winter">winter 2012</a> lectures. The course consists of a sequence of 10 lessons, which correspond to the 10 chapters of the book. The co-author Art Friedman is computer scientist who attended the lessons. The books can be considered as the elaborated lecture notes for Susskind's courses.</p>
<p>
The name of the series and subtitle <em>what you need to know to start doing physics</em> explains the concept. This is intended for the amateur scientist who really wants to learn and understand the subject. It is Susskind's conviction that this can only be achieved if one masters the mathematics that describe the physics. In this second volume it is assumed that the reader is familiar with elements about classical mechanics explained in volume 1 of the series. Furthermore, the reader should not be afraid of the abstraction of mathematics. The ideal situation would be that the reader is familiar with the basic mathematics at the level of a bachelor university degree. On the other hand, introducing the necessary mathematics is exactly what this course is about, so in principle no mathematics of this level is assumed. But if mathematics scares you off, then this is not for you, which implies, if we believe Susskind, that you will never be able to properly understand quantum physics.</p>
<p>
</p>
<p>
Susskind obviously is addressing the reader who has some (classical) physics background. He starts from the intuitive physical description, and then illustrates that this can be perfectly described by introducing the necessary mathematical concepts. For example: what is spin? Not just an ordinary 3-vector. If you measure it there are only two possible outcomes, say +1 and -1. But before it is measured, it can be in different (coherent) states. The outcome of the measurement will depend on this state and on the "orientation" of the measuring apparatus. So the reader is taken along on an exploration tour and it takes a while (in fact 3 chapters) before a proper mathematical concept sublimates. In this case, the Pauli matrices. Given an orientation for the measuring apparatus, it will be possible to combine the Pauli matrices to a matrix and its eigenvalue decomposition will allow you to compute the probabilities of either outcome of the experiment. So the mathematics do not come gratuitously, but come to the help of the physicist who just needs it at some point.</p>
<p>
On the other hand, if you are a mathematician, then the content may somewhat disappoint you because the mathematics is relatively elementary knowledge. Since I assume that a mathematician feels comfortable with abstraction, he or she might have preferred a more axiomatic approach. For example, the above state vector is a 3D unit vector and hence has two degrees of freedom. The Pauli matrices are 2x2 unitary matrices with eigenvalues +1 and -1 and the eigenvectors are orthonormal basis vectors. Of course, the line of thought that Susskind uses, has the advantage of explaining why a certain mathematical concept is needed to catch a particular physical phenomenon. So, if the mathematical abstraction is the more challenging part for the physicist, for a mathematician it also requires an effort to detach from the oh so familiar laws of mechanics in our everyday life and to accept the sometimes paradoxical physical interpretation imposed by the mathematics of the quantum mechanical game.</p>
<p>
To summarize, this is more about mathematics for physicists than about physics for mathematicians. Whatever the approach or background, quantum mechanics remains a difficult subject because it is often counter intuitive. The reason is that humans normally observe the world they live in at a scale that is hugely different from the scale needed to describe the physics at a quantum level. As Susskind claims at some point: conceptually quantum mechanics should be the first approach to describe mechanical phenomena because that corresponds to reality and classical mechanics is a simplification that is a good approximation only at a much larger scale.</p>
<p>
However, as is usually the case, that top-down approach is not the best way to learn things. It is much better to start with a simple special case and when that is properly understood, one may step up to generalizations. However, supposing you can assimilate all the material as it is intended by the authors, you will not be at a level where you can directly involve in current research on quantum mechanics. You will be at an elementary level still, and it will take much more mathematics to reach the level of current research. There are many more advanced courses available on the courses website, so one may expect several more volumes in the <em>Theoretical Minimum</em> book series to come.</p>
<p>
Let's go quickly through the contents of the book. It takes three lessons/chapters to explain the notion of a state of a system. For this, Susskind uses the easiest example of a system of just one simple observable: the spin or a qubit. While explaining this concept, Susskind introduces on the mathematical side the complex numbers, vector spaces, orthogonal basis vectors and Gram-Schmidt orthogonalization, (Hermitian) operators and their eigenvalues, and of course the bra-ket notation that was introduced by P. Dirac.<br />
The next two chapters deal with time dependency. Here the notion of Lie bracket (the authors prefer to use commutator as an alias), Hamiltonian, Schrödinger equation, Cauchy-Schwartz, unitary operators, and the general uncertainty principle is introduced.<br />
The two following chapters are on observation and state of a combination of systems. Again, the simplest case is observing two spins. Susskind shows that the tensor product of the bases for the states of the separate systems does not form a complete basis for the states of the combined system. Hence some extra basis vectors (singlet and triplets) are needed to describe the whole state space. These vectors are responsible for entanglement. The mathematical concepts here are the outer product, density, and correlation.<br />
The remaining chapters are about particles and waves. This duality is probably the best known aspect of quantum mechanics: a photon behaves as a wave and at the same time it is like a particle. I believe the reader who has been hanging on till this point of the course will have to fasten seat belts and shift to a higher gear now. The first thing is to move from eigenvectors of a matrix to eigenfunctions of an operator, and from finite sums to integrals. Position, velocity, momentum, Hamiltonian all become operators and can best be studied in the Fourier domain, the latter resulting for example in the classical Heisenberg uncertainty principle. To know how particles move, one has to reconsider time dependency in this continuous setting. The equation of motion with kinetic and potential energy and the classical form of the Schrödinger equation, waves, wave packages and the harmonic oscillator are the results with which this course comes to an end. Several exercises are inserted where the reader is asked to prove some of the properties or to work out some formulas. Usually they are not very hard and they help to assimilate the material. In an appendix some of the basic formulas are summarized so that they can be easily looked up, for example when going through the formulas to solve the exercises. Also the subject index is handy when studying the material.</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>
This is the second volume in the <em>Theoretical Minimum</em> series to accompany a lecture series given by L. Susskind. The material covers only the basics and is intended for the layman, but a layman that is interested in understanding the mathematics needed to go beyond a general descriptive approach of quantum physics.</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/leonard-susskind" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">leonard susskind</a></li><li class="vocabulary-links field-item odd"><a href="/author/art-friedman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">art friedman</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/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic books</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">978-0-4650-3667-7 (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">£17.99</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">384</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.basicbooks.com/full-details?isbn=9780465036677" title="Link to web page">http://www.basicbooks.com/full-details?isbn=9780465036677</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/81-quantum-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81 Quantum 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/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li></ul></span>Tue, 06 May 2014 06:05:35 +0000Adhemar Bultheel45560 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/quantum-mechanics#commentsLove and Math. The heart of hidden reality
https://euro-math-soc.eu/review/love-and-math-heart-hidden-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>
Ever since he grew up as a boy in Kolomna (Russia), Frenkel was fascinated by elementary particles and quantum physics. It was pointed out to him that to understand these, he should start learning mathematics. So he started reading mathematics in his free time. An obvious choice would be to study at the department of Mechanics and Mathematics (<em>Mekh-Mat</em>) of the Moscow State University (MGU). However, back in 1984, his father being Jewish, this was impossible by the ruling anti-Semitism. His second choice was the Institute of Oil and Gas (<em>Kerosinka</em>), but he sneaked into the GMU to attend some courses and seminars by Gelfand. On the side he worked on a problem of braid groups proposed by D. Fuchs which resulted in his first paper published in <em>Funct. Anal. Appl.</em> at the age of 20. This brought him to study symmetry, (braid) groups and curves over finite fields. Further work brought him straight to the <em>Langlands Program</em> that was proposed by <em>Robert Langland</em> in 1967 and more formally in 1970. It is based on an earlier idea of <em>André Weil</em> who, while imprisoned in 1940 (having a disagreement with the French authorities), wrote a letter to his sister explaining the idea of a mathematical <em>Rosetta Stone</em> which would allow to translate results between three seemingly different fields in mathematics into each other: number theory, curves over finite fields, and Riemann surfaces. Exploring this connection has been shown successful by the proof of Fermat's Last Theorem. This connection is the mathematical analog to what the theoretical physicist call the <em>Grand Unifying Theory</em> in their study of quantum physics. The mathematical or physical aspects are just two different interpretation of the same theory. So quantum physics is like a fourth column to be added to Weil's <em>Rosetta Stone</em>. Frenkel's work with B. Feigin on Kac-Moody algebras came just in time because he got an invitation to spend a semester at Harvard in 1989 at the very time that <em>perestroika</em> was emerging. Because of the worsening situation in Russia with an unclear outcome, he decided after his 3 months stay, that it was better not to return home. So he stayed at Harvard where he got his PhD in 1991. Later he became professor of mathematics at UC Berkley. In 2003 he got directly involved in a multi-million DARPA grant to work out more elements of Weil's Rosetta Stone. Since then, his mathematical career is largely devoted to building the bits and pieces of this <em>Grand Unifying Theory</em>.</p>
<p>
Frenkel makes it crystal clear that he is a passionate lover of mathematics and that his enthusiasm for the <em>Langlands Program</em> is immense. This love and passion is what he wants to convey to the reader. The math that most people learn in school is like learning to paint a fence in an art class, while true painting is about creating master pieces like Da Vinci or Picasso did. Mathematics is also a moral duty. Our world is ruled by mathematics that are hidden to most of us. The financial crisis in 2008 was caused by applying mathematics by people that were not controlled in a democratic way because our society does not care about mathematics and most people tend to stay away from it as far as possible. Mathematics should not be restricted to the "initiated few" but it should be shared by everybody. There is nothing more democratic than mathematics. There are no patents for formulas, its a universal language, and a correct formula can only represent truth, the universal truth.</p>
<p>
With this conviction, Frenkel wants to transfer not only his love for mathematics but he also wants to show us the beauty of the mathematics that he is devoting most of his life to, and not just the "fence painting" bits. Of course reading this book will not make you a mathematician, but he succeeds by describing his life (at least the part related to his mathematical career) and gradually taking the reader along in his conquest of the mathematics he needed. So he explains symmetries, groups, finite fields, SU(3), manifolds, Galois groups, Lie algebras, sheaves, supersymmetry, strings, branes, etc. All things that are far beyond the low-fi kind of math that one usually finds in popular science books. Of course this is not easy, but I can imagine that his charismatic account will make some readers regret that they are not mathematicians, rather than the usual conviction that mathematics is a natural habitat where only nerds can survive. Many of the more technical details are removed from the main body as (sometimes quite extensive) notes that are collected at the end of the book. For a mathematical reader, they are of course useful, but others may want to skip them and still follow the essence of Frenkel's Conquest of Paradise.</p>
<p>
But Frenkel is not only a mathematician. The last chapter of the book is still about mathematics and love, but now revealing the artistic talents of Frenkel. After a visit to Paris, he got the idea to make a film about math. With his neighbour, the author T. Farber, he wrote a screen-play called <em>The two-body problem</em> about two men in the South of France, one is a writer, the other a mathematician. They exchange their experiences, their passion for their profession and for women. It was published as a book in 2010. Before starting on the movie project, he wanted to get some cinematographic experience at a smaller scale and decided to produce a short movie. During another visit to France, he joined in with Reine Graves, a young film director. Inspired by a Japanese film of Y. Mishima <em>Rites of Love and Death</em> in which a lieutenant commits a ritual suicide together with his wife. Frenkel and Graves imitate the movie more or less. It shows a man (Frenkel) and a women (K.I. May) with in the back a poster with the text istina (Russian for truth). The man tattoos a mathematical formula (the formula of love) on the body of the women. The film is called <a href="http://ritesofloveandmath.com/"><em>Rites of Love and Math</em></a>. It was well received, and you will find pictures on the Web of Frenkel teaching in Berkeley, but also where he shows up at the Cannes film festival. In fact by different media, Frenkel tries to transmit the same message: a mathematical formula or mathematics in general can be a thrilling thing of beauty, it can give you goose bumps, one may fall in love with it, it represents the ultimate truth, and it is worth committing your life to. The return you get from it is overwhelming.</p>
<p>
One final remark. It is of course a side remark after Frenkel's plea for beauty, but I do not think that the cover design of the book is a success. It shows text in slightly tilted rectangles on a background image that is a detail of Van Gogh's <em>The Starry Night</em> painting. The symbolism is obviously well chosen, but it looks terribly chaotic, and I would have preferred a more stylish design representing the mathematical purity and beauty of its contents.</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>
Partly a story of his life, partly an introduction to the essence of his work: the Langlands Program. Frenkel displays his enthusiasm, and love for mathematics and in particular for this "Grand Unifying Theory" of mathematics and quantum physics by taking the reader along on this journey from his first contact with SU(3) to what he now is, a leading mathematician at the forefront of this exciting development in mathematics and quantum physics.</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/edward-frenkel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">edward frenkel</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/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic books</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">2013</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">978-0-465-05074-1 (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">27,99 USD (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">304</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.basicbooks.com/full-details?isbn=9780465050741" title="Link to web page">http://www.basicbooks.com/full-details?isbn=9780465050741</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/81-quantum-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81 Quantum 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/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li></ul></span>Wed, 12 Feb 2014 06:06:55 +0000Adhemar Bultheel45553 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/love-and-math-heart-hidden-reality#comments