Book reviews
http://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenThe Art of Logic
http://euro-math-soc.eu/review/art-logic
<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 is a book about applied logic. Books that popularize mathematics (and logic) often have chapters about paradoxes, or logic puzzles. If you were expecting something in that style, then it will soon become clear that you are mistaken. Eugenia Cheng who combined in her previous book <em>Cakes, Custard + Category Theory</em> her profession as a mathematician specialised in category theory and her hobby of cooking. In <em>Beyond Infinity</em> she discussed the infinitely small and the infinitely large, which is also rather mathematical, but it can also lead to paradoxes such as the Hilbert Hotel.</p>
<p>
In the current books she discusses the roots that govern all mathematics: the rules of logic and axioms that lay at the origin. Close as this may be to the heart of a mathematician, she considers here however how logic also applies to our daily life, although in a much fuzzier way and lacking the mathematical formalism. As a consequence misunderstandings will occur. These will result in endless and unsolvable discussions, because the opponents apply different rules or different axioms and so both claim to be correct in coming to opposing conclusions.</p>
<p>
Cheng applies this to unravel some of the currently hot topics that roam the media like (political) discussions about health care, or racism and sexism. She clearly explains why the different parties can not come to an agreement. People come to their own version of "The Truth" by making logical mistakes. For example a negation is mistaken for a contraposition, or they use a false premise which logically allows to imply anything, or people apply the rules to different classes of subjects, etc.</p>
<p>
To be able to point out where things go wrong in many practical situations, Cheng of course needs to explain some rules and terms form logic that are much more clearly defined in a mathematical situation. Mathematicians will agree on what is true or correct because they are arguing within a much more abstract and unambiguous universe, using generally accepted rules, even though they need not make all the details of their logical deduction steps explicit. As long as their peers will be able to see how the gaps need to be filled, they will accept the result. Only if the gaps are too large, a referee will require more details.</p>
<p>
So the first part of this book is explaining what logic actually is and how it is experienced every day by anyone. Using many examples from social discussions, political disagreement, or just parent-kid discussions, Chen introduces the different terms, using some necessary abstraction, to talk properly about the terms that are used in more formal proposition logic, including quantifiers, Venn diagrams, truth tables, negation of implications, equivalence, but also fuzzy logic (the world has many "grey zones"),... When there is a discussion about who is to blame for some unfortunate event, then one should first see how it is possible that the event came about, and those previous events are caused in turn by some other events, etc. So there can be a very complex network of causal coincidences that have eventually led to the event that is the subject of the dispute (Chen uses all these relations as a pretext to smuggle in some of her beloved category theory).</p>
<p>
In a second part she explains the limits of logic. In practice there is no peer reviewing process of some person's argument like in a mathematical environment of publishing a paper. Which mechanisms (correct or inappropriate) are used to convince people? Perhaps (Internet) memes are assumed correct while they are not. When one comes to paradoxes, some alarm should go off to revise the system applied. In other situations, logic will not be useful like in emergencies, or when we do not have all the information to act logically, in which cases we may perhaps just follow a reflex, a gut feeling, or trust the judgement of others.</p>
<p>
The third and last part is called beyond logic. This is where one should agree on axioms, the things that are accepted without a (logic) proof. Then there are of course the many grey zones where binary logic is not the proper tool to use. What universe is one talking about (all humans, all men, all women, all white women, all rich and white women,...?) Things may be considered equivalent (the same) for somebody, but not for the adversary. And then there are of coarse emotions that are important factors in everything we do or say.</p>
<p>
In this book Chen is strongly engaged in social justice, minority groups, gurus, religion, climate issues, the role of science, etc. So in her last chapter she somehow summarizes how logic can help you to be a reasonable and intelligent person. There should be some framework that one believes in, and one should be sceptical towards charismatic "superstars". You should realize that there are a lot of grey zones and that you are not alone so that reaching a joint objective can be more rewarding than reaching your own. Correct and reasonable logical arguments should be used, even in a world that is not always logical.</p>
<p>
This is an engaging book, that should be read by everyone. It will help solving disagreements, or direct discussions away from and "it is - it isn't" arguments, and help you focus on the underlying cause of the dispute. Of course real life is not mathematics, boundaries are fuzzy, and obviously, it can not prevent that people disagree, and they should if for the proper reasons and when using the correct arguments, and this is the main message 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>
In this book, Cheng illustrates how by using logic, one can become a better, reasonable, and intelligent human. She describes the possibilities, the limitations, and the pitfalls of logic when it is applied beyond the abstract context of mathematics. Can it define what is right or wrong or help to resolve a deadlock in political or social discussions about subjects such as solidarity principles, climate issues, racism, or sexism?</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/eugenia-cheng" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eugenia Cheng</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/profile-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">profile 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">978-1-78816038-4 (hbk); 978-1-78283442-7 (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">£14.99 (hbk); £12.99 (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><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://profilebooks.com/the-art-of-logic-hb.html" title="Link to web page">https://profilebooks.com/the-art-of-logic-hb.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/03-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03-01</a></li>
<li class="field-item odd"><a href="/msc-full/97a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97a40</a></li>
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Thu, 03 Jan 2019 08:09:55 +0000adhemar48975 at http://euro-math-soc.euAlice and Bob Meet the Wall of Fire
http://euro-math-soc.eu/review/alice-and-bob-meet-wall-fire
<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>
Thomas Lin was science editor for the online <em>New York Times</em> before he decided in 2012 to become the founder and editor in chief of the online <a href="https://www.quantamagazine.org" target="_blank"><em>Quanta Magazine</em></a>. By now it has become a highly appreciated freely available online source for journalists to find information about (hard) science: physics, biology, computer science, and mathematics. Quanta is sponsored by the <em>Simons Foundation</em>, an organization for the promotion of science created by James Simons, the billionaire mathematician and hedge-fund founder. The editorial policy is to cover in about five to ten pages the cutting edge topics that fall somewhat outside the interest of the mainstream media. The texts are engaging newspaper style stories that are rigorous without being really technical. It should catch the attention of any science minded reader. The authors are mostly reporters that have talked to or interviewed the scientists. Only in very exceptional cases, it is written by the scientists themselves.</p>
<p>
After the first five years of Quanta's existence, Lin has made a "Best of"-selection of the texts in two volumes. One is entitled <em>The prime number conspiracy</em> and collects papers dealing with mathematical subjects. That involves obviously prime numbers, but is not restricted to number theory. The remaining subjects (physics, computer science, and biology) is covered in the present volume. The texts are only slightly edited to add the latest news. Clearly some are older while others are more recent. By reading subsequent texts on the same or a closely related topic it is seen how science advances.</p>
<p>
The present collection consists of 38 texts, grouped in eight parts. Clearly it is not possible to cover each of the subjects here in detail, but the titles of the eight parts can give an idea of what is covered. Note that they are all formulated as questions, which reflects that they relate to some of the "big questions" that humans naturally ask and that scientists have been trying to solve, often replacing them by new, even more challenging ones. What follows is a selective survey.</p>
<ol><li>
<em>Why doesn't our universe make sense?</em><br />
This is all about cosmology, space-time, multiverse collision, etc. It contains the article that delivered the book's title. Alice and Bob are the usual persons used in thought experiments. The wall of fire is how an outside observer would see the event horizon of a black hole, if Hawking radiation is accepted, but there are still paradoxes connected to black holes that could not be solved yet.</li>
<li>
<em>What is quantum reality, really?</em><br />
Although quantum theory was conceived about a century ago, it is still not completely understood. It is very real as confirmed by experiments over and over again. So there are still attempts to provide new explanations or old ones are revived. For example various multiverse concepts have their believers and non-believers based on different arguments. We can read about the amplituhedron, a geometrical object that should simplify the quantum field theoretical computations, a considerable improvement over Feynman diagrams. Noteworthy is also a text by Nobel Prize winner Frank Wilczek about quantum entanglement (he also had a contribution in the previous part about Feynman diagrams). He is one of the three authoring scientists in this collection (Robbert Dijkgraaf, director of the IAS in Princetion is another exception, with a contribution in the last part).</li>
<li>
<em>What is time?</em><br />
Time is in many aspects an "outsider" in physical quantities. Physicists have developed several theories about what it is and what is causing it. It is intimately related to an increase of entropy described by the second law of thermodynamics. Entropy is a measure of information. It quantifies the amount of uncertainty, and hence directly links to quantum theory. The preferred laboratory to investigate time (and other quantum physical effects) in extreme circumstances are black holes. Mathematically, time just stops at the singularity of a black hole like it popped into existence with the Big Bang. Quantum entanglement comes into the picture because entanglement happens in space-time, and hence there can also be this "spooky action" at a distance in time which makes causality questionable, but it may explain the evaporation of black holes that Hawking predicted.</li>
<li>
<em>What is life?</em><br />
A lot of progress has been made in cell biology up to the tiniest scale, and that has sparked some hypotheses about the origin of life. Life seems to counteract the second law of thermodynamics, creating structure from chaos. Again, the intimate relation between entropy and information can bring insight. External energy can make self-replication possible, but is it life? Should a sharp boundary between living and non-living be erased? Artificial life, editing and generating new DNA became reality. Animal life with asexual self-replication was discovered. And there is debate when in the course of evolution neurons where developed. All of these questions are discussed in this part.</li>
<li>
<em>What makes us human?</em><br />
The brain is still one of the most complex and least understood organs. There are speculations of why about 3 million years ago the brain of humans started to quadruple in size although size is not the only thing that counts. Why do we have an evolutionary aversion to loneliness? Why do we sometimes make bad decisions, and neuroscientists investigate how the brain of a child changes into the brain of an adult. This part is connected to the next one where machines simulate how the brain operates.</li>
<li>
<em>How do machines learn?</em><br />
Here it is explained how computers are programmed to win in chess or Go from humans. However, this is a machine programmed by humans who feed the rules of the game. In this setting a machine can beat a human only because it is faster. The proper learning machine is however obtained by neural nets where deep learning and reinforced learning are the driving mechanisms that make the machine learn on its own. It will be clumsy in the beginning, but it never gets tired and hence can learn much faster than humans.</li>
<li>
<em>How will we learn more?</em><br />
Here we are back into cosmology and quantum theory. Since the LIGO has measured gravitational waves, a whole new area has opened to scientists. The waves emerge from colliding black holes, but how did these black holes come about and why did they collide? Also a pair of neutron stars can collapse and how did that happen?</li>
<li>
<em>Where do we go from here?</em><br />
It was hoped or expected that the LHC at CERN would detect new particles, but except for the Higgs boson, which was predicted, none other new particle has been observed. So what about the speculative string theory? Will there ever be evidence for some of the, by now many, versions of string theory? Can we ever arrive at a Theory of Everything, and at explaining quantum gravity, or will a completely new vision emerge? Hawking was very optimistic at first to have a ToE at the beginning of the 21st century, but he eventually had admit that it will take a while longer.</li>
</ol><p>
</p>
<p>
Of course most of the topics covered in this book rely on mathematics, fundamental or applied. However, because of the purpose of these texts is to inform the non-specialist about the latest developments, the mathematics are left out and reference to the underlying mathematics is only rarely made. Nevertheless, I believe that also mathematicians, certainly mathematicians, will and should be interested. These applied topics is where mathematical tools are needed that may not be available yet. Here models and simulations of ever higher complexity are required, and more complex abstract tools should be developed. Anyone can read this to stay informed about recent developments in science, but young mathematicians may find here inspiration on which applied direction they want to build their career.</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 collection of texts from the first five years (2012-2017) of the highly appreciated <em>Quanta Magazine</em>. The articles deal with cutting edge achievements from physics, biology, and computer science brought in a thorough, yet generally accessible form.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/thomas-lin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Thomas Lin</a></li>
<li class="field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT 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">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">9780262536349 (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">GBP 14.99 (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">328</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://mitpress.mit.edu/books/alice-and-bob-meet-wall-fire" title="Link to web page">http://mitpress.mit.edu/books/alice-and-bob-meet-wall-fire</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li>
<li class="field-item odd"><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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li>
<li class="field-item odd"><a href="/msc-full/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li>
<li class="field-item even"><a href="/msc-full/85-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-01</a></li>
<li class="field-item odd"><a href="/msc-full/68-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68-01</a></li>
<li class="field-item even"><a href="/msc-full/98-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">98-01</a></li>
</ul>
</span>
Thu, 03 Jan 2019 08:00:25 +0000adhemar48974 at http://euro-math-soc.euAlexandre Grothendieck: A mathematical profile
http://euro-math-soc.eu/review/alexandre-grothendieck-mathematical-profile
<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 pays homage to the mathematical life and achievements of one of the greatest mathematicians ever, a unique genius: Alexandre Grothendieck. Surely he changed Mathematics, and Mathematics would not be the same without him. EGA, SGA, FGA, schemes, flatness, strict localization, fundamental group of a scheme, Picard scheme, change of base point, descent, infinitesimal lifting, relative geometry, completeness theorem, standard conjectures, sites, topos, derived functors, étale cohomology, higher categories, stacks, motives… The book consists of 13 papers by mathematicians who knew him, as his students and/or collaborators. These papers discuss various aspects of Grothendieck’s work: his approach to mathematical problems, his goals and visions of Mathematics, his devotion and generosity in research for the sake of science, and also biographical recollections. In different degrees each paper presents technical concepts and results, discloses metamathematics behind the scenes and comments personal remembrances. Surely some parts will be somehow out of reach for a non-specialist, but even then they are a highly profitable reading. One finds here an enthralling presentation by Schneps of the Grothendieck-Serre correspondence, some quite private considerations by Mumford and Cartier, what Diestel explains about the transformation of Banach spaces theory under Grothendieck’s spell, the track of Grothendieck’s inspiration sources by Oort, the discussions by Simpson of the phylosophy of descent and then its higher and higher in crescendo, Kleiman historical account of the Picard group back to Clebsch, Murre on the fundamental group, Illusie on cohomology étale, Karoubi on K-theory, Raynaud, Manin, Hartshorne! And everywhere the impressive depth and transcendency of Grothendieck visions of Mathematics. Even in one does not understand completely the music, the noise is wonderful. This reviewer can only say: of course, anyone must read this, and close with three quotations:</p>
<p>[Cartier] When one follows Grothendieck’s work throughout its development, one has exactly this impression of rising step by step toward perfection. </p>
<p>[Schneps] Grothendieck devoted his life to the pursue of the absolute. </p>
<p>[Grothendieck] If it is not obvious, it is probably false.</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">Jesus M Ruiz</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 book is devoted to the mathematical life and achievements of a unique genius who changed Mathematics: Alexandre Grothendieck. It consists of 13 papers dealing with different aspects of his work, covering in different degrees thecnical aspects for specialists, historical references and ideas behind the scenes.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/leila-schneps-editor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Leila Schneps (Editor)</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/international-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">International 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">9781571462824</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">84 EUR</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">324</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="https://intlpress.com/site/pub/pages/books/items/00000416/index.html" title="Link to web page">https://intlpress.com/site/pub/pages/books/items/00000416/index.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li>
</ul>
</span>
Mon, 31 Dec 2018 17:16:47 +0000JMRz48969 at http://euro-math-soc.euThe Calculus Gallery
http://euro-math-soc.eu/review/calculus-gallery
<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 is a slightly corrected reprint of the book originally published in 2005. The fact that it is now made available in the <em>Princeton Science Library</em> series as a cheaper version is a confirmation of its quality.</p>
<p>
Dunham has chosen to tell the history of calculus from its origin, as conceived by Leibniz and Newton, till the moment that Lebesgue redefined Riemann's concept of an integral. Of course there exist several books on the history of mathematics, but Dunham has chosen to tell the story as if he is the intendant of a mathematical art exhibition. He chose a number of key results that he discusses in some detail, that means including the ideas of the original proofs (although translated in a for us readable form). These are the stepping stones that tell us about the evolution taking place. So Dunham walks with the reader through the historical museum and tells us why a particular result is important in the chain of ideas that brought us to our current understanding of the subject, and eventually how the current abstraction became a necessity. The museum where his exhibition is displayed has twelve period rooms corresponding to as many chapters in the book, named after the artist-mathematicians who, if not produced, at least published the result(s). So each chapter has a short introductory biographical sketch but the emphasis lies on the discussion of the mathematics and why these are important in an historical perspective. The museum has also two lounge rooms, two interludes, where there is time to summarize the history so far, looking at remaining problems and at what is ahead, and where a somewhat broader bird's-eye view is given because the twelve mathematicians selected are of course not the only ones that have shaped the history of mathematics.</p>
<p>
The names of the twelve chapters chosen to support the evolution are Newton, Leibniz, Jakob and Johann Bernoulli, Euler, Cauchy, Riemann, Liouville, Weierstrass, Cantor, Volterra, Baire, and Lebesgue. This includes obviously some of the usual suspects but a somewhat surprising name in the list is Baire and one may wonder why Liouville and Volterra are featuring while for example Gauss is not. So Dunham justifies his choice in the introduction. To answer the question which functions were continuous, differentiable, or integrable, one needs to know something about the continuum of the real numbers. Here Liouville was important for the discussion about irrational (algebraic, transcendental) numbers and how close these could be approximated by rationals, somewhat similar to what Weierstrass did for the approximation of continuous functions by polynomials. Volterra was instrumental in helping to answer the question of how irregular a function can be and still be (Riemann-)integrable. He was able to construct some pathological example that had everywhere a bounded derivative and yet was not integrable. Baire fits in this story because with his category theory, functions were finally classified with respect to their irregularity, which settled the discussion.</p>
<p>
Because Dunham digs into primary sources, we learn how also these brilliant pioneers who paved the way, had their struggles with concepts and approaches that for us seem clumsy. But we should realize that our calculus courses are the results of many years of filtration, polishing and reshaping of these original ideas. For example we know how to deal with infinitesimals as quantities that go to zero in the limit, but in the early days, without limits, serious resistance against the new ideas of calculus was raised because the infinitesimals were non-zero at some points and were replaced by zero at others. Manipulations that were considered by opponents to be all but sound mathematics. This issue was only solved with the introduction of the limit by d'Alembert.</p>
<p>
We also see that although Newton's fluxion stands for the derivative, both Newton's and Leibniz' approach was via integration, heavily relying on series expansions for small perturbations. The role of the integral for the origin of calculus can be seen in an historical context where geometry was dominant in solving mathematical problems and computing a surface area is a geometric problem. But calculus gradually moves away from geometry as we read on. Series however remained important issues in the early days. The Bernoulli's as well as Euler have analysed their convergence or divergence, but Cauchy was the one to formulate sound convergence criteria, while Riemann later showed the importance of differentiating between absolute and conditional convergence.</p>
<p>
With Riemann we are back to integration. Integrability was however related to the construction of pathological functions which were often of "ruler type" like being equal to 1 for <em>x</em> rational and 0 for <em>x</em> irrational. Weierstrass could construct a function continuous everywhere and yet nowhere differentiable. So this goes hand in hand with a discussion about algebraic and transcendental irrational numbers (hence the Liouville chapter). With this fundamental discussion of the number system, set theory enters the scene with Cantor's fundamental contributions and Dedekind's cuts. Topological aspects such as density of a subset of an interval has eventually triggered Lebesgue to redefine the concept of the integral to circumvent the problems raised when using Riemann's concept. With this evolution, for the finer details of calculus one has to leave not only geometry but also algebra to take off in a more abstract topological realm.</p>
<p>
Many generations of students are currently instructed in calculus courses, more or less advanced. Some may feel annoyed with the abstraction and may not see why it is needed. This book will reveal how and why their modern calculus course was shaped into its current form. This book is unique in its content because it is not a full history book, and it is not a calculus course. There are however many proofs that require some knowledge of (modern) calculus, and some of them are quite involved. But by restricting the discussion to functions of one real variable, the mathematics stay within the reach of students familiar with a basic calculus course at the level of a first year at the university. The nice thing about these proofs is that they follow the original ideas. Also Dunham's style is pleasant and much more entertaining than a formal course text. Princeton University Press has made a proper choice by promoting this book to their <em>Science Library</em> series and making it in this cheaper form available to a broader readership. My warm recommendation is only appropriate.</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 paperback reprint of the book originally published in 2005. It sketches the history of calculus from Newton and Leibniz till Lebesgue by a selection of key results during the evolution from a geometric/algebraic approach to a more abstract topological framework that was needed to cope with pathological cases when dealing with derivatives and integrals of functions. By restricting the discussion to functions of one real variable the book should be readable for students familiar with a basic calculus course.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/william-dunham" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">William Dunham</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">978-0-691-18285-8 (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">£ 24.00</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">256</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="https://press.princeton.edu/titles/14169.html" title="Link to web page">https://press.princeton.edu/titles/14169.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/26-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-03</a></li>
<li class="field-item odd"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li>
<li class="field-item even"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li>
</ul>
</span>
Thu, 13 Dec 2018 14:40:33 +0000adhemar48936 at http://euro-math-soc.euThe Best Writing on Mathematics 2018
http://euro-math-soc.eu/review/best-writing-mathematics-2018
<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>
Here is number nine in this series in which Pitici collects previously published papers about mathematics and how it relates to society. See also the reviews of the previous volumes on this EMS site. As usual the papers are harvested from magazines and journals that are not in the mainstream of specialised mathematical research journals. Although they are about mathematics, they do not really involve new mathematical results as such. They are rather about mathematicians and the way they do mathematics. The topics are familiar for those who know (some of the) previous volumes. The papers deal with for example the teaching and the history of mathematics, or things that inspire(d) mathematicians. Others are about paradoxes, games, or recreational aspects. Eighteen papers were reprinted in a uniform way for this volume, all published originally in 2017. What follows is a quick survey.</p>
<p>
The first paper is a contribution by <strong>Francis Edward Su</strong>, past president of the MAA and strongly involved in methods to transfer mathematics to students. His paper is a plea arguing that practising mathematics cultivates virtues that bring humans to completion (he uses the term "human flourishing") because mathematics answers to five basic human desires: play, beauty, truth, justice, and love.</p>
<p>
<strong>Margaret Wertheim</strong> argues that people often think of mathematics as being formulas and abstractions, i.e., fabrications of human intelligence. However, much of our mathematical inspiration comes from nature. Nature does not have the advanced human intelligence, but it is just <em>"living the mathematics"</em>. It does expose all the juicy mathematical charateristics such as patterns, fractals, hyperbolic geometry, etc.<br />
On a more philosophical level is a contribution by <strong>Robert Thomas</strong> who explains that the aesthetics of mathematics is not only beauty. Just being <em>interesting</em> is also an important aesthetic category. If your paper is not interesting, it will not get published.</p>
<p>
Computers certainly have influenced the way we do mathematics. <em>Satisfiability</em> (or SAT) is a concept of computer science. A Boolean formula is satisfiable if it is possible to find a model or interpretation of the variables that makes the formula true. Here computers can come to the rescue giving automated proofs that recently solved open problems where traditional proofs are infeasible because too many cases have to be considered separately. <strong>Marijn Heule and Oliver Kullmann</strong> give an introduction to SAT and make a point of the fact that this may give insight in the complexity of a problem and it may open some perspectives for Ramsey theory. So, what is needed is that the proofs themselves should become objects of investigation. Also in the realm of computer science is the paper by <strong>Peter Denning</strong> who gives his definition of the recent paradigm of <em>computational thinking</em> and discusses its advantages and limitations.</p>
<p>
Mathematics can be inspired by physics, as <strong>Robbert Dijkgraaf</strong> illustrates with quantum physics. The concept of <em>mirror symmetry</em> such as particle and wave interpretations of the same (quantum) mechanical phenomenon may have consequences for mathematics. These different interpretations require different mathematical tools and methods. So perhaps mirror symmetry also connects different mathematical disciplines as two sides of the same reality.<br />
But also the playfulness of mathematicians can lead to interesting problems. Three examples can be found here. <strong>Erik Demaine et al.</strong> discuss planar configurations that can be produced in <em>Tangle toy</em> by joining quarter circle pieces together to form a closed loop. Like in other papers in which father and son Demaine are involved, there are some interesting mathematical questions that can be asked about something that starts from playful amusement. And these are just the planar problems, predicting much more involved questions since the loops can also be constructed in 3D. <strong>James Grimm</strong> describes the design of dice to play a game where the winning strategy is nontransitive, meaning that one state winning over another may lead to circular arrangements like in rock, paper, scissors. And <strong>Arthur Benjamin et al.</strong> analyse the statistics of a kind of Bingo game to explain a paradoxical outcome. In the same vein but less straightforward is the analysis of the <em>Sleeping Beauty problem</em> by <strong>Peter Winkler</strong>. The problem was formulated around mid 1980's and has caused a lot of controversy since. The Sleeping Beauty has to undergo a sleeping experiment, with a final result that depends on a coin toss. She has to estimate the probability of the outcome of a coin toss when she wakes up at the end of the experiment (the details can be looked up on the web). Winkler does not favour one solution but he gives arguments to support the different possible probabilities that have been proposed in the past.</p>
<p>
There are also papers of a more historical nature. <strong>José Ferreiros</strong> discusses the paper by E. Wigner in which he introduces his often quoted "unreasonable effectiveness of mathematics in natural sciences".<br /><strong>Daniel Mansfield and N. Wildberger</strong> show that the Babylonians knew Pythagoras' theorem long before Pythagoras (which is probably not a surprise), but also that their study of (rectangular) triangles was based on ratios of the sides of the triangle. Since they used a number system in base sixty, they were able to produce some rather accurate tables that we now call goniometric tables, but the surprise is that they did it independently of the angles.<br />
Isidore of Seville (560-636 CE) was a Spanish bishop and scholar. He authored the encyclopedic <em>Etymology</em>. In his time mathematics was part of the priesthood education. It is via sources like this that classical Greek mathematics was passed on to the Middle Ages. <strong>Isabel Serrano et al.</strong> discuss what sources he may have used for his description of the classical <em>Quadrivium</em>.<br /><strong>Michael Barany</strong> shows how after World War II <em>mathematical awareness</em> became an issue (and it still is today) and that Mina Rees has been instrumental as the first president of the <em>American Association for the Advancement of Science</em> (AAAS) to promote the idea.</p>
<p>
In the vein of education and teaching, there are four papers: Methods, concepts, and techniques of <em>interdisciplinary teaching</em> is discussed by <strong>Chris Arney</strong>. <strong>Nancy Emmerson Kres</strong> goes through a list of essential questions to ask when solving a problem. <strong>Benjamin Braun et al.</strong> explain what <em>active learning</em> means when teaching mathematics. And finally <strong>Caroline Yoon</strong> wants to remove the dichotomy between mathematics and languages (sometimes almost considered to be opposites of each other) by illustrating that there is a clear parallel between them: writing a text can be seen as a task of modelling (getting insight), of problem solving (reorganizing the text), and even of giving a proof (convincing your reader).</p>
<p>
Pitici gives also a list of around sixty interesting books, and a long list of "notable writings" (journal papers, or special issues, book reviews, teaching tips, interviews, biographies, obituaries). The selection from this vast amount of the eighteen papers included here is obviously a personal choice made by Pitici. If you are interested in just one of the broad set of topics that were touched upon in this volume and you want to know more, then you will certainly find a lot of inspiration to read. There is certainly enough material to keep you reading till the next volume of "The Best Writing in Mathematics" to which I am already looking forward to now.</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>
For the ninth time, Pitici has collected papers about mathematics and its relation to society. This volume contains the harvest of 2017. As in previous anthologies, the papers do not really contain mathematics but they discuss topics such as history, philosophy, education, games, and recreational mathematics.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/mircea-pitici-ed" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">mircea pitici (ed.)</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">978-0-691-18276-6 (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">USD 24.95</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">270</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="https://press.princeton.edu/titles/14178.html" title="Link to web page">https://press.princeton.edu/titles/14178.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00b15</a></li>
</ul>
</span>
Fri, 30 Nov 2018 15:21:17 +0000adhemar48887 at http://euro-math-soc.euWavelet Analysis on the Sphere: Spheroidal Wavelets
http://euro-math-soc.eu/review/wavelet-analysis-sphere-spheroidal-wavelets
<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>Wavelets are a powerful tool in many fields, such as numerical analysis, signal and image processing or data analysis. As the title suggests, the main objective of this book is to present the construction of wavelets for functions defined on the sphere.</p>
<p>The book is organised in 6 chapters. The first one is an introduction to the topic and a guide throughout the next five chapters. In chapters 2,3, and 4, the authors study some mathematical tools useful in wavelet analysis. First of all, they present the notion and basic properties of orthogonal polynomials. They also introduce three different methods for their construction: Rodrigues formula, recurrence rules and orthogonal polynomials as solutions to ODEs. Then they apply these results to the construction of classical families of orthogonal polynomials such as Legendre, Laguerre, Hermite, Chebyshev and Gegenbauer. In the third chapter, they present the homogeneous polynomials and their interaction with harmonic analysis on the sphere. They begin with the study of the spherical harmonics in $\mathbb{S}^{2}$ as the solutions of the Laplace equation and then they develop the general theory. Chapter 4 is devoted to the study of special functions, such as beta, gamma, Bessel or Legendre functions among others. They detail their definitions and properties and provide some graphic illustrations as well. </p>
<p>The last two chapters focus on wavelets. First, the authors study wavelets related to orthogonal polynomials such as Chebyshev, Gegenbauer or Cauchy wavelets; next they concentrate on spherical wavelets. Finally in Chapter 6, they show some examples of wavelets' applications to numerical solutions of PDEs, integrodifferential equations, image and signal processing and time-series processing.</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">Blanca Fernández</div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/sabrine-arfaoui" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Sabrine Arfaoui</a></li>
<li class="field-item odd"><a href="/author/imen-rezgui" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Imen Rezgui</a></li>
<li class="field-item even"><a href="/author/anouar-ben-mabrouk" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Anouar Ben Mabrouk</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/walter-de-gruyter-berlin-boston" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Walter de Gruyter Berlin-Boston.</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">2017</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">978-3-11-048109-9</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">144</div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/42-fourier-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42 Fourier analysis</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/42c40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42C40</a></li>
</ul>
</span>
Fri, 30 Nov 2018 10:27:20 +0000blanferbe48886 at http://euro-math-soc.euPtolemy's Philosophy
http://euro-math-soc.eu/review/ptolemys-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>
Claudius Ptolemy (c 100 - c 160 AD) is best known for his <em>Almagest</em>, a collection of 13 books that formed a treasure trove for astronomers and historians because of the historical collection of astronomical data they contain. He also wrote the <em>Tetralibris</em> consisting of four books about astrology, and in his <em>Planetary Hypotheses</em> he describes the universe as a set of spheres defining the movements of the celestial bodies either via an epicyclic or an eccentric model. Circular motion was essential as it represented mathematical and hence divine-like perfection. He was not only an mathematician/astronomer since he also left us his <em>Geographia</em> with geographic coordinates of the world as it was known in his days, and in the <em>Optics</em> he studies properties of light.</p>
<p>
All these texts are used as sources by Feke although her book is not about the astronomy or the mathematics, but about the philosophy that Ptolemy is conveying in his books. He lived in the midst of what is often considered to be a period of philosophical eclecticism where elements were freely imported from Aristotle, Plato, Epicurus, or Stoics. Elements of all these are indeed found in Ptolemy's vision, however with a twist of his own, as Feke clearly explains. In that respect the beginning of the <em>Almagest</em> and even more so his book <em>Harmonics</em>, in which he mainly discusses the application of mathematics to music, are crucial elements to understand the role of mathematics for his philosophical and ethical views. These are the ones that Feke cites most. The perfect ratios of the harmonic pitches, are reflected in the harmony of the stars in heaven and that divine commensurable organized constancy should also be a guide for the human soul. Mathematics is the only way to gain knowledge and mathematical ethics will furnish a good life since it will transform the soul into a godlike condition by mimicking the perfection of the stars. Let me quote the strong statement by which Feke summarizes Ptolemy's vision in the last sentences of her book:</p>
<blockquote><p>
[...] according to Ptolemy, the benefits mathematics provides are epistemological and ethical. Mathematics is the only field of enquiry through which human beings can acquire knowledge, and it is the only path to the good life. In Ptolemy's philosophy, the best way an individual can live his life is to live the mathematical way of life.</p></blockquote>
<p>
To come to such a conclusion, some explanation is in order.</p>
<p>
</p>
<p>
According to Ptolemy, theoretical philosophy has three parts: theology, physics, and mathematics. The object of theology is the immaterial "Prime Mover" who governs the motion of the stars but who is imperceptible. Physical objects are of course perceptible but mathematical objects are in between, since the latter can be observed with your senses or thought of as abstractions. Mathematics is however the only way to generate knowledge because the Prime Mover is too far out and cannot be perceived, while in physics, one has to measure and make observations, which are necessary to define the parameters of the mathematical model but observations are always inaccurate. Thus theology and physics can only lead to conjectures, while mathematics generates knowledge. Moreover, mathematics is useful since if can help to predict the outcome of astronomical and physical phenomena.</p>
<p>
Feke claims that the ethical aspects of Ptolemy's vision are the result of his solution for a contemporary debate over the relation between theoretical and practical philosophy. Ptolemy wrote a pamphlet about this, known as <em>On the Kritêrion and Hêgemonikon</em>. It gives a criterion for truth, explains how knowledge can be acquired and describes the structure of the soul, of which the <em>hêgemonikon</em> is the ruling part, located in the brain. His authorship is somewhat controversial since it is not completely in line with what he wrote elsewhere. So Feke assumes that it was written at an earlier stage before his <em>Harmonics</em> in which his mathematical methodology was fully developed. Ptolemy considers the soul to be something mortal and physical, with faculties in different parts of the body, only it consists of much finer particles than the body, so that it evaporates much faster after death. The same harmony of perfect ratios of the musical pitches, and the mathematical study of this harmony will generate the beauty in mathematical objects. This has ethical and practical applications because mathematical models will transform physical matter into a perfect form, and likewise, when applied to the soul, it will lead to an orderly, calm, commensurable good life that, as was quoted above, resembles the astronomical properties of the stars, and hence approximate a perfect divine status.</p>
<p>
What Ptolemy describes in his <em>Harmonics</em> are the rational and unavoidable rules that govern musical pitches, the heavenly bodies and the human soul alike. These harmonic relationships and the role of mathematics is what Feke explores in further detail in the subsequent chapters discussing Ptolemy's astronomy, the human soul, astrology, and cosmology. I will not report on all her fine dissections, but I will just mention how she sees Ptolemy's view on the role, the interplay, and the relative importance of the classical subjects of the quadrivium: arithmetic, geometry, music, and astronomy.<br />
According to Ptolemy, geometry and arithmetic are not mathematics but they are just methods. Geometry can be applied to study optics via observations, arithmetic can handle the harmonic ratios that are provided by mathematical theory. Astronomy relies on both geometry and arithmetic: it requires geometry for the optical observations, while arithmetic can deal with the harmonics that are intrinsically derived from the theoretical model.<br />
Cosmology (the study of the heavenly bodies) and astrology (the effect of the movement of the stars on sublunary bodies) are both physical sciences that Ptolemy bases on astronomy which he thus considers superior to cosmology and astrology. Therefore the results of the latter two are predictive but remain just like in physics only conjectural,</p>
<p>
In a book like this, the precise meaning and nuances of a word are important and for that Feke gives precise references and quotations from the work of Ptolemy. Often the translation is in English in the text, while the Greek text from the source is in a footnote. She also relates meticulously the interpretation that Ptolemy attached to some concept and she discusses the difference or the similarity to corresponding notions of Aristotle, Plato or other of the ancient philosophers. Occasionally she also refers to interpretations given by more modern authors who wrote on Ptolemy's work and ancient Greek philosophy. That does not happen very often because her book is mostly historical. Only in her concluding chapter she briefly discusses Ptolemy's influence in later centuries. Even the way she is harvesting an idea of Ptolemy from his different books is impressive. One can read sentences like "Ptolemy used this word at only one other place namely...". Ptolemy's philosophical ideas are indeed scattered all over his mathematical books and it requires a thorough knowledge of all the texts to crystallise them into clear English sentences.</p>
<p>
As I said before, Feke shows that mathematics is for Ptolemy the science that rules everything and to which humans should submit to lead a good life, but her book is far from dealing with mathematics as such. This is a book written by a philosopher mainly for fellow philosophers. Thus if the reader is a mathematician who is not so familiar with the philosophical jargon and writing style, it may require some pages to adapt, but Feke does a good job to make it a comfortably readable text also for an interested non-philosopher. I am not a philosopher myself, so I may not be aware of all the professional philosophical literature, but my search on the Web seems to confirm that this is indeed the first fully-fledged book bringing the philosophy of Ptolemy to the foreground (as it is announced in the blurb on the back cover). Neither have I the background to agree or disagree with the conclusions that Feke distils from the available sources, so I leave that to her peers. As a mathematician, I can only be warmed by a tingling ASMR experience while reading the conclusion that she arrived at in the quote formulated above. I am so pleased by Ptolemy's tap on my shoulder reassuring me that it has not been a waste of time to have devoted the larger part of my life to mathematics.</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>
By a careful analysis of the philosophical elements in Ptolemy's mathematical work, Feke shows that mathematics is the driving force for his philosophical and ethical views that can be summarized as follows. Only through mathematics one can acquire knowledge, and the mathematics that can analyse the rations of harmonic pitches in music is equally applicable to the stars in heaven, to predict the behaviour of physical objects, and to the human soul. Hence mathematics will lead humans to live a good life.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/jacqueline-feke" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jacqueline Feke</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">978-0-69117958-2 (hbk); 978-0-69118403-6 (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">£ 30.00 (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">256</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="https://press.princeton.edu/titles/13256.html" title="Link to web page">https://press.princeton.edu/titles/13256.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
<li class="field-item odd"><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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A20</a></li>
<li class="field-item odd"><a href="/msc-full/97e20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97E20</a></li>
</ul>
</span>
Thu, 08 Nov 2018 09:23:52 +0000adhemar48815 at http://euro-math-soc.euA short course in Differential Topology
http://euro-math-soc.eu/review/short-course-differential-topology
<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>There are several very good books on Differential Topology, the first to mention Milnor's gem that everybody knows and loves. However, it is difficult to use those books as a text for an introductory one term 60 hours course where you want to teach the basics form scratch (as far as possible, taking into account the nature of Maths degrees nowadays) and get in the end to some interesting applications. This being the situation one confronts often in the real classroom today, this text by Professor Dundas is a wonderful proposal to welcome. Indeed, the topics (summarized in the short description below) are presented in a very friendly and appealing way to motivate abstract notions; this includes many nice examples and assorted references to web links worth clicking. Quite useful too, some parts are explicitely marked when they can be omitted to have all the same a good program. There are in addition some pluses to stress: (1) the use of germs to discuss tangencies, (2) a presentation of vector bundles in general, (3) an introduction to Morse functions and (4) a very nice proof of Ehresmann theorem. Another remarkable feature is the care taken to provide a good bunch of meaningful exercises.</p>
<p>Summing up, the book is highly recommendable for all publics. An interested student can very well go through it quite by her/himself and learn a lot. A professor surely can discover useful aspects that may have skipped her/his attention before. But most important, Professor Dundas offers us a very enjoyable reading!</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">Jesus M Ruiz</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 text designed for a real course on Differential Topology for postgraduate students. The topics are as expected: smooth manifolds, tangent spaces and derivatives, regular values and transversality, vector bundles, tangent fields and flows and some global basic facts. There are many examples carefully chosen and described for motivation, as well as many exercises with hints to help the student. A very good book on the matter.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/bjorn-ian-dundas" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bjorn Ian Dundas</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/cambridge-university-press-cambridge-textbooks" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Cambridge University Press: Cambridge Textbooks</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">978-1-108-42579-7</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">50 EUR</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">263</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="https://www.cambridge.org/core/books/short-course-in-differential-topology/F961891A6076D7B5EDC1A12A29D8F5B1" title="Link to web page">https://www.cambridge.org/core/books/short-course-in-differential-topology/F961891A6076D7B5EDC1A12A29D8F5B1</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
<li class="field-item odd"><a href="/imu/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/58-global-analysis-analysis-manifolds" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">58 Global analysis, analysis on manifolds</a></li>
</ul>
</span>
Mon, 05 Nov 2018 21:47:53 +0000JMRz48809 at http://euro-math-soc.euMillions, Billions, Zillions
http://euro-math-soc.eu/review/millions-billions-zillions
<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>
Journalists reporting on somebody else's results may easily be mistaken in citing the numbers that are not theirs, Perhaps authors change numbers on purpose to bias their arguments. In such cases unprepared and naive readers are easily deceived. In this book you can learn how to defend yourself from mistakes as an author and from deceptions as a reader.</p>
<p>
Kernighan is the guide on a tour where he shows all the number traps that people can easily fall into. There are of course the big numbers like millions and billions mentioned in the title. It is hard to get a mental idea of what they actually mean. As a consequence an interchange of millions and billions is a mistake easily made without being noticed. One should also be aware of what the really big numbers actually stand for when they are indicated by prefixes like mega, giga, tera, and peta. Even exa, zetta, and maybe yotta can come into the picture. In modern texts these terms regularly occur referring to the large amounts of digital data stored in for example the Deep Web. At the other end of the spectrum we should know something about micro, nano, pico, femto, atto etc. when reading about the tiny parts of the hardware on which these data are stored, or even further down the scale when reading about the Theory of Everything where the natural playground is at a subatomic level. Like in mathematics the super large and the super small often match to keep everything within finite boundaries.</p>
<p>
To avoid errors of course the units have to be right. Mistaking barrels for gallons, years for months, or hours for seconds may give quite unexpected and hard to believe interpretations of the numbers that are on display. Besides mistakes in scales, there is the extra complication that there are different systems of units like American miles, gallons, or degrees Fahrenheit that should somehow match with European kilometres, litres and degrees Celsius. Even a mile can have many different meanings in different contexts. All this requires carefully introducing the proper conversions. Just picking up a number from a website may easily lead to such mistakes if the proper conversion is not made. Another typical mistake is to confuse a square mile and a mile squared. This means that you should be aware that if you double the length of the sides of a square you get a surface four times as large. For a volume it is even more dramatic because that will give a volume or mass that is eight times as large.</p>
<p>
The previous examples are possible sources of mistakes. How should we detect them and how could we protect ourselves against malicious attempts to deceive us? We could compare different sources or different ways of computing. When the results are approximately the same we can probably trust the numbers. It is of course useful to check with some numbers from your own experience or numbers that you know like for example the population of your country. Approximate rounded values for these numbers can be used in crude and simple computations to get at least an idea about the magnitude of the number that should be expected. If the number presented to you deviates considerably, either it is fake or your computations are wrong. For example <em>Little's Law</em> is applicable for a simple check in many situations. Given the population of your country and an average human life span, Little's Law allows you to estimate for example the number of people that will turn 64 twenty years from now.</p>
<p>
Usually numbers appear in non-scientific texts with approximate rounded values. If one spots a specious number that is given with many digits, then it is probably the result of a computation or conversion and only the most significant digits or rounded values are actually meaningful. They are probably the result of some conversion, like a mountain over 4.000 meter high should not be referred to as over 13.123 feet. Statistics is another possible source of deception: the median is not the average, a correlation does not imply causation. It might also be interesting to know who did the statistics and the sampling or polling. Results may be consistently biassed towards the results of which some lobbyist or pressure group wants to convince you. Another well known trick is to fiddle with the scales used in the graphical representation of the numbers in pie or bar charts. Numbers representing a percent are again possible pitfalls that can put you on the wrong foot. A percent is definitely different from a percent point and you should also be aware that a percent increase is computed in terms of the lower number, while a percent decrease is referring to a percent of the larger number: a 50% decrease can only be compensated by a 100% increase.</p>
<p>
Kernighan gives many examples of all these issues, mostly from newspapers and websites. He also keeps his readers alert by continuously pushing them to do some mental calculation to estimate some results for themselves. As some kind of a test at the end of the book he gives many such problems that one should be able to approximately solve (he also gives his own estimates): How many miles did Google drive to get the pictures for Street View (for your country)? How long did that take? How much did it cost? Or, if you have a garden with some trees in it, how many leaves do you have to rake every autumn? And there are many of these so called <em>Fermi problems</em> throughout the book. Kernighan gives some tricks to solve them, hence the "test" at the end. However it certainly requires a lot of practising and training which the reader has to do for him or herself to acquire some routine in this,</p>
<p>
In this time of "fake news" and in a society that is more and more spammed by numbers, it seems like problems of numeracy among a general public is gaining interest and public awareness. More books devoted to different aspects of this issue seem to appear lately. Among the earlier examples are the books by John Allen Paulos <em>Innumeracy</em> (1988) and <em>A mathematician reads the newspaper</em> (1996). Kernighan mentions them in his survey of "books for further reading". However several more books appeared since 2010. Almost simultaneously with this one I received <em>Is This a Big Number?</em> (2018) by Andrew Elliott which is also reviewed <a href="/review/big-number" target="_blank">here</a>. Elliott has a more positive approach to the problem: how should I interpret the numbers presented to me (assuming they are correct) while Kernighan is more defensive: how to to stay away from mistakes or deceiving numbers.</p>
<p>
It should also be noted that Kernighan uses American units most of the time, and his examples are mostly related to the American situation, or from the American newspapers. His advise is of course generally applicable, but it might be an extra hurdle to take for European readers who are used to the metric system of metre-litre-gram. It is surprising that Kernighan does not discuss the difference between the short scale (a billion is $10^9$) and long scale (a billion is $10^{12}$) hence also missing the milliard ($10^9$) and billiard ($10^{15}$) and giving different meaning to trillions, and nomenclature higher up. These are also obvious ways to get the wrong numbers cited.</p>
<p>
This book does not need mathematics to read and it is actually not about mathematics at all. There is not a formula made explicit, even though the rule of 72 is explained (it takes 72/x units of time to double your capital when it has a compounding interest of x percent) and an idea is given about what it means to grow exponentially or by powers of 10 or powers of 2. The style of Kernighan is fluent and casual, but not particularly funny. The charm sits in his continuous teasing to make you think of these Fermi problems, and of course in his ample illustrations of how often authors are mistaken in citing numbers and how easily a reader can be deceived.</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 book about numeracy learns you how to defend yourself against making mistakes with numbers or to recognize incorrectly cited numbers by explaining what the possible sources of these errors are. On the positive side, one learns how to solve Fermi problems, that is to make rough numerical estimates of certain quantities, using little available data and where the few computations can me made "on the back of an envelope".</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/brian-w-kerninghan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Brian W. Kerninghan</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">978-0-691-18277-3 (hbk), 978-0-691-19013-6 (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">£ 17.99 (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">176</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="https://press.princeton.edu/titles/14171.html" title="Link to web page">https://press.princeton.edu/titles/14171.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li>
<li class="field-item odd"><a href="/msc-full/01a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A80</a></li>
</ul>
</span>
Wed, 24 Oct 2018 12:08:48 +0000adhemar48773 at http://euro-math-soc.euIs That a Big Number?
http://euro-math-soc.eu/review/big-number
<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>
Mathematics is much more than numbers, but in an historical as well as in an educational context numbers are certainly essential for the development of mathematical skills. Counting is one of the primitive mathematical activities, but as societies became more complex, numbers became much more essential for the socio-economical fabric. Sometimes the balance tips the wrong way by reducing something or even someone to just a number expressing an amount of a particular something that can be associated with them. It has become more and more important to have an idea of what a number means or what it stands for. Whether it is small or big in the context it is used. Think of the kcal of the food we eat, the GNP or the debt per capita or the happiness index of your country, the meaning of Olympic records, or the area of forest destroyed by wildfire. This book is about the numeracy that people should gain to be able to understand the meaning of numbers in our daily life so that they can discuss about them knowing what they are talking about. This understanding of numbers has been promoted before by J.A. Paulos in his book <em>Innumeracy</em> (Hill and Wang, 2001) and his semi-autobiographical sequel <a href="/review/numerate-life-mathematician-explores-vagaries-life-his-own-and-probably-yours" target="_blank"> <em>A Numerate Life</em></a> (Prometheus, 2015).</p>
<p>
To get an idea of what is small and what is big, one should be able to get a mental picture of what a number means. So the book starts with enumerating five ways to attach a meaning to the adjectives "big" or "small".</p>
<ol><li>
<em>Landmark numbers</em> are numbers that one should memorise so that they are readily available for comparison</li>
<li>
<em>Visualisation</em> means to use your imagination to get a mental picture of the amount</li>
<li>
<em>Divide and conquer</em>: break a larger number up into smaller parts (e.g. x rows of y columns having stacks that are z high)</li>
<li>
<em>Rates and ratios</em> are usually smaller and often more relevant for what the numbers mean</li>
<li>
<em>Log scales</em> will bring numbers varying over a wide range to within reasonable bounds and separates them better in dense parts of the range</li>
</ol><p>
These ideas are illustrated abundantly with an incredible number of facts. Some are marked in frames containing possible landmark numbers, others are enumerated in lists of random alignments or number ladders. Random alignments are lists of (unrelated) numbers that happen to appear in almost perfect ratios (like for example the height of St. Paul's in London which is 100 times the height of R2D2 from Star Wars). Such lists are usually found at the end of the chapters. Number ladders are lists of increasing or decreasing numbers that run through a whole scale (for example the weight of animals ranging from an Indian flying fox (1 kg) to a blue whale (110 tons)). Each chapter also starts with a multiple choice question for the reader to test his/her numeracy (like which is largest in a list of four populations). The answers and references for the latter are given at the end of the book.</p>
<p>
</p>
<p>
The chapters are grouped into four parts that reflect the context in which the numbers appear. They are also somewhat arranged in historical order. There is in fact also a lot of history and etymology about the origin of names that have been invented to indicate units by which quantities are measured, or the origin of terms used in for example our monetary system. The metric system has simplified a lot, but before that, there were many different words for units to measure a quantity, and it may be a surprise to read how many are still in use that have escaped metric standardisation.</p>
<p>
Part 1 is about counting, which is our oldest use of numbers. When it gets to really big numbers (how many stars in the sky?) or in other instances (what percentage of alcohol in your beer?) it involves more than "just counting 1,2,3,...". The reader is instructed how to "visualise" numbers mainly by the first two methods of the list above. Methods 3 and 4 of the list are better illustrated in the second part.</p>
<p>
In that second part, we learn about measures. A spatial dimension was historically often measured using body parts. This still resounds in our names for units of length like inches, feet, fathoms, and other less known units. Time has for obvious astronomical reasons escaped the decimal subdivision that is now used in the naming of space dimensions. It is the reason why twelve or sixty (which happen to have many divisors) have played a basic role in early mathematical cultures, and we still use the terms "dozen" and "gross" today (strange that Elliott didn't mention these two terms). The hierarchical subdivision of (pre)history in time spans like ages < epochs < periods < eras < eons (where x < y means that y consists of several x's) is an illustration of the fact that a divide and conquer technique is a way to get an idea of our geological time scale. To discuss history, it is important to get at least a rough idea about the rise and decline of the different civilisations that directly or indirectly had an influence on the age we are currently living in. So both historical and prehistorical periods of time get much attention in the book. In this part it is also illustrated that to measure areas, size needs to be squared and for volumes (and hence also for mass and weight) size must be cubed. It explains why the legs of an elephant are relatively much thicker than the legs of a mosquito. The strength depends on the cross section (size squared) and the weight on the volume (size cubed). That is why a giant Godzilla cannot be real. Scales are used to quantify wind speed and hurricanes or earth quakes. These scales are actually logarithmic. So log-scales are first introduced in a (lightly) technical intermezzo (which also mentions Benford's law, the slider rule and mortality rates).</p>
<p>
Numbers get much bigger in the third part dealing with numbers in science. For example naming astronomical distances, measuring energy or capacity of a digital memory, or quantifying the information content of a text. Also measuring the complexity of solving combinatorial problems dives into the large numbers very fast. The fourth and final part is probably the most important for the general public since it treats numbers in a political and socio-economical context. This means among other things money (exchange rates) and economy (GDP), population (quantities and dynamics), quality of life (Millennium Development Goals and happiness index), etc.</p>
<p>
The book illustrates well that knowledge (being numerate) is power. So we should be able to unveil the true value of a number used in an argument and know whether it has to be considered alarmingly big or small, and hence defend ourselves against deceit, fake news, and false arguments which has become an essential skill in our current society. This book is an essential tool if you want to work on this skill for personal use.</p>
<p>
However, what the book does not discuss is that one should be careful with just ranking the numbers. Numbers do represent something, but it may not always be clear what that "something" really is. GNP is considered a measure for the wealth of a nation, but is it? Should military expenses, included in the GNP, be taken into account to measure wealth? Is an IQ really a measure of intelligence? A number will represent the result of a test or a poll, but what the test or poll is supposed to measure is not always clear because it may not correspond to what is actually measured. Moreover, these usually refer to just an sample while more general statements about a much wider population are concluded. In a world where everything is being managed, numbers are used to manipulate and define strategies, but often reality can not be caught in just a number. Believing that the number stands for some effect may be a generally accepted hoax. So the numeracy discussed in this book is just one aspect of being knowledgeable about a topic. Knowing whether a number is small or big is only one element to be taken into account for the interpretation of numbers. Even more important is to know what exactly these numbers measure and how they were obtained. This is not so important for the first three pars of this book since most of the numbers there concern quantities that can be objectively measured. Only in the fourth part when numbers are discussed in a social context, one should be careful with their interpretation and drawing conclusions.</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>
he book aims at promoting numeracy for everybody. When using numbers in arguments, one should have a good idea what these numbers mean and one should know when they are to be considered "small" or "big". An overwhelming set of numeric facts are provided together with some techniques that you can train to obtain a certain level of numeracy. Numbers and facts are situated in the context of simple applications such as counting and measuring and later in more complex situations that can be encountered in science and society.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/andrew-elliott" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Andrew Elliott</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford 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">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">978-0-198-82122-9 (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">£18.99 (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">352</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="https://global.oup.com/academic/product/is-that-a-big-number-9780198821229" title="Link to web page">https://global.oup.com/academic/product/is-that-a-big-number-9780198821229</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li>
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</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Wed, 03 Oct 2018 09:18:51 +0000adhemar48716 at http://euro-math-soc.eu