European Mathematical Society - 00-01
https://euro-math-soc.eu/msc-full/00-01
enHumble Pi
https://euro-math-soc.eu/review/humble-pi
<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>
Matt Parker was a math teacher who has set himself as a life goal to be a math and science communicator. He is well known as a guest in tv-shows, by his stage shows and Numberphile videos on Youtube, and as the author of <a href="/review/things-make-and-do-fourth-dimension" target="_blank"> <em>Things to make and do in the fourth dimension</em></a> (2014). The current book got the subtitle "<em>A comedy of maths errors</em>", and a slapstick comedy it is, because of all the things "that went not exactly as planned" because of the "mathematical" errors behind the wrong decisions made. Unfortunately some of the consequences were rather dramatic and caused a loss of lives, money, and energy. Parker warns us in the introduction that he has deliberately left three errors in the book for the reader to detect. Also the cover of the book represents a funny Boeing plane with the wings wrongly attached. That is very unlikely to happen in reality, but unfortunately accidents with planes happen all too often and it is sometimes just a tiny programming or construction error that has caused the death of many passengers and crew in the history of commercial aviation.</p>
<p>
This book is not the first to describe a collection of errors with numbers or mathematical errors. For example, read about math errors made by students in <em>Magnificent Mistakes in Mathematics</em> (2013) by A.S. Posamentier and I. Lehman while J.A. Paulos wrote his book <em>Inumeracy</em> (1988) to warn the man in the street not to be blind for blatant mathematical misconceptions, and more on the informatics side, there is <em>Weapons of math destruction</em> (2016) by O'Neil and the more recent <em>Bits and Bugs</em> (2019) by T. Huckle and T. Neckel. Also Parker includes many software errors in his collection. So the "maths errors" have to be understood in a broader sense. The fact that if integers are represented by one byte or 8 bits and thus can at most be 11111111 or 255 in decimal notation, will cause an overflow problem when 256 is reached much like the dreaded (but anticipated) millennium bug. The 256 overflow problem however has often been overlooked and has been a source of many mistakes and disasters when a count suddenly dropped from the maximal 255 to zero. This caused Twitter, Minecraft, and Pac-man to break when reaching level 256 but it also can turn an X-ray medical instrument into a deadly weapon.</p>
<p>
Other tangentially mathematical errors are related to the unwanted interpretation that is made by Excel of what you type: the leading zeros of telephone numbers are removed when inserted as a number, if an hexadecimal number contains an E it can be interpreted as a scientific notation for a decimal number, it may think to recognize a date in strings of biological information with names for enzymes like MARCH5 or SEP15, or in strings of the form 6/11, etc. It is obviously a bad idea to misuse excel sheets with many formulas. Parker makes an analysis of a large set of excel sheets that are interconnected by a large set of formulas with an inevitable daunting amount of errors. Calenders have changed in the course of history and this has been the reason why the Russian delegation arrived two weeks late at the Olympic Games in 1908 because in Russia the Julian calendar was still in use while the rest of the world had switched to the Gregorian calendar. The different units for weights or volumes, and distances nearly crashed a plane that fell out of fuel in mid-air or another one was heavily overweighted and just made it to its destination (kg and lb are not the same). But there are of course the genuine mathematical problems when engineers do not make the correct computations (or make last moment changes and neglect to redo the computations) during the construction of a bridge or a building. The video of the Tacoma bridge is legendary, but there are many other bridges that had stability problems like the wobbly Millennium bridge in London. The Walkie-Talkie is a nickname for a London skyscraper with parabolic glass facade that reflected sunlight and melted things and set carpets on fire in the focal point. Parker is also the man behind a petition protesting against the UK road signs pointing to football stadiums with the wrong football logo. The classic football is an inflated truncated icosahedron with 20 white hexagons and 12 black pentagons. The logo consisted completely of hexagons. Rounding and statistical errors have upset the financial markets, and were misused in politics. Non-causal correlations are often used as arguments by activists or pressure groups. It's not all strictly mathematics, but Parker keeps going on and on. It's misunderstanding and misinterpretation galore, and often a sequence of coincidences for which Parker borrows the image of a Swiss cheese from James Reason: there are holes in every level of the security checks, like holes in every slice of cheese, but the holes have to line up to let the error slip past them all. Unfortunately that happens from time to time.</p>
<p>
On the nerdy side: The pages of the book are numbered from 314 down to 0 and then switches to count down from 4,294,967,295 which is 232-1 for the appendix with the list of illustrations and the index, an example of underflow in binary countdown. A similar type of error played an important role in the first disaster example where the time was stored in 4-bytes and all electrical power was shut down in the Boeing 787 plane when the counter reached 2,147,483,647. The index of the book is automatically produced by a code written by Andrew Taylor. The code selects interesting couples of two successive words from the text. References are given to pages and lines where these couples appear, like for example Richard Feynman: 154.95522, 222.00000-223.29851 meaning that the text "Richard Feynman" appears on the bottom line of page 154 and is discussed on pages 222 top line to page 223 line 11 from the top. This allows to figure out that lines are numbered as multiples of about 2985 starting from line 0 at the top. The more precise value 2985.074662686... is probably a conversion into some units that I have not figured out. There is an exception for the index entry "oddly specific" where lines are shown with 13 digits instead of 5. For example the first is pointing to 117.2089552238806 and "oddly specific" appears indeed on page 117, line 8 from the top, which should on other pages correspond to 117.20896. Another funny and deliberately vague entry is "deliberately vague: somewhere between 7 and 10 and maybe 74". There you can indeed find the phrase "deliberately vague" in connection with the hush up of some errors in the system. It is fun figuring out for yourself how the index is produced. More details below after the "Spoiler alert". Whether the title of the book has any relation with the name of the supergroup of Humble Pie of the late 60's is not clear either. In any case, if pi stands for mathematics, then engineers, programmers, and anyone applying mathematics as a humble tool in their great creative design should be aware of the importance of this humble tool that can safe the life of many that should not die because of a miscalculation.</p>
<p>
All of these mistakes with amusing and not so amusing consequences are told by Parker in general in a funny way, even though there are a lot of people dying in this book. Notwithstanding the long list of examples of what went wrong, probably many other mistakes were never discovered or were swept under the rug after investigation, so that the public doesn't even know they ever happened. Newly discovered mistakes create new safety regulations. But new boundaries will always be explored, that is just human's nature, and humans will always make mistakes. We just have to learn to be alert and do the mathematics properly. Read the book! You'll enjoy it! Mathematics is required in life, but not for enjoying this book.</p>
<p>
<strong>Spoiler alert</strong> If you scroll down, you might read things that you prefer to find out for yourself.</p>
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There is some hint in the "oddly specific" item of the index on how the lines are numbered. That entry refers to places 139.1194029850746 and 117.2089552238806 where the text appears on line 5 of page 139 and line 8 of page 117. Dividing 1194029850748 by 4 gives 298507462687, which is to be the oddly specific value of the distance between lines (multiplied by 10,000,000). The bottom line (line 33) corresponds to 298507462687 × 32 = 9552238805984, which is rounded to 95522. The first (hard cover) edition of <em>Things to make and do in the fourth dimension</em> had no index. But in the paperback edition an index was added, also produced by Andrew Taylor. There the references were given in 3D: first the page number and then the coordinates of a square like on a map: columns are indicated with a letter (A,B,C) and rows with a number (1-5), giving 15 squares per page to approximately locate the place where the item of the index can be found.<br />
Another bit worth noting is that in the hardback edition of <em>Things to make and do</em>, Brady Haran in the acknowledgements was misspelled as Bradley Haran. Parker promised to have the error corrected in the paperback edition. In this book he thanks again Bradley Haran to which he adds "Consider this a sign of my appreciation, mate". Haran popularized the Parker Square, which Parker produced on Numberphile. It was supposed to be a 3 x 3 kind of magic square containing 9 unique squared numbers with the same sum along rows, columns and diagonals, an unsolved problem in mathematics. Parker is however proud to have found one sloppy approximation in which squared numbers are repeated and one diagonal sum fails. This became known as the Parker Square, an inside joke on Numberphile.</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 his well known funny way, Matt Parker is collecting an impressive list of mishaps and coincidences where things went wrong because of (sometimes tiny) mistakes made by engineers, programmers, or just by any link in the chain leading to a (near) disaster. In the background it is always a bad number or an incorrect formula or just the wrong logic that is applied. So the "maths" in the subtitle "a comedy of maths errors" has to be understood as broadly as in the expression "do the math". It's pretty obvious that the mistake should not have happened, if someone just had thought about if for a sec.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/matt-parker" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matt Parker</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/penguin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Penguin</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">9780241360231 (hbk), 9780241360194 (pbk), 9780141989136 (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">£20.00 (hbk), £9.99 (pbk), £16.99 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">336</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.penguin.co.uk/books/300640/humble-pi/9780241360231.html" title="Link to web page">https://www.penguin.co.uk/books/300640/humble-pi/9780241360231.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li></ul></span>Thu, 22 Aug 2019 08:23:32 +0000Adhemar Bultheel49658 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/humble-pi#commentsAlice and Bob Meet the Wall of Fire
https://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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/thomas-lin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Thomas Lin</a></li><li class="vocabulary-links field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/85-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/68-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68-01</a></li><li class="vocabulary-links 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 +0000Adhemar Bultheel48974 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/alice-and-bob-meet-wall-fire#commentsIs That a Big Number?
https://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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/andrew-elliott" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Andrew Elliott</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Wed, 03 Oct 2018 09:18:51 +0000Adhemar Bultheel48716 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/big-number#commentsApplied Mathematics: A Very Short Introduction
https://euro-math-soc.eu/review/applied-mathematics-very-short-introduction
<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>
If you are a mathematician, try to define what exactly is applied mathematics and in what sense is it different form (pure) mathematics, and you will realize that it is not that easy. Existing definitions are not univocal. Although in most cases you recognize it when you see it. To that conclusion comes also Alain Goriely, who is an applied mathematician himself, in the introduction of this booklet. Yet he isolates three key elements characterizing the topic. First there is the modelling: some phenomenon is (approximately) described by choosing variables and parameters brought together in equations. Then there is of course a whole mathematical machinery to support and analyse the model theoretically, and finally there are the theoretical as well as the algorithmic and computational methods that solve the equations. The digital computers that emerged after WW II have certainly contributed to the development of applied mathematics bifurcating from pure mathematics. These three elements (model, theory, methods) form the framework for the rest of this (very short) introduction to applied mathematics which is intended for a mathematically interested outsider. Like the other booklets in this series it is a compact (17 x 11 x 0.6 cm) pocket book that is entertaining to read, even on a commuting train or during some idle moments.</p>
<p>
The data that an applied mathematician has to deal with are numbers, but these numbers have a certain dimension (length, weight, time,...) and they need to be expressed in proper units (like mks) and at a proper scale. Only when all this has been taken care of in a proper way, one can start building a model to, for example, predict the cooking time of poultry as a function of their weight or try to solve the inverse problem: how fast mammals can loose heat. With the answer to the latter problem, one may deduce something about their metabolism as a function of their volume. Keeping track of the proper dimensions throughout the modelling and the computations is called <em>dimensional analysis</em>.</p>
<p>
Choosing a model is a matter of deciding which are the most influential variables. The finer the model, the more computing time it will require while its predictive power or insight will not increase correspondingly. A simple mechanism to arrive at a model is illustrated with the model describing our solar system. First there was the geocentric system, but anomalies in the observations made Copernicus propose his heliocentric alternative. The more precise observations provided when telescopes were being used (a lot of data were provided by Tycho Brahe), allowed Kepler to derive his laws which fit the data, but it was only Newton's gravitational theory that gave the eventual explanation, not only for Kepler's laws, but for gravitation in general. Nowadays, models are constructed in a similar although a more interactive and more complex way. Observations lead to simple models, that are checked by experiments, which require subsequently refinements of the simple model, which is then checked against new observations, etc.</p>
<p>
Once the model is shaped in the form of equations, it requires theory to analyse under what conditions there exist solutions and what properties these solutions will have. For example one may analyse when they have an explicit solution (in terms of simple functions). If not, the equations can be considered as defining equations for new (less elementary) functions. The celestial gravitational problem of two mutually attracting bodies was generalized to the three-body problem, which was only solved partially by Henri Poincaré who, by doing so, created chaos theory. A deterministic world view had to be left behind and a qualitative analysis of (nonlinear) differential equations was born. The Lotka-Volterra equations describes a prey-predator model has periodic solutions, but with three species involved they will have chaotic solutions. Also the Lorenz equations, a set of three simple differential equations, originally describing an atmospheric convection problem is a famous model generating chaotic solutions.</p>
<p>
When it comes to periodicity, then the wave equation is the example that pops to mind. However when non-linearities are involves, like with seismic P-waves that travel trough the earth mantle, or phenomena like rogue waves, then solitons are involved, which are bump-like shapes that travel along without changing shape. They have a particle-like behaviour, and thus they have potential as carriers of digital information in optical communication, which is an exciting recent research field.</p>
<p>
The applications mentioned in the remaining chapters are computer tomography, the discovery of DNA, and the use of wavelets in JPEG2000 for image compression. Other examples are illustrating that what originally were theoretical developments, eventually turned out to be of the highest importance for applications like complex numbers, quaternions, and octonions (this line of complication was eventually replaced by the concept of a vector space), and knot theory (which found application in DNA modification). Finally large networks and big data are fairly recent topics that are used for describing global phenomena. Even with the complexity and magnitude of these networks, they are still inferior to what a human brain is capable of. Accurate modelling of our brain is momentarily still a (distant) target exceeding our current computational capacity but we are closing the gap.</p>
<p>
The previous enumeration is just a selection of some of the topics discussed that should illustrate what applied mathematics is about. Of course this limited booklet cannot be exhaustive. The approach is partially historical and still manages to refer to topics of current research. While examples are rather elementary in the beginning, towards the end, the topics tend to be more advanced. But even when discussing these more advanced subjects, Goriely tries to convince the reader that even if math is not always simple, still it is fun to do. The many quotations from the Marx brothers (most of them from Groucho) sprinkled throughout the text are funny of course. Goriely even provides a play-list of pop music that you could play in the background while reading (at least some of these he used while writing). This makes it clear that he has enjoyed writing the book and some of this joy radiates from the text when you read it.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In this short survey, Goriely gives examples (rather than a precise definition) of how applied mathematics relates to and interacts with pure mathematics. Applied mathematics fills the gap between the abstraction of pure mathematics and the world we live in. He describes historical models as well as more recent applications and even reaches out to future targets.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alain-goriely" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alain Goriely</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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">978-0-1987-5404-6 (pbk), 978-0-1910-6888-1 (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">9.99 € (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">168</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/applied-mathematics-a-very-short-introduction-9780198754046" title="Link to web page">https://global.oup.com/academic/product/applied-mathematics-a-very-short-introduction-9780198754046</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span>Mon, 02 Jul 2018 09:27:22 +0000Adhemar Bultheel48570 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/applied-mathematics-very-short-introduction#commentsInfinity: A Very Short Introduction
https://euro-math-soc.eu/review/infinity-very-short-introduction
<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 one of the small booklets (literally pocket books: 174 x 111 mm) appearing in the Oxford series <em>Very Short Introductions</em> that treat diverse subjects from accounting to zionism. Infinity, is a concept mainly of importance and practically useful in mathematics, but it has also philosophical and even religious aspects. Stewart is as broad as "a very short introduction" allows and adds a lot of history to his discussion. So much is to be told on only 143 small pages. Although there is obviously a lot of overlap, Stewart's treatment is wider than Marcus Du Sautoy's <a href="/review/how-count-infinity" target="_blank">How to Count to Infinity</a> and Eugenia Chen's <a href="/review/beyond-infinity-expedition-outer-limits-mathematics" target="_blank">Beyond Infinity</a> who stay more on a mathematical playground.</p>
<p>
Infinity, and certainly the infinitely large, has long been something fuzzy that was discussed on a philosophical basis. The Greek were arguing over a distinction between an actual (existing) infinity and a potential version, i.e. that "something" that is beyond all natural numbers, which is never reached by enumeration. They got away with the infinitely small by their concept of commensurability in what was mainly a geometric approach to mathematics. The infinitely small was beyond any possible subdivision of a finite length. Their fundamental common measure was thus finite and that led Zeno to his paradoxes. The infinitely small was somehow tackled when calculus was developed by Newton and Leibniz in the eighteenth century introducing infinitesimals. They represented something almost zero but not quite. When used in calculations one could divide by them, since they were not zero, but at some point, when suitable for the result, they were assumed to be zero. Not very rigorous mathematics that was. It was not until towards the end of the nineteenth century that Georg Cantor brought more insight into the nature of the infinitely large. Stewart guides us through this history and illustrates how the concept of infinity has played a role in several disciplines that all have somehow contributed to how we think of the concept today.</p>
<p>
With a first chapter, Stewart puts forward some puzzles or paradoxes that involve infinity to illustrate that it is not sufficient to say that infinity is that "something" that is beyond all numbers. More precise definitions are needed for the infinitely large as well as for the infinitely small. Examples are the processes that hide irrational numbers like a staircase approximating the diagonal of a square converging to a straight line when its steps become finer and finer, and the regular polygon converging to the circle as it gets more edges. These demonstrate the problem of evaluating $0\times\infty$ in a sensible way. Hilbert's hotel is illustrating that a more precise definition of the infinitely large is required, and Stewart gives some other examples. These puzzles and paradoxes are first raised as questions for the reader to think about. Stewart's explanations of all these confusing statements are given afterwards.</p>
<p>
The second chapter illustrates that infinity is not hidden away in higher mathematics but that it is also embedded into elementary calculus. Gabriel's horn is obtained by revolving $1/x$ for $x>1$ around the $x$-axis. This has the surprising property that its volume is finite even though the surface is infinite. Of course infinity is also hidden in 0.9999... being equal to 1, a fact that astonishes many an undergraduate student, and of course infinity resonates in the decimal representation of irrational numbers. Distinguishing discrete from continuous would not be possible without infinity. Here as in the other chapters Stewart gives quite some attention to history: Dedekind defining the real numbers as sections which are essentially infinite objects, Lambert who proved the irrationality of $\pi$. In the Jain religion of India (600 BCE), people distinguished infinity from enormously large numbers, etc.</p>
<p>
Chapter three is further exploring the historical views of infinity. Space and time were traditionally assumed to be infinite, but when looking at the infinitely small, the situation is different. People had difficulty in dealing properly with infinitely small things. Zeno's paradoxes are examples that illustrate that a sum of infinitely many nonzero numbers can be finite. Since the ancient Greek there has been a distinction between an actual infinity and a potential infinity, a discussion that has continued throughout the centuries among philosophers. Even some theologians claimed that God was the only existing impersonation of something infinite. Some proofs for the existence of God were based on this belief. For mathematics, this distinction is not essential. Mathematical existence is abstract and does not coincide with physical or actual existence.</p>
<p>
The next chapter is a discussion of the infinitely small and how this has triggered the development of calculus. The original historical concept of infinitesimals is now replaced by the concept of a limit. The infinitesimals where revived when in the 1960's Abraham Robinson developed non-standard analysis.</p>
<p>
In geometry, infinity is where the horizon is. It led to the development of perspective in the Renaissance. This is extensively discussed in chapter six, explaining why a ship seems to become smaller as it approaches the horizon, and how this has led to the concept of a point or a line at infinity. The Euclidean plane can be modelled as a disk where infinity is represented by its boundary. More concretely, the line at infinity makes it easy to produce perspective drawings. Eventually this discussion ends in ideas of projective geometry and the mapping of the plane to a sphere and vice versa by stereographic projection, the point at infinity corresponding to the North Pole on the sphere.</p>
<p>
Infinity is a useful concept in mathematics, but how does it appear in a physical world? That is what the next chapter is about. In physical sciences, infinity often leads to a nasty singularity. Stewart discusses three examples. The analysis of the rainbow phenomenon is an optical example. If light is incident at a certain angle, then the intensity of the rainbow would be infinite according to ray optics. This singularity entailed that light had to be reconsidered as a wave. In Newton's gravitation theory a singularity occurs when the distance between particles becomes zero and the potential becomes infinite. For example Zhihong Xia proved in 1988 that by solving equations in a five-body problem, dramatically non-physical solutions are obtained after a singularity. Black holes are singularities in general relativity theory and in cosmology the Big Bang is obviously a singularity. Stewart also explains here why cosmologists are wrong when they use curvature as a parameter that determines whether our universe is finite or not.</p>
<p>
Te last chapter is the discussion of how Cantor came to his proof that the real number are not countable and how this has led to set theory and his transfinite numbers, and how this resulted in a revision of the foundations of mathematics. This story is best known by mathematicians or anyone who is a bit familiar with this kind of mathematical background literature. But again here Stewart follows the historical evolution of who did what and why in brewing up the eventual result.</p>
<p>
This is a lot of information and because of the compact presentation, it will not always be casual reading for a general reader. There are a few references provided per chapter, which might be of interest if the reader wants to look up more details. Some aspects are elaborated more than what is needed for explaining the impact of infinity (e.g. the computation of the angle of the rainbow, the geometry of perspective) but these topics are of course interesting in their on right, and they are usually not found in other treatments of infinity. If you are interested in only the strict mathematical concept of infinity, then Du Sautoy's or Chen's treatises that were mentioned above might be simpler alternatives. But in this booklet, even the experienced reader may have more occasion to learn something new. Some of these non-essential but nevertheless flashes of a that's-interesting-I-didn't-know-that experience will make it worthwhile 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">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 booklet wants to introduce a general reader to the concept of infinity. With a lot of historical, philosophical, and occasionally theological background Stewart shows how the concepts of the infinitely small and the infinitely large were eventually settled in a mathematical setting towards the end of the nineteenth and early twentieth century when the current foundations of mathematics were established.<br />
</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/ian-stewart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ian stewart</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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">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">978-0-1987-5523-4 (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">£7.99 (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">154</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/infinity-a-very-short-introduction-9780198755234" title="Link to web page">https://global.oup.com/academic/product/infinity-a-very-short-introduction-9780198755234</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li></ul></span>Tue, 03 Apr 2018 06:35:02 +0000Adhemar Bultheel48365 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/infinity-very-short-introduction#commentsThe Turing Guide
https://euro-math-soc.eu/review/turing-guide
<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>
Jack Copeland is a professor at the University of Canterbury, NZ, director of the <a href="http://www.alanturing.net/">Turing Archive for the History of Computing</a>, co-director of the <a href="http://www.turing.ethz.ch/" target="_blank">Turing Center of the ETH Zürich</a>, and he has written or edited several books about Turing and his work. So he seems to be also the driving force behind this new collection of papers devoted to the life and the legacy of Alan Turing. Only four authors are explicitly mentioned on the cover of this book, but the collection contains 42 papers authored by 33 persons with very diverse backgrounds. Fifteen of the 42 papers were (co)authored by Copeland. Four of the papers by older authors (three of them have known or collaborated with Turing) are published posthumously.</p>
<p>
Alan Turing (1912-1954) hardly needs any introduction. Most people will know him as a codebreaker of the German Enigma at Bletchley Park during the second World War. They probably also have heard of his tragic death covered by a veil of uncertainty: was it an accident or suicide. He was convicted in 1952 to chemical castration for having a gay relationship. Only in 2013 he was rehabilitated by a royal pardon. Some may also have an idea of what a Turing Test is. A mathematician or a computer scientist will almost certainly also know that he proved independently but almost simultaneously with Alonso Church that Hilbert's <em>Entscheidungsproblem</em> was unsolvable. Turing proved it by reducing it to a halting problem which is undecidable on a universal Turing Machine. Many books and even films tell the story of Turing and of all the activities at Bletchley Park. The Turing Centenary Year 2012 which triggered the publication of many more and the recent (loosely biographical) film <em>The Imitation Game</em> (2014) have spread the knowledge about Turing in a broader audience. Bletchley Park may now be a major tourist attraction park, but the confidentiality that was kept by the British authorities about what was developed there during the war concerning cryptanalysis and the early digital computers has delayed the historical disclosure of the role played by Turing and other scientists in that period. Somewhat less known, but very familiar to biologists is Turing's work on morphogenesis which he developed during a later stage in his life. The book has eight parts that cluster papers about eight different aspects of Turing's life and legacy.</p>
<p>
Thus Turing was much more than just a codebreaker. His universal machine was an essential theoretical model in proving results about the foundations of mathematics, logic, and computer science. Because of his work at Bletchley Park while the first digital computing machines were being assembled during and just after the war, he was intensively involved in writing original software, a user's manual, and he has even contributed to the design of circuits and hardware. The introduction of machines that could be instructed to perform less trivial tasks raised concern about the future of Artificial Intelligence and Turing contributed with several variants of his Turing test in an attempt to define what intelligence really meant. He called his ultimate version of 1950 the 'imitation game'.</p>
<p>
It should not be forgotten, that, even though his scientific interest and contributions are broad, Turing was fundamentally a mathematician. It is less known that his Kings College Fellow Dissertation (1935) involved a proof of the Central Limit Theorem. It was little known that this was proved already in 1920 by Jarl Lindeberg and so Turing's result was never published. He also worked on group theory, in particular the word problem, on number theory (the Riemann hypothesis and normal numbers) and of course the code breaking involved statistical analysis and hypothesis testing. Turing exploited these statistics in his algorithms Banburismus and later Turingery. After the war he was also doing numerical analysis (LU decomposition, error analysis,...). His work on morphogenesis was also mathematical and involved diffusion equations that model the random behaviour of the morphogenes.</p>
<p>
This collection of papers is produced for an interested but general audience. Formulas are kept to a minimum and technical discussion is maintained at an accessible level. It may not be the best choice to read as a first introduction to Turing and his work. Better introductions that are less chopped up in different papers are available. On the other hand, if you have read already several books about Turing and his work, I am sure you will find here some anecdotes and historical facts that you did not know yet in each of the eight parts of the book.</p>
<p>
A first part is biographical. The timeline by Copeland is useful to place everything in a proper historical sequence. There is a testimony of Sir John Dermot Turing, Alan's nephew, and another by the late Peter Hilton an Oxford professor who worked with Turing at Bletchley Park.<br />
Part two is more history in which Copeland explains about the Universal Turing Machine conceived by Turing to solve the Entscheidungsproblem. It has also a noteworthy contribution by Stephen Wolfram, the creator Mathematica and Wolfram-alpha, who praises Turing for initiating computer science.<br />
The third part is the most extensive one and puts the codebreaking and Bletchley Park in the spotlight. Some of the texts are by people who worked there and who give an account of how everyday life was during the war, other papers are explaining how the Enigma machine worked and how it could be broken.<br />
In part four the first computers as they developed after the war are in the focus. The Colossus machines were computers that were used since 1943 for codebreaking, These facts were only declassified in 2000 so that one got the impression that the original ideas and prototypes came from von Neumann at Princeton who developed the ENIAC and the EDVAC. However, the University of Manchester had a small scale computer <em>Baby</em> (1948) that was running a few months before the ENIAC and Turing at the National Physical Laboratory developed the Automatic Computing Engine (ACE) that was operational in 1950. Turing even wrote a manual on how to program the machine to play musical notes.<br />
The fifth part is about computers and the mind: chess computers, neural computing, and the working of the human brain. It also has a remarkable text by novelist David Leavitt about Turing and the paranormal.<br />
The next two parts are about Turing's biological (morphogenesis) and mathematical (cf. supra) contributions. The final part has two papers contemplating the Turing thesis (1936) which claims that a Turing machine can do any task a human computer can do. Similar claims were made by Zuse and Church, but whether the whole universe can be seen as a computer, obviously depends on what you call a computer.<br />
In the last chapter about Turing's legacy in different disciplines we find many references to books and other media that can be consulted for further information.</p>
<p>
The remaining pages offer a short biography of the contributors, references to some books about Turing, and a list of published papers by Turing. The many references and notes from the contributions are also gathered at the end. The book ends with a very detailed index, which is of course very welcome and obviously non-trivial with that many different authors.</p>
<p>
In summary, this is a welcome addition to the existing generally accessible literature that gives additional testimony of the brilliant mind of Alan Turing. There is historical as well as technical material that will be appreciated also by specialists whatever their discipline: history, mathematics, biology, computer science, or philosophy.</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 papers about Alan Turing, his life and legacy. It has biographical and historical details and explains the influence of Turing on codebreaking, artificial intelligence, computer science, mathematics, biology, and philosophy.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/jack-copeland" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jack Copeland</a></li><li class="vocabulary-links field-item odd"><a href="/author/jonathan-bowen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jonathan Bowen</a></li><li class="vocabulary-links field-item even"><a href="/author/mark-sprevak" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mark Sprevak</a></li><li class="vocabulary-links field-item odd"><a href="/author/robin-wilson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robin Wilson</a></li><li class="vocabulary-links field-item even"><a href="/author/et-al" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">et. al</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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">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">978-0-1987-4782-6 (hbk), 978-0-1987-4783-3 (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">£ 75.00 (hbk), £ 19.99 (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">576</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/the-turing-guide-9780198747833" title="Link to web page">https://global.oup.com/academic/product/the-turing-guide-9780198747833</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/03d10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03D10</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03b07" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03B07</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/68-06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68-06</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/68q05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68Q05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/92c15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92C15</a></li></ul></span>Tue, 13 Mar 2018 07:33:47 +0000Adhemar Bultheel48322 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/turing-guide#commentsThe Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love
https://euro-math-soc.eu/review/calculus-happiness-how-mathematical-approach-life-adds-health-wealth-and-love
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The subtitle explains more clearly what the book is about: <em>How a mathematical approach to life adds up to health, wealth, and love</em>. It is thus one of these books showing to the layperson how mathematics <em>can</em> be used in everyday life (not necessarily how it <em>is</em> used in practice). Therefore the mathematics are really elementary. Unlike similar books, written with the same purpose, here the health, wealth and love take up some serious part of the pages, and give only little mathematics in return.</p>
<p>
Let's start with the health. That subject has two chapters: one on the calories you take in and burn and one about the composition of your diet. What you get to digest for mathematics is a weighted linear sum of components such as your age, weight, and height that are influencing your metabolic rate, your calorie burning, or your cholesterol ratio. A simple quadratic defines your maximal heart rate as a function of age, and the expected years of life loss as a function of waist to height ratio.</p>
<p>
The second part has the promising title <em>A mathematician's guide to manage your money</em>. This also has two chapters. One is about managing your budget and the second about financial transactions like saving and investing. The mathematics we learn here is that taxes are computed in a linear way but only within certain intervals, so that it is actually a piecewise linear function. Also we learn what a compound interest rate is (or inflation rate in this case) and this leads to Euler's constant e and consequently also to the logarithm. A glimpse at the financial markets is the occasion to introduce some statistical concepts like average and standard deviation.</p>
<p>
The 'love' part introduces a formula to compute the number of possible dating candidates, and the well known 37 percent rule which states that if you need to select the best one (for example partner among the candidates) in a sequence, then you should first register the best candidate among the first 37% of the sequence and then take the first one that is better than that one. It also describes the Gale-Shapley algorithm to solve the stable matching problem. The last chapter is mathematically the most involved one of the book and analyses the relation between two persons as a dynamical system described by two simple differential equations. Also the Nash bargaining problem is discussed in which the optimalization of the quadratic Nash product has to be found when the couple has to come to a joint decision.</p>
<p>
Most of the mathematical derivations and computations are removed from the text and are summarized in appendices and if you want to apply it to your own life, you don't even need a pocket calculator because the publisher's web page has a link to online apps that will evaluate the formulas for you when you introduce your data. Each chapter also ends with a summary of the mathematical and nonmathematical takeaways. If you are interested in one of the topics, further reading is provided. Indeed, all the equations and methods described here are abstractions and usually drastic simplifications of reality. Therefore I would also like to refer to a don't-try-this-at-home type of warning that Fernandez provides in the introduction: if you want to implement major changes in your life based on the methods presented in this book, be sure there is an expert (like for example your medical doctor) to assist you and give good advise.</p>
<p>
I doubt that the noble hope of the author, which is that by reading this book the reader will adopt a mathematical approach to life, shall be fulfilled. The mathematics are really precalculus, while the problems like composing a diet, financial investment, and finding a partner for life, do not seem like the problems one is facing at the age one is brought in contact with the required precalculus. Somehow I think that the level of the applications and the level of the mathematics do not match well. There are however still wise lessons to learn from the book which anybody (certainly journalists and politicians) should know. For example one should have the numeracy to know that doubling the price of a sandwich over 10 years, does not mean that the inflation is 10% per year. Also the mathematical techniques shown here do not only apply to the three main topics enumerated above, but they are also applicable in many other situations, like an optimal selection of a secretary or the best way to subdivide a pizza among a number of hungry children.</p>
<p>
I believe it would take a student already interested in mathematics to be sincerely attracted to reading the book. On the other hand, teachers may find inspiration in some of the examples to use these as illustrations in their teaching. Or perhaps the mathematics that are used in the book may be an inspiration for them to apply it in perhaps similar applications that are more adapted to their particular set of students.</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 another book showing the use of mathematics in everyday life. The mathematics are rather elementary and include simple functions like linear, quadratic, or cubic at most (in relation with calorie consumption, or composing a diet), the computation of interest or inflation and the logarithm as well as mean and standard deviation (in connection with managing a budget or investment) and the 37% rule for making an optimal selection in a sequence, an algorithm for the stable matching problem and the Nash bargaining problem (to solve partnership and relational problems).</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/oscar-e-fernandez" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Oscar E. Fernandez</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">9780691168630 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">19.95 £ (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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10952.html" title="Link to web page">http://press.princeton.edu/titles/10952.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Thu, 20 Apr 2017 13:35:20 +0000Adhemar Bultheel47635 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-happiness-how-mathematical-approach-life-adds-health-wealth-and-love#commentsBeyond Infinity: An Expedition to the Outer Limits of Mathematics
https://euro-math-soc.eu/review/beyond-infinity-expedition-outer-limits-mathematics
<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>
Eugenia Cheng is a professor of mathematics whose research field is higher dimensional category theory. She has made it one of her missions to counter mathphobia. Her credo is that mathematics is not the difficult part to deal with in life, but that on the contrary it is life that is difficult and mathematics helps us to make it simpler and manageable. She has tried to illustrate this by combining her love for cooking and her passion for mathematics in her previous book <em>Cakes, Custard + Category Theory</em> (reviewed <a href="/review/cakes-custard-category-theory" target="_blank">here</a>). In that book she gave attention to mathematics alright, but there were also proper recipes for cooking. The latter are interesting if you love cooking yourself and they are a springboard to make a link towards mathematics, but they do not really help to understand category theory.</p>
<p>
In the present book however she is explaining a really important mathematical concept: infinity, and it is far from being the simplest one to explain for non-mathematicians. The approach here is that she does just that. Not like in her previous book where she placed cookery next to the mathematics. Here of course Chen is still Chen and she still can't hide her love for cooking and category theory. However cooking is now only used as an anecdote or as and introduction to a chapter, just like perhaps a hiking experience, of a concert she attended, can be.</p>
<p>
So what is this book about? The first part is intended to explain what infinity really is, and it soon becomes clear that it is not as simple as saying it is larger than any number one can imagine. It cannot be a number since the usual arithmetic rules do not work as with finite numbers. And then there are the paradoxes like the well known Hilbert hotel with infinitely many rooms that can always accommodate infinitely many more guests, even when it is fully booked. So Chen uses a more systematic approach introducing the simplest number systems: natural numbers, integers, and rationals. She goes even further and defines the natural numbers in the set theoretic style of Frege, only she does not use the abstract concept of a set, but she uses 'bags' instead. So 0 corresponds to the empty bag, 1 to the bag containing only the empty bag, 2 to the bag containing the two previous bags, etc. Also concepts like injection, surjection, and countable are introduced here.</p>
<p>
Then a stumble stone is blocking the development. It turns out that there are more than countably many real numbers. The reals are not properly defined yet, but using Cantor's diagonal argument, and using a binary representation, Chen shows that there are more irrational numbers than natural numbers. Thus there are gradations of infinity, at which point $\aleph_0$ is introduced. The 'smallest' infinity of a countable set, but there exist higher forms like $\aleph_1=2^{\aleph_0}$ the number of reals, and this can be iterated $\aleph_k=2^{\aleph_{k-1}}$. The continuum hypothesis is briefly touched upon, and it is noted that it can't be proved (Cohen) or disproved (Gödel). The distinction between ordinal and cardinal numbers clarifies the difficulty that infinity gives with the usual arithmetic operations.</p>
<p>
All this work in the first part of the book, leading to an understanding of what infinity actually is, is like a journey uphill. In a second part Chen points to the sights that are possible from the top of the hill. With the recursive definition of the natural numbers, a proof by induction is within reach and one can solve all sorts of counting problems and even evaluate infinite sums. Although the latter needs more careful consideration. She also introduces higher dimensions, i.e., larger than 2 or 3. It may even grow to infinity for a continuum. When a relation or a property is associated with a dimension, this brings her to her beloved research subject: higher dimensional category theory. Perhaps, this doesn't fit so well in the otherwise rather elementary exposition, but it is a nice, be it a somewhat unusual example, of a higher dimensional mathematical object.</p>
<p>
The move is then from the infinitely large to the infinitely small, leading back to infinite sums of diminishing terms and Zeno paradoxes. What is needed here is the concept of a limit. She however explains it essentially avoiding to use that name. Instead she illustrates the idea with hitting a target that becomes smaller and smaller. This way it can be explained what infinitesimals are and how they are applied. It can now be proved that the harmonic series diverges, and eventually also that irrational numbers do exist, which is done by approaching Dedekind's definition of the reals.</p>
<p>
I find this a very pleasing way of introducing some elementary, but also some less elementary, mathematical concepts to the layperson. Taking infinity as the carrot to lead the reader uphill is an interesting choice. This is the most essential concept needed when moving from algebra to analysis. Chen is an excellent guide to show the reader the way uphill. With many analogies and illustrations and reformulations it seems like the reader is carried to the top, no toiling required. The story is told fluently. Side remarks, historical notes, or a slightly more advanced remark are inserted as a framed boxes in the text. I guess it will be too elementary for mathematicians of mathematics students, but it is warmly recommended for secondary school pupils. In fact anyone who has the slightest interest in what infinity actually means should read it. The word is used lightly in common language, but you will learn what it means in a more exact sense and thus what it means to a mathematician. It turns out that it triggered the development of calculus and it has shaken the foundations of mathematics as recently as in the early 20th century.</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>
Eugenia Cheng continues her crusade against mathphobia. In this book she explores the meaning of infinity. To properly define infinity she has to define the cardinality of the natural numbers, and thus also the definition of the latter. That includes solving the inconsistencies with arithmetic operations and the paradoxes that result. However it turns out that the real numbers are not countable, so that there are gradations of infinity and hence also the definition of the reals is needed. That requires to consider the infinitely small, which leads to infinitesimals that form the onset of calculus. All is brought to the reader avoiding the usually boring technical approach of mathematics, but using many analogies and elementary everyday language.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/eugenia-cheng" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eugenia Cheng</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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">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">978-1781252857 (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 12.99 (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">316</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://profilebooks.com/beyond-infinity.html" title="Link to web page">https://profilebooks.com/beyond-infinity.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Thu, 20 Apr 2017 13:12:50 +0000Adhemar Bultheel47634 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/beyond-infinity-expedition-outer-limits-mathematics#commentsA Mathematics Course for Political & Social Research
https://euro-math-soc.eu/review/mathematics-course-political-social-research
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Nowadays, the research in political or social sciences is unavoidably connected with some mathematics. And very frequently, this mathematics are not just a mere presentation of data (a work that already requires some mathematical skills) but the construction of models, the elaboration of predictions or the delimitations of patters.<br />
This situation justifies the existence of a collection of math references tailored for researchers, or students, in political and social sciences. The book under review is one of them, specially designed to provide the reader the basic tools needed to tackle the initial problems in these fields. This basic approach can be also seen as the first step for those who will need to study further mathematical techniques from other sources, to be applied in more sophisticated investigations.<br />
The manual is divided into five parts. The first gives the basic vocabulary and building blocks in mathematics. The second explains the calculus in one dimension. The third is devoted to probability. The fourth explains basic linear algebra and the final part, multivariate calculus and Optimization. Each part is divided into several chapters, which are endowed with a very similar structure: They present motivation for the tools that are going to be introduced by means of examples in political and social sciences. Then, the techniques are introduced in the simplest way, avoiding unnecessary abstraction. Proofs are omitted. Each chapter ends with a collection of exercises. In addition, every section of the chapters is ended with a subsection called “Why Should I Care?”. These subsections represent a continuous bridge between the mathematics and the real situations to which the techniques can be applied. The authors, professors of political sciences, make a remarkable effort to motivate the theory with the problems that can be solved from them.</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">Marco Castrillon Lopez</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 contains a course on basic mathematics designed for students and researchers on political and social sciences.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/will-h-moore" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Will H. Moore</a></li><li class="vocabulary-links field-item odd"><a href="/author/david-siegel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David A. Siegel</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2013</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-15995-9</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">430</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span>Mon, 26 Dec 2016 12:28:48 +0000Marco Castrillon Lopez47350 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-course-political-social-research#commentsElements of Mathematics: From Euclid to Gödel
https://euro-math-soc.eu/review/elements-mathematics-euclid-g%C3%B6del
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In this book Stillwell explores the boundary between elementary mathematics and advanced mathematics. This boundary is not strict and some elementary topics are close to the boundary and some advanced topics are just across, while others are way beyond. So it makes sense to look for what exactly makes a topic advanced or not, even though the classification can be the subject of discussion making it a bit fuzzy in some cases, Stillwell has a rather clear vision on it. There is clearly also an historical aspect to this. Mathematics evolved and it is interesting to see where, in hindsight, the line was crossed. When it comes to foundations of mathematics or to more subtle differences, there is also a philosophical component to this. So Stillwell gives a survey (or rather a limited selection) of elementary mathematics, proving basic theorems with elementary tools, but from time to time he has to cross the line and he also treats some topics just across to explore where the boundary lies and to explain why some theorems or concepts are more advanced than others. The sections dealing with advanced elements are clearly indicated with a * in their title.</p>
<p>
In a first chapter he gives an introduction to the rest of the book, introducing the eight mathematical domains that form the successive subjects of the next chapters: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and finally logic. In a concluding chapter, some advanced topics in each of these subjects are discussed. Each chapter is concluded with historical and philosophical remarks. Although logic forms the foundation of mathematics, it does not come up first, but it is the last in the row because it was logic of the previous century that lifted mathematics in many different ways to a much more advanced level. On the other hand, computation comes up early in the list at the second place because it has become so important in modern society.</p>
<p>
Looking at educational systems, there are three main parameters that help distinguish between elementary and advanced: infinity, abstraction, and proof. About the latter Stillwell advocates to include at least some proofs in the elementary curriculum. Mathematics, even elementary without proof is a delusion. Mathematics without abstraction is unthinkable. One should be able to work with symbols instead of numbers. Much of algebra is elementary, yet assumes some abstraction. Infinity is a bit dual. There is a soft form of infinity that cannot be avoided. There are infinite processes in Euclidean mathematics, but infinite is not an existing object, it is just representing something going on indefinitely. The infinity of analysis relying on set theory is much harder. This is Stillwell's main criterion to draw the line. Whenever a concept relies directly or indirectly on infinity (the hard version like in the concept of real numbers) it is considered to be advanced. This is what he does in his last chapter. Some of the topics already discussed in the domains of the eight previous chapters are reconsidered and pushed a but further usually by allowing some concept relying on infinity which lifts them to the advanced side of the boundary.</p>
<p>
I will give some illustrations of where Stillwell draws the line in the eight chapters:<br />
Elementary <em>arithmetic</em> contains topics such as prime numbers, finite arithmetic, Gaussian integers, and the Pell equation.<br />
Under <em>computation</em> Stillwell starts with addition, multiplication and exponentiation, possibly in binary arithmetic. These are obviously very elementary. Also the P-NP problem, and Turing machines are still elementary but close to the boundary, while universal Turing machines and analysis of unsolvable problems cross the line.<br />
Elementary <em>algebra</em> contains rings, fields (number fields and polynomial rings) and vector spaces. Groups (drops commutativity of the multiplication from the ring, hence is less natural) and the fundamental theorem of algebra (requires reals or complex numbers) are advanced.<br />
In Euclid' <em>geometry</em>, it is about constructions with compass and straightedge, but proofs about irrational numbers could be given geometrically. Doubling an square (area) is possible, but not for the cube (volume), which means that a volume is 'more advanced' or 'less elementary' than area. Linear algebra arithmetizes geometry and it requires a vector space with inner product to do elementary geometry. Also constructible number fields are elementary. However non-Euclidean and projective geometry are not.<br />
<em>Calculus</em> obviously needs infinity, but only the `soft version' in its elementary part. Infinite series, and the concept of derivative and elementary integration but basically in their geometric meaning of tangent and area under a curve are elementary. Stillwell is carefully about integration which in its elementary version is only restricted to integrals of rational functions which give all the elementary functions and their inverses (logarithm, exponentials, trigonometric,...). The fundamental theorem of calculus introducing the primitive function is advanced, but a simple proof of the irrationality of e is elementary. Anything relying on the completeness of the reals (e.g. continuity) is advanced.<br />
Under <em>combinatorics</em> we find topics usually described as discrete mathematics. Proving there are infinitely many primes, Fermat's little theorem, binomial coefficients, Fibonacci numbers and generating functions are all elementary, as are several elements from graph theory, including Euler's polyhedron formula. It becomes advanced when the graphs become infinite and one has to deal with Kőnig's infinity lemma, Bolzano-Weierstrass theorem, and Brouwer's fixed point theorem. All these need the `hard version' of infinity.<br />
<em>Probability</em> theory starts from combinatorics (the binomial coefficients gives approximations of the bell shape), random walk, mean, variance and standard deviation and the law of large numbers for coin tossing and random walks are all on the elementary side. You stay there as long as there is no concept of limit or measure.<br />
Finally, <em>logic</em> can include propositions, quantifiers, Boolean algebra and induction. The latter basically relies on the definition of the natural numbers and can therefor be considered elementary, but most of the concepts introduced here are advanced like Peano arithmetic, the real numbers, countability, infinity, set theory. In fact reverse mathematics is an advanced topic that studies which axioms are needed for proving a theorem. This implies that it can somehow classify why some theorems are more advanced than others if they require more or more complicated axioms.</p>
<p>
Clearly the subject to be discussed very broad, and there is definitely a limited selection of what can be said in just one book about defining a line bordering elementary mathematics. The line will probably remain fuzzy anyway. Since there is no comparable book, there is no list of references for reading on this boundary subject. There are few occasions where the reader is referred to for more information on a specific topic, but the long list of references consists mostly of original historical references. The subject index is also very complete, which is very much appreciated since there is obviously interference between the different chapters. Writing a mathematics book without a single typo is probably impossible. I found a blatant one on page 238 where it is said that $x^n$ goes to zero when $|x|<0$ where it should obviously be $|x|<1$. There probably are some more as in any first edition, but that hardly diminishes the overall quality of the text. It's a privilege and a pleasure to follow the pros and cons of Stillwell's arguments.</p>
<p>
Most of what is described as elementary will be familiar if the reader has had a good mathematical secondary school course. Some of the advanced topics probably require some mathematics from a first year university course. It may give such a student an idea of what else there is to follow in his or her further mathematical education. However, I think the book is at least also (or perhaps even more so) of interest to professional mathematicians. The views and approaches that are presented by Stillwell are sometimes rather unconventional, and I am certain there will be some new unexpected connections or proofs to be discovered. The historical endnotes of the chapters are interesting but standard. In my opinion, the philosophical endnotes are often very interesting because this is where I recognize mostly Stillwell's idea that he borrows from Klein: "elementary mathematics from an advanced standpoint" and that he used as a <em>leitmotif</em> for this 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 Stillwell tries to define a boundary between what may be considered elementary and what is advanced in mathematics. It is Stillwell's opinion that infinity and all concepts depending on it, and that includes e.g. a definition of the real numbers, seems to be a good parameter to identify a topic as advanced.<br />
</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/john-stillwell" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Stillwell</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691171685 (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">USD 39.95 (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">440</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10697.html" title="Link to web page">http://press.princeton.edu/titles/10697.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97d99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97D99</a></li></ul></span>Tue, 05 Jul 2016 13:41:44 +0000Adhemar Bultheel47037 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/elements-mathematics-euclid-g%C3%B6del#comments