European Mathematical Society - Ron Aharoni
https://euro-math-soc.eu/author/ron-aharoni
enCircularity. A Common Secret to Paradoxes, Scientific Revolutions and Humor
https://euro-math-soc.eu/review/circularity-common-secret-paradoxes-scientific-revolutions-and-humor
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The title of this book refers to circular reasoning. The snake that bites its own tail. This may play tricks on us with the logic of our arguments and it will almost always result in a paradox. In our binary Western culture we are used to things being either true or false. Circular arguments make us believe that our arguments result in a proposition being true, but that then implies that the same proposition must be false which implies that it must be true etc ad infinitum. A paradox. The simplest examples of such statements are "This sentence is false" or "I am always lying". These are examples that everybody knows and where the circularity if blatantly obvious. However, the same kind of circularity can be much more subtle and well hidden, which often leaves us perplexed by a confusing paradox.</p>
<p>
Circularity can be both a curse and a blessing. It is a curse when it creates problems that are actually no problems or definitions that are non-definitions so that the wrong conclusions are drawn. On the other hand, when a paradox is a challenge that pokes the minds of bright philosophers or scientists to develop the proper framework to solve the paradox, it can be a blessing. In this book, Aharoni shows us both sides, the dark and the illuminating side of circularity.</p>
<p>
Aharoni starts by giving several examples of paradoxes and points to the (sometimes hidden) circularity. What is tricking us is the self-reference. The solution of the paradox is often that we should not look at the proposition from the inside but we should place ourselves at the outside. It may then become clear that the proposition does not refer to itself but to something that is different from itself like in the above examples or a more explicit example: "the smallest number not defined by this sentence". More paradoxes can arise from logical errors, like if X is not true, then the implication "if X then Y" is always true, whether or not Y is true. It is wrong to argue that if Y is true, then X must also be true, just because the implication holds. This is an unpermitted inversion of the given implication. Zeno's paradox is based on the wrong assumption that an infinite sum of finite numbers must be infinite. The self-reference can be clarified by an analogy. Start with one grain of sand, which certainly is not a heap of sand. Keep adding one grain at a time. There is not a particular point where the set of grains magically turns into a heap of sand. So heaps of sand do not exist. Obviously a wrong conclusion. With every grain added, the definition of heap keeps shifting in our head. That's where the circularity is. Tricked by circularity, one can prove anything: the existence of the monster of Loch Ness or whatever. Some have tried to prove in this way the existence of God.</p>
<p>
Next Aharoni introduces the reader to the problem of free will by starting with the Newcomb paradox. You have a 100 dollar bill and can drop it into a deep well or not. A never failing oracle says that it has predicted what you will do and acted accordingly in the past. A 1000 dollar was deposited on your bank account if you drop the 100 dollar. Nothing has been deposited if you keep it. What should you do? Can one change the past by doing something now? If not, you should keep the bill, if you believe the oracle, you should drop it. The origin of the dilemma was a problem in game theory. Formulated as above, it directly leads to the dilemma whether everything is predetermined and thus nobody is responsible for whatever he or she does (fatalism and the idle argument), or is there actually free will. Where is the circularity? Aharoni explains that past and future are linked through decisions that we make in the present. What causes the paradox is that we are trying to make a decision about the process of making a decision, which is self-referencing. Only Baron von Munchausen could pull himself up by his own hair. The deliberation is about a causal chain involving the deliberation itself.</p>
<p>
The mind-body problem is similar. How one should link the non-physical sensation of pain and the physical event of the pricking of a needle? Here the circularity of all the philosophers discussing the mind-body gap is that they are talking about sensations in other people's minds, hence they refer to consequences or externalization of the sensation, which is quite different from the sensation itself. The sensation exists only in your own mind. It can only be experienced from the inside. You may observe a person having a sensation, which is different from observing the sensation into yourself. When the observed person and the observer are the same, then the observation of the sensation becomes the sensation itself.</p>
<p>
So far for the dark side of paradoxes that play tricks on us. The second half of the book is about the bright aspects by bringing an account of stories about theories that resulted from successful attempts to resolve a paradox. These theories are better known to mathematicians and do not need an extensive clarification here. Aharoni however is addressing a general public. So he carefully explains countability and the diagonalization process to show that the rational numbers are countable and that the reals are not. Also the foundation of set theory was clarified thanks to the well known self-referencing Russell paradox of the set of all sets that are not a member of themselves. Zermelo and Fraenkel came to the rescue of the barber who cuts the hair of all the villagers who do not cut their own.</p>
<p>
Pushing the limit somewhat further, after the introduction of Boolean algebra and the formality added by Frege, the question rose to mechanically derive all possible theorems. A genuine target it seems and Russell and Whitehead devotedly set to the task of writing their <em>Principia Mathematica</em>. It was however the incompleteness theorem of Gödel that caused a Copernican revolution in mathematics. Solving another case of self-reference: the (mathematical) reasoning about (mathematical) reasoning. The liar's paradox, equivalent with "this sentence is false" became "this sentence is not provable". Gödel's theorem is closely related to the mechanical (algorithmic) verification of the validity of a proposition, which links it to the Halting Problem. There the question is whether the the machine stop verifying the validity of its input after a finite number of steps or will it run forever? This was solved by Turing (with his Turing machine) and independently by Church (using Lambda calculus). The Turing machine is the theoretical machine that later became the general purpose computer. A most exciting serendipity if this may be considered the consequence of solving a paradox.</p>
<p>
In an appendix some further explanation is given for those who feel a bit more mathematically inclined, explaining in more technical terms the diagonalization procedure and the proof that the reals are not countable. It is also clarified that there is no contradiction between Gödel's completeness theorem (of the Frege-Russell-Whitehead system) and his incompleteness theorem (of the set of Peano axioms).</p>
<p>
The whole book is written in small chunks of only a few pages that introduce simple concepts, each one easily digestible, but at the subtle points one has to stay focussed not to loose track. No mathematical knowledge is assumed, not even for the appendices, but of course a straight mind to make sound logical deductions is needed. Much more than in the brief outline given above, Aharoni relates the material discussed to historical ideas and philosophers. It is pleasant reading, with a whiff of humor. The humor coagulates especially in a longer section where Aharoni discusses why this self-reference shows up in a particular type of jokes, and why we think of it as funny.</p>
<p>
I did enjoy reading (and re-reading) this book very much. Reading it deserves a warm recommendation not only for mathematicians but for anybody (especially if you have the slightest interest in logic). Aren't we all are reasonable beings that are supposed to handle according to some logic most of the time. I can safely end with a self-referencing conclusion: this book makes you think about how and what you think you are thinking. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The circularity of the title refers to circular reasoning or self-reference which leads to paradoxes. Aharoni uses this to solve the problem of determinism vs free will and for the mind-body problem. Some paradoxes have lead to scientific breakthroughs like Cantor's set theory, Gödel's incompleteness theorem, and Turing's solution of the Halting Problem. He even analyzes the humor in self-referencing jokes. </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/ron-aharoni" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ron Aharoni</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4723-68-8 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">23.00 GBP (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">180</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></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/9805" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9805</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/03-mathematical-logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03 Mathematical logic and foundations</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/03-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03-01</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/03d10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03D10</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03d05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03D05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/03f40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03F40</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11u05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11U05</a></li></ul></span>Mon, 08 Aug 2016 12:12:33 +0000Adhemar Bultheel47100 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/circularity-common-secret-paradoxes-scientific-revolutions-and-humor#commentsMathematics, Poetry and Beauty
https://euro-math-soc.eu/review/mathematics-poetry-and-beauty
<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 idea that there is a parallel between mathematics and art is not new and many have tried to express the experience of beauty in both, but it is not easy to make it explicit and to convey to nonbelievers. For visual arts, beauty is related to order, patterns, and symmetry, but not too much either, so that it doesn't become dull and boring. Also the link between music and mathematics has been explored a lot ever since mathematics was around. The link with poetry is less explored, and this is what this book is about.</p>
<p>
Clearly, if this book should be readable by mathematicians <em>and</em> poets, the mathematics can't be too complicated. What we find here are often the usual topics that one may find in books on popular, hence easily understandable, mathematics such as games, history, and paradoxes. There is however also some probing in more difficult topics like topology, transcendental numbers, Gödel's incompleteness theorems, etc., <em>and</em> there are even proofs. Brilliantly elusive ones! Of course, each time a parallel is drawn between some mathematical concept or tool and the corresponding poetical one.</p>
<p>
There are three parts in the book: (1) Order, (2) How mathematicians and poets think, (3) Two levels of perception.<br />
The first part about order is clearly an attempt to define what arises our sense of pleasure or beauty in mathematics and in poetry. One element is that sometimes a seemingly complicated problem all of a sudden can become an stunningly simple solution by looking at it from a totally different angle. But the opposite is also true: sometimes a very simple conjecture may require a proof that is totally out of proportion. Historical examples of the latter are the proofs for trancendence of some real numbers, but also sphere packing, or as yet unsolved ones like the Collatz conjecture, and twin prime conjecture.<br />
In this part one may also find a statement that not everybody may agree with. It is a classical subject of discussion: is mathematics discovered or invented? Aharoni says that mathematics is discovered and poetry is invented. His argument is that if a mathematician misses a theorem or a proof, then sooner or later another one will discover it. However, if a poem is not written, then it will be lost forever. Hence Aharoni is a supporter of formalism like Einstein, Hilbert, Cantor, etc., but there are equally illustrious mathematicians like Hardy, Penrose, and Gödel who thought otherwise. Personally, I am more a believer that mathematics is a result of both components like also Mario Livio does in <em>Is God A Mathematician?</em> (2009).</p>
<p>
In the second and most extensive part, Aharoni explores the minds of both mathematicians and poets. Both their minds use images, sometimes use an oblique approach, compress statements, and they play a ping-pong game between the concrete and the abstract until some leap of insight erupts. Some mathematical proofs rely on a law of conservation like for example the number of transpositions in a sequence being even does not change no matter how much exchanges you make. The `conservation of truth' in poetry is related to fate like the inevitable fatum in Greek tragedies. The metaphor is a form of cross fertilization mixing concepts from different senses. It is quite often used in poetry, but this kind of insights can also be obtained in mathematics for example by solving a problem in discrete mathematics by applying a theorem from topology. Fantasy, imagination, and analogies are main ingredients of poetry as well as mathematics. However, care must be taken in mathematics, not to generalize too easily based on intuition and analogy. Tautologies and symmetries are are familiar to both the poet and the mathematician. With the hyperbole, one approaches the impossible, the infinitely small or infinitely large, so that paradoxes and oxymorons may be imminent, like the ones dealt with by Cantor and Gödel.</p>
<p>
In the last short part, the tension between what we know and what we do not completely understand is explored. A poem may be confusing, extracting beauty from the strange and the unknown. Progress in mathematics is only possible by leaving the familiar and the habitual. When the known framework leads to results causing alianation and estragement, a leap of faith into the unknown becomes inevitable. After a while however, it will turn out to be a step that brings us one rung higher on the ladder of understanding.<br />
After all these parallels between mathematicians and poets have been explored, Aharoni places ithis a bit in perspective because one obvious difference between both is that mathematicians most often collaborate and learn from each other, while creating a poem is a lonely and individual activity.</p>
<p>
The book is the English translation of the original Hebrew edition from 2008. Of course there are many poems or quotes from poems included, most often in an English translation. Several of them are analyzed in view of the point that the author wants to illustrate. Although he has written several books with a philosophical inclination, he is a mathematician and I believe this book is in the first place written by a mathematician for mathematicians. In any case the mathematical aspects treated in the book are more elaborated than the poetical aspects. It is not a deep theoretical philosophical work, but basically a collection of exemplary illustrations of the point the author wants to make. As I mentioned above, many of the examples are also found in other popular math books, but it is of course most inspiring to see them placed against a background of poetical analogs and vice versa. Apart from the fact that the details and the history of the mathematical stories are most informative, and without them, the book would be skinny and probably not very enjoyable, I do not always see their direct relevance to make the link with poetry. There are some appendices explaining some of the mathematical terminology and poetical terms and figures of speech, but since there are so many different facts and persons appearing, it is difficult to trace them if you pick up the book at a later stage. It would have been nice to have a subject index to facilitate that. But I very much enjoyed reading the book the first time though.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
A parallel is drawn between mathematics and poetry. Both have order, symmetry, analogies, etc., and there are similarities in the way a mathematician and a poet think. This is a translation of the Hebrew original of 2008.</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/ron-aharoni" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ron Aharoni</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4602-93-8 (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">GBP 55.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">300</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/9160" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9160</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</a></li></ul></span>Thu, 05 Mar 2015 07:47:54 +0000Adhemar Bultheel46085 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-poetry-and-beauty#comments