logic
https://plus.maths.org/content/taxonomy/term/514
enWhy we want proof
https://plus.maths.org/content/brief-introduction-proofs
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Marianne Freiberger </div>
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<p>What are mathematical proofs, why do we need them and what can they say about sheep?</p>
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<p>A proof is a logical argument that establishes, beyond any doubt, that something is true. How do you go about constructing such an argument? And why are mathematicians so crazy about proofs?</p>
<h3>Which way around?</h3>
<div class="rightimage" style="width: 300px;"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/articles/2015/proof/sheep.jpg" alt="Sheep" width="300" height="300" />
<p>What can maths prove about sheep?</p><p><a href="https://plus.maths.org/content/brief-introduction-proofs" target="_blank">read more</a></p>https://plus.maths.org/content/brief-introduction-proofs#commentsFP-carousellogicproofFri, 10 Apr 2015 08:33:03 +0000mf3446328 at https://plus.maths.org/contentIs the Universe simple or complex?
https://plus.maths.org/content/universe-simple-complex
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Faye Kilburn </div>
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<p>On the face of it the Universe is a fairly complex place. But could mathematics ultimately lead to a simple description of it? In fact, should simplicity be a defining feature of a "theory of everything"? We ponder the answers.</p>
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<p>There are at least 500 billion planets in the Milky Way.</p><p><a href="https://plus.maths.org/content/universe-simple-complex" target="_blank">read more</a></p>https://plus.maths.org/content/universe-simple-complex#commentsmathematical realitycomplexityemergent behaviourlogicphilosophy of mathematicsMon, 14 Jan 2013 09:29:55 +0000mf3445851 at https://plus.maths.org/contentThis is not a carrot: Paraconsistent mathematics
https://plus.maths.org/content/not-carrot
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Maarten McKubre-Jordens </div>
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Paraconsistent mathematics is a type of mathematics in which contradictions may be true.
In such a system it is perfectly possible for a statement <em>A</em> and its negation <em>not A</em> to both be true. How can this be, and be coherent? What does it all mean? </div>
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<p>Paraconsistent mathematics is a type of mathematics in which contradictions may be true.
In such a system it is perfectly possible for a statement <em>A</em> and its negation <em>not A</em> to both be true. How can this be, and be coherent? What does it all mean?
And why should we think mathematics might actually be paraconsistent? We'll look
at the last question first starting with a quick trip into mathematical history.</p><p><a href="https://plus.maths.org/content/not-carrot" target="_blank">read more</a></p>https://plus.maths.org/content/not-carrot#commentsmathematical realityGödel's Incompleteness Theoremhalting problemimpossible objectlogicphilosophy of mathematicsRussell's Paradoxwhat is impossibleWed, 24 Aug 2011 07:42:07 +0000mf3445522 at https://plus.maths.org/contentThe philosophy of applied mathematics
https://plus.maths.org/content/philosophy-applied-mathematics
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Phil Wilson </div>
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<p>We all take for granted that mathematics can be used to describe the world, but when you think about it this fact is rather stunning. This article explores what the applicability of maths says about the various branches of mathematical philosophy.</p>
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<p>I told a guest at a recent party that I use mathematics to try to understand migraines. She thought that I ask migraine sufferers to do mental arithmetic to alleviate their symptoms. Of course, what I really do is use mathematics to understand the biological causes of migraines.</p><p><a href="https://plus.maths.org/content/philosophy-applied-mathematics" target="_blank">read more</a></p>https://plus.maths.org/content/philosophy-applied-mathematics#commentsmathematical realityconstructivist mathematicsinfinitylogicphilosophy of mathematicsplatonismwhat is impossibleFri, 24 Jun 2011 09:35:32 +0000mf3445497 at https://plus.maths.org/contentTuring's papers stay at home
https://plus.maths.org/content/turings-papers-stay-home
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<p>Almost nothing tangible remains of the legendary Bletchley Park codebreaker Alan Turing. So when an extremely rare collection of papers relating to his life and work was set to go to auction last year, an ambitious campaign was launched to raise funds to purchase them for the Bletchley Park Trust and its Museum. The Trust has announced today that the collection has been saved for the nation as the National Heritage Memorial Fund (NHMF) has stepped in quickly to provide £213,437, the final piece of funding required.</p>
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<p><a href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Turing.html">Alan Turing</a> was one of the twentieth century's most influential mathematicians. He's regarded as the father of modern computer science, played a vital part in breaking the Germans' Enigma code during WW2, fundamental to the Allied victory, and his work in mathematical logic penetrated to the very foundations of maths. Arguably, his work has touched more lives than that of most other mathematicians. </p><p><a href="https://plus.maths.org/content/turings-papers-stay-home" target="_blank">read more</a></p>https://plus.maths.org/content/turings-papers-stay-home#commentsAlan TuringBletchley Parkcomputer scienceEnigmalogicFri, 25 Feb 2011 10:59:47 +0000mf3445432 at https://plus.maths.org/contentPicking holes in mathematics
https://plus.maths.org/content/picking-holes-mathematics
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Marianne Freiberger </div>
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<p>In the 1930s the logician Kurt Gödel showed that if you set out proper rules for mathematics, you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite contrived. But are they about to enter mainstream mathematics?</p>
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<p>Kurt Gödel</p><p><a href="https://plus.maths.org/content/picking-holes-mathematics" target="_blank">read more</a></p>https://plus.maths.org/content/picking-holes-mathematics#commentsmathematical realityaxiombinary treeGödel's Incompleteness Theoremgraphgraph theorylogicphilosophy of mathematicstreewhat is impossibleZermelo-Fraenkel axiomatisation of set theoryWed, 23 Feb 2011 10:00:00 +0000mf3445413 at https://plus.maths.org/contentSpaceships are doing it for themselves
https://plus.maths.org/content/spaceships
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<p>It requires only a little processing power, but it's a giant leap for robotkind: engineers at the University of Southampton have developed a way of equipping spacecraft and satellites with human-like reasoning capabilities, which will enable them to make important decisions for themselves.</p>
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<div class="packagebacklink">Back to the <a href="https://plus.maths.org/content/ingenious-constructing-our-lives">Constructing our lives package</a></div><br clear="all"><p>It requires only a little processing power, but it's a great leap for
robotkind: engineers at the University of Southampton have developed a
way of equipping spacecraft and satellites with human-like reasoning
capabilities, which will enable them to make important decisions for
themselves.<p><a href="https://plus.maths.org/content/spaceships" target="_blank">read more</a></p>https://plus.maths.org/content/spaceships#commentsmathematical realityartificial intelligenceengineeringlogicspace explorationMon, 21 Feb 2011 10:00:00 +0000mf3445425 at https://plus.maths.org/contentSearching for the missing truth
https://plus.maths.org/content/searching-missing-truth
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Marianne Freiberger </div>
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<p>Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. But it's not quite like that. You can actually build different versions of maths in which statements are true or false depending on your preference. So is maths just a game in which we choose the rules to suit our purpose? Or is there a "correct" set of rules to use? We find out with the mathematician Hugh Woodin.</p>
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<p>Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. </p><div class="rightimage" style="width: 150px;"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/articles/2011/woodin/woodin.jpg" alt="" width="150" height="224"
<p>Hugh Woodin.</p><p><a href="https://plus.maths.org/content/searching-missing-truth" target="_blank">read more</a></p>https://plus.maths.org/content/searching-missing-truth#commentsmathematical realitycontinuum hypothesisGödel's Incompleteness Theoreminfinitylogicphilosophy of mathematicsset theorywhat is impossiblewhat is infinityZermelo-Fraenkel axiomatisation of set theoryFri, 28 Jan 2011 19:09:07 +0000mf3445398 at https://plus.maths.org/contentConstructive mathematics
https://plus.maths.org/content/constructive-mathematics
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Phil Wilson </div>
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If you like mathematics because things are either true or false, then you'll be worried to hear that in some quarters this basic concept is hotly disputed. In this article <b>Phil Wilson</b> looks at <i>constructivist mathematics</i>, which holds that some things are neither true, nor false, nor anything in between. </div>
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<p>Before the world awoke to its own finiteness and began to take the need for recycling seriously, one of the quintessential images of the working mathematician was a waste paper basket full of crumpled pieces of paper. The mathematician sits behind a large desk, furrowed brow resting on one hand, the other hand holding a stalled pencil over yet another sheet of paper soon to be crumpled and
discarded.</p><p><a href="https://plus.maths.org/content/constructive-mathematics" target="_blank">read more</a></p>https://plus.maths.org/content/constructive-mathematics#comments49binary logicconstructivist mathematicsintuitionist mathematicslaw of excluded middlelogicphilosophy of mathematicswhat is impossibleMon, 01 Dec 2008 00:00:00 +0000plusadmin2349 at https://plus.maths.org/contentCantor and Cohen: Infinite investigators part I
https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-i
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Richard Elwes </div>
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What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? In this article <b>Richard Elwes</b> explores how these questions brought triumph to one man and ruin to another, ventures to the limits of mathematics and finds that, with infinity, you're spoilt for choice. </div>
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<h1>The axiom of choice</h1>
<p><i>This is one half of a two-part article telling a story of two mathematical problems and two men: Georg Cantor, who discovered the strange world that these problems inhabit, and Paul Cohen (who died last year), who eventually solved them. The first of these problems — the axiom of choice — is the subject of this article, while the <a href="/issue47/features/elwes2">other article</a>
explores what is known as the continuum hypothesis.<p><a href="https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-i" target="_blank">read more</a></p>https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-i#comments47axiomaxiom of choicehistory of mathematicsinfinitylogicphilosophy of mathematicsRussell's Paradoxset theorywhat is infinityZermelo-Fraenkel axiomatisation of set theoryMon, 02 Jun 2008 23:00:00 +0000plusadmin2329 at https://plus.maths.org/contentCantor and Cohen: Infinite investigators part II
https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii
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Richard Elwes </div>
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<b>Richard Elwes</b> continues his investigation into Cantor and Cohen's work. He investigates the <i>continuum hypothesis</i>, the question that caused Cantor so much grief. </div>
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<h1>The continuum hypothesis</h1>
<p><i>This is one half of a two-part article telling a story of two mathematical problems and two men: Georg Cantor, who discovered the strange world that these problems inhabit, and Paul Cohen (who died last year), who eventually solved them. This article explores what is known as the continuum hypothesis, while <a href="/issue47/features/elwes1">the other article</a> explores the axiom
of choice.<p><a href="https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii" target="_blank">read more</a></p>https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii#comments47axiomcontinuum hypothesishilbert problemshistory of mathematicsinfinitylogicphilosophy of mathematicsset theorywhat is infinityZermelo-Fraenkel axiomatisation of set theorySun, 01 Jun 2008 23:00:00 +0000plusadmin2330 at https://plus.maths.org/contentOuter space: Venn you can't use Venn
https://plus.maths.org/content/outer-space-venn-you-cant-use-venn
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John D. Barrow </div>
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When the famous diagram fails </div>
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<p>The Reverend <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Venn.html">John Venn</a> was Master of Caius College, Cambridge and the inventor of a machine for bowling cricket balls, but it is for neither of these distinctions that his name is (almost) a household word. In 1880 he created a handy diagram to help him teach and explain logical possibilities.<p><a href="https://plus.maths.org/content/outer-space-venn-you-cant-use-venn" target="_blank">read more</a></p>https://plus.maths.org/content/outer-space-venn-you-cant-use-venn#comments47logicouterspacevenn diagramSat, 31 May 2008 23:00:00 +0000plusadmin4804 at https://plus.maths.org/contentWe must know, we will know
https://plus.maths.org/content/we-must-know-we-will-know
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Rebecca Morris </div>
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<b>Runner up in the general public category</b>. Great minds spark controversy. This is something you'd expect to hear about a great philosopher or artist, but not about a mathematician. Get ready to bin your stereotypes as <b>Rebecca Morris</b> describes some controversial ideas of the great mathematician David Hilbert. </div>
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<p style="color:purple;"><b><i>This article is a runner-up in the general public category of the Plus new writers award 2006.</i></b></p>
<p><i>"It seems that there was a mathematician who had become a novelist. 'Why did he do that?' people in Göttingen marvelled. 'How can a man who was a mathematician write novels?' 'But that is completely simple,' Hilbert said. 'He did not have enough imagination for mathematics, but he had enough for novels.' "</i></p><p><a href="https://plus.maths.org/content/we-must-know-we-will-know" target="_blank">read more</a></p>https://plus.maths.org/content/we-must-know-we-will-know#comments41axiomEuclidean geometryGödel's Incompleteness Theoremhilbert problemshistory of mathematicslogicphilosophy of mathematicsFri, 01 Dec 2006 00:00:00 +0000plusadmin2295 at https://plus.maths.org/contentGödel and the limits of logic
https://plus.maths.org/content/goumldel-and-limits-logic
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John W Dawson </div>
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When Kurt Gödel published his <i>incompleteness theorem</i> in 1931, the mathematical community was stunned: using maths he had proved that there are limits to what maths can prove. This put an end to the hope that all of maths could one day be unified in one elegant theory and had very real implications for computer science. <b>John W Dawson</b> describes Gödel's brilliant work and troubled
life. </div>
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<p>Kurt Gödel. Photograph by Alfred Eisenstaedt, taken from the Gödel Papers courtesy of <a href='http://www.princeton.edu/main/'>Princeton University</a> and <a href='http://www.ias.edu/'>Institute for Advanced Study</a>.</p><p><a href="https://plus.maths.org/content/goumldel-and-limits-logic" target="_blank">read more</a></p>https://plus.maths.org/content/goumldel-and-limits-logic#comments39Gödel's Incompleteness Theoremhistory of mathematicslogicphilosophy of mathematicswhat is impossibleWed, 31 May 2006 23:00:00 +0000plusadmin2284 at https://plus.maths.org/contentA bright idea
https://plus.maths.org/content/os/issue36/features/nishiyama/index
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Yutaka Nishiyama </div>
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What do computers and light switches have in common? <b>Yutaka Nishiyama</b> illuminates the connection between light bulbs, logic and binary arithmetic. </div>
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<div class="pub_date">September 2005</div>
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<p><i>A light switch is a simple object: it's either on or off. But this simple binary system is enough to illustrate Boolean algebra, on which all modern computers are based. <b>Yutaka Nishiyama</b> illuminates the connection between light bulbs, logic and binary arithmetic.</i></p><p><a href="https://plus.maths.org/content/os/issue36/features/nishiyama/index" target="_blank">read more</a></p>https://plus.maths.org/content/os/issue36/features/nishiyama/index#comments36artificial intelligenceboolean algebracomputer sciencelogiclogic gatetruth tableWed, 28 Sep 2005 16:38:18 +00002273 at https://plus.maths.org/content