philosophy of mathematics
http://plus.maths.org/content/taxonomy/term/309
enComputers, maths and minds
http://plus.maths.org/content/computers-maths-mind
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Alan Aw </div>
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Most of us have a rough
idea that computers are
made up of complicated hardware and software. But perhaps few of us
know that the concept of a computer was envisioned long before these
machines became ubiquitous items in our homes, offices and even
pockets. </div>
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<p>Many of us own computers, and we (well, most of us) have a rough
idea that computers are
made up of complicated hardware and software. But perhaps few of us
know that the concept of a computer was envisioned long before these
machines became ubiquitous items in our homes, offices and even
pockets. And as we will see later, some have even suggested that our
own brains are embodiments of this theoretical concept.
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<p><a href="http://plus.maths.org/content/computers-maths-mind" target="_blank">read more</a></p>http://plus.maths.org/content/computers-maths-mind#commentsAlan Turingcomputer scienceneurosciencephilosophy of mathematicsTuring MachineTue, 04 Feb 2014 09:16:52 +0000mf3446032 at http://plus.maths.org/contentCognition, brains and Riemann
http://plus.maths.org/content/cognition-brains-and-riemann
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Joselle DiNunzio Kehoe </div>
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<p>Are number, space and time features of the outside world or a result of the brain circuitry we have developed to live in it? Some interesting parallels between modern neuroscience and the mathematics of 19th century mathematician Bernard Riemann.</p>
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<p><em>Modern neuroscience suggests that number, space and time aren't so much features of the outside world but more a result of the brain circuitry we evolved to move around in it. And this circuitry is all about judging less than/greater than relationships. In the 19th century the mathematician Bernard Riemann suggested that the mathematical ideas of space, quantity and measure should not depend on the outside world, but defined abstractly and in relation to each other. Joselle DiNunzio Kehoe finds some interesting parallels between these two ideas.</em></p><div class="field field-type-number-integer field-field-hidden">
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<p><a href="http://plus.maths.org/content/cognition-brains-and-riemann" target="_blank">read more</a></p>http://plus.maths.org/content/cognition-brains-and-riemann#commentsmathematical realityneurosciencephilosophyphilosophy of mathematicspsychologyTue, 09 Jul 2013 05:29:55 +0000mf3445915 at http://plus.maths.org/contentIs the Universe simple or complex?
http://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="http://plus.maths.org/content/universe-simple-complex" target="_blank">read more</a></p>http://plus.maths.org/content/universe-simple-complex#commentsmathematical realitycomplexityemergent behaviourlogicphilosophy of mathematicsMon, 14 Jan 2013 09:29:55 +0000mf3445851 at http://plus.maths.org/contentIs the Universe simple or complex? Part II
http://plus.maths.org/content/universe-simple-or-complex-part-ii
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Faye Kilburn </div>
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<p>In this, the second part of this series, we look at a mathematical notion of complexity and wonder whether the Universe is just too complex for our tiny little minds to understand.</p>
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<p><em>In search for an answer to this question the <a href="http://plus.maths.org/content/universe-simple-complex">first part</a> of this article led us to Occam's razor and, rather surprisingly, we ended up thinking about God. But is there a more objective assessment of simplicity and complexity?</em></p><p><a href="http://plus.maths.org/content/universe-simple-or-complex-part-ii" target="_blank">read more</a></p>http://plus.maths.org/content/universe-simple-or-complex-part-ii#commentsmathematical realitycomplexityphilosophy of mathematicsMon, 14 Jan 2013 06:36:20 +0000mf3445852 at http://plus.maths.org/contentThis is not a carrot: Paraconsistent mathematics
http://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="http://plus.maths.org/content/not-carrot" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentThe philosophy of applied mathematics
http://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="http://plus.maths.org/content/philosophy-applied-mathematics" target="_blank">read more</a></p>http://plus.maths.org/content/philosophy-applied-mathematics#commentsmathematical realityconstructivist mathematicsinfinitylogicphilosophy of mathematicsplatonismwhat is impossibleFri, 24 Jun 2011 09:35:32 +0000mf3445497 at http://plus.maths.org/contentPicking holes in mathematics
http://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="http://plus.maths.org/content/picking-holes-mathematics" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentSearching for the missing truth
http://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="http://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="http://plus.maths.org/content/searching-missing-truth" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentBiology's next microscope, mathematics' next physics
http://plus.maths.org/content/biologys-next-microscope-mathematics-new-physics
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Rachel Thomas </div>
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<p>It is thought that the next great advances in biology and medicine will be discovered with mathematics. As biology stands on the brink of becoming a theoretical science, Thomas Fink asks if there is more to this collaboration than maths acting as biology's newest microscope. Will theoretical biology lead to new and exciting maths, just as theoretical physics did in the last two centuries? And is there a mathematically elegant story behind life?</p>
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<div style="position: relative; left: 50%; width: 70%"><font size="2"><i>Back to the <a href="http://plus.maths.org/content/do-you-know-whats-good-you-maths-next-microscope">Next microscope package </a><p><a href="http://plus.maths.org/content/biologys-next-microscope-mathematics-new-physics" target="_blank">read more</a></p>http://plus.maths.org/content/biologys-next-microscope-mathematics-new-physics#commentsmathematical realitybiologyevolutiongeneticsgraph theorymedicine and healthphilosophy of mathematicsphysicstheoretical biologyWed, 22 Sep 2010 10:04:05 +0000Rachel5314 at http://plus.maths.org/contentConstructive mathematics
http://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="http://plus.maths.org/content/constructive-mathematics" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentUnreasonable effectiveness
http://plus.maths.org/content/unreasonable-effectiveness
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Mario Livio </div>
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When it comes to describing natural phenomena, mathematics is amazingly — even unreasonably — effective. In this article <b>Mario Livio</b> looks at an example of strings and knots, taking us from the mysteries of physical matter to the most esoteric outpost of pure mathematics, and back again. </div>
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<p><i>Mario Livio's book, <a href="/issue49/reviews/book5/index.html">Is God a mathematician</a> is reviewed in this issue of Plus.</i></p><p><a href="http://plus.maths.org/content/unreasonable-effectiveness" target="_blank">read more</a></p>http://plus.maths.org/content/unreasonable-effectiveness#comments49history of mathematicsknotknot theoryphilosophy of mathematicswhat is impossibleMon, 01 Dec 2008 00:00:00 +0000plusadmin2348 at http://plus.maths.org/contentCantor and Cohen: Infinite investigators part I
http://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="http://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-i" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentCantor and Cohen: Infinite investigators part II
http://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="http://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentWe must know, we will know
http://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="http://plus.maths.org/content/we-must-know-we-will-know" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentGödel and the limits of logic
http://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
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<div class="pub_date">June 2006</div>
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<div class="rightimage" style="width: 250px;"><img src="/issue39/features/dawson/Godel_alone.jpg" alt="Portrait of Kurt Gödel " width="250" height="378" />
<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="http://plus.maths.org/content/goumldel-and-limits-logic" target="_blank">read more</a></p>http://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 http://plus.maths.org/content