what is impossible
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enThis 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/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/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/contentUnreasonable effectiveness
https://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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<div class="pub_date">December 2008</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="https://plus.maths.org/content/unreasonable-effectiveness" target="_blank">read more</a></p>https://plus.maths.org/content/unreasonable-effectiveness#comments49history of mathematicsknotknot theoryphilosophy of mathematicswhat is impossibleMon, 01 Dec 2008 00:00:00 +0000plusadmin2348 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/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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<div class="pub_date">June 2006</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/contentMathematical mysteries: The Barber's Paradox
https://plus.maths.org/content/mathematical-mysteries-barbers-paradox
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Helen Joyce </div>
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<p>Suppose you walk past a barber's shop one day, and see a sign that says</p>
<p>"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."<br />
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself?</p>
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<div class="pub_date">May 2002</div>
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<p>Suppose you walk past a barber's shop one day, and see a sign that says</p><p><a href="https://plus.maths.org/content/mathematical-mysteries-barbers-paradox" target="_blank">read more</a></p>https://plus.maths.org/content/mathematical-mysteries-barbers-paradox#comments20Barber's ParadoxlogicMathematical mysteriesphilosophy of mathematicsRussell's Paradoxset theoryTheory of Typeswhat is impossibleZermelo-Fraenkel axiomatisation of set theoryTue, 30 Apr 2002 23:00:00 +0000plusadmin4757 at https://plus.maths.org/content