Russell's Paradox
http://plus.maths.org/content/taxonomy/term/840
enThis is not a carrot: Paraconsistent mathematics
http://plus.maths.org/content/not-carrot
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Maarten McKubre-Jordens </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/1_aug_2011_-_1757/icon.jpg?1312217855" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
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>
</div>
</div>
<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/contentVisual curiosities and mathematical paradoxes
http://plus.maths.org/content/visual-curiosities-and-mathematical-paradoxes
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Linda Becerra and Ron Barnes </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/21%20Oct%202010%20-%2016%3A38/icon.jpg?1287675519" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<p>When your eyes see a picture they send an image to your brain, which your brain then has to make sense of. But sometimes your brain gets it wrong. The result is an optical illusion. Similarly in logic, statements or figures can lead to contradictory conclusions, which we call paradoxes. This article looks at examples of geometric optical illusions and paradoxes and gives explanations of what's really going on.</p>
</div>
</div>
</div>
<p>When your eyes see a picture they send an image to your brain, which your brain then has to make sense of. But sometimes your brain gets it wrong. The result is an optical illusion. Similarly in logic, statements or figures can lead to contradictory conclusions; appear to be true but in actual fact are self-contradictory; or appear contradictory, even absurd, but in fact may be true. Here again it is up to your brain to make sense of these situations. Again, your brain may get it wrong. These situations are referred to as paradoxes.<p><a href="http://plus.maths.org/content/visual-curiosities-and-mathematical-paradoxes" target="_blank">read more</a></p>http://plus.maths.org/content/visual-curiosities-and-mathematical-paradoxes#commentsarchitectureBanach-Tarski paradoxBarber's Paradoxeschergeometryimpossible objectoptical illusionparadoxPenrose staircasePenrose triangleperspectiveRussell's ParadoxWed, 17 Nov 2010 14:06:13 +0000mf3445337 at http://plus.maths.org/contentCantor and Cohen: Infinite investigators part I
http://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-i
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Richard Elwes </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/issue47/features/elwes1/icon.jpg?1212274800" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
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>
</div>
</div>
<div class="pub_date">June 2008</div>
<!-- plusimport -->
<br clear="all" />
<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/contentMathematical mysteries: The Barber's Paradox
http://plus.maths.org/content/mathematical-mysteries-barbers-paradox
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Helen Joyce </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/30%20Jun%202010%20-%2014%3A58/icon-2.jpg?1277906288" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<div class="pub_date">May 2002</div>
<!-- plusimport -->
<h2>A close shave for set theory</h2>
<p><!-- FILE: include/leftfig.html --></p>
<div class="leftimage" style="width: 280px;"><img src="/issue20/xfile/barber.jpg" alt="" width="280" height="280" /></div>
<!-- END OF FILE: include/leftfig.html -->
<!-- drawn and copyright Charles Trevelyan -->
<p>Suppose you walk past a barber's shop one day, and see a sign that says</p><p><a href="http://plus.maths.org/content/mathematical-mysteries-barbers-paradox" target="_blank">read more</a></p>http://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 http://plus.maths.org/contentThe origins of proof III: Proof and puzzles through the ages
http://plus.maths.org/content/origins-proof-iii-proof-and-puzzles-through-ages
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Jon Walthoe </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="110" height="110" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/issue9/features/proof3/icon.jpg?936140400" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
For millennia, puzzles and paradoxes have forced mathematicians to continually rethink their ideas of what proofs actually are. <strong>Jon Walthoe</strong> explains the tricks involved and how great thinkers like Pythagoras, Newton and Gödel tackled the problems. </div>
</div>
</div>
<div class="pub_date">September 1999</div>
<!-- plusimport -->
<!-- ====================================================================== -->
<p>In the Millennia since Euclid, people's conceptions of mathematical proof have been revolutionised. From the discovery of Calculus and the rise of abstract mathematics, to Gödel's amazing discovery. There have been many changes and a few surprises along the way.</p><p><a href="http://plus.maths.org/content/origins-proof-iii-proof-and-puzzles-through-ages" target="_blank">read more</a></p>http://plus.maths.org/content/origins-proof-iii-proof-and-puzzles-through-ages#comments9axiomcalculusdeductionGödel's Incompleteness Theoreminductionirrational numberparadoxproofrational numberRussell's ParadoxTue, 31 Aug 1999 23:00:00 +0000plusadmin2394 at http://plus.maths.org/content