axiom
https://plus.maths.org/content/taxonomy/term/310
enPicking holes in mathematics
https://plus.maths.org/content/picking-holes-mathematics
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Marianne Freiberger </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="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/15_feb_2011_-_1802/icon.jpg?1297792933" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<div class="rightimage" style="width: 200px;"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/articles/2011/woodin/godel.jpg" alt="" width="200" height="269"
<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/contentAutomated mathematics
https://plus.maths.org/content/automated-mathematics
<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="https://plus.maths.org/content/sites/plus.maths.org/files/latestnews/sep-dec08/proof/icon.jpg?1227571200" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
Human versus machine: who's better at proving theorems? </div>
</div>
</div>
<div class="pub_date">25/11/2008</div>
<!-- plusimport --><br clear="all"></br>
<!-- END OF FILE: newinclude/news_header.html -->
<p>Does 1 plus 1 really equal 2? Only mathematicians could ask such a strange question and only mathematicians could (and, as you'll be relieved to hear, have) come up with a rigorous proof based on a meticulously worked-out definition of the whole numbers.<p><a href="https://plus.maths.org/content/automated-mathematics" target="_blank">read more</a></p>https://plus.maths.org/content/automated-mathematics#commentsaxiomcomputer sciencefour-colour theoremproofTue, 25 Nov 2008 00:00:00 +0000plusadmin2457 at https://plus.maths.org/contentCantor and Cohen: Infinite investigators part I
https://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="https://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="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
<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="https://plus.maths.org/content/sites/plus.maths.org/files/issue47/features/elwes2/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">
<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>
</div>
</div>
<div class="pub_date">June 2008</div>
<!-- plusimport -->
<br clear="all" />
<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/contentWe must know, we will know
https://plus.maths.org/content/we-must-know-we-will-know
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Rebecca Morris </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="https://plus.maths.org/content/sites/plus.maths.org/files/issue41/features/morris/icon.jpg?1164931200" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
<div class="pub_date">December 2006</div>
<!-- plusimport -->
<br clear="all" />
<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/contentThe origins of proof IV: The philosophy of proof
https://plus.maths.org/content/origins-proof-iv-philosophy-proof
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Robert Hunt </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="130" height="130" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue10/features/proof4/icon.jpg?946684800" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<strong>Robert Hunt</strong> concludes our Origins of Proof series by asking what a proof really <i>is</i>, and how we know that we've actually found one. One for the philosophers to ponder... </div>
</div>
</div>
<div class="pub_date">January 2000</div>
<!-- plusimport -->
In this final article in our series on Proof, we examine the philosophy of mathematical proof.
<p>What precisely <em>is</em> a proof? The answer seems obvious: starting from some <a href="/issue7/features/proof1/index.html#axiom">axioms</a>, a proof is a series of logical <a href="/issue7/features/proof1/index.html#deduction">deductions</a>, reaching the desired conclusion.<p><a href="https://plus.maths.org/content/origins-proof-iv-philosophy-proof" target="_blank">read more</a></p>https://plus.maths.org/content/origins-proof-iv-philosophy-proof#comments10axiomFermat's Last Theoremfour-colour theoremminimal criminalphilosophy of mathematicsproofSat, 01 Jan 2000 00:00:00 +0000plusadmin2162 at https://plus.maths.org/contentThe origins of proof III: Proof and puzzles through the ages
https://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="https://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="https://plus.maths.org/content/origins-proof-iii-proof-and-puzzles-through-ages" target="_blank">read more</a></p>https://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 https://plus.maths.org/contentThe origins of proof
https://plus.maths.org/content/origins-proof
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Kona Macphee </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="109" height="110" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue7/features/proof1/icon.jpg?915148800" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
Starting in this issue, PASS Maths is pleased to present a series of articles about proof and logical reasoning. In this article we give a brief introduction to deductive reasoning and take a look at one of the earliest known examples of mathematical proof. </div>
</div>
</div>
<div class="pub_date">January 1999</div>
<!-- plusimport -->
<p><b>What is proof?</b> Philosophers have argued for centuries about the answer to this question, and how (and if!) things can be proven; no doubt they will continue to do so! Mathematicians, on the other hand, have been using "working definitions" of proof to advance mathematical knowledge for equally long.</p>
<p>Starting in this issue, PASS Maths is pleased to present a series of articles introducing some of the basic ideas behind proof and logical reasoning and showing their importance in mathematics.</p><p><a href="https://plus.maths.org/content/origins-proof" target="_blank">read more</a></p>https://plus.maths.org/content/origins-proof#comments7axiomdeductionEuclid's ElementspremiseproofFri, 01 Jan 1999 00:00:00 +0000plusadmin2385 at https://plus.maths.org/content