proof
http://plus.maths.org/content/taxonomy/term/308
enAutomated mathematics
http://plus.maths.org/content/automated-mathematics
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Human versus machine: who's better at proving theorems? </div>
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<div class="pub_date">25/11/2008</div>
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<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="http://plus.maths.org/content/automated-mathematics" target="_blank">read more</a></p>http://plus.maths.org/content/automated-mathematics#commentsaxiomcomputer sciencefour-colour theoremproofTue, 25 Nov 2008 00:00:00 +0000plusadmin2457 at http://plus.maths.org/contentOmega and why maths has no TOEs
http://plus.maths.org/content/omega-and-why-maths-has-no-toes
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Gregory Chaitin </div>
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Kurt Gödel, who would have celebrated his 100th birthday next year, showed in 1931 that the power of maths to explain the world is limited: his famous incompleteness theorem proves mathematically that maths cannot prove everything. <b>Gregory Chaitin</b> explains why he thinks that Gödel's incompleteness theorem is only the tip of the iceberg, and why mathematics is far too complex ever to be
described by a single theory. </div>
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<div class="pub_date">December 2005</div>
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<p><i>Over the millennia, many mathematicians have hoped that mathematics would one day produce a Theory of Everything (TOE); a finite set of axioms and rules from which every mathematical truth could be derived. But in 1931 this hope received a serious blow: Kurt Gödel published his famous Incompleteness Theorem, which states that in every mathematical theory, no matter how extensive, there will
always be statements which can't be proven to be true or false.</i></p><p><a href="http://plus.maths.org/content/omega-and-why-maths-has-no-toes" target="_blank">read more</a></p>http://plus.maths.org/content/omega-and-why-maths-has-no-toes#comments37binary codeGödel's Incompleteness Theoremphilosophy of mathematicsproofThu, 01 Dec 2005 00:00:00 +0000plusadmin2278 at http://plus.maths.org/contentWelcome to the maths lab
http://plus.maths.org/content/welcome-maths-lab
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Has mathematics become an experimental science? </div>
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<div class="pub_date">22/11/2004</div>
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<p>Has mathematics become an experimental science? Yes, according to the most prestigious journal of mathematics - at least that part of maths that involves computer proofs.<p><a href="http://plus.maths.org/content/welcome-maths-lab" target="_blank">read more</a></p>http://plus.maths.org/content/welcome-maths-lab#commentsKepler's conjecturepacking problemsproofMon, 22 Nov 2004 00:00:00 +0000plusadmin2511 at http://plus.maths.org/content1089 and all that
http://plus.maths.org/content/1089-and-all
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David Acheson </div>
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Why do so many people say they hate mathematics, asks <b>David Acheson</b>? The truth, he says, is that most of them have never been anywhere near it, and that mathematicians could do more to change this perception - perhaps by emphasising the element of surprise that so often accompanies mathematics at its best. </div>
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<div class="pub_date">September 2004</div>
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<h2>The element of surprise in mathematics</h2>
<p><i>Why do so many people say they hate mathematics?</i></p>
<p>All too often, the real truth is that they have never been allowed anywhere near it, and I believe that mathematicians like myself could do more, if we wanted, to bring some of the ideas and pleasures of our subject to a wide public.</p>
<p>And one way of doing this might be to emphasise the element of <i>surprise</i> that often accompanies mathematics at its best.</p><p><a href="http://plus.maths.org/content/1089-and-all" target="_blank">read more</a></p>http://plus.maths.org/content/1089-and-all#comments31ellipseFermat's Last Theoremfocalpointsgeometrykeplerleibnizmathematics and magicpendulumPiproofTue, 31 Aug 2004 23:00:00 +0000plusadmin2250 at http://plus.maths.org/contentThe origins of proof IV: The philosophy of proof
http://plus.maths.org/content/origins-proof-iv-philosophy-proof
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Robert Hunt </div>
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<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>
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<div class="pub_date">January 2000</div>
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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="http://plus.maths.org/content/origins-proof-iv-philosophy-proof" target="_blank">read more</a></p>http://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 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
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Jon Walthoe </div>
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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>
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<div class="pub_date">September 1999</div>
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<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/contentThe origins of proof II : Kepler's proofs
http://plus.maths.org/content/origins-proof-ii-keplers-proofs
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J.V. Field </div>
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Johannes Kepler (1571-1630) is now chiefly remembered as a mathematical astronomer who discovered three laws that describe the motion of the planets. <b>J.V. Field</b> continues our series on the origins of proof with an examination of Kepler's astronomy. </div>
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<div class="pub_date">May 1999</div>
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<p>As we explained in <a href="/issue7/features/proof1/index.html">The Origins of Proof, Part I</a> in <a href="http://plus.maths.org/content/issue/7">Issue 7</a> of PASS Maths, the concept of a "proof" was developed in the field of geometry by the Greeks. The Pythagoreans and Euclid were among the mathematicians who developed the idea of abstract deduction. But during the Renaissance the philosophy
of nature increasingly came to rely upon mathematics to help to explain the Universe and its workings.</p><p><a href="http://plus.maths.org/content/origins-proof-ii-keplers-proofs" target="_blank">read more</a></p>http://plus.maths.org/content/origins-proof-ii-keplers-proofs#comments8astronomyellipseerrorgeometrygravityhistory of mathematicsKepler's three laws of planetary motionproofFri, 30 Apr 1999 23:00:00 +0000plusadmin2389 at http://plus.maths.org/contentThe origins of proof
http://plus.maths.org/content/origins-proof
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Kona Macphee </div>
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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>
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<div class="pub_date">January 1999</div>
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<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="http://plus.maths.org/content/origins-proof" target="_blank">read more</a></p>http://plus.maths.org/content/origins-proof#comments7axiomdeductionEuclid's ElementspremiseproofFri, 01 Jan 1999 00:00:00 +0000plusadmin2385 at http://plus.maths.org/content