philosophy of mathematics
https://plus.maths.org/content/taxonomy/term/309
enThe future of proof
https://plus.maths.org/content/future-proof
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Marianne Freiberger and Rachel Thomas </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/5/9_apr_2015_-_1552/map_icon.png?1428591122" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<p>Will computers ever replace human mathematicians?</p>
</div>
</div>
</div>
<p>Are mathematicians ever going to be replaced by computers? If maths
was all about routine calculations, then the answer would most
definitely be yes. But if you've ever
tried to come up with a mathematical proof, or even played with a logic
puzzle, you know this involves intuition and leaps of imagination
you'd think are beyond any computer. Even just
deciding which kind of questions are mathematically interesting, and
which are boring or beyond reach, seems to be something that needs
human input.</p><p><a href="https://plus.maths.org/content/future-proof" target="_blank">read more</a></p>https://plus.maths.org/content/future-proof#commentsBMC2015computer scienceFP-top-storyphilosophy of mathematicsproofFri, 10 Apr 2015 14:50:59 +0000mf3446345 at https://plus.maths.org/contentWhat is information?
https://plus.maths.org/content/what-information
<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/5/5_aug_2014_-_1033/info_icon.jpg?1407231186" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<p>Books, brains, computers — information comes in many guises. But what exactly <em>is</em> information?</p>
</div>
</div>
</div>
<div class="rightimage" style="width: 260px;"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p><em>This article is part of our <a href="https://plus.maths.org/content/information-about-information">Information about information project</a>, run <a href="https://plus.maths.org/content/information-about-information#fqxi">in collaboration with FQXi</a>. Click <a href="https://plus.maths.org/content/information-about-information">here</a> to see other articles from the project. </em></p><p><a href="https://plus.maths.org/content/what-information" target="_blank">read more</a></p>https://plus.maths.org/content/what-information#commentsDNAinformation about informationInformation theoryphilosophy of mathematicsTue, 05 Aug 2014 08:09:55 +0000mf3446132 at https://plus.maths.org/contentMade of maths?
https://plus.maths.org/content/world-made-maths
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Jeremy Butterfield </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/5/16_may_2014_-_1045/icon.jpg?1400233539" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
Mathematics is incredibly good at describing the world we live
in. So much so that some people have argued that maths is not just a
tool for describing the world, but that the world <em>is</em>
itself a mathematical structure. Does his claim stand up to scrutiny? </div>
</div>
</div>
<p><em>Mathematics is incredibly good at describing the world we live
in. So much so that some people have argued that maths is not just a
tool for describing the world, but that the world <em>is</em>
itself a mathematical structure. It's a claim that will tempt
anyone who loves maths, but does it stand up to scrutiny? We asked
philosopher of physics Jermemy Butterfield for his opinion, and here
is what he told us.</em></p><p><a href="https://plus.maths.org/content/world-made-maths" target="_blank">read more</a></p>https://plus.maths.org/content/world-made-maths#commentsphilosophy of mathematicsTue, 20 May 2014 14:55:42 +0000mf3446093 at https://plus.maths.org/contentComputers, maths and minds
https://plus.maths.org/content/computers-maths-mind
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Alan Aw </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/5/29_jan_2014_-_1232/brain_icon.jpg?1390998728" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
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>
</div>
</div>
<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.
</p><p><a href="https://plus.maths.org/content/computers-maths-mind" target="_blank">read more</a></p>https://plus.maths.org/content/computers-maths-mind#commentsAlan Turingcomputer scienceneurosciencephilosophy of mathematicsTuring MachineTue, 04 Feb 2014 09:16:52 +0000mf3446032 at https://plus.maths.org/contentCognition, brains and Riemann
https://plus.maths.org/content/cognition-brains-and-riemann
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Joselle DiNunzio Kehoe </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/5/26_jun_2013_-_1254/icon-4.jpg?1372247682" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<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><p><a href="https://plus.maths.org/content/cognition-brains-and-riemann" target="_blank">read more</a></p>https://plus.maths.org/content/cognition-brains-and-riemann#commentsmathematical realityneurosciencephilosophyphilosophy of mathematicspsychologyTue, 09 Jul 2013 05:29:55 +0000mf3445915 at https://plus.maths.org/contentIs the Universe simple or complex?
https://plus.maths.org/content/universe-simple-complex
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Faye Kilburn </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/5/4_jan_2013_-_1237/milky_icon.jpg?1357303065" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<div class="rightimage" style="width: 300px"><img src="/sites/plus.maths.org/files/articles/2013/kilburn/milkyway.jpg" alt="Milky Way" width="300px" height="300px" />
<p>There are at least 500 billion planets in the Milky Way.</p><p><a href="https://plus.maths.org/content/universe-simple-complex" target="_blank">read more</a></p>https://plus.maths.org/content/universe-simple-complex#commentsmathematical realitycomplexityemergent behaviourlogicphilosophy of mathematicsMon, 14 Jan 2013 09:29:55 +0000mf3445851 at https://plus.maths.org/contentIs the Universe simple or complex? Part II
https://plus.maths.org/content/universe-simple-or-complex-part-ii
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Faye Kilburn </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/5/14_jan_2013_-_1249/solar_icon.jpg?1358167758" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<p><em>In search for an answer to this question the <a href="https://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="https://plus.maths.org/content/universe-simple-or-complex-part-ii" target="_blank">read more</a></p>https://plus.maths.org/content/universe-simple-or-complex-part-ii#commentsmathematical realitycomplexityphilosophy of mathematicsMon, 14 Jan 2013 06:36:20 +0000mf3445852 at https://plus.maths.org/contentThis is not a carrot: Paraconsistent mathematics
https://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="https://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="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
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Phil Wilson </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/5/8_jun_2011_-_1618/icon.jpg?1307546325" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<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
<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/contentSearching for the missing truth
https://plus.maths.org/content/searching-missing-truth
<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/5/10%20Jan%202011%20-%2015%3A02/icon.jpg?1294671756" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<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/contentBiology's next microscope, mathematics' next physics
https://plus.maths.org/content/biologys-next-microscope-mathematics-new-physics
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Rachel Thomas </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="Icon" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/4/7%20Oct%202010%20-%2011%3A56/icon.jpg?1286449017" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<div style="position: relative; left: 50%; width: 70%"><font size="2"><i>Back to the <a href="https://plus.maths.org/content/do-you-know-whats-good-you-maths-next-microscope">Next microscope package </a><p><a href="https://plus.maths.org/content/biologys-next-microscope-mathematics-new-physics" target="_blank">read more</a></p>https://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 https://plus.maths.org/contentConstructive mathematics
https://plus.maths.org/content/constructive-mathematics
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Phil Wilson </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/issue49/features/wilson/icon.jpg?1228089600" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
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>
</div>
</div>
<div class="pub_date">December 2008</div>
<!-- plusimport -->
<br clear="all" />
<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/contentUnreasonable effectiveness
https://plus.maths.org/content/unreasonable-effectiveness
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Mario Livio </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/issue49/features/livio/icon.jpg?1228089600" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
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>
</div>
</div>
<div class="pub_date">December 2008</div>
<!-- plusimport -->
<br clear="all" />
<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/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/content