tag:theconversation.com,2011:/global/topics/fermats-last-theorem-19561/articlesFermat's Last Theorem – The Conversation2016-06-01T20:16:40Ztag:theconversation.com,2011:article/601682016-06-01T20:16:40Z2016-06-01T20:16:40ZWill computers replace humans in mathematics?<figure><img src="https://images.theconversation.com/files/124546/original/image-20160531-13810-16flhes.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Computers are coming up with proofs in mathematics that are almost impossible for a human to check.</span> <span class="attribution"><span class="source">Shutterstock/Fernando Batista</span></span></figcaption></figure><p>Computers can be valuable tools for helping mathematicians solve problems but they can also play their own part in the discovery and proof of mathematical theorems.</p>
<p>Perhaps the first major result by a computer came 40 years ago, with proof for the <a>four-color theorem</a> – the assertion that any map (with certain reasonable conditions) can be coloured with just four distinct colours.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">No more that four colours are needed in this picture to make sure that no two touching shapes share the same colour.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Four_Colour_Map_Example.svg">Wikimedia/Inductiveload</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>This was first proved by computer in 1976, although flaws were later found, and a <a href="http://www.ams.org/notices/200811/tx081101382p.pdf">corrected proof</a> was not completed until 1995.</p>
<p>In 2003, Thomas Hales, of the University of Pittsburgh, published a computer-based proof of <a href="http://experimentalmath.info/blog/2014/08/formal-proof-completed-for-keplers-conjecture-on-sphere-packing/">Kepler’s conjecture</a> that the familiar method of stacking oranges in the supermarket is the most space-efficient way of arranging equal-diameter spheres.</p>
<p>Although Hales published a proof in 2003, many mathematicians were not satisfied because the proof was accompanied by two gigabytes of computer output (a large amount at the time), and some of the computations could not be certified.</p>
<p>In response, Hales produced a <a href="http://experimentalmath.info/blog/2014/08/formal-proof-completed-for-keplers-conjecture-on-sphere-packing/">computer-verified formal proof</a> in 2014.</p>
<h2>The new kid on the block</h2>
<p>The latest development along this line is the <a href="http://www.nature.com/news/two-hundred-terabyte-maths-proof-is-largest-ever-1.19990">announcement this month in Nature</a> of a computer proof for what is known as the Boolean Pythagorean triples problem. </p>
<p>The assertion here is that the integers from one to 7,824 can be coloured either red or blue with the property that no set of three integers a, b and c that satisfy a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> (Pythagoras’s Theorem where a, b and c form the sides of a right triangle) are all the same colour. For the integers from one to 7,825, this cannot be done.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=360&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=360&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=360&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=452&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=452&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=452&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Pythagoras’s theorem for a right-angled triangle.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
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<p>Even for small integers, it is hard to find a non-monochrome colouring. For instance, if five is red then one of 12 or 13 must be blue, since 5<sup>2</sup> + 12<sup>2</sup> = 13<sup>2</sup>; and one of three or four must also be blue, since 3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup>. Each choice has many constraints.</p>
<p>As it turns out, the number of possible ways to colour the integers from one to 7,825 is gigantic – more than 10<sup>2,300</sup> (a one followed by 2,300 zeroes). This number is far, far greater than the number of fundamental particles in the visible universe, which is a mere <a>10<sup>85</sup></a>. </p>
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<a href="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=524&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=524&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=524&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=658&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=658&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=658&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The numbers one to 7,824 can be coloured either red or blue so that no trio a, b and c that satisfies Pythagoras’s theorem are all the same colour. A white square can be either red or blue.</span>
<span class="attribution"><span class="source">Marijn Heule</span></span>
</figcaption>
</figure>
<p>But the researchers were able to sharply reduce this number by taking advantage of various symmetries and number theory properties, to “only” one trillion. The computer run to examine each of these one trillion cases required two days on 800 processors of the University of Texas’ <a>Stampede supercomputer</a>.</p>
<p>While direct applications of this result are unlikely, the ability to solve such difficult colouring problems is bound to have implications for coding and for security.</p>
<p>The Texas computation, which we estimate performed roughly 10<sup>19</sup> arithmetic operations, is still not the largest mathematical computation. A 2013 <a href="http://www.ams.org/notices/201307/rnoti-p844.pdf">computation</a> of digits of pi<sup>2</sup> by us and two IBM researchers did twice this many operations. </p>
<p>The Great Internet Mersenne Prime Search (<a href="http://www.mersenne.org">GIMPS</a>), a global network of computers search for the largest known prime numbers, routinely performs a total of <a href="https://www.sciencedaily.com/releases/2016/01/160120084917.htm">450 trillion calculations per second</a>, which every six hours exceeds the number of operations performed by the Texas calculation. </p>
<p>In computer output, though, the Texas calculation takes the cake for a mathematical computation – a staggering 200 terabytes, namely 2✕10<sup>14</sup> bytes, or 30,000 bytes for every human being on Earth.</p>
<p>How can one check such a sizeable output? Fortunately, the Boolean Pythagorean triple program produced a solution (shown in the image, above) that can be checked by a much smaller program.</p>
<p>This is akin to factoring a very large number c into two smaller factors a and b by computer, so that c = a ✕ b. It is often quite difficult to find the two factors a and b, but once found, it is a trivial task to multiply them together and verify that they work.</p>
<h2>Are mathematicians obsolete?</h2>
<p>So what do these developments mean? Are research mathematicians soon to join the ranks of <a href="http://www.nytimes.com/1997/05/12/nyregion/swift-and-slashing-computer-topples-kasparov.html">chess grandmasters</a>, <a href="http://www.nytimes.com/2011/02/17/science/17jeopardy-watson.html">Jeopardy champions</a>, <a href="http://www.geekwire.com/2016/more-layoffs-at-nordstrom/">retail clerks</a>, <a href="https://www.theguardian.com/technology/2016/feb/10/black-cab-drivers-uber-protest-london-traffic-standstill">taxi drivers</a>, <a href="http://www.cnet.com/news/driverless-truck-convoy-platoons-across-europe/">truck drivers</a>, <a href="http://www.huffingtonpost.com/entry/ibm-watson-radiology_us_55cbccf9e4b0898c48867c56">radiologists</a> and other professions threatened with obsolescence due to rapidly advancing technology?</p>
<p>Not quite. Mathematicians, like many other professionals, have for the large part embraced computation as a new mode of mathematical research, a development known as experimental mathematics, which has far-reaching implications.</p>
<p>So what exactly is experimental mathematics? It is best defined as a mode of research that employs computers as a “laboratory,” in the same sense that a physicist, chemist, biologist or engineer performs an experiment to, for example, gain insight and intuition, test and falsify conjecture, and confirm results proved by conventional means.</p>
<p>We have written on this topic at some length elsewhere – see our <a href="http://www.experimentalmath.info/books/">books</a> and <a href="https://www.carma.newcastle.edu.au/jon/papers.html#PAPERS">papers</a> for full technical details.</p>
<p>In one sense, there there is nothing fundamentally new in the experimental methodology of mathematical research. In the third century BCE, the great Greek mathematician Archimedes <a href="https://books.google.com/books?id=Vvj_AwAAQBAJ&pg=PA314#v=onepage">wrote</a>:</p>
<blockquote>
<p>For it is easier to supply the proof when we have previously acquired, by the [experimental] method, some knowledge of the questions than it is to find it without any previous knowledge.</p>
</blockquote>
<p>Galileo once reputedly wrote:</p>
<blockquote>
<p>All truths are easy to understand once they are discovered; the point is to discover them.</p>
</blockquote>
<p>Carl Friederich Gauss, 19th century mathematician and physicist, frequently employed computations to motivate his remarkable discoveries. He once wrote:</p>
<blockquote>
<p>I have the result, but I do not yet know how to get [prove] it.</p>
</blockquote>
<p>Computer-based experimental mathematics certainly has technology on its side. With every passing year, computer hardware advances with <a href="http://www.intel.com/content/www/us/en/silicon-innovations/moores-law-technology.html">Moore’s Law</a>, and mathematical computing software packages such as Maple, Mathematica, Sage and others become ever more powerful.</p>
<p>Already these systems are powerful enough to solve virtually any equation, derivative, integral or other task in undergraduate mathematics.</p>
<p>So while ordinary human-based proofs are still essential, the computer leads the way in assisting mathematicians to identify new theorems and chart a route to formal proof.</p>
<p>What’s more, one can argue that in many cases computations are more compelling than human-based proofs. Human proofs, after all, are subject to mistakes, oversights, and reliance on earlier results by others that may be unsound. </p>
<p><a href="http://www.intel.com/content/www/us/en/silicon-innovations/moores-law-technology.html">Andrew Wiles’</a> initial proof of <a href="http://simonsingh.net/books/fermats-last-theorem/the-whole-story/">Fermat’s Last Theorem</a> was later found to be flawed. This was fixed later.</p>
<p>Along this line, recently Alexander Yee and Shigeru Kondo computed <a href="http://www.numberworld.org/misc_runs/pi-12t/">12.1 trillion digits of pi</a>. To do this, they first computed somewhat more than 10 trillion base-16 digits, then they checked their computation by computing a section of base-16 digits near the end by a completely different algorithm, and compared the results. They matched perfectly.</p>
<p>So which is more reliable, a human-proved theorem hundreds of pages long, which only a handful of other mathematicians have read and verified in detail, or the Yee-Kondo result? Let’s face it, computation is arguably more reliable than proof in many cases.</p>
<h2>What does the future hold?</h2>
<p>There is every indication that research mathematicians will continue to work in respectful symbiosis with computers for the foreseeable future. Indeed, as this relationship and computer technology mature, mathematicians will become more comfortable leaving certain parts of a proof to computers. </p>
<p>This very question was discussed in a June 2014 <a href="http://experimentalmath.info/blog/2014/11/breakthrough-prize-recipients-give-math-seminar-talks/">panel discussion</a> by the five inaugural <a href="https://breakthroughprize.org/?controller=Page&action=news&news_id=18">Breakthrough Prize in Mathematics</a> recipients for mathematics. The Australian-American mathematician Terence Tao expressed their consensus in these terms:</p>
<blockquote>
<p>Computers will certainly increase in power, but I expect that much of mathematics will continue to be done with humans working with computers.</p>
</blockquote>
<p>So don’t toss your algebra textbook quite yet. You will need it!</p><img src="https://counter.theconversation.com/content/60168/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jonathan Borwein (Jon) receives funding from the Australian Research Council.</span></em></p><p class="fine-print"><em><span>David H. Bailey does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Computers are increasingly used to prove mathematical theorems. So does that mean human mathematicians will become obselete?Jonathan Borwein (Jon), Laureate Professor of Mathematics, University of NewcastleDavid H. Bailey, PhD; Lawrence Berkeley Laboratory (retired) and Research Fellow, University of California, DavisLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/524912015-12-21T12:47:59Z2015-12-21T12:47:59ZA purported new mathematics proof is impenetrable – now what?<figure><img src="https://images.theconversation.com/files/106736/original/image-20151219-27875-yjavw6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Wait, what was that? You lost me.</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=306402011&src=id">Notations image via www.shutterstock.com.</a></span></figcaption></figure><p>What happens when someone claims to have proved a famous conjecture? Well, it depends. When a paper is submitted, the journal editor will pass it off to a respected expert for examination. That referee will then scan the paper looking for a significant new idea. If there isn’t one, then the whole argument is unlikely to get much more scrutiny. </p>
<p>But if there is a kernel of a new approach, it will be checked carefully. Additional experts may be consulted. Eventually the mathematics community may reach consensus that the argument is correct and the conjecture becomes a theorem. This can happen outside the formal refereeing process thanks to preprint servers such as the <a href="http://arxiv.org">arXiv</a>, but in the end, enough expert referees have to give the work their imprimatur before the paper is finally published in a journal.</p>
<p>In my mathematical career, there have been a few such big announcements, the most well-known being Andrew Wiles’ <a href="https://en.wikipedia.org/wiki/Wiles'_proof_of_Fermat's_Last_Theorem">solution</a> of Fermat’s Last Theorem in 1994. Grigori Perelman’s <a href="https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture">proof</a> in 2003 of the Poincaré Conjecture comes to mind as well. Now a reclusive yet respected Japanese mathematician has put forth a solution to another notorious problem.</p>
<p>In those earlier examples, the stature of the mathematicians involved made other experts interested in verifying their results. But what if the proposed solution is impenetrable? What if it reads, as University of Wisconsin Math Professor Jordan Ellenberg <a href="https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/">put it on his blog</a>, like mathematics from the future, full of new concepts and definitions that are disconnected from current language and techniques? If the author is relatively unknown it may be dismissed, or even ignored. But if the mathematician has a reputation for being careful and producing solid results, what then?</p>
<h2>The ABC conjecture</h2>
<p>Shinichi Mochizuki of the <a href="http://www.kurims.kyoto-u.ac.jp/en/index.html">Research Institute for Mathematical Sciences</a> at Kyoto University is such a mathematician. In August 2012, he posted a series of four papers on his personal web page claiming to prove <a href="https://en.wikipedia.org/wiki/Abc_conjecture">the ABC conjecture</a>, an important outstanding problem in number theory. A proof would have Fermat’s Last Theorem as a consequence (at least for large enough exponents), and given the difficulty of Wiles’ proof of Fermat’s Last Theorem, we should expect a proof of the ABC conjecture to be similarly opaque.</p>
<p>The conjecture is fairly easy to state. Suppose we have three positive integers <em>a,b,c</em> satisfying <em>a+b=c</em> and having no prime factors in common. Let <em>d</em> denote the product of the distinct prime factors of the product <em>abc</em>. Then the conjecture asserts roughly there are only finitely many such triples with <em>c > d</em>. Or, put another way, if <em>a</em> and <em>b</em> are built up from small prime factors then <em>c</em> is usually divisible only by large primes.</p>
<p>Here’s a simple example. Take <em>a=16</em>, <em>b=21</em>, and <em>c=37</em>. In this case, <em>d = 2x3x7x37 = 1554</em>, which is greater than <em>c</em>. The ABC conjecture says that this happens almost all the time. There is plenty of numerical evidence to support the conjecture, and most experts in the field believe it to be true. But it hasn’t been mathematically proven – yet.</p>
<p>Enter Mochizuki. His papers develop a subject he calls <a href="https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory">Inter-Universal Teichmüller Theory</a>, and in this setting he proves a vast collection of results that culminate in a putative proof of the ABC conjecture. Full of definitions and new terminology invented by Mochizuki (there’s something called a Frobenioid, for example), almost everyone who has attempted to read and understand it has given up in despair. Add to that Mochizuki’s odd refusal to speak to the press or to travel to discuss his work and you would think the mathematical community would have given up on the papers by now, dismissing them as unlikely to be correct. And yet, his previous work is so careful and clever that the experts aren’t quite ready to give up.</p>
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<h2>A meeting at Oxford</h2>
<p>The <a href="http://www.claymath.org/">Clay Mathematics Institute</a> and the <a href="https://www.maths.ox.ac.uk/">Mathematical Institute</a> at Oxford recently sponsored a <a href="https://www.maths.nottingham.ac.uk/personal/ibf/files/iut-sch1.html">meeting</a> about Mochizuki’s work. He was not in attendance, but many of the world’s leading number theorists and arithmetic geometers were. The goal was not to verify the proof of the ABC conjecture, but rather to equip experts in the field with enough background and information to at least begin to read through the papers carefully. There are many summaries of the meeting online (Stanford Math Professor <a href="http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/">Brian Conrad’s</a> is particularly detailed and illuminating), and some attendees <a href="https://twitter.com/search?q=mochizuki%20shinichi&src=tyah">tweeted</a> about it.</p>
<p>The general feeling was one of frustration, especially during the last two days when audience members repeatedly asked for illustrative examples, were promised they were coming, but then they never materialized. Mathematicians have little patience for being led down a rabbit hole, but the potential payoff in this case may persuade some to at least go in a little deeper.</p>
<h2>Prognosis</h2>
<p>It’s not clear what the future holds for Mochizuki’s proof. A small handful of mathematicians claim to have read, understood and verified the argument; a much larger group remains completely baffled. The December workshop reinforced the community’s desperate need for a translator, someone who can explain Mochizuki’s strange new universe of ideas and provide concrete examples to illustrate the concepts. Until that happens, the status of the ABC conjecture will remain unclear.</p>
<p>There’s a general sense among nonmathematicians that the subject is either right or wrong, and the truth is easily discovered. While our discipline does insist on rigorous, logical proof of correctness, we often argue over the details. This is good for mathematics since it generally leads to better exposition and streamlined proofs.</p>
<p>These arguments have happened before. Wiles’ proof of Fermat’s Last Theorem was scrutinized thoroughly, and an error was found which had to be corrected. Perelman’s work on the Poincaré Conjecture was only a detailed sketch of a proof which required hard work on the part of others to be made rigorous. Mochizuki’s work may eventually pass the test, but it could take many years before we get to a clean version that can be more widely understood.</p><img src="https://counter.theconversation.com/content/52491/count.gif" alt="The Conversation" width="1" height="1" />
A Japanese mathematician says he's proved a famous unsolved conjecture. The problem is, nobody can understand the solution he's put forth.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/450852015-08-18T10:06:28Z2015-08-18T10:06:28ZIn the push for marketable skills, are we forgetting the beauty and poetry of STEM disciplines?<figure><img src="https://images.theconversation.com/files/92112/original/image-20150817-5110-17nt3z4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">There is beauty in mathematical ideas and proofs.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/lucapost/694780262/in/photolist-24oVY3-diC14-4f3Jaz-4JqiwG-a5sw5-2RsTt1-geDfL-agNTbS-bz6igw-4f7GCA-aZLKD4-acJS5w-zdTJr-o8nVHc-6GsoZ-A3oZS-cd1WBC-8BMbiL-jXn1k8-jy4a28-4ikigj-usq3wD-6zjnBu-oo7TWg-anDsYW-2RsUqE-rzSR2m-pktg1Y-6aBPfC-qzzDXg-akeS8f-LfcF1-wdC58y-fkp13e-e9XnEF-73kFqy-d4AxJs-97N2Vr-baxAc-ugXsf-oqbq-8hDsUX-acJS9E-cVnd-pnMLBq-acJS7o-vhEJ3-6Mbpka-pDrCn8-5XbUTK">lucapost</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Thousands of students are preparing to begin their job searches with newly earned STEM (science, technology, engineering and mathematics) degrees in hand, eagerly waiting to use the logical, analytical and practical skills they’ve acquired.</p>
<p>However, as qualified as they might be, they could be missing one critical component of the STEM field – art.</p>
<p>I pursued an education and career in computer science and mathematics. And I know only too well that in the field of computer science, there is often an emphasis on elegance and beauty alongside sheer practicality. Indeed, programming itself is sometimes referred to as an <a href="http://ruben.verborgh.org/blog/2013/02/21/programming-is-an-art/">art</a>.</p>
<p>It is the same in related fields. The discipline of mathematics has long championed beauty as an important quality of ideas and proofs. And, of course, many engineers value elegance and beauty as important components in their designs and solutions.</p>
<h2>Poetry is at the heart of technology</h2>
<p>As many seasoned programmers and mathematicians will tell you, there is <a href="http://www.i-programmer.info/news/200-art/6808-writing-code-as-poetry-poetry-as-code.html">poetry in technology</a>. In fact, some regard such poetry as being at the heart of what they do. </p>
<p>In the 1980s, <a href="http://www.tracykidder.com/">Tracy Kidder</a> wrote The Soul of a New Machine, a book about the <a href="https://www.nytimes.com/books/99/01/03/specials/kidder-soul.html%22">pressure and effort</a> of building a next-generation computer. But more importantly, that account opened a lot of people’s eyes to the passion and beauty in creating these machines.</p>
<p>Many of the engineers in the book repeatedly explained that they didn’t work for the money, but rather for the gratification of invention and design – in essence, the beauty of it.</p>
<p>Indeed, reading that book was especially meaningful to me as I began my own studies in computing. </p>
<p>As I know through experience, constructing something poetic/beautiful is very fulfilling to the practitioner. Computer scientists value elegance and beauty in the creation of algorithms and computer programs. </p>
<h2>Mathematical beauty around us</h2>
<p>Similarly, ideas of beauty and poetry have always been important in mathematics.</p>
<p>Prominent mathematicians and computer scientists have long embraced elegance, beauty, poetry and literacy in the code that they write and the theorems that they prove. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/92115/original/image-20150817-25727-jkcswp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/92115/original/image-20150817-25727-jkcswp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=361&fit=crop&dpr=1 600w, https://images.theconversation.com/files/92115/original/image-20150817-25727-jkcswp.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=361&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/92115/original/image-20150817-25727-jkcswp.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=361&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/92115/original/image-20150817-25727-jkcswp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=453&fit=crop&dpr=1 754w, https://images.theconversation.com/files/92115/original/image-20150817-25727-jkcswp.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=453&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/92115/original/image-20150817-25727-jkcswp.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=453&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Beauty is important in programming as well.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/seeminglee/8921779798/in/photolist-eAosiC-eM7Nt3-8fUHJ5-dJJaMa-75vxwE-7YmQp6-nVVCfx-7FiU1M-88RAmR-qcSVGW-73zoNZ-o9rumE-o9jEvD-o9CnNi-kJz7fa-75tWag-6sr9f1-74T8P4-9fK69A-pVwTQC-8zPfCv-75vGtW-6h7DLk-ao4yLs-6KosGR-9NnUjX-75y7TJ-9NqG1N-8KuSmz-9hYhPp-9NqGR3-9bavnP-qXct22-imxhyi-9NnXnt-4qfL61-byidYK-9wCAyR-e9VDki-8w1M8R-9fK64C-gDR3aL-bCL9xk-dDNWKm-73Arz1-81JzLb-j7vKq2-mocVVz-9s18nz-bBYzuW">See-ming Lee</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>These ideas, in fact, have been around for millennia. Indeed, the extreme separation of the disciplines is relatively new in Western history.</p>
<p>Those doing science (natural philosophy) and mathematics were also often doing poetry and music. Many of today’s disciplines were subsumed as philosophy. So contemporary surprise at the idea that science and mathematics could be poetic is a somewhat recent phenomenon.</p>
<p>For example, Pythagoras was a philosopher/scientist/mystic/mathematician who <a href="http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html">explored beauty</a> in art and music.</p>
<p>This attention to beauty and pattern continued through Fibonacci and beyond.</p>
<p>Fibonacci (13th century), considered to be the leading mathematician in the Middle Ages, is probably best known for the <a href="https://www.mathsisfun.com/numbers/fibonacci-sequence.html">Fibonacci Sequence</a> named after him: a number in the sequence is the sum of the previous two numbers (eg, start with 1, 2; then add to get 3. Then add 2, 3 to get 5, and it goes on: 1,2,3,5,8,13,21,34,…). Fibonacci <a href="http://io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature">discovered</a> that much else that we regard as beautiful follows this elegant pattern.</p>
<p>This technical, mathematical beauty is evident in all of nature – from flower petals and shells to spiral galaxies and hurricanes,</p>
<h2>Discoveries come through intuition</h2>
<p>Intuition and discovery, rather than a kind of routine analysis, are important in computing as well as mathematics and science. Significant insights are known to come through intuition.</p>
<p>Intuition is deemed to be so important that developing “computer intuition” is one of the goals in the subfield of artificial intelligence.</p>
<p>So, in computing, there is really no “standard” way to write complex, interesting and aesthetically pleasing programs. Little surprise then that Stanford Professor Emeritus <a href="http://www-cs-faculty.stanford.edu/%7Euno/taocp.html">Donald Knuth’s</a> four-volume masterpiece is titled The Art of Computer Programming.</p>
<p>Similarly, years ago a colleague in the arts told me about the <a href="http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html">PBS show</a> on Andrew Wiles’s proof of <a href="http://mathworld.wolfram.com/FermatsLastTheorem.html">Fermat’s Last Theorem</a>. Wiles, a British mathematician, devoted much of his career to proving Fermat’s Last Theorem, a problem that no one had been able to solve for 300 years.</p>
<p>My colleague confided that she was moved to tears during the program. Until that show she had thought that mathematics was cold, dry, absolute and passionless. That show completely changed her view so that she could finally see the passion and the poetry that permeates the STEM fields.</p>
<h2>STEM versus liberal arts?</h2>
<p>Many STEM graduates today spend their college years enrolling only in courses they believe will benefit them in their field, zeroing in on skills that will make them more marketable in the digital age, while overlooking social sciences, humanities and the arts. Of course, likewise, many humanities students try to avoid taking science and mathematics courses. </p>
<p>And it shouldn’t be this way; but that’s a discussion for another day.</p>
<p>It’s projected that <a href="http://www.bls.gov/emp/ep_table_101.htm">685,000 new employment opportunities</a> will be created by 2022 in computer and mathematical occupations.</p>
<p>But today’s students need to remember that technology is not just a matter of rote procedure – completing the task according to set protocol; that would not be particularly elegant. </p>
<p>As sciences, technology and computing become ever more powerful forces in the world, it’s important that the people piecing these things together are ethical and bring in the human attributes that are central to a liberal arts education.</p>
<p>We need thinkers, visionaries and creative minds. As the technology industry grows – and with it, employment opportunities – we need more candidates who are rooted in thought and fewer who can simply carry out a task.</p>
<p>For those students graduating with a liberal arts degree, who are unsure where their job hunt will take them, we welcome you with open arms to technology, mathematics and computing.</p>
<p>And for those in technology, celebrate your humanity and the the available cultural riches; become aware of the intuition and the poetry in what you do. Bring with you your love for beauty, passion and artistry, and be prepared to use them.</p><img src="https://counter.theconversation.com/content/45085/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Paul Myers does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Poetry is at the heart of technology. Did not Pythagoras find the connections between beautiful music and mathematics?Paul Myers, Chair of Computer Science , Trinity UniversityLicensed as Creative Commons – attribution, no derivatives.