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-<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN" "http://www.w3.org/TR/REC-html40/loose.dtd"> <html><body> <p>Math is really a beautiful system (framework?) from which we can derive ideas of beauty through symmetry and geometry, study of complex systems through calculus, and also of creativity.</p> <ul> <li>How would I define creativity? o Imagine you’re an enthusiastic pursuer of truth. In front of you are a some numbers and two operators: + (ordinary addition) and * (ordinary multiplication). Science, or the discovery of knowledge, is analogous to discovering new numbers. You randomly add two numbers, 1 and -1 and discover 0. Then you add 1 to 1 and obtain 2. Repeating this process you discover all the integers. From God’s point of view, addition and multiplication over integers is closed in this set, therefore you can’t discover anything beyond the horizon of integers, even though you can discover a lot. You can find patterns, such as properties of commutativity, transitivity and distributivity. o Creativity, in this scenario, would be proposing another operator such as division. Having this new operator in your arsenal of tools to perceive reality, you can now compute a new class of numbers, the rationals and thus discover the set of all real numbers. o The infiniteness of numbers is analogous to the infinity of knowledge.  Further creativity would be introducing the square root, and this tool extends our perceptual horizon to encompass the realm of complex/imaginary numbers.  A different civilization, using a different set of symbols to encode numbers, may make similar discoveries, i.e., numbers obtained following the same principles of computation using + and *, discovering commutativity, transitivity and distributivity. This underlies/represents the universality of knowledge. o I’ve ignored letters, phonemes, words to simplify this overly-idealized world to get my ideas on what is creativity across.</li> </ul> </body></html>
+<!DOCTYPE html> <html lang="en"> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <meta charset="utf-8"> <meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <title>A mathematical analogy for understanding creativity | Xuelong An Wang</title> <meta name="author" content="Xuelong An Wang"> <meta name="description" content="Using numbers to illustrate an intuitive definition of what creativity is"> <meta name="keywords" content="jekyll, jekyll-theme, academic-website, portfolio-website"> <link href="https://cdn.jsdelivr.net/npm/bootstrap@4.6.1/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha256-DF7Zhf293AJxJNTmh5zhoYYIMs2oXitRfBjY+9L//AY=" crossorigin="anonymous"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/mdbootstrap@4.20.0/css/mdb.min.css" integrity="sha256-jpjYvU3G3N6nrrBwXJoVEYI/0zw8htfFnhT9ljN3JJw=" crossorigin="anonymous"> <link defer rel="stylesheet" href="https://unpkg.com/bootstrap-table@1.21.4/dist/bootstrap-table.min.css"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@fortawesome/fontawesome-free@6.4.0/css/all.min.css" integrity="sha256-HtsXJanqjKTc8vVQjO4YMhiqFoXkfBsjBWcX91T1jr8=" crossorigin="anonymous"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/academicons@1.9.1/css/academicons.min.css" integrity="sha256-i1+4qU2G2860dGGIOJscdC30s9beBXjFfzjWLjBRsBg=" crossorigin="anonymous"> <link rel="stylesheet" type="text/css" href="https://fonts.googleapis.com/css?family=Roboto:300,400,500,700|Roboto+Slab:100,300,400,500,700|Material+Icons"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jwarby/jekyll-pygments-themes@master/github.css" media="" id="highlight_theme_light"> <link href="https://cdn.rawgit.com/afeld/bootstrap-toc/v1.0.1/dist/bootstrap-toc.min.css" rel="stylesheet"> <link rel="shortcut icon" href="/assets/img/brain-fractal-3.jpg"> <link rel="stylesheet" href="/assets/css/main.css?d41d8cd98f00b204e9800998ecf8427e"> <link rel="canonical" href="https://awxlong.github.io/blog/2023/creativity/"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jwarby/jekyll-pygments-themes@master/native.css" media="none" id="highlight_theme_dark"> <script src="/assets/js/theme.js?96d6b3e1c3604aca8b6134c7afdd5db6"></script> <script src="/assets/js/dark_mode.js?9b17307bb950ffa2e34be0227f53558f"></script> </head> <body class="fixed-top-nav "> <header> <nav id="navbar" class="navbar navbar-light navbar-expand-sm fixed-top"> <div class="container"> <a class="navbar-brand title font-weight-lighter" href="/"><span class="font-weight-bold">Xuelong </span>An Wang</a> <button class="navbar-toggler collapsed ml-auto" type="button" data-toggle="collapse" data-target="#navbarNav" aria-controls="navbarNav" aria-expanded="false" aria-label="Toggle navigation"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar top-bar"></span> <span class="icon-bar middle-bar"></span> <span class="icon-bar bottom-bar"></span> </button> <div class="collapse navbar-collapse text-right" id="navbarNav"> <ul class="navbar-nav ml-auto flex-nowrap"> <li class="nav-item "> <a class="nav-link" href="/">about</a> </li> <li class="nav-item active"> <a class="nav-link" href="/blog/">blog<span class="sr-only">(current)</span></a> </li> <li class="nav-item "> <a class="nav-link" href="/publications/">research</a> </li> <li class="nav-item "> <a class="nav-link" href="/projects/">projects</a> </li> <li class="nav-item "> <a class="nav-link" href="/cv/">cv</a> </li> <li class="toggle-container"> <button id="light-toggle" title="Change theme"> <i class="fas fa-moon"></i> <i class="fas fa-sun"></i> </button> </li> </ul> </div> </div> </nav> <progress id="progress" value="0"> <div class="progress-container"> <span class="progress-bar"></span> </div> </progress> </header> <div class="container mt-5"> <div class="row"> <div class="col-sm-3"> <nav id="toc-sidebar" class="sticky-top"></nav> </div> <div class="col-sm-9"> <div class="post"> <header class="post-header"> <h1 class="post-title">A mathematical analogy for understanding creativity</h1> <p class="post-meta">November 2, 2023</p> <p class="post-tags"> <a href="/blog/2023"> <i class="fas fa-calendar fa-sm"></i> 2023 </a>   ·   <a href="/blog/tag/food-for-thought"> <i class="fas fa-hashtag fa-sm"></i> food-for-thought</a>     ·   <a href="/blog/category/blog-post"> <i class="fas fa-tag fa-sm"></i> blog-post</a>   </p> </header> <article class="post-content"> <div id="markdown-content"> <p>Math is really a beautiful framework with which we can analyze several ideas. For instance, beauty can be seen through the lenses of math using symmetry and geometry, and the study of complex systems can be cast through calculus and probability theory.</p> <p>Given this premise, can we use math to understand what creativity is. Here is my attempt:</p> <h1 id="how-can-creativity-be-defined">How can creativity be defined?</h1> <p>Let us run the following thought experiment. Imagine you’re an enthusiastic pursuer of truth. The whole universe is simply an empty space. In this void, in front of you are a some numbers and two operators: + (ordinary addition) and * (ordinary multiplication)<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>.</p> <p>Science, or the discovery of knowledge, is analogous to discovering new numbers. You randomly add two numbers, 1 and -1 and discover 0. Then you add 1 to 1 and obtain 2. Repeating this process, you end up discovering all the integers. Furthermore, you can find patterns that lead to theorems among the integers, such as properties of commutativity, transitivity and distributivity.</p> <p>However, from a metaphysical point of view, addition and multiplication over integers is closed in this set, therefore you can’t discover anything beyond the horizon of integers.</p> <h1 id="can-you-discover-anything-else">Can you discover anything else?</h1> <p>Creativity, in this scenario, would be proposing another operator such as division. Having this new operator in your mind’s eye to perceive reality, you can now compute a new class of numbers, the rationals and thus discover the set of all real numbers.</p> <p>In either case, prior and after the discovery of the division operator, the infiniteness of numbers is analogous to the infinity of knowledge.</p> <p>Creativity can also materialize by introducing the square root, and this new operator extends our perceptual horizon to encompass the realm of complex/imaginary numbers.</p> <h1 id="more-analogies">More analogies</h1> <p>A different civilization from another point in space, using a different set of symbols to encode numbers, may make similar discoveries, i.e., numbers obtained following the same principles of computation using + and *. Moreover, they may also discover commutativity, transitivity and distributivity. This analogy represents the universality of knowledge.</p> <p>In the above thought experiment, I’ve ignored letters, phonemes, and words to simplify this overly-idealized world to get my ideas on what is creativity across.</p> <figure> <img src="/assets/img/creativity.jpg" alt="Sorry, an unanticipated error occured and the image can't load." width="100%" height="auto"> <figcaption id="creativity"> The thumbnail is generate by Dall-E powered MS Bing Image Generator given the prompt "your interpretation of what creativity is". </figcaption> </figure> <p>PS: Demis Hassabis, Deepmind CEO, also comments on another definition of <a href="https://www.youtube.com/watch?v=Gfr50f6ZBvo" rel="external nofollow noopener" target="_blank">creativity, or AGI</a>. This is going beyond statistical interpolation and extrapolation of training data. An AGI model that is creative should be able to give an output to an instruction such as “design a game that has simple rules, but is extremely hard to master, and can be played for hours”.</p> <div class="footnotes" role="doc-endnotes"> <ol> <li id="fn:1" role="doc-endnote"> <p>Perhaps these numbers and operators existed there a priori, maybe they emerged from a Creator, or perhaps they emerged from a singularity. <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p> </li> </ol> </div> </div> </article> <br> <hr> <br> <ul class="list-disc pl-8"></ul> <h2 class="text-3xl font-semibold mb-4 mt-12">Enjoy Reading This Article?</h2> <p class="mb-2">Here are some more articles you might like to read next:</p> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2020/miRNA/">miRNA: General Blood Biomarkers</a> </li> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2023/nesy-summer-school/">Reflections on the Neurosymbolic Summer School 2023</a> </li> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2023/tale-two-worlds/">A tale of two worlds: the natural and the mathematical</a> </li> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2020/math-uncertainty/">Math, certainty and uncertainty</a> </li> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2021/yuval-orig-math/">Review on Sapiens and Homo deus by Yuval Harari</a> </li> <div id="disqus_thread" style="max-width: 800px; margin: 0 auto;"> <script type="text/javascript">var disqus_shortname="al-folio",disqus_identifier="/blog/2023/creativity",disqus_title="A mathematical analogy for understanding creativity";!function(){var e=document.createElement("script");e.type="text/javascript",e.async=!0,e.src="//"+disqus_shortname+".disqus.com/embed.js",(document.getElementsByTagName("head")[0]||document.getElementsByTagName("body")[0]).appendChild(e)}();</script> <noscript>Please enable JavaScript to view the <a href="http://disqus.com/?ref_noscript" rel="external nofollow noopener" target="_blank">comments powered by Disqus.</a> </noscript> </div> </div> </div> </div> </div> <footer class="fixed-bottom"> <div class="container mt-0"> © Copyright 2023 Xuelong An Wang. 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		<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN" "http://www.w3.org/TR/REC-html40/loose.dtd"> <html><body> <p>Math is really a beautiful system (framework?) from which we can derive ideas of beauty through symmetry and geometry, study of complex systems through calculus, and also of creativity.</p> <ul> <li>How would I define creativity? o Imagine you’re an enthusiastic pursuer of truth. In front of you are a some numbers and two operators: + (ordinary addition) and * (ordinary multiplication). Science, or the discovery of knowledge, is analogous to discovering new numbers. You randomly add two numbers, 1 and -1 and discover 0. Then you add 1 to 1 and obtain 2. Repeating this process you discover all the integers. From God’s point of view, addition and multiplication over integers is closed in this set, therefore you can’t discover anything beyond the horizon of integers, even though you can discover a lot. You can find patterns, such as properties of commutativity, transitivity and distributivity. o Creativity, in this scenario, would be proposing another operator such as division. Having this new operator in your arsenal of tools to perceive reality, you can now compute a new class of numbers, the rationals and thus discover the set of all real numbers. o The infiniteness of numbers is analogous to the infinity of knowledge.  Further creativity would be introducing the square root, and this tool extends our perceptual horizon to encompass the realm of complex/imaginary numbers.  A different civilization, using a different set of symbols to encode numbers, may make similar discoveries, i.e., numbers obtained following the same principles of computation using + and *, discovering commutativity, transitivity and distributivity. This underlies/represents the universality of knowledge. o I’ve ignored letters, phonemes, words to simplify this overly-idealized world to get my ideas on what is creativity across.</li> </ul> </body></html>
		<!DOCTYPE html> <html lang="en"> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <meta charset="utf-8"> <meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <title>A mathematical analogy for understanding creativity \| Xuelong An Wang</title> <meta name="author" content="Xuelong An Wang"> <meta name="description" content="Using numbers to illustrate an intuitive definition of what creativity is"> <meta name="keywords" content="jekyll, jekyll-theme, academic-website, portfolio-website"> <link href="https://cdn.jsdelivr.net/npm/bootstrap@4.6.1/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha256-DF7Zhf293AJxJNTmh5zhoYYIMs2oXitRfBjY+9L//AY=" crossorigin="anonymous"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/mdbootstrap@4.20.0/css/mdb.min.css" integrity="sha256-jpjYvU3G3N6nrrBwXJoVEYI/0zw8htfFnhT9ljN3JJw=" crossorigin="anonymous"> <link defer rel="stylesheet" href="https://unpkg.com/bootstrap-table@1.21.4/dist/bootstrap-table.min.css"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@fortawesome/fontawesome-free@6.4.0/css/all.min.css" integrity="sha256-HtsXJanqjKTc8vVQjO4YMhiqFoXkfBsjBWcX91T1jr8=" crossorigin="anonymous"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/academicons@1.9.1/css/academicons.min.css" integrity="sha256-i1+4qU2G2860dGGIOJscdC30s9beBXjFfzjWLjBRsBg=" crossorigin="anonymous"> <link rel="stylesheet" type="text/css" href="https://fonts.googleapis.com/css?family=Roboto:300,400,500,700\|Roboto+Slab:100,300,400,500,700\|Material+Icons"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jwarby/jekyll-pygments-themes@master/github.css" media="" id="highlight_theme_light"> <link href="https://cdn.rawgit.com/afeld/bootstrap-toc/v1.0.1/dist/bootstrap-toc.min.css" rel="stylesheet"> <link rel="shortcut icon" href="/assets/img/brain-fractal-3.jpg"> <link rel="stylesheet" href="/assets/css/main.css?d41d8cd98f00b204e9800998ecf8427e"> <link rel="canonical" href="https://awxlong.github.io/blog/2023/creativity/"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jwarby/jekyll-pygments-themes@master/native.css" media="none" id="highlight_theme_dark"> <script src="/assets/js/theme.js?96d6b3e1c3604aca8b6134c7afdd5db6"></script> <script src="/assets/js/dark_mode.js?9b17307bb950ffa2e34be0227f53558f"></script> </head> <body class="fixed-top-nav "> <header> <nav id="navbar" class="navbar navbar-light navbar-expand-sm fixed-top"> <div class="container"> <a class="navbar-brand title font-weight-lighter" href="/"><span class="font-weight-bold">Xuelong </span>An Wang</a> <button class="navbar-toggler collapsed ml-auto" type="button" data-toggle="collapse" data-target="#navbarNav" aria-controls="navbarNav" aria-expanded="false" aria-label="Toggle navigation"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar top-bar"></span> <span class="icon-bar middle-bar"></span> <span class="icon-bar bottom-bar"></span> </button> <div class="collapse navbar-collapse text-right" id="navbarNav"> <ul class="navbar-nav ml-auto flex-nowrap"> <li class="nav-item "> <a class="nav-link" href="/">about</a> </li> <li class="nav-item active"> <a class="nav-link" href="/blog/">blog<span class="sr-only">(current)</span></a> </li> <li class="nav-item "> <a class="nav-link" href="/publications/">research</a> </li> <li class="nav-item "> <a class="nav-link" href="/projects/">projects</a> </li> <li class="nav-item "> <a class="nav-link" href="/cv/">cv</a> </li> <li class="toggle-container"> <button id="light-toggle" title="Change theme"> <i class="fas fa-moon"></i> <i class="fas fa-sun"></i> </button> </li> </ul> </div> </div> </nav> <progress id="progress" value="0"> <div class="progress-container"> <span class="progress-bar"></span> </div> </progress> </header> <div class="container mt-5"> <div class="row"> <div class="col-sm-3"> <nav id="toc-sidebar" class="sticky-top"></nav> </div> <div class="col-sm-9"> <div class="post"> <header class="post-header"> <h1 class="post-title">A mathematical analogy for understanding creativity</h1> <p class="post-meta">November 2, 2023</p> <p class="post-tags"> <a href="/blog/2023"> <i class="fas fa-calendar fa-sm"></i> 2023 </a> · <a href="/blog/tag/food-for-thought"> <i class="fas fa-hashtag fa-sm"></i> food-for-thought</a> · <a href="/blog/category/blog-post"> <i class="fas fa-tag fa-sm"></i> blog-post</a> </p> </header> <article class="post-content"> <div id="markdown-content"> <p>Math is really a beautiful framework with which we can analyze several ideas. For instance, beauty can be seen through the lenses of math using symmetry and geometry, and the study of complex systems can be cast through calculus and probability theory.</p> <p>Given this premise, can we use math to understand what creativity is. Here is my attempt:</p> <h1 id="how-can-creativity-be-defined">How can creativity be defined?</h1> <p>Let us run the following thought experiment. Imagine you’re an enthusiastic pursuer of truth. The whole universe is simply an empty space. In this void, in front of you are a some numbers and two operators: + (ordinary addition) and * (ordinary multiplication)<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>.</p> <p>Science, or the discovery of knowledge, is analogous to discovering new numbers. You randomly add two numbers, 1 and -1 and discover 0. Then you add 1 to 1 and obtain 2. Repeating this process, you end up discovering all the integers. Furthermore, you can find patterns that lead to theorems among the integers, such as properties of commutativity, transitivity and distributivity.</p> <p>However, from a metaphysical point of view, addition and multiplication over integers is closed in this set, therefore you can’t discover anything beyond the horizon of integers.</p> <h1 id="can-you-discover-anything-else">Can you discover anything else?</h1> <p>Creativity, in this scenario, would be proposing another operator such as division. Having this new operator in your mind’s eye to perceive reality, you can now compute a new class of numbers, the rationals and thus discover the set of all real numbers.</p> <p>In either case, prior and after the discovery of the division operator, the infiniteness of numbers is analogous to the infinity of knowledge.</p> <p>Creativity can also materialize by introducing the square root, and this new operator extends our perceptual horizon to encompass the realm of complex/imaginary numbers.</p> <h1 id="more-analogies">More analogies</h1> <p>A different civilization from another point in space, using a different set of symbols to encode numbers, may make similar discoveries, i.e., numbers obtained following the same principles of computation using + and . Moreover, they may also discover commutativity, transitivity and distributivity. This analogy represents the universality of knowledge.</p> <p>In the above thought experiment, I’ve ignored letters, phonemes, and words to simplify this overly-idealized world to get my ideas on what is creativity across.</p> <figure> <img src="/assets/img/creativity.jpg" alt="Sorry, an unanticipated error occured and the image can't load." width="100%" height="auto"> <figcaption id="creativity"> The thumbnail is generate by Dall-E powered MS Bing Image Generator given the prompt "your interpretation of what creativity is". </figcaption> </figure> <p>PS: Demis Hassabis, Deepmind CEO, also comments on another definition of <a href="https://www.youtube.com/watch?v=Gfr50f6ZBvo" rel="external nofollow noopener" target="_blank">creativity, or AGI</a>. This is going beyond statistical interpolation and extrapolation of training data. An AGI model that is creative should be able to give an output to an instruction such as “design a game that has simple rules, but is extremely hard to master, and can be played for hours”.</p> <div class="footnotes" role="doc-endnotes"> <ol> <li id="fn:1" role="doc-endnote"> <p>Perhaps these numbers and operators existed there a priori, maybe they emerged from a Creator, or perhaps they emerged from a singularity. <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p> </li> </ol> </div> </div> </article> <br> <hr> <br> <ul class="list-disc pl-8"></ul> <h2 class="text-3xl font-semibold mb-4 mt-12">Enjoy Reading This Article?</h2> <p class="mb-2">Here are some more articles you might like to read next:</p> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2020/miRNA/">miRNA: General Blood Biomarkers</a> </li> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2023/nesy-summer-school/">Reflections on the Neurosymbolic Summer School 2023</a> </li> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2023/tale-two-worlds/">A tale of two worlds: the natural and the mathematical</a> </li> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2020/math-uncertainty/">Math, certainty and uncertainty</a> </li> <li class="my-2"> <a class="text-pink-700 underline font-semibold hover:text-pink-800" href="/blog/2021/yuval-orig-math/">Review on Sapiens and Homo deus by Yuval Harari</a> </li> <div id="disqus_thread" style="max-width: 800px; margin: 0 auto;"> <script type="text/javascript">var disqus_shortname="al-folio",disqus_identifier="/blog/2023/creativity",disqus_title="A mathematical analogy for understanding creativity";!function(){var e=document.createElement("script");e.type="text/javascript",e.async=!0,e.src="//"+disqus_shortname+".disqus.com/embed.js",(document.getElementsByTagName("head")[0]\|\|document.getElementsByTagName("body")[0]).appendChild(e)}();</script> <noscript>Please enable JavaScript to view the <a href="http://disqus.com/?ref_noscript" rel="external nofollow noopener" target="_blank">comments powered by Disqus.</a> </noscript> </div> </div> </div> </div> </div> <footer class="fixed-bottom"> <div class="container mt-0"> © Copyright 2023 Xuelong An Wang. 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