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====== Resolutions of mathematical conjectures over time ======

// Published 14 April, 2020; last updated 26 March, 2021 //

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<p>Conditioned on being remembered as a notable conjecture, the time-to-proof for a mathematical problem appears to be exponentially distributed with a half-life of about 100 years. However, these observations are likely to be distorted by various biases.</p>
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===== Support =====


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<p>In 2014, we found conjectures referenced on Wikipedia, and recorded the dates that they were proposed and resolved, if they were resolved. We updated this list of conjectures in 2020, marking any whose status had changed. We then used a Kaplan-Meier estimator<span class="easy-footnote-margin-adjust" id="easy-footnote-1-2409"></span><span class="easy-footnote"><a href="#easy-footnote-bottom-1-2409" title='“Kaplan–Meier Estimator.” Wikipedia. Wikimedia Foundation, April 1, 2020. &lt;a href="https://en.wikipedia.org/w/index.php?title=Kaplan–Meier_estimator&amp;amp;oldid=948523181"&gt;https://en.wikipedia.org/w/index.php?title=Kaplan–Meier_estimator&amp;amp;oldid=948523181&lt;/a&gt;.'><sup>1</sup></a></span> to approximate the survivorship function.<span class="easy-footnote-margin-adjust" id="easy-footnote-2-2409"></span><span class="easy-footnote"><a href="#easy-footnote-bottom-2-2409" title='“Survival Function.” Wikipedia. Wikimedia Foundation, October 8, 2019. &lt;a href="https://en.wikipedia.org/w/index.php?title=Survival_function&amp;amp;oldid=920310453"&gt;https://en.wikipedia.org/w/index.php?title=Survival_function&amp;amp;oldid=920310453&lt;/a&gt;.'><sup>2</sup></a></span></p>
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<p>The results of this exercise are recorded <a href="https://docs.google.com/spreadsheets/d/119VtkbzNWGdAhGsDx-IjYODkwCz0GHRNKDkipRGqjf8/edit?usp=sharing">here</a>.<span class="easy-footnote-margin-adjust" id="easy-footnote-3-2409"></span><span class="easy-footnote"><a href="#easy-footnote-bottom-3-2409" title='The ‘Data’ tab of &lt;a href="https://docs.google.com/spreadsheets/d/119VtkbzNWGdAhGsDx-IjYODkwCz0GHRNKDkipRGqjf8/edit?usp=sharing"&gt;this spreadsheet&lt;/a&gt; contains the list of conjectures we used and their sources. The ‘Kaplan Meier’ tab contains the calculation of the survival function. &lt;a href="https://docs.google.com/spreadsheets/d/119VtkbzNWGdAhGsDx-IjYODkwCz0GHRNKDkipRGqjf8/edit#gid=380799079&amp;amp;range=K2"&gt;The cell next to the cell marked ‘Exponential trendline’&lt;/a&gt; contains our calculation for the exponential function fitting our Kaplan-Meier estimator.'><sup>3</sup></a></span> Figure 1 below shows the survivorship function for the mathematical conjectures we found. The data is fit closely by an exponential function with a half-life of 117 years.<span class="easy-footnote-margin-adjust" id="easy-footnote-4-2409"></span><span class="easy-footnote"><a href="#easy-footnote-bottom-4-2409" title="When fitting our exponential, we did not count the last point at 750 years, because it had a y-value of 0, which the Google Sheets LOGEST function would not accept when generating a best-fit curve. Nonetheless, Figure 1 suggests that the last point seems to fit our exponential reasonably well."><sup>4</sup></a></span></p>
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<figure class="wp-block-image size-large is-resized">
<img alt="" class="wp-image-2410" height="358" sizes="(max-width: 580px) 100vw, 580px" src="https://aiimpacts.org/wp-content/uploads/2020/04/image-1024x633.png" srcset="https://aiimpacts.org/wp-content/uploads/2020/04/image-1024x633.png 1024w, https://aiimpacts.org/wp-content/uploads/2020/04/image-300x186.png 300w, https://aiimpacts.org/wp-content/uploads/2020/04/image-768x475.png 768w, https://aiimpacts.org/wp-content/uploads/2020/04/image-1536x950.png 1536w, https://aiimpacts.org/wp-content/uploads/2020/04/image.png 1646w" width="580"/>
<figcaption>
                  Figure 1: Survivorship function of mathematical conjectures over time, also known as the fraction of mathematical conjectures unresolved at time t after being posed.
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===== Biases =====


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<p>We are using resolution times for remembered conjectures as a proxy for resolution times for all conjectures. Resolution time for remembered conjectures might be biased in several ways: old conjectures are perhaps more likely to be remembered if they are solved than if they are not, very recently solved conjectures are probably more likely to be remembered (though this only matters because the rate of conjecture posing has probably changed over time), and conjectures that were especially hard to solve might also be more notable. The latter hundred years contains few data points, which makes it particularly easy for it to be inaccurate.</p>
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===== Relevance =====


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<p>To the extent that open theoretical problems in AI are similar to math problems, time to solve math problems may be informative for forming a prior on time to solve AI problems.</p>
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<p><em>Corresponding author: Asya Bergal</em></p>
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===== Notes =====


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<ol class="easy-footnotes-wrapper">
<li><div class="li">
<span class="easy-footnote-margin-adjust" id="easy-footnote-bottom-1-2409"></span>“Kaplan–Meier Estimator.” Wikipedia. Wikimedia Foundation, April 1, 2020. <a href="https://en.wikipedia.org/w/index.php?title=Kaplan%E2%80%93Meier_estimator&amp;oldid=948523181">https://en.wikipedia.org/w/index.php?title=Kaplan–Meier_estimator&amp;oldid=948523181</a>.<a class="easy-footnote-to-top" href="#easy-footnote-1-2409"></a>
</div></li>
<li><div class="li">
<span class="easy-footnote-margin-adjust" id="easy-footnote-bottom-2-2409"></span>“Survival Function.” Wikipedia. Wikimedia Foundation, October 8, 2019. <a href="https://en.wikipedia.org/w/index.php?title=Survival_function&amp;oldid=920310453">https://en.wikipedia.org/w/index.php?title=Survival_function&amp;oldid=920310453</a>.<a class="easy-footnote-to-top" href="#easy-footnote-2-2409"></a>
</div></li>
<li><div class="li">
<span class="easy-footnote-margin-adjust" id="easy-footnote-bottom-3-2409"></span>The ‘Data’ tab of <a href="https://docs.google.com/spreadsheets/d/119VtkbzNWGdAhGsDx-IjYODkwCz0GHRNKDkipRGqjf8/edit?usp=sharing">this spreadsheet</a> contains the list of conjectures we used and their sources. The ‘Kaplan Meier’ tab contains the calculation of the survival function. <a href="https://docs.google.com/spreadsheets/d/119VtkbzNWGdAhGsDx-IjYODkwCz0GHRNKDkipRGqjf8/edit#gid=380799079&amp;range=K2">The cell next to the cell marked ‘Exponential trendline’</a> contains our calculation for the exponential function fitting our Kaplan-Meier estimator.<a class="easy-footnote-to-top" href="#easy-footnote-3-2409"></a>
</div></li>
<li><div class="li">
<span class="easy-footnote-margin-adjust" id="easy-footnote-bottom-4-2409"></span>When fitting our exponential, we did not count the last point at 750 years, because it had a y-value of 0, which the Google Sheets LOGEST function would not accept when generating a best-fit curve. Nonetheless, Figure 1 suggests that the last point seems to fit our exponential reasonably well.<a class="easy-footnote-to-top" href="#easy-footnote-4-2409"></a>
</div></li>
</ol>
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