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<!DOCTYPE html>
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<meta name="author" content="Ioannis Konstantoulas" />
<title>Deciding on Pure Mathematics</title>
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<body>
<header id="title-block-header">
<h1 class="title">Deciding on Pure Mathematics</h1>
<p class="author">Ioannis Konstantoulas</p>
</header>
<nav id="TOC" role="doc-toc">
<ul>
<li><a href="#sec:the_beginnings" id="toc-sec:the_beginnings"><span
class="toc-section-number">1</span> The Beginnings</a></li>
<li><a href="#sec:investigations_in_number_theory_and_analysis"
id="toc-sec:investigations_in_number_theory_and_analysis"><span
class="toc-section-number">2</span> Investigations in Number theory and
Analysis</a></li>
</ul>
</nav>
<h1 data-number="1" id="sec:the_beginnings"><span
class="header-section-number">1</span> The Beginnings</h1>
<p>When I was a high school student in the early 2000s, my relatives
wanted me to go into programming as a safe and stable career with many
employment opportunities. I loved programming and had amassed some
experience in C, assembly and the like writing coders/decoders and
cryptographic routines on my 386 machine running MS-DOS (and later
Windows 95). But I had already been enamored by pure mathematics, which
I had encountered trying to understand the underpinnings of
cryptography.</p>
<p>As I explored Computer Science and its applied aspects more and more,
I became increasingly worried that I would end up solving problems that
are not fundamental, but rather contingent on very specific and
ephemeral technologies. Needless to say, if I had been aware of the
Javafication of corporate coding<a href="#fn1" class="footnote-ref"
id="fnref1" role="doc-noteref"><sup>1</sup></a> and the subsequent
Webdev revolution, I wouldn’t have touched programming with a ten-foot
pole. Despite this, it was not those fears that made me choose
Mathematics; it was the encounter with one mathematical object.</p>
<p>The circumstances which led to my encounter with that object are
quite complicated and deserving a document of their own; or rather, the
main character, Professor Bill Transue of the University of Alabama,
does. That document is still in the works, as I am finding it hard to
cover the extent of his impact on my development. For now, it will
suffice to note him as my first teacher of mathematics and my first
mathematical friend.</p>
<h1 data-number="2"
id="sec:investigations_in_number_theory_and_analysis"><span
class="header-section-number">2</span> Investigations in Number theory
and Analysis</h1>
<p>Bill and I met online in the late 90s, I think 1998 in particular. We
were part of a fascinating US program that matched American university
teachers with school students from all over the world to explore and
solve a mathematical problem of our choice over email. Ours was the
following question: <em>find the probability that a randomly chosen
positive integer is prime</em>. We had to make proper mathematical sense
of the question, gather the necessary mathematical tools and attack the
problem with all we had.</p>
<p>In those days, and especially for a poor student from Greece with
very limited connectivity, quality information was very hard to find
online<a href="#fn2" class="footnote-ref" id="fnref2"
role="doc-noteref"><sup>2</sup></a>. I scavenged information about
programming from disparate sources like random tutorials, old Phrack
articles, a limited selection of books from the local library and tons
and tons of experimentation.</p>
<p>For higher mathematics the situation was much worse; I could not find
much of anything rigorous and complete. Part of it was that I didn’t
know where or how to search; Bill would teach me as much as he could
over email during our once-a-week correspondence, but there were hard
limits to how much information he could convey. Nevertheless, with his
help I built up an understanding of probability distributions, densities
and the distribution of discrete sequences, basic complex analysis (I
already knew some real analysis) and aspects of number theory I had not
seen in my cryptographic studies.</p>
<p>After a long journey of trying and disposing of ideas, I managed to
put the problem into a form that could be attacked by powerful analytic
methods. Very roughly, the probability of a number being prime could be
approximated in finite intervals by an expression whose dominant term
“should” be <span class="math display">\[\left(
1 - \frac{1}{2}
\right)
\left(
1 - \frac{1}{3}
\right)
\ldots
\left(
1 - \frac{1}{p}
\right)\ldots\]</span></p>
<aside id="footnotes" class="footnotes footnotes-end-of-document"
role="doc-endnotes">
<hr />
<ol>
<li id="fn1"><p>I did write a hideous, over-abstracted C++ codebase in
University and I am still traumatized by the experience.<a
href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn2"><p>As an amusing aside, my first experience with
programming languages, long before I got a computer, was a book on COBOL
I/O; a story <em>definitely</em> deserving its own telling.<a
href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</aside>
</body>
</html>