Agents, Agents, Agents - idmontie's Portfolio

I was researching the origin of the Basel problem to write a short blog post about
how Euler approached the problem (only to get beaten by <a href="https://www.youtube.com/watch?v=jmtiWGnj5os">anime characters talking
about it</a>). I however wanted to
dive deeper into the context of why Euler wanted to even approach this problem,
so I investigated the work of Petri Mengoli.
The publication of his work “On the Addition of Fractions” is available via the
<a href="https://archive.org/details/bub_gb_PrKgVx1LcUUC/mode/1up">Internet Archive in its original Latin</a>.
Petri Mengoli’s work focuses on the harmonic series in the preface and shows
that it diverges. The harmonic series is written in modern notation as:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mn>1</mn><mi>n</mi></mfrac><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots</annotation></semantics></math>n=1∑∞​n1​=1+21​+31​+41​+…
His proof relies on finding groupings of fractions that sum greater to one
(which his paper refers to as “unity”). First, look at the first 3 terms after
“unity”: <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn><mo>+</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>3</mn><mo>+</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/2 + 1/3 + 1/4</annotation></semantics></math>1/2+1/3+1/4. When added, we see that when we set common denominators:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>12</mn><mo>+</mo><mn>4</mn><mi mathvariant="normal">/</mi><mn>12</mn><mo>+</mo><mn>3</mn><mi mathvariant="normal">/</mi><mn>12</mn><mo>=</mo><mn>13</mn><mi mathvariant="normal">/</mi><mn>12</mn><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">6/12 + 4/12 + 3/12 = 13 / 12 &gt; 1</annotation></semantics></math>6/12+4/12+3/12=13/12&gt;1
And in fact, for any three terms:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>&gt;</mo><mfrac><mn>3</mn><mi>n</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{n - 1} + \frac{1}{n} + \frac{1}{n+1} &gt; \frac{3}{n}</annotation></semantics></math>n−11​+n1​+n+11​&gt;n3​
Next Mengoli shows that there is another group of fractions that adds up to
greater than one, and another set of fractions after that set that also adds up
to one. He demonstrates that for any given number, there is a finite set of
fractions that can be added to get a value greater than that. Therefore the
series must diverge.
His approach here is inspired by Archimedes’ &quot;Quadrature of the Parabola&quot;,
and focuses on using the “method of exhaustion”, rather than manipulating
infinite series or dealing with limits.
From this work, comes the original quote of what became known as the Basel
problem:
<blockquote class="border-l-4 border-gray-300 pl-4">
Ab huius fractionum dispositionis contemplatione feliciter
expeditus, ad aliam progrediebar dispositionem, in qua
singulæ unitates numeris quadratis denominantur.
Hæc speculatio fructus quidem laboris rependit, nondum tamen
effecta est solvendo, sed ingenii ditioris postulat adminiculum,
ut præcisam dispositionis, quam mihimetipst proposui, summam
valeat reportare.
</blockquote>
Roughly translating to:
<blockquote class="border-l-4 border-gray-300 pl-4">
Having successfully concluded my contemplation of this arrangement
of fractions, I proceeded to another arrangement, in which
each unit is denominated by square numbers.
This speculation indeed rewards labor with fruit, but it has not
yet been completed in solving and demands the assistance of
greater ingenuity so that the precise sum of the arrangement
I proposed to myself may be recovered.
</blockquote>
This is the work that perhaps sparked the inspiration for Eulers proof that the
sum of the reciprocal of the squares adds up to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>π</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>6</mn></mrow><annotation encoding="application/x-tex">\pi^2 /6</annotation></semantics></math>π2/6 . Euler required much
more advanced mathematics and insight in order to come up with this equality, and
he revisited the sum in multiple letters with slightly different arguments to
validate his work.
For a full write up on this, see my self-published work here: <a href="/docs/on-the-addition-of-fractions.pdf">PDF Download</a>.

I was researching the origin of the Basel problem to write a short blog post about
how Euler approached the problem (only to get beaten by [anime characters talking
about it](https://www.youtube.com/watch?v=jmtiWGnj5os)). I however wanted to
dive deeper into the context of why Euler wanted to even approach this problem,
so I investigated the work of Petri Mengoli.

The publication of his work “On the Addition of Fractions” is available via the
[Internet Archive in its original Latin](https://archive.org/details/bub_gb_PrKgVx1LcUUC/mode/1up).
Petri Mengoli’s work focuses on the harmonic series in the preface and shows
that it diverges. The harmonic series is written in modern notation as:

$$
\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots
$$

His proof relies on finding groupings of fractions that sum greater to one
(which his paper refers to as “unity”). First, look at the first 3 terms after
“unity”: $1/2 + 1/3 + 1/4$. When added, we see that when we set common denominators:

$$
6/12 + 4/12 + 3/12 = 13 / 12 > 1
$$

And in fact, for any three terms:

$$
\frac{1}{n - 1} + \frac{1}{n} + \frac{1}{n+1} > \frac{3}{n}
$$

Next Mengoli shows that there is another group of fractions that adds up to
greater than one, and another set of fractions after that set that also adds up
to one. He demonstrates that for any given number, there is a finite set of
fractions that can be added to get a value greater than that. Therefore the
series must diverge.

His approach here is inspired by Archimedes’ "Quadrature of the Parabola",
and focuses on using the “method of exhaustion”, rather than manipulating
infinite series or dealing with limits.

From this work, comes the original quote of what became known as the Basel
problem:

> Ab huius fractionum dispositionis contemplatione feliciter
expeditus, ad aliam progrediebar dispositionem, in qua
singulæ unitates numeris quadratis denominantur.
Hæc speculatio fructus quidem laboris rependit, nondum tamen
effecta est solvendo, sed ingenii ditioris postulat adminiculum,
ut præcisam dispositionis, quam mihimetipst proposui, summam
valeat reportare.
>

Roughly translating to:

> Having successfully concluded my contemplation of this arrangement
of fractions, I proceeded to another arrangement, in which
each unit is denominated by square numbers.
This speculation indeed rewards labor with fruit, but it has not
yet been completed in solving and demands the assistance of
greater ingenuity so that the precise sum of the arrangement
I proposed to myself may be recovered.
>

This is the work that perhaps sparked the inspiration for Eulers proof that the
sum of the reciprocal of the squares adds up to $\pi^2 /6$ . Euler required much
more advanced mathematics and insight in order to come up with this equality, and
he revisited the sum in multiple letters with slightly different arguments to
validate his work.

For a full write up on this, see my self-published work here: [PDF Download](/docs/on-the-addition-of-fractions.pdf).

On the Addition of Fractions, by Petri Mengoli

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In 2008, Steve Ballmer once famously said “<a href="https://www.youtube.com/watch?v=Vhh_GeBPOhs">Developers, developers, developers</a>” in a highly meme-able video. He was trying to emphasize the importance of software developers in business and that their importance was only going to continue to grow. Microsoft shifted to try to support software developers, noticing that if they supported developers, they would build better software for Windows, which in turn would make people want to use Windows.
The 2010s saw companies like “Twilio” have guiding questions like Jeff Lawson’s “Why Isn’t That and API?”. This guiding principle was based around the idea that if you build great APIs for software developers, they would want to use them, and in doing so, spend money on those services. For that entire decade, companies would start, build for engineers, and use the fly-wheel effect to become multi-billion dollar companies.
<img alt="Why isn&#x27;t that an API?" src="/media/2025-04-21-agents-agents-agents/why-isnt-that-an-api.png" style="max-height:500px;margin:auto;text-align:center"/>
Now, in the age of LLMs (Large Language Models) and AI, my assumption about the future is that we are going to find that “AI Agents” are the next Big Thing. We will probably see software engineers coding not to make APIs that other developers can use, but so that AI Agents can use. With <a href="https://openai.com/index/introducing-operator/">OpenAI introducing Operator</a>, their AI that can do tasks for you autonomously, the trajectory looks clear for companies that want to partner in this space: make products that will interface with AI Agents*.
And this isn’t going to only apply to software engineering. Even those in marketing are starting to think about how their content can be better consumed by the crawlers that feed into the training data for these LLMs. In “<a href="https://benjamincongdon.me/blog/2025/01/25/AI-Slop-Suspicion-and-Writing-Back/">AI Slop Suspicion and Writing Back</a>”, the author hypothesizes about the future of SEO. Will we be writing not for other people to reach our content via search, but to see our content as part of an LLM’s answer to a question? If people are starting to transition to asking LLMs like ChatGPT a question first before they even Google, then wouldn’t getting your name and your thoughts ingested by the massive corpus of training data be beneficial in a sense, similar to a web-crawler indexing for search?
These are questions I think we will see answered very soon, and some of the answers may not be ones we like. We may very well be starting to write content, apps, and APIs not for other people, but for machines.
*Of course, not everyone is going to jump on this bandwagon. There are companies right now that make concerted efforts to make sure only humans use their site. They don’t have APIs, they stick their entire site behind a Cloudflare captcha check. These aren’t the companies I’m talking about as they weren’t trying to bridge into the larger software developer ecosystem.


In 2008, Steve Ballmer once famously said “[Developers, developers, developers](https://www.youtube.com/watch?v=Vhh_GeBPOhs)” in a highly meme-able video. He was trying to emphasize the importance of software developers in business and that their importance was only going to continue to grow. Microsoft shifted to try to support software developers, noticing that if they supported developers, they would build better software for Windows, which in turn would make people want to use Windows.

The 2010s saw companies like “Twilio” have guiding questions like Jeff Lawson’s “Why Isn’t That and API?”. This guiding principle was based around the idea that if you build great APIs for software developers, they would want to use them, and in doing so, spend money on those services. For that entire decade, companies would start, build for engineers, and use the fly-wheel effect to become multi-billion dollar companies.

![Why isn't that an API?](/media/2025-04-21-agents-agents-agents/why-isnt-that-an-api.png)

Now, in the age of LLMs (Large Language Models) and AI, my assumption about the future is that we are going to find that “AI Agents” are the next Big Thing. We will probably see software engineers coding not to make APIs that other developers can use, but so that AI Agents can use. With [OpenAI introducing Operator](https://openai.com/index/introducing-operator/), their AI that can do tasks for you autonomously, the trajectory looks clear for companies that want to partner in this space: make products that will interface with AI Agents*.

And this isn’t going to only apply to software engineering. Even those in marketing are starting to think about how their content can be better consumed by the crawlers that feed into the training data for these LLMs. In “[AI Slop Suspicion and Writing Back](https://benjamincongdon.me/blog/2025/01/25/AI-Slop-Suspicion-and-Writing-Back/)”, the author hypothesizes about the future of SEO. Will we be writing not for other people to reach our content via search, but to see our content as part of an LLM’s answer to a question? If people are starting to transition to asking LLMs like ChatGPT a question first before they even Google, then wouldn’t getting your name and your thoughts ingested by the massive corpus of training data be beneficial in a sense, similar to a web-crawler indexing for search?

These are questions I think we will see answered very soon, and some of the answers may not be ones we like. We may very well be starting to write content, apps, and APIs not for other people, but for machines.

*Of course, not everyone is going to jump on this bandwagon. There are companies right now that make concerted efforts to make sure only humans use their site. They don’t have APIs, they stick their entire site behind a Cloudflare captcha check. These aren’t the companies I’m talking about as they weren’t trying to bridge into the larger software developer ecosystem.
