How was I going to bring back the excitement of sucking at something?

Study the hardest subject in college. Obviously.

While at MIT, I almost failed the Discrete Mathematics course: 6.042.

It was a hard pill to swallow that changed my whole trajectory in college. This course is strictly easier than any of the introductory math major courses and I scored 30 points below the average of 95 on the first exam.

My exam session was also labeled too easy so the other session’s average (higher than my score) was bumped up to match my sessions average. Ouch. But, luckily, by the end of the class my final exam score bumped me up into a passing C.

My only C other than Senior Spring Chemistry.

I came to realize recently why. MIT gave me a week to catch up. I never took any logic or proofs fundamentals course beforehand. Proofs seemed to be pulled out of thin air.

Recently, I started the MIT Math curriculum as an independent study.

However, this time around, I am expanding the first week of each of the courses into a pre-course book.

Helpful books to onramp into pure math

For proofs, I found How to Read and Do Proofs. A little orange book that would have changed my major.

The best part is that the book’s density increases over time. The appendix, although most people probably skip it, is much closer to the texts I struggled with in my first attempt at pure math.

It reads really easily and covers the first lecture of most of the other pre course books I am currently reading.

Abstract Algebra by Judson is another pre book. MIT uses Artin for 18.701, which is very hard to read in my opinion. I will be going through it next, I am on chapter 16 of Judson’s. It is perfect for software engineer’s looking to learn Abstract Algebra. Coding Theory was covered in College but from the applied side. Burnside’s Counting Theorem was super interesting.

It also covered RSA, which I learned in 6.042, however, the clarity of Euclid’s Algorithm and fast modular exponentiation algorithm gave me a deeper understanding. I implemented the whole RSA chapter in python to make sure I understood it. I will be sharing a deep dive in a future post.

Elementary Number Theory by Burton was recommended to me by a Cryptographer friend. I am on chapter 8. Frankly, it was a hard read at first. I first attempted it over two years ago. And it prompted me to look for a pre book. Luckily after How to Read and Do Proofs I was able to make more progress on it. It is still a harder read than Abstract Algebra but given how dense it is, every few pages sends me on a mathematical tangent.

Number Systems and the Foundations of Analysis by Mendelson was recommended by a Quora rabbit hole I went down. It reminds me of How to Read and Do Proofs but it is centered on the preparations for Analysis. The book is a slower take on the foundations of 18.100a. 18.100a, Analysis at MIT, has an amazing lecture series that I have been enjoying as well.

Outside of books I tend to distract myself with trying to reformulate extremely hard famous problems.

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Mathematical Tangents

• I became slightly obsessed with representing perfect powers as series. I wrote induction proofs for n**2

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• And my continual exploration of Euler’s Phi function and its connection to Goldbach’s Conjecture. I will go deeper into this later but the crux of the idea is that the relative primes in the Euler Totient sum up to the target number. For 104:

  

  • By pigeonhole principle this implies that there is a prime pair. My program found a few portions of numbers where the property may not be holding, I need to verify it further but it seemed like an interesting spin on a problem. I am researching that formulation applied to numbers of the form 2**n*p and looking into whether there are number theoretic functions for a lower bound on composite pairs of that form.

Computer guided independent study

At first, I mostly spent my time using pen and paper to study. After a while I wanted feedback and found Lean4. It has been a whole new element to my study practice. If I find a pattern in a sequence, I am able to use induction to prove it. But, what if I made a mistake? I have no advisor to reach out to.

That is where Lean4 has helped. Lean4 is a coding language with a built in Theorem checker. I checked a few of my induction proofs and am using it to formalize proofs that I want to build on. I will share more about Lean4 and how I use it to build out my own Math world. One of my next books is How To Prove It by Velleman because of it’s integration with Lean4.

I am also almost done with the Natural Number Game, and will also look into more levels on https://adam.math.hhu.de/#/g/leanprover-community/nng4

Python for conjecture validation has helped me expand on or disprove my ideas quickly. I can also visualize ideas where, writing them out by hand would take some time. Here is a table I made to find patterns in composite pairs within Euler’s Totient for 2**n*13. My program found that an invariant I thought would hold might not work for higher powers of 2. I need to verify the code further to see why the property does not hold.

Each visualization I make will help me explore future problems. It is also much easier to build nowadays. For now, I am reformulating elementary questions. Over time, I will expand my scope and hopefully find a little corner of research to contribute to.

Regardless, I am excitedly learning new Math each day. Code is helping me explore ideas early on. And I am allowing myself to take pre-courses to prepare for books that are currently out of my range.

If you are interested in following along as I cover the Pure Math curriculum for Number Theory subscribe for more updates and learnings.

Keep Exploring.

To “pass” my self paced class I needed to find the notes within 2 seconds. This test is common amongst the guitar learning community. To generate the next option I was told to buy a 6 sided and a 12 sided dice.

As a developer I figured this should be coded up so I started on the task. Mind you, I have not built a website since college. Even then, it was tedious.

In comes AI. I used O1 instead of gpt4o and made my first attempt in 10 seconds. By just defining the product.

Completely buggy, but it started to look like something. The frets correspond to the 6 notes I needed to recall quickly in 5 positions on the guitar. I can go into more details on the specifics but for now, lets assume notes are somewhere, and I need to know what note is where.

A few prompts later and I had something that looked ok. Looks, unfortunately, were all that mattered. The rest I could fix myself.

Better.

Almost, but it really had a funny take on the guiding dots.

This take on a guitar nut was hilarious. And, honestly, it was the hardest visual to pin down. I assume I could have dived into the css at this point but I was intent on not touching html/css until absolutely necessary.

Still no guitar nut after 5 iterations. Luckily, the iterations are quick.

I still was not done but within 30 minutes, I went through 20 iterations before I started looking into the code.

This was my last generated UI. It was really close to the final goal. Each of the frets is clickable and I added some UI components after for a current final version.

It works. I use it to practice and I hosted it for really cheap so I can share it for free at Songs from Scratch

This experience was a bit jarring. On the one hand, I made my tool really quickly. And I am sure many people can make simple websites hosted on GitHub pages that solve their problems for 0 to 4 dollars a month. That is amazing. On the other hand, I did not learn anything during the experience.

I am reading through the html and css after the fact. A few days after completing the task. Since I want to create a whole suite of tools, it makes sense to familiarize myself with the fundamentals. But, I am not sure if most people do.

Also, AI leads to really awful git practice. Given the AI’s limitations, I asked for a single page website to start. And went through 24 versions. All of which had too many changes in-between them. I suspect Cursor or Kline may help with this so I will look into them for my next UI work, but, overall, it led to bad practices.

Luckily, I learned to code before all of these fancy tools worked. So I can refactor a working product and dive into breaking bugs. For example, the notes were really odd to start. So I explained to the AI that representing notes as numbers and doing modular arithmetic makes it easier. However, the AI knew guitar strings are reversed just handled it incorrectly.

So, after it generated a full page of code I kept noticing the notes being off. After digging in I realized they were mirrored, I fixed the issue. BUT, only locally. The next iteration the issue came back. Until I explicitly told the AI it reversed one but not the other. This led to it not reversing anything. Which was also wrong.

AI felt like a very enthusiastic dev student. Not a developer yet, but willing to get its hands dirty. For me, this fits perfectly with the UI code that has stopped me from pursuing entrepreneurship. I will read through every line of code, but, only retroactively. No need to build domain knowledge on how to put a footer right above the end of the website instead of in the middle.

The AI does not help with the core issue yet. It setup the scaffolding quickly, but the core problem tends to require my input. This has led me to send back the full file each prompt. It is definitely an expensive use of tokens, but it is the only way I was able to maintain a reasonable git log. I also tell the AI to make the minimal changes needed to have less scope creep.

All in all. AI is useful but not all powerful, for now. I will build many tools for a lot less effort than I would have needed a few years ago. I built this widget in 1 hour after not having worked on web for 8 years.

Next stop, processing acoustics. This has always been a fascinating topic that is related to but not exactly like the NLP work I love. For now I am reading through some research on how sound is represented digitally. My next tool will help me practice playing a note on any of the 6 strings. It will measure the time it takes me to find and play the note.

During that time many refuges were brought to the Freedom Towers in Miami as step one. After some documentation was completed, they would sleep in the Orange Bowl and receive 50 bucks or so and a bus ticket anywhere. Both of my parents, independently, found themselves in Jersey. Many winters later, they would drive down to Miami. Hialeah to be exact.

Although they both have similar stories, lets focus in on my dad’s.

Single parent immigrants face an even steeper uphill battle than usual. One person cannot handle a families worth of problems in most scenarios. Especially without the language. This led to my dad working since he was 11. Getting kicked out of school for working in his teens. Forcing him to get his GED and search for work. A family friend mentioned working in Insurance while he was finally enrolled in an associates degree. The pay was too good to pass up. He never finished school, he became a licensed insurance agent. Worked in that for 30 years, until he couldn’t.

Now what?

He did not choose insurance. And for most of my adulthood he seemed bored. His job was too easy for him.

But how could he know that before he started?

All he knew is that the pay was more than he currently made. How could he know the pay would peak around 2010 and slowly lower ever since? Or that he would spend the next 30 years of his life in that field.

At first the work was enjoyable because he could help people out of a jam. 1000s of contacts in his phone would call him throughout the day. Mostly for insurance related help but sometimes for car related help.

I can only imagine a good few also called him to chat. He could never turn down a nice conversation.

But as time went by the field became less tolerant of his connections. One acquisition led to him losing all of his contacts. Forwarding the people he helped for years to a commission hungry associate. There was no room for humanity in his job anymore.

For 20 of those years his kids were home and exhaustingly active, in a good way I guess. Once we left, with no one to help, he grew bored. Unknowingly though, that is just how life was for him.

Fight to get the bills paid. Enjoy the little time you’re children give you. Try to have fun when you can.

For most of those years he did not even receive proper holidays.

Weekends? Nope.

Christmas? Nope

New years? Not even the lunar one.

Insurance was brutal on his schedule.

I say this not to argue against work. Cultural battles around work are misplaced sometimes. Humans need mental stimulation, alongside a lot more. My dad needed more interesting, fulfilling, and higher level mental stimulation. But he was not born into a situation where he could find it early on.

I was on the other hand. We were not well off growing up but I was a kid for 18 years. Thanks to my parents. I became a software engineer and helped out where I could. But after 5 years I decided to buy some of my own time and go home. Helping where I could while learning something new. Coincidentally, my dad gained some free time while I still had the bandwidth to spend time with him.

At first he was going mad. With good reason, free time is more a curse than a gift to someone who has never had it. At first. It gives you almost the opposite of imposter syndrome. You wonder if you fit in with the people that do not help others. Lazy. Broken. Vagabundo as my family quips from time to time.

Either way, I tried to find my dad a project to help fill his free time. After 2 months of searching I asked my dad if he would give coding a chance. As a software engineer, I have heard and said “learn to code” one too many times. This time, I asked him if he was interested.

Heard his main fear: what if I cannot understand it? Then proceeded to give him a real world example of how organizing people, a thing he knows how to do, is about as hard as his first computer science class would be.

We chatted about search and sort options for ABC order. The ABCs of coding. But we double clicked on anything he let me. Reaching an English version of merge sort. Merge sort is basically how you can get a group of items in order by first grouping the items into pairs. Making sure the pairs are in order. Then grouping those pairs until you have one group, of fully in order items.

Lastly, we found a person in the alphabetized list by asking the middle person if they were before or after the name we were looking for.

Coding, to me, is a language. People teach it with typing, visual GUIs, etc. But what my little human centered example did was expel the myth that coding is solely a thing done by computers. Understanding what you want to do comes first. Picky computer language come second.

Soon after our chat, he started Code Academy Coding Foundations. I chose it because it isolates learning for now. Later I will set my dad up with real world projects, but I am taking it slower this time. My promise to him is that I will unblock him if he gets stuck. Give it a shot first but no matter what I will help you through it. The same advice I give to any of my reports.

He is learning to code from scratch. Literally 0 preparation. He does not type well. Within a week his computer broke. I gave him my old one. Which he did not know how to turn on or off.

My dad is going from struggling to change his Netflix password to coding.

It will take a while. I will patiently be there to help guide him through it. My goal is to help him learn how to learn on his own. Each time he gets stuck, I share tips and tricks on how to debug and use resources to get around it for next time.

After he completed his first course. Coding foundations in about two weeks. He told me something I could relate with. He started dreaming again. It had been years, but finally, when he slept at night he would dream.

I lost my ability to dream when most of my time aligned with something that I was already good at. It is an efficient life, but not a fulfilling one. Productivity has its place, but novelty nurtures the soul more than big wins. At least for me. And I guess my dad too.

Keep exploring.

Eddie

P.S. If you are thinking about learning how to code I strongly encourage you to. Comment below any of my articles if you get stuck and need some advice. It may lead to a job, it may not. Regardless, it will unlock creativity and help you better understand how digital products work. Like the program you are reading this from.