The Long Road to Understanding - Part III

If the first phase was curiosity, and the second was learning how to think, the third has been about learning how to work.

The change was not dramatic. There was no moment where everything suddenly became easy or clear. The way I approached problems stopped depending on who was guiding me and started becoming something I could reproduce on my own.

I began to notice patterns in how I made progress. When something felt confusing, it was usually because I had skipped a definition or was trying to move too quickly. Slowing down helped. Writing things out helped. Returning to first principles helped. These were simple ideas, but applying them consistently made a difference.

Over time, this became a method. Not a rigid system, but a way of working that I could rely on. I would start by grounding myself in the basic objects of a subject, understanding what they were and how they behaved. From there, I would work through problems, not just to get answers, but to see how the ideas moved. When something broke, I would trace it back rather than patch over it.

This approach carried across subjects. Linear algebra, probability, number theory. The details were different, but the process was the same. Progress was still slow at times, but it was no longer directionless. Each piece fit into something larger.

Alongside this, coding remained a constant. Where mathematics required careful construction, programming allowed me to test ideas quickly and build systems that did something tangible. The two began to reinforce each other. Math gave structure to my thinking. Coding gave it a way to express and test that structure.

What feels different now is not that the gaps have disappeared. It is that I know how to work with them. When I encounter something I do not understand, it no longer feels like a dead end. It feels like a signal to slow down and rebuild.

Looking back, the path has not been linear. It has involved false starts, resets, and long periods of confusion. But those phases were not wasted. They shaped the way I approach learning now.

I no longer think of myself as someone trying to catch up. I think of myself as someone building, layer by layer, with a process that holds.

That is what I carry forward.

Reasoning with sticks

One of the recurring patterns in my number theory study has been getting trapped in circular reasoning without noticing it. I read an explanation, restate it in my own words, and feel as if I have understood something. But when I try to justify the idea, the entire argument often turns out to be feeding itself.

This happened most clearly when I was trying to understand why Euclid’s Algorithm works. Textbooks say, “Replacing a pair $(a,b)$ with $(b, \mathrm{a }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>b)$ does not change the gcd.” I kept trying to explain this using statements like “the gcd stays the same when you subtract multiples,” or “the algorithm preserves common divisors.” Both sounded correct, but both were really just the conclusion restated in slightly different language. Nothing in the reasoning actually began from a place that did not already assume the result.

The loop broke the moment I pictured the numbers as two sticks: one long stick of length $a$ and a shorter stick of length $b$. If you measure the longer stick using the shorter one, the piece left over after laying down the shorter stick repeatedly is exactly $\mathrm{a }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d </mi></mrow>b$. Any stick that measures both $a$ and $b$ will also measure the leftover piece, because that piece is created by repeating the shorter stick until it no longer fits. The idea becomes physical at that point. It is not an abstract rule but a direct observation about lengths.

The reverse direction also becomes obvious. If a stick divides $b$ and the leftover piece, then by assembling the leftover pieces and full pieces of $b$, it will divide $a$ as well. The equivalence is suddenly clear, and the reasoning does not run in a circle anymore. There is a concrete starting point that does not depend on the conclusion.

I had a similar experience with linear Diophantine equations. I knew the statement: the equation $ax+by=c$ has a solution exactly when $c$ is divisible by $\gcd (a,b)$. I tried to motivate it by saying that combinations of $\mathrm{an}$ and $b$ somehow determine what is possible, and therefore the gcd must divide $c$. That felt intuitive but it was another loop. I was assuming the existence of the combination in order to justify the condition for the combination.

The picture that fixed it was again based on sticks. Consider all lengths you can create by cutting and taping together sticks of lengths $a$ and $b$ Those lengths do not appear randomly. They form a neat pattern, and the smallest positive one you can create is the gcd. Everything else is a multiple of that smallest value. If you want to create length $c$, it must fit into that pattern. If it does not, no rearrangement of the pieces will ever give you that exact length. This explanation begins somewhere real, so the loop disappears.

What I have learned is that circular reasoning often shows up when I keep everything inside symbols. Symbols can hide gaps because they feel precise even when the underlying logic is not. When I switch to a picture or an analogy, the reasoning becomes grounded, and the loop cannot sustain itself.

Now, when an explanation feels too smooth, I check whether it actually begins from a stable point. If it does not, I translate the idea into something I can visualize. Every time I do that, the reasoning stops circling and finally lands.

Also, here's a painting I made

Why Coding Came Easy (And What That Says About Math)

My coding journey started at 14, when I took computer science as an optional subject in school and learned BASIC. For my final project, I tried to build a simple car racing game. I got stuck trying to generate random obstacle positions and detect collisions and eventually switched to a Hangman game instead. Even then, something about programming clicked.

After school, I moved toward biology, but coding kept reappearing. I took a C++ course in college, then a bioinformatics course and a computational cognitive neuroscience course in graduate school. Over time, programming shifted from being an interest to a necessity. Modern biology, especially bioinformatics and large-scale data analysis, depends heavily on code.

I taught myself Python and R, and during my postdocs in computational biology, both became tools I used fluently. At that point, I was not just using them for my own work but also helping others. Coding had become a natural way for me to think and solve problems.

What has been more difficult to understand is the contrast with my experience in mathematics. Coding and math are often described as closely related. Both require precision, structure, and logical thinking. But my experience of learning them has been very different. Coding became usable and intuitive relatively quickly, while mathematics took much longer to develop into something I could work with comfortably.

My current view is that the difference lies in the type of abstraction involved. Programming often allows you to anchor ideas in concrete systems. You write code, run it, and observe what happens. The feedback loop is immediate, and even complex systems can be built incrementally from smaller, testable pieces.

Mathematics, especially in its more abstract forms, demands a different kind of engagement. The objects are less tangible, and the feedback loop is slower. Progress depends more heavily on internal consistency and precise reasoning without the same level of external grounding.

This difference became clearer when I encountered theoretical computer science. Topics like automata theory, formal languages, and the structure of computation felt much closer to mathematics than to programming. The ease I felt with coding did not carry over. Instead, I had to approach these subjects in the same deliberate way I approached math, building understanding step by step.

Looking back, the contrast is less about ability and more about where I started. Coding allowed me to operate effectively within concrete systems early on. Mathematics required me to develop a deeper level of abstraction over time.

That layer is something I am still building, but now with a clearer understanding of what it demands.

Learning Math: What Worked, What Didn't

Like everyone, I started learning math in school. I wasn't bad at it, but I was no prodigy either. I decided my path lay in biology and only returned to math in 2016. You can read about that journey in my previous post. Today I want to write about the how of learning math: what worked, what didn't, and what I wish I'd known earlier.

I think there is something I'd call a mathematical gaze, a style of thinking that some people access easily and others don't. Everything in learning math, at its core, revolves around developing this gaze. Another word for it is intuition.

What is mathematical intuition, exactly? Does it have anything to do with the real world? For a long time I was convinced that solving enough problems would develop it automatically. The truth is more complicated. Problem-solving alone isn't enough. You need a teacher who already has intuition and can show you how to think like them. That transmission from one mind to another is the only reliable way to generalize beyond the specific equations you've been drilling.

A related insight came to me while thinking about mathematical modelling. People in applied mathematics spend a great deal of time building models of real-world phenomena: abstracting out the parts they care about and expressing relationships between them. Finding the right representations is how they define success. To do that well, you need a solid grasp of the objects mathematics deals with and how they relate, and for that, there is no substitute for practice.

One trick I've found surprisingly powerful: asking which seemingly different problems are actually the same? The right mapping can turn an "impossible" problem into an easy one in seconds.

I also used to binge-watch advanced math lectures on YouTube to "build intuition." You might laugh, but I think it actually works. I'd go in expecting to understand less than 1% of what was being said. Even so, watching people with genuine mathematical intuition think out loud gave me something: a faint but real sense of how math is done, not just the pattern-matching of homework problems, but the branching, exploratory quality of real mathematical thought.

What I've found, in the end, is that for a determined but learner, there is a real path to mathematical understanding, intuition included. It takes only two things: practice, and a teacher who genuinely has the gaze.

My Mathematical Journey

Around early 2016, I decided to learn math. The trigger was a comment I came across about someone who had struggled with math but was able to rebuild their understanding through proofs. That idea stayed with me. I had always been comfortable with biology, and it became clear that a deeper command of mathematics would open up entirely new ways of thinking about it.

I started with enthusiasm but very little structure. Like many beginners, I jumped too far ahead and picked up a book on stochastic methods. That attempt did not go far. At the time, I was finishing graduate school, and the effort faded.

The real shift began during my first postdoc, where I worked with both a physicist and a biologist. My physicist advisor suggested I learn linear algebra and pointed me to Strang’s lectures. I supplemented this with visual material like 3Blue1Brown. That phase gave me an initial intuition for the subject, but it was still shallow. I was beginning to see patterns, but without enough precision to reliably work with them.

For a while, my approach remained inconsistent. I revisited calculus, explored different areas, and often tried to move ahead faster than my foundations allowed. In hindsight, the pattern was clear. I was prioritizing exposure over consolidation.

Around 2023, two things changed. I began a second postdoc at TIFR in Mumbai, and I started learning regularly with a close friend who had trained in physics. This introduced something I had been missing steady pacing and feedback at the right level. Instead of jumping topics, I focused on building depth.

I returned to linear algebra with more discipline, working through Strang’s lectures again and using problem sets to reinforce the material. This time, the concepts became operational rather than just familiar. I also began studying probability and statistics in a more structured way, following recommendations from faculty at TIFR. With consistent practice, I developed a working understanding of both.

Later, I explored number theory using Silverman’s A Friendly Introduction to Number Theory, on the recommendation of a professor. This was challenging in a different way. It required a level of abstraction and precision that exposed remaining gaps in my thinking. Working through it has been slow, but it has also been one of the most valuable parts of the process.

Looking back, the main change has been in how I learn. Early on, I chased exposure and intuition without enough grounding. Now I focus on building foundations, testing understanding through problems, and progressing layer by layer.

Today, I have a solid base in linear algebra, probability, and statistics, along with exposure to other areas. More importantly, I have a process that works. The trajectory is steady, and each layer builds on the last.

I continue from there.

The Library Bar and Sherlock Holmes

 

The Library Cafe and Bar, Madison, WI, USA

Back in grad school, I used to frequent a bar called “The Library.” They had books, food, and drinks, and the music was always great. There was nothing I enjoyed more than showing up there after a long day at work, finding a spot under the bright lights, and reading Sherlock Holmes. The walls were lined with actual bookshelves, but I usually stuck to the hard wooden chairs with my own worn paperbacks.

If you would have told me back then that I would eventually be seriously attempting to learn math that wasn't essential for my work, I would’ve laughed in your face. My way of thinking felt completely incompatible with the rigid world of mathematics. I saw myself as a person of prose and puzzles, not proofs and polynomials.

My perspective has changed over the last five years. I have slowly begun to realize that the thrill of a Holmesian deduction is not so different from the thrill of a mathematical discovery.