The 10 Most Important Lessons 20 Years of Mathematics Taught Me
#5. There are no shortcuts to mastery.
One. Breaking the rules is often the best course of action.
I can’t even count the number of math-breaking ideas that propelled science forward by light-years.
We have set theory because Bertrand Russell broke the notion that “sets are just collections of things.” We have complex numbers because Gerolamo Cardano kept the computations going when encountering √−1, refusing to acknowledge that it doesn’t exist. We have non-Euclidean spaces because János Bolyai did not accept that, given a line and an external point, there’s only one parallel line that intersects the point.
A triangle of three right angles breaks one space, but creates another.
No assumption should be set in stone, and I’m not afraid to challenge them. You shouldn’t be afraid either.
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Two. You have to deeply understand the rules to successfully break them.
Let’s be honest: breaking rules is a cliché.
All of my examples of successful rule-breakings above were executed by experts in their fields: Russell was a master of logic, Cardano of algebra, and Bolyai of geometry.
Think of doing mathematics science as holding a large marble orb in your hands, with treasure inside. You want to open it, but the marble is so sturdy that you can’t just smash it open.
What you can do, however, is to obsessively study it in detail until you find a tiny crack, giving you a way in.
If you don’t know what you are doing, you are just an elephant in a porcelain store.
This holds true outside of math. Miles Davis, legendary jazz musician, famously stated that
“Once is a mistake, twice is jazz.”
Mistakes are easy. Jazz is hard.
Three. Understanding happens when you take things slow.
At the university, most of my classes were in the classic chalk + blackboard style, and let me tell you, taking notes on formulas, definitions, and theorems by hand from a blackboard is infinitely more beneficial than relying on a PowerPoint presentation.
We were thinking and working together with the lecturers instead of just hanging along for the ride.
Something magical happens when you copy formulas by hand. Your mind automatically wraps itself around the concepts, bending them, studying them.
It’s not just true for math but for any topic.
That’s why typing out a solution from an LLM response (or StackOverflow if you are old-school) is better than copying and pasting it.
Four. The best way to learn is to solve problems.
I just said that “understanding happens when you take things slow.”
Well, the easiest way to take things slow is to be forced to by pushing your skills and knowledge to the limit.
This happens when you are working with tools you don’t know how to use, on problems you don’t know how to solve.
Until I implemented my own neural network library from scratch, my understanding was limited to the user level. I prepared my datasets, ran a couple of epochs, evaluated the model, and repeated the process until I liked what I saw.
Soon, it wasn’t enough. I wanted to dig deeper, so I built a neural network framework from scratch. And another one. And another one that became mlfz, the cornerstone of my next book.
Consuming YouTube tutorials is an extremely inefficient way to improve at anything. If you want to get good, get your hands dirty and start solving problems.
Think.
Build.
Take it apart and put it back together again.
Five. There are no shortcuts to mastery.
Once upon a time, the legendary Greek mathematician Euclid was summoned by King Ptolemy I Soter (not to be confused with the astronomer Ptolemy), who wanted to study geometry.
Judging from the fact that, two thousand years later, we still call classical geometry Euclidean, you have probably guessed that Euclid was pretty good. However, the king soon got impatient and asked if there was a shortcut.
Euclid replied that
“There is no royal road to geometry.”
The man invented the axiomatic approach to mathematics, wrote the second-most printed book of all time, and told a king to f**k off. That’s a pretty impressive resume.
But what does no royal road mean?
That there are no shortcuts. You can’t pay your way to knowledge; you must put in the work for math and anything else that’s worth doing.
Six. Always tackle one issue at a time. Look at special cases, then add complexity step by step.
I’m quite an impatient person. Even though my recklessness has gotten better with age, sometimes I still add two new features to a single commit. I just can’t help myself.
However, if you are a frequent reader of mine, you probably heard me say, “We’ll add complexity one layer at a time,” or something along these lines.
This is not an accident.
Whenever I want to understand a new concept, I know that mentally juggling with more than one ball thought is a recipe for failure, so I avoid it at all costs. Once I’m familiar with the basics, I add more and more details.
Think of the very first time you drove a car. Personally, my head was spinning so hard I didn’t even know whether I was coming or going. Letting off the gas and gently pressing the brake with one leg, smashing the clutch with the other, finding the right gear with my right hand, and holding the steering wheel steady with the other. That’s just slowing down before a turn. (I’m European, so I learned to drive on a manual.)
After twenty years of experience, I can do all this in my sleep, and I learned it one step at a time.
Driving on autopilot is the problem now, but that’s a topic for another day.
Seven. Finding the right perspective is half the success.
The all-time most-read post of The Palindrome is titled Matrices and Graphs, and it is secretly about this principle. By looking at matrices as graphs, we can immediately prove complex and profound results. Like in computer science, each implementation has its pros and cons. The proper choice of data structure can make or break a problem.
Matrices on one hand, graphs on the other.
Algebraic expressions like a + ib on one hand, vectors on the Euclidean plane on the other.
Driving your startup into the ground on one hand, a priceless learning opportunity on the other.
Success and failure are matters of perspective: you win, or you learn.
Eight. Asking questions is a superpower.
“It’s not that there are no stupid questions,” the words of my professor echo in my ears, “it’s that not asking your questions is stupid.”
You probably don’t know this about me, but I’m quite shameless. I never had a problem raising my hand and asking my question during a lecture, no matter what it was.
My mid-(and sometimes post)-lecture discussions with the professors followed a notable trajectory of improvement, from “What does that ∀ symbol on the board mean?” to discussing open problems and eventually solving a couple of them.
The key to that improvement was my blatant disregard for others’ perceptions of me, daring to play some wrong notes to master my instrument.
Nine. Talent is just the icing on the cake. The rest is hard work and perseverance.
When people discover that I have a PhD in mathematics, one of the most common reactions is, “you must be a genius.”
Let me tell you, this could not be further from the truth.
I was never a mathlete or a brilliant researcher. I am a slow thinker. I have problems with mental arithmetic.
However, three redeeming qualities helped me get where I am today. I am
emotionally resilient,
curious,
and a hard worker.
Without these, no amount of brilliance can put you at the top or even in the middle of the pack.
Ten. Don’t give too much credit to advisors and professors. They are people like you, and experience is the only thing they have over you.
I follow quite a few notable people who have successfully realized their vision, solving problems to move our world forward. Their audiences are full of the young and motivated, yearning to carve out their piece of the pie, failing to realize that the only path they can walk is their own.
What worked for Elon Musk or Paul Erdős might not work for you.
Why? Because you have a different personality, a different socio-economic background, and potentially a different zeitgeist.
“No man ever steps in the same river twice, for it’s not the same river and he’s not the same man.”
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I love this post. I will suggest a 4a or 11 for you: the very best way to learn a thing is to teach that thing.
Teaching curious and courageous students, who will ask challenging questions, requires the teacher to understand the problems and the solutions more deeply than is possible in any other scenario in my experience and in my opinion.
Thank you for sharing your thinking.
Loved this write up! Incidentally I was talking about “working the math problems at a reduced pace” just yesterday with my son.
my only £0.02 is the first point could have been worded as Break the assumptions is the first course of action… but I guess that was the hook as you had covered it up with the second point! 💪