Daniel O'Donnell

@TivadarDanka @threadreaderapp unroll

The 10 Most Important Lessons 20 Years of Mathematics Taught Me: 1. 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. 2. 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 the 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. 3. Understanding happens when you take things slow. At the university, most of my classes were in classic chalk + blackboard style, and let me tell you, taking notes of 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 around the concepts, bending them, studying them. It’s not just for math but for any topic. That’s why typing out a solution from an LLM response (or StackOverflow if you are old school, like me) is better than copying and pasting it. 4. 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 by being 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. Build. Create. Take it apart and put it back together again. 5. 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. 5. 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 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 have learned it one step at a time. Driving on autopilot is the problem now, but that’s a topic for another day. 7. 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 the one hand, and vectors on the Euclidean plane on the other. Driving your startup to the ground on the one hand, a priceless learning opportunity on the other. Success and failure are matters of perspective: you win, or you learn. 8. Asking questions is a superpower. “It’s not that there are no stupid questions,” the words of my professor echo in my ear, “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 with raising my hand and asking my question during a lecture, no matter what it was. My mid-(and sometimes after)-lecture discussions with the professors had 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. 9. 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 that “you must be a genius.” Let me tell you, this could not be further from the truth. I was never a mathlete nor 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. 10. 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. As an entrepreneur, 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 Nikita Bier and Pieter Levels 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.”

@mazeincoding This you?

just got rejected from yc for using all lowercase in our application

@AGoldclaim69300 @Atomicrod The grid power that Eilson AFB no longer uses will then be available to the residents of Fairbanks AK.

@Atomicrod Manufactured out of what? Cladding is used to build cores and fuel rods. Maybe eileson will get more power but not the rest of Alaskans.

The Air Force is diligently pursuing a project to build a 5 MWe nuclear plant to augment the power supply for Eielson AFB. They’ve spent 18 months working through a contract protest filed by a competitor. (Another example of self damaging behavior in the nuclear industry.) Success at Eielson could lead AF to purchase 8 or more plants for other bases. As the below linked article reminds us, the Air & Space Forces have rapidly growing data center and sensor power requirements. They are also seeking a level of grid independence.

@HackingButLegal x.com/MarkHertling/s…

Deportation is the wrong word. The correct term is rendition: Moving an individual by force to another country to engage in activities which would be illegal in yours.

@jtrebach @threadreaderapp unroll

🧵hi, medical toxicologist and poisoning expert here let's talk about pong-pong (“the suicide tree”) from The White Lotus #whitelotus #thewhitelotus 🪷 (yes it's real, yes it's deadly, and YES I HAVE OPINIONS)

@Mylovanov Little Marco is not exactly Elvis though, is he? youtube.com/watch?v=WWVMXL…

Less talk and more action, says Rubio Europe must develop concrete proposals on Ukraine security

@highbrow_nobrow Doesn't Sen. Grassley have one of those gentleman hobby farms that gives him a huge tax break? Why yes, yes he does. So maybe he is just protesting for himself.

'Inflation Day' 11/18: Even Trump’s own party is panicking. Senator Chuck Grassley, a stalwart Republican, just introduced legislation to rein in Trump’s tariff power, fearing what this could do to Iowa’s farmers. Globally, the reaction has been just as severe. Singapore’s defense minister likened America to “a landlord seeking rent.” Germany’s economics minister rebranded the day as “Inflation Day.” Yaroslav Trofimov of The Wall Street Journal reported that alliances are collapsing, partners are defecting, and the U.S. is no longer seen as a reliable power. youtube.com/watch?v=o2bV4_…

Liberation Day: Trump Imposes Largest Tax Hike in World History “This is Liberation Day. April 2, 2025, will forever be remembered as the day American industry was reborn, the day America's destiny was reclaimed, and the day that we began to make America wealthy again.” — President Donald J. Trump, Rose Garden Address 🧵1/18: That’s what the President of the United States said, standing in the White House Rose Garden, flanked by American flags draped behind him—declaring April 2nd “Liberation Day.” But what, exactly, is being liberated? Not the American people. Not the working class. Not struggling families crushed under inflation. Not small businesses trying to survive. Not farmers in Iowa or retirees on fixed incomes. No. youtube.com/watch?v=o2bV4_…

1977: Ron Rivest first introduced Alice and Bob in the paper "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems".

@SundaeDivine That bird would also poop in its own nest.

If you put Donald Trump's brain in a bird it would fly backwards.

Every secret plan needs a secret code. Even Calvin and Hobbes know this, and they are a 5 year old and a stuffed animal.

@novayagazeta_en @threadreaderapp unroll

In order to circumvent strict international sanctions, Moscow has assembled a vast “shadow fleet” of ageing oil tankers sailing under the flag of poorly regulated third countries to continue exporting oil. 🧵

@HopfJames @threadreaderapp unroll

More nuclear generation means less use of coal and gas. Always did. Article link in reply. Korea's nuclear share has increased, due to higher capacity factors and the start up of a new reactor. In January, 2025, nuclear's share was ~35%, a 20% increase vs. January, 2024. This has reduced Korea's use of expensive imported coal and gas (LNG). Use of those fuels has negative economic, environmental, and geopolitical impacts. Nuclear's benefits are large and multi-faceted.

@binarybits @threadreaderapp unroll

I took a close look at the 38 most serious crashes Waymo experienced between July 2024 and February 2025. Almost all of them involved a Waymo following the rules and a human-driven car... not doing that.

@AP About 6 weeks prior to the invasion something cut the fiber link between Norway and Svalbard, the location of the biggest satellite communications antenna far in the world. (It handles comms with satellites in polar orbit.) Timing was/is suspicious. nrk.no/tromsogfinnmar…

Western officials have accused Russia and its proxies of staging dozens of attacks and other incidents across Europe since the invasion of Ukraine, according to data collected by The Associated Press. apnews.com/article/russia…

@GregTSargent @threadreaderapp unroll

Horrifying: After the contract tracking Ukrainian children abducted by Russia was terminated, the underlying evidence of these potential war crimes seems to be missing, a bipartisan group of Reps just charged in letter to Rubio. Two GOPers signed. Details: newrepublic.com/article/192924…

@pickover In their urban design the French seem to favor the hub-and-spoke, while the British seem to favor the grid.

Symmetry, roadways, humanity, mathematics, beauty. Arc de Triomphe.

@rshereme He lasted about 7.5 Mooches. Not bad!

U.S. Special Representative Kellogg has been removed from peace negotiations at the Kremlin’s request due to a “too pro-Ukrainian stance.” — NBC News So, I presume that Ukraine can also request removal of some people from U.S. administration due to a “too pro-russian stance.” I have a candidate in mind.

@HackingButLegal @threadreaderapp unroll

1/ 🚨 NEW: Treasury just launched what might be the most sweeping financial surveillance program in U.S. history. In 30 border ZIP codes, ANY money transfer over $200 will trigger automatic government reporting. Previously this threshold was $10,000. That isn't a typo! #privacy

@HackingButLegal @threadreaderapp unroll

Trump's new policy of financial surveillance of American citizens and residents– My story is live. open.substack.com/pub/hackingbut…

@vtchakarova Professor of Respecting Red Lines more like...

Right on April Fool’s Day