Paul Borrill

Someone should make a systems programming language that has a syscall monad. Easy to tell at a glance what code is deterministic and what isn't. As an added bonus, it would upset the "fUnCtIoN cOlOuR" people.

“No one shall expel us from the paradise that Cantor has created for us.” In 1928, one of the most influential mathematicians of the twentieth century, David Hilbert, moved to remove another leading mathematician, L. E. J. Brouwer, from the editorial board of Mathematische Annalen, one of the world's most prestigious mathematical journals. Behind this extraordinary confrontation was a fundamental question: What actually counts as a valid mathematical proof? Their disagreement became known as the Grundlagenstreit, or dispute over the foundations of mathematics. Brouwer, the founder of intuitionism, challenged the unrestricted use of the law of excluded middle: P ∨ ¬P In classical logic, every proposition is either true or false. Brouwer argued that, particularly for statements involving infinite mathematical objects, proving that something cannot be false does not necessarily establish its truth. Mathematical assertions, in his view, required constructive justification. Hilbert strongly defended classical reasoning and sought to secure its foundations through rigorous finitary consistency proofs. In his famous 1925 lecture On the Infinite, he declared: “No one shall expel us from the paradise that Cantor has created for us.” The disagreement became increasingly personal during the 1928 International Congress of Mathematicians in Bologna. Brouwer opposed the unequal treatment of German mathematicians, while Hilbert attended as leader of the German delegation. On 25 October 1928, Hilbert wrote to Brouwer that their cooperation was impossible because of “the incompatibility of our views on fundamental matters.” Brouwer protested his removal. Albert Einstein refused to support Hilbert's action, and the controversy divided the journal's editors. Eventually, the editorial board was dissolved and reconstituted without Brouwer. Einstein and Constantin Carathéodory declined to join the new board. Neither mathematical viewpoint disappeared. Gödel's incompleteness theorems of 1931 revealed fundamental limitations of Hilbert's original programme, while Brouwer's ideas continued to influence intuitionistic logic, constructive mathematics, proof theory, and computer science. Brouwer subsequently founded Compositio Mathematica, which began publication in 1934.

If you want to make the most progress you've ever made in your life, you need to be alone. You need an extended period of isolation, not influenced by friends or media, spent building your future. You need space to learn, think, experiment, and fail. Give yourself 6 months. Focus solely on the development of your mind, body, spirit, and finances. This is how you get ahead of most people.

If you want to get into formal verification, listen to everything this man says

I have decided to leave math academia. Not because of burnout, the academic job market, or a loss of love for mathematics. Surprisingly, it is because of how much mathematics I could do with LLMs over the last few months. For most of my life, pursuing mathematics felt like the obvious choice. It gave me a sense of purpose, rooted in the search for hidden truths in God’s Book. Over the past few months, that conviction has been shaken. With LLMs, I could get several breakthroughs on problems I care deeply about and had worked on for years during my PhD. This progress might have taken years of my academic life if not for LLMs. This could have strengthened my conviction that mathematics was my calling. Instead, it undermined it. With each passing week, my role in the discovery loop seemed to grow smaller. What made mathematics meaningful to me was never just the final answer. It was the struggle: months or years of exploration, failed approaches, and deep thought before a hidden structure finally became visible. That struggle gave me a sense that the result was truly mine. Prompting my way to answers, without the struggle that once defined the process, no longer felt like the same vocation. The answer might be just as beautiful, but it did not feel earned in the same way. I now believe that we are rapidly approaching a world in which most answers from God’s Book will be only a prompt away. Once I truly internalized that possibility, dedicating much of my life to finding those answers slightly earlier no longer felt as meaningful as it once had. But one problem kept bothering me: How do we know the oracle is right? Intelligence (and hence the amount of math papers) is becoming abundant and cheaper by the day. Trust is not. The proofs produced by these systems can be highly sophisticated, and their errors can be extremely subtle. Determining whether an apparent breakthrough was actually correct sometimes took me days. Until it is verified, a beautiful proof is not different from slop. That gap drew me toward formal verification and eventually toward formalizing math proofs in Lean, including proofs of several longstanding conjectures and Erdős problems discovered with the help of LLMs. I began to see verification as a central intellectual bottleneck. Intelligence is useful only when its outputs can be trusted. So I am leaving math academia. I am shifting my attention from discovering answers to building systems that can certify them. I am very excited to be joining @PramaanaLabs to work on this challenge. Over the next few years, I hope to help build a future in which increasingly powerful systems are not merely intelligent, but provably correct and reliable across problems far beyond mathematics.

A permanently crewed lunar science base would be far more impressive than a repeat of what was already done incredibly well by Apollo in 1969

Never call an API in a transaction. In relational databases, it's best practice to keep transactions as short-lived as possible. Longer transactions lead to more resource contention. If each transaction acquires row locks, being longer-lived means higher likelihood of lock waits and execution delays. Then there's MVCC. Postgres keeps multiple versions of rows for MVCC, and long-running transactions keep old rows from becoming autovacuum candidates. MySQL handles this with an undo log. When there are lots of long transactions, the log has to be kept around longer, log length and performance. If you're using a connection pooler like PgBouncer in transaction pooling mode, long transactions can easily result in pool exhaustion producing failures on your clients. The database is the most core part of your stack. Keep it as contention-free as possible.