Turns out Gauss wasn’t a fan of the concept of infinity either [1]. "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already" Henri Poincare also objected to the idea of infinity [2]. 1.

Controversy over Cantor's theory - Wikipedia

2. https://link.springer.com/article/10.1007/s40329-013-0007-z View quoted note →

*Long note on math* ZF(C) set theory is the basis of modern math and includes the axiom of infinity; something we grant as obvious. "There's no number so big that you can't add 1 to it." Yet nothing we can observe or measure in reality is infinite. Not the size of the observable universe nor all observable particles in it nor even all possibles states of all observable particles (hot take; see below for explanation). Should we really use that axiom? Even the digits of pi can’t be fully observed. Spend all available resources building as many computers as you can and use all matter for computation from now until the heat death of the universe and nothing and nobody will ever experience or observe more than a finite number of digits of pi. You could argue that we need irrational numbers for algebraic closure. Fine. But even then, in practice on computers or in hand written solutions only finite-precision representations exist. Not a single engineering project in existence relies on full precision of pi or the square root of 2. Yes, even the LHC and LIGO. "But the ratio of the circumference of a circle and its diameter is pi. That ratio has infinite precision." Well, almost! That assumes a Platonic ideal circle; you're introducing new assumptions even in such a basic concept. Good luck creating such a perfect circle in reality (nothing against the band with MJK). The total curvature theorem that gives us some multiple of pi for any closed loop rewrites integrating over a continuous set so implicitly assumes infinity (ie. infinitely divisible). What about quantum mechanics? Hilbert spaces as far as the eyes can see! We need a state space rich enough to represent continuous observables like momentum. But that's a circular argument! We claim the observables are continuous but as far as I can tell that's just an assumption. Our theories break down below the planck length which means we can't just assume spacetime is continuous; at best it's a useful approximation. Continuous spacetime also creates problems like black hole singularities. I think most physicists agree singularities are a flaw in the models and don't represent reality. Quoting Oppenheimer: "Physically such a singularity would mean that the expression used for the energy-momentum tensor does not take account of some essential physical fact which would really smooth the singularity out. Further ... it is impossible for a singularity to develop in a finite time" - 'On Continued Gravitational Contraction' And according to Hawking eventually black holes dissipate thus removing the singularity. You can't go from infinite to finite density in finite time yet that's the best our theories have come up with. We have tools like renormalization to cut out the infinities in some sense and to be honest I don't fully understand the logic there, but it ultimately boils down to using infinities to cancel out other infinities that pop up in our models that, yeah you guessed it, are built on continuous space and therefore more infinities! In conclusion, as far as I can tell all uses of infinities are useful approximations to make math easier but have quite possibly led us astray.

AI is getting better at finding answers/solutions over time. For example AlphaZero, RAG, Claude Code, etc. The bottleneck now seems to be in knowing which questions to ask, what problems to solve next, etc.

Interested in learning more about #quantum computation and quantum information theory? You can read one of the most influential books in the space for free here [1]. There's also this repo that includes code for quantum algorithms and problem sets [2]. With all the quantum talk these days it may be useful to sit down with your favorite LLM and go through the book. I plan to do that over the next few months (emphasis on plan 😅). 1. 📄.pdf 2.

GitHub

GitHub - JackHidary/quantumcomputingbook: Companion site for the textbook Quantum Computing: An Applied Approach

Companion site for the textbook Quantum Computing: An Applied Approach - JackHidary/quantumcomputingbook

Been seeing a lot of negative sentiment about Bitcoin lately. Why? I transferred fiat between two banks accounts the other day and it’s still pending. Can’t use it at all for a few more days. Want to send more than $1k to a family member in need? Can’t do it with zelle or venmo in just one payment. That fiat money you shoved under your mattress back in 2021? It buys fewer goods and services now but Bitcoin buys more. Maybe it didn’t outperform the AI bubble but that’s fine. Savings isn’t the same as investment; never has been. Bitcoin is a better payment rail and it’s not even close. It’s a better savings technology for those who plan to live beyond the next 4 years. For the first time we have digital money we can actually own! Nobody said a monetary revolution would be easy. Don’t cry about it. Appreciate all the cypherpunks out there who gave us options.

This. Q-day comes before BIP 360 is finalized? Riding this all the way to 0. Saylor market dumps and becomes a Fartcoin maxi? Riding to 0. Turns out Satoshi was literally Hitler? Damn that sucks. Riding. To. Zero. Cold storage sats are for use as money one day or for nothing at all. Not trading back to fiat regardless of price. View quoted note →

Wall St entered the game when Bitcoin was at $42k. If they’re getting rinsed there’s a chance we’ll retest those levels. Not my base case. Not financial advice. And as always GFY. View quoted note →