@paulg: Explained to 14 yo how you can multiply numbers by adding their logarithms (and divide by subtracting them), and how be…
Summary
Paul Graham explains to a 14-year-old how logarithms allowed multiplication via addition and were a critical hack for calculations before calculators.
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@paulg: Exponential growth in 5000 BC.
Paul Graham shares a link about exponential growth observed as early as 5000 BC.
@HowToAI_: Researchers proved that every single elementary function, sin, exp, log, sqrt, comes from one single binary operator. I…
A paper proves that all elementary functions like sin, exp, log, sqrt can be generated from a single binary operator eml(x,y)=exp(x)-ln(y), similar to how NAND gates unify digital logic. This could simplify AI architectures by enabling a single trainable node for continuous mathematics.
The Discrete-Log Clock: How a Transformer Learns Modular Multiplication
This paper demonstrates that when transformers grok modular multiplication, the dense Fourier spectrum observed in previous work is an artifact of using the additive Fourier transform; using the multiplicative character transform reveals a sparse representation, leading to a reverse-engineered 'Discrete-Log Clock' algorithm analogous to the clock algorithm for modular addition.
@rohanpaul_ai: Brilliant. This feels like one of those cases where the math idea finally arrived at the right timing, because AI infer…
The tweet praises a mathematical idea timed well for AI inference's arithmetic profile and expresses interest in seeing results on reasoning models during long generation runs.
@rewind02: A Stanford professor just gave a public lecture on exactly how GPT, Claude, and LLaMA are built under the hood no insid…
A Stanford professor delivered a public lecture providing a comprehensive breakdown of how modern LLMs like GPT, Claude, and LLaMA are built under the hood, making advanced architecture accessible to the public.