Not content with shapes in two or three dimensions, mathematicians like to explore objects in any number of spatial dimensions. Now they have discovered shapes of constant width in any dimension, which roll like a wheel despite not being round.
By Alex Wilkins
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The universe creates complexity out of simplicity, but despite many attempts at understanding how, scientists still have not figured it out. We do know that complexity relies on the emergence of new features and laws, but then again we don’t understand emergence either. The first step must be to clearly define what we are talking about and to measure it. A group of scientists now put forward a way to do exactly this. Let’s have a look.
Paper here: https://arxiv.org/abs/2402.
Correction to what I say at 04:07 \.
👉 Researchers at the Shanghai Artificial Intelligence Laboratory are combining the Monte Carlo Tree Search (MCTS) algorithm with large language models to improve its ability to solve complex mathematical problems.
Integrating the Monte Carlo Tree Search (MCTS) algorithm into large language models could significantly enhance their ability to solve complex mathematical problems. Initial experiments show promising results.
While large language models like GPT-4 have made remarkable progress in language processing, they still struggle with tasks requiring strategic and logical thinking. Particularly in mathematics, the models tend to produce plausible-sounding but factually incorrect answers.
In a new paper, researchers from the Shanghai Artificial Intelligence Laboratory propose combining language models with the Monte Carlo Tree Search (MCTS) algorithm. MCTS is a decision-making tool used in artificial intelligence for scenarios that require strategic planning, such as games and complex problem-solving. One of the most well-known applications is AlphaGo and its successor systems like AlphaZero, which have consistently beaten humans in board games. The combination of language models and MCTS has long been considered promising and is being studied by many labs — likely including OpenAI with Q*.
Posted in cosmology, information science, mathematics, physics, singularity
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Help translate our videos!
Data assimilation is a mathematical discipline that integrates observed data and numerical models to improve the interpretation and prediction of dynamical systems. It is a crucial component of Earth sciences, particularly in numerical weather prediction (NWP).
Officials of the U.S. Defense Advanced Research Projects Agency (DARPA) in Arlington, Va., issued a broad agency announcement (HR001124S0029) for the Artificial Intelligence Quantified (AIQ) project.
AIQ seeks to find ways of assessing and understanding the capabilities of AI to enable mathematical guarantees on performance. Successful use of military AI requires ensuring safe and responsible operation of autonomous and semi-autonomous technologies.
What does “equals” mean? For mathematicians, this simple question has more than one answer, which is causing issues when it comes to using computers to check proofs. The solution might be to tear up the foundations of maths.
By Alex Wilkins
Emergent phenomena: large-scale patterns and organization arise from innumerable interactions between component parts.
The behavior of a complex system might be considered emergent if it can’t be predicted from the properties of the parts alone.
The puzzle of emergence asks how regularities emerge on macro scales out of uncountable constituent parts. A new framework has researchers hopeful that a solution is near.
Fields Medalist Terence Tao explains how proof checkers and AI programs are dramatically changing mathematics.
Mathematics is traditionally a solitary science. In 1986 Andrew Wiles withdrew to his study for seven years to prove Fermat’s theorem. The resulting proofs are often difficult for colleagues to understand, and some are still controversial today. But in recent years ever larger areas of mathematics have been so strictly broken down into their individual components (“formalized”) that proofs can be checked and verified by computers.
This is a series of videos that I decided to make on Georg Cantor’s groundbreaking works published in 1,895 and 1,897 titled Contributions to the Founding of the Theory of Transfinite Numbers.
This work could probably be counted among the most influential and significant works in mathematical history — Cantor’s transfinite numbers changed the face of mathematics completely (although, not to everyone’s pleasure). The impact of Cantor’s work can’t be underestimated.
In this series of videos I will go through the definitions of aggregate, cardinal numbers, simply ordered aggregates, ordinal types and ordinal numbers amongst others. I will also go through some of the properties of these objects including arithmetical operations of cardinal numbers and ordinal types and culminating in the arithmetic of the ordinal numbers of the second number class.
