Mathematics

73 items

Terence Tao blog RSS Sep 14, 2026
This essay by Nestor Guillen argues that large language models (LLMs) should be understood as a kind of cultural and social technology, like markets, bureaucracies, and the scientific literature, which aggregates and compresses human accumulated information, so that when an LLM output contains a new mathematical idea it should be received as the fruit of mathematics' shared heritage rather than as a defeat for mathematicians; drawing on Farrell, Gopnik, Shalizi, and Evans's claim that large models are cultural technologies, the author proposes the metaphor of a 'convex hull of ideas,' suggesting that whether an LLM can solve a given mathematical problem depends largely on the state of the mathematical literature at the time rather than merely on model scale, illustrating this with the Kryl
This essay by Nestor Guillen argues that large language models (LLMs) should be understood as a kind of cultural and social technology, like markets, bureaucracies, and the scientific literature, which aggregates and compresses human accumulated information, so that when an LLM output contains a new mathematical idea it should be received as the fruit of mathematics' shared heritage rather than as a defeat for mathematicians; drawing on Farrell, Gopnik, Shalizi, and Evans's claim that large models are cultural technologies, the author proposes the metaphor of a 'convex hull of ideas,' suggesting that whether an LLM can solve a given mathematical problem depends largely on the state of the mathematical literature at the time rather than merely on model scale, illustrating this with the Kryl
Happy, Those Able to Know the Causes of Things: Reading LLMs as Cultural Technologies Rather Than Agents That Replace Mathematicians
This essay by Nestor Guillen argues that large language models (LLMs) should be understood as a kind of cultural and social technology, like markets, bureaucracies, and the scientific literature, which aggregates and compresses human accumulated information, so that when an LLM output contains a new mathematical idea it should be received as the fruit of mathematics' shared heritage rather than as a defeat for mathematicians; drawing on Farrell, Gopnik, Shalizi, and Evans's claim that large models are cultural technologies, the author proposes the metaphor of a 'convex hull of ideas,' suggesting that whether an LLM can solve a given mathematical problem depends largely on the state of the mathematical literature at the time rather than merely on model scale, illustrating this with the Kryl
This essay by Nestor Guillen argues that large language models (LLMs) should be understood as a kind of cultural and social technology, like markets, bureaucracies, and the scientific literature, which aggregates and compresses human accumulated information, so that when an LLM output contains a new mathematical idea it should be received as the fruit of mathematics' shared heritage rather than as a defeat for mathematicians; drawing on Farrell, Gopnik, Shalizi, and Evans's claim that large models are cultural technologies, the author proposes the metaphor of a 'convex hull of ideas,' suggesting that whether an LLM can solve a given mathematical problem depends largely on the state of the mathematical literature at the time rather than merely on model scale, illustrating this with the Kryl
arXiv Jul 29, 2026
This work extends a bottom-up, statistics-first linear-algebraic framework previously developed for prepare-measure scenarios to operational scenarios with sequential transformations of an arbitrary number of stages, provides a full decision procedure for contextuality in such scenarios together with a complexity analysis (linearly exponential in the minimum generalized probabilistic theory dimension and polynomial in the number of procedures), demonstrates the approach through multiple examples including Spekkens' toy theory and the 8-state single-qubit stabilizer theory, and constructs an operational theory in which contextuality manifests itself only in the sequential structure of the transformations.
This work extends a bottom-up, statistics-first linear-algebraic framework previously developed for prepare-measure scenarios to operational scenarios with sequential transformations of an arbitrary number of stages, provides a full decision procedure for contextuality in such scenarios together with a complexity analysis (linearly exponential in the minimum generalized probabilistic theory dimension and polynomial in the number of procedures), demonstrates the approach through multiple examples including Spekkens' toy theory and the 8-state single-qubit stabilizer theory, and constructs an operational theory in which contextuality manifests itself only in the sequential structure of the transformations.
Linear Algebra of Generalized Contextuality in All Prepare-Transform-Measure Scenarios
This work extends a bottom-up, statistics-first linear-algebraic framework previously developed for prepare-measure scenarios to operational scenarios with sequential transformations of an arbitrary number of stages, provides a full decision procedure for contextuality in such scenarios together with a complexity analysis (linearly exponential in the minimum generalized probabilistic theory dimension and polynomial in the number of procedures), demonstrates the approach through multiple examples including Spekkens' toy theory and the 8-state single-qubit stabilizer theory, and constructs an operational theory in which contextuality manifests itself only in the sequential structure of the transformations.
This work extends a bottom-up, statistics-first linear-algebraic framework previously developed for prepare-measure scenarios to operational scenarios with sequential transformations of an arbitrary number of stages, provides a full decision procedure for contextuality in such scenarios together with a complexity analysis (linearly exponential in the minimum generalized probabilistic theory dimension and polynomial in the number of procedures), demonstrates the approach through multiple examples including Spekkens' toy theory and the 8-state single-qubit stabilizer theory, and constructs an operational theory in which contextuality manifests itself only in the sequential structure of the transformations.
Nature Communications Nov 26, 2023
This work introduces TopCap and TopNN, which use time-delay embedding and persistent homology to extract topological features (such as maximal persistence and its birth time) from speech time series; on voiced versus voiceless consonant classification, TopCap reaches accuracy comparable to some state-of-the-art neural networks on small datasets while offering greater efficiency and interpretability, and TopNN, which concatenates topological features with gated recurrent unit features, achieves higher accuracy, steadier performance, and stronger noise robustness than standard neural networks across multiple datasets and signal-to-noise ratios.
This work introduces TopCap and TopNN, which use time-delay embedding and persistent homology to extract topological features (such as maximal persistence and its birth time) from speech time series; on voiced versus voiceless consonant classification, TopCap reaches accuracy comparable to some state-of-the-art neural networks on small datasets while offering greater efficiency and interpretability, and TopNN, which concatenates topological features with gated recurrent unit features, achieves higher accuracy, steadier performance, and stronger noise robustness than standard neural networks across multiple datasets and signal-to-noise ratios.
Topology-Enhanced Machine Learning for Speech Signal Processing
This work introduces TopCap and TopNN, which use time-delay embedding and persistent homology to extract topological features (such as maximal persistence and its birth time) from speech time series; on voiced versus voiceless consonant classification, TopCap reaches accuracy comparable to some state-of-the-art neural networks on small datasets while offering greater efficiency and interpretability, and TopNN, which concatenates topological features with gated recurrent unit features, achieves higher accuracy, steadier performance, and stronger noise robustness than standard neural networks across multiple datasets and signal-to-noise ratios.
This work introduces TopCap and TopNN, which use time-delay embedding and persistent homology to extract topological features (such as maximal persistence and its birth time) from speech time series; on voiced versus voiceless consonant classification, TopCap reaches accuracy comparable to some state-of-the-art neural networks on small datasets while offering greater efficiency and interpretability, and TopNN, which concatenates topological features with gated recurrent unit features, achieves higher accuracy, steadier performance, and stronger noise robustness than standard neural networks across multiple datasets and signal-to-noise ratios.

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Happy, Those Able to Know the Causes of Things: Reading LLMs as Cultural Technologies Rather Than Agents That Replace Mathematicians

Linear Algebra of Generalized Contextuality in All Prepare-Transform-Measure Scenarios

Topology-Enhanced Machine Learning for Speech Signal Processing