The Math Behind the Intuition: How LLMs Understand Abstract Concepts

How does a large language model "understand" something as abstract as truth, time, or a bridge in San Francisco? In this deep dive, we uncover the mathematics behind that intuition: how neural networks encode meaning as geometry. We start with the distributional hypothesis and word2vec, see how prediction connects co-occurrence with vector geometry, and follow the path to transformers, attention and the residual stream. Then we explore the linear representation hypothesis, linear probes that reveal maps of the world and timelines inside Llama-2, superposition in high-dimensional spaces, and sparse autoencoders that extracted up to 34 million features from Claude 3 Sonnet. Finally, we steer models by adding vectors (Golden Gate Claude), examine how removing a refusal-related direction changes behavior, discover circular representations of weekdays, and look at evidence of multi-step processing and planning in selected Claude 3.5 Haiku examples. All formulas are shown on screen with a short explanation; the narration focuses on the intuition behind each one. ⏱️ Chapters 00:00 Cold open: a bridge inside the machine 00:44 What you will learn 01:34 The distributional hypothesis 02:20 From tokens to vectors 03:07 Embedding lookup 03:55 Cosine similarity 04:41 word2vec and the famous analogy 05:25 Vector arithmetic for analogies 06:10 Why prediction creates geometry (skip-gram & PMI) 06:54 The problem of static meaning 07:36 Attention as information routing 08:19 Scaled dot-product attention 09:06 The residual stream 09:49 Residual stream update 10:34 Feed-forward layers as memories 11:18 MLP as key-value memory 12:01 The linear representation hypothesis 12:45 Finding a concept direction 13:27 Linear probes 14:11 A map and a clock inside the model 14:54 Truth directions and world models 15:39 Too many concepts, too few dimensions 16:20 Near-orthogonality & Johnson–Lindenstrauss 17:04 Superposition and polysemantic neurons 17:49 Sparse autoencoders 18:32 The SAE objective 19:18 Scaling Monosemanticity (34M features) 20:01 Activation steering 20:44 One direction for refusal 21:27 When concepts are circles 22:12 Hierarchies and a language-free core 22:54 Reasoning and planning inside the model 23:38 Limits and open questions 24:23 Summary 📚 Papers & References Harris (1954): Distributional Structure Firth (1957): A Synopsis of Linguistic Theory 1930–1955 Mikolov et al. (2013): Efficient Estimation of Word Representations in Vector Space — arXiv:1301.3781 Mikolov et al. (2013): Distributed Representations of Words and Phrases and their Compositionality — arXiv:1310.4546 Levy & Goldberg (2014): Neural Word Embedding as Implicit Matrix Factorization (NeurIPS) Nissim, van Noord & van der Goot (2020): Fair is Better than Sensational — Computational Linguistics Vaswani et al. (2017): Attention Is All You Need — arXiv:1706.03762 Brown et al. (2020): Language Models are Few-Shot Learners (GPT-3) — arXiv:2005.14165 Geva et al. (2021): Transformer Feed-Forward Layers Are Key-Value Memories — arXiv:2012.14913 Park, Choe & Veitch (2023): The Linear Representation Hypothesis and the Geometry of LLMs — arXiv:2311.03658 Gurnee & Tegmark (2023): Language Models Represent Space and Time — arXiv:2310.02207 Li et al. (2023): Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task (Othello-GPT) — arXiv:2210.13382 Nanda, Lee & Wattenberg (2023): Emergent Linear Representations in World Models of Self-Supervised Sequence Models — arXiv:2309.00941 Marks & Tegmark (2023): The Geometry of Truth: Emergent Linear Structure in Large Language Model Representations of True/False Datasets — arXiv:2310.06824 Elhage et al. (2022): Toy Models of Superposition — transformer-circuits.pub Bricken et al. (2023): Towards Monosemanticity — transformer-circuits.pub Templeton et al. (2024): Scaling Monosemanticity — transformer-circuits.pub/2024/scaling-monosemanticity Arditi et al. (2024): Refusal in Language Models Is Mediated by a Single Direction — arXiv:2406.11717 Engels et al. (2024): Not All Language Model Features Are Linear — arXiv:2405.14860 Park et al. (2024): The Geometry of Categorical and Hierarchical Concepts in Large Language Models — arXiv:2406.01506 Lindsey et al. (2025): On the Biology of a Large Language Model — transformer-circuits.pub/2025/attribution-graphs/biology.html Huh et al. (2024): The Platonic Representation Hypothesis — arXiv:2405.07987 🔔 Subscribe to MartinHanderPhD for in-depth AI & machine learning explanations.