Have you ever wondered what actually happens inside GPT-4 when you ask it "What is 37 + 58?"
It doesn't add numbers like a calculator. It maps the tokens "37" and "58" into vectors in a high-dimensional space, and through a series of attention mechanisms and matrix operations, ultimately "predicts" the token "95."
But here is a fascinating question: Different language models — Transformers, LSTMs, Linear RNNs, even the ancient Word2Vec — use completely different architectures, different training data, and different optimizers, yet they all learn to represent numbers in almost the same way.
This is no coincidence. That's the message of a recent arXiv paper.
Core Findings
The paper, "Convergent Evolution: How Different Language Models Learn Similar Number Representations," comes from a team at UCSB, UCSD, University of Washington, and UIUC.
Their striking discovery: All language models encode numbers using periodic features.
Specifically, if you apply a Fourier transform to a model's learned number representations, you see clear peaks at periods T=2, T=5, and T=10.
What does that mean? An analogy:
Imagine using a spring stretched to different lengths to represent different numbers. That's not what models do. Models are more like metronomes — they "tick" at different rhythms, and combine rhythms to distinguish numbers.
- Period T=2: separates odd and even numbers (0,1,0,1,0,1...)
- Period T=5: separates residues mod 5 (0,1,2,3,4,0,1,2,3,4...)
- Period T=10: separates digits (0,1,2,...,9,0,1,2,...,9...)
- Uncovers a cross-architecture phenomenon and analyzes it with rigorous mathematical tools
- Distinguishes "sensing" from "exploiting," avoiding oversimplified conclusions
- The two learning pathways are highly illuminating
- Focuses mainly on integer representations; floats, negatives, and scientific notation are unexplored
- Experiments are on English corpora; whether number representations differ across languages remains unknown
- Title: Convergent Evolution: How Different Language Models Learn Similar Number Representations
- Authors: Deqing Fu, Tianyi Zhou, Mikhail Belkin, Vatsal Sharan, Robin Jia
- arXiv: https://arxiv.org/abs/2604.20817
- Project page: https://convergent-evolution.github.io
Two Levels of Hierarchy: "Sensing" vs. "Classifying"
The paper's most elegant contribution is distinguishing two hierarchical levels of features:
1. Fourier Sparsity: the model's number representations show clear periodic peaks in the frequency domain. Nearly all models have this property.
2. Geometric Separability: whether a linear classifier can accurately determine a number's value mod T. Only some models achieve this.
The authors prove a mathematical theorem: Fourier sparsity is necessary but not sufficient for geometric separability.
In plain terms: merely "sensing" the periodic structure of numbers is not enough — a model must also learn to exploit it for classification. It's like hearing a musical beat but not necessarily being able to dance to it.
Two Learning Pathways
How do models actually learn these number representations? The paper identifies two distinct pathways:
Pathway 1: Learning from natural language
While reading large amounts of text, models pick up on co-occurrence patterns between numbers and words. For example, "1st place" and "11th place" tend to appear in similar contexts, while "1st place" and "2nd place" do not. These complementary co-occurrence signals help the model learn the periodic structure of numbers.
Pathway 2: Drilling on arithmetic
When a model encounters multi-token addition problems like "37 + 58 = 95," it is forced to build precise numeric representations. Interestingly, single-token addition (e.g., "3+5=8") does not help — it's too easy, so the model can just memorize the answers.
Why This Matters
The paper's significance goes far beyond "how language models understand numbers."
It reveals a deeper phenomenon: Convergent Evolution.
In biology, dolphins (mammals) and sharks (fish) evolved along completely different paths, yet both evolved streamlined bodies and dorsal fins — because these are optimal solutions for efficient swimming.
Likewise, language models with radically different "brain structures" all evolve similar number representations — because this is the optimal solution for processing numeric information in text.
This means some knowledge structures may be universal and unavoidable: whatever architecture you use, learning converges to similar representations. That has important implications for interpreting model internals and designing better architectures.
An Honest Assessment
Strengths:
Weaknesses: