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What is a Number, That a Large Language Model May Know It?

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Large language models represent numbers as a blend of digit-string and magnitude structure, and that blend causes measurable errors in quantitative decisions.

desk verdict Behavioral result is solid and new; the 'representation' claim is softer than advertised, so conditional acceptance is right. read the letter →

arxiv 2502.01540 v1 pith:KBPMUJ5J submitted 2025-02-03 cs.CL cs.AI

Raja Marjieh , Veniamin Veselovsky , Thomas L. Griffiths , Ilia Sucholutsky This is my paper · ORCID
classification cs.CLcs.AI
keywords numberrepresentationlargelanguagemodelssimilarityjudgmentsLevenshteineditdistancelog-linearstringbiastokenizationquantitativereasoning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Large language models must treat the same digit sequence as both a number and a string, and this paper asks what representation actually arises. The authors elicited pairwise similarity ratings over integers 0 to 999 from six modern large language models and found that the resulting similarity patterns are well described by a weighted combination of Levenshtein edit distance and a psychological Log-Linear numerical distance. Explicitly telling the model whether the tokens are int() or str() pushed the pattern in the predicted direction but did not remove the mixture. The same entanglement shows up in the model's internal embeddings, and a test-tube concentration choice shows that the string bias can cause wrong answers in a realistic quantitative decision. If correct, this means models do not have a clean 'number sense': numerical competence is entangled with surface string structure, and context only partially resolves the ambiguity.

What carries the argument

The paper's central instrument is the similarity-judgment paradigm: a neutral prompt that asks the model to rate pairwise similarity, converted into a symmetric similarity matrix over the integers 0 through 999. The theoretical yardsticks are the Levenshtein edit distance, the minimum number of character insertions, deletions, or substitutions between two digit strings, and the Log-Linear distance $d_{\mathrm{Log}}(x,y) = 1 - \exp(-|\log(x+\epsilon) - \log(y+\epsilon)|)$ with $\epsilon = 10^{-4}$, a psychological magnitude measure with logarithmic compression. The argument runs by linear regression of the elicited matrix on the two normalized distances, and by linear probes that decode each distance from the last-token residual of Llama-3.1-8b. The combination of the two distances is what carries the 'entanglement' claim.

What would settle it

Run the same similarity protocol in the int() condition at zero temperature across the six models; if any model's Levenshtein regression weight over the Log-Linear-only model falls within a 95% confidence interval that includes zero, the claim that the string component is irreducible fails.

Watch

Extended reading notes

Core claim

On its own terms, this paper establishes that the representation of integer tokens inside large language models is an entanglement of a string-like code and a magnitude code. Across six models, similarity matrices elicited by asking 'How similar are the two numbers?' are regressed by a linear combination of Levenshtein edit distance and Log-Linear numerical distance, explaining on average 72.6% of the variance in the default prompt; each component contributes on its own, and replacing Log-Linear with absolute difference lowers the fit. Writing int() or str() around the tokens shifts the weights in opposite directions, but the Levenshtein contribution never disappears. Internal linear probes on Llama-3.1-8b decode both distances from the embeddings, and in the integer-probe subspace the string-induced sub-diagonals persist. In a forced-choice scenario with compound concentrations, a model with string bias can choose the numerically wrong tube: for three-digit concentrations the maximal error rate is 36.9% for Llama-3.1-8b (with GPT-4o at 0.02%), and for five-digit numbers the maximal error rises to 47.0%.

Load-bearing premise

The whole argument assumes that asking a model 'how similar are these two numbers?' reports the model's internal representation of numbers, rather than a task-specific habit of comparing strings that the prompt itself triggers.

Editorial extensions

If this is right

  • Explicit type disambiguation via int() or str() shifts but does not abolish the entanglement: the Levenshtein component stays above zero in every tested context, so prompt design can mitigate but not remove string bias.
  • In decision tasks where magnitude matters, string similarity between digit strings can override numerically closer options, and the error rate grows with digit count for the most biased models.
  • The representational blend is visible at the embedding level, not only in prompted answers, so it is a property of the learned representation rather than a quirk of the similarity prompt.
  • Replacing the Log-Linear distance with a simple absolute difference degrades the fit, supporting a logarithmically compressed magnitude code in these models, paralleling human approximate number representations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension the paper does not test: applying the same protocol to reasoning-oriented models or chain-of-thought settings should shrink the Levenshtein component if the string bias is a fast surface heuristic; if it does not, the blend is more deeply structural.
  • The similarity protocol could be used to probe tokenization effects directly: numbers with group separators (e.g., '1,234' vs '1234') should shift the edit-distance component if token boundaries drive the string effect.
  • The concentration-choice task could be turned into a cheap behavioral calibration instrument: the ratio of Levenshtein to Log-Linear regression weights might predict a model's error rate on magnitude comparisons, giving a testable predictor beyond the paper's reported results.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper asks whether large language models (LLMs) represent numbers using a mixture of string-like (Levenshtein edit distance) and numerical (Log-Linear magnitude) codes. The authors elicit pairwise similarity ratings for integers 0–999 from six LLMs, regress the ratings on the two theoretical distances, and report that a combined linear model achieves an average R^2 of 0.726 in the default context. They further manipulate context with int()/str() wrappers and number-base changes, probe the internal activations of Llama-3.1-8b to decode the two distances, and construct a test-tube selection scenario where a string-distance bias produces objectively wrong choices. The paper concludes that LLM number representations are an entangled blend of string and numerical structure that can be reduced but not eliminated by context.

Significance. If the result holds, this is a valuable and methodologically interesting contribution. The behavioral regression is carefully executed with bootstrap confidence intervals and a linear-distance control, the context manipulations (int/str, base 4/8) are strong controls, and the decision scenario gives the work practical relevance. The paper also provides code, model versions, and an appendix with temperature robustness checks, which supports reproducibility. The central finding—that elicited similarity judgments over integers are well described by a combination of Levenshtein and Log-Linear distances—is falsifiable and re-runnable. The main risk is that the paper interprets this finding as evidence about internal representations, and that interpretation rests on load-bearing assumptions that are not fully validated.

major comments (4)
  1. [4.3 and 5.1] The two predictor distances (Levenshtein and Log-Linear) are highly correlated for numbers, but the paper reports only R^2 for the combined and separate models, not partial correlations or a variance-partitioning analysis. Without partial R^2 values, beta weights, or a commonality analysis, the claim that the two distances contribute uniquely is underdetermined. Please report standardized regression coefficients with confidence intervals and a decomposition of the combined R^2 into unique and shared components.
  2. [4.4 and 5.2] The internal-probe experiment does not independently confirm an entangled representation. The probes are trained on the last-token residual after the full similarity prompt ('Rating:') and their targets are exactly the two theoretical metrics used in the behavioral regression. A linear probe can decode any feature that is useful for producing the rating, so the correlations in Table 2 are consistent with a task-specific response heuristic (e.g., 'compare digit overlap and length') rather than with a stable internal number representation. To support the representational claim, the probes should be trained on activations from a neutral context that does not mention similarity (e.g., the number tokens alone or next-token prediction), or the claims in Section 5.2 should be restricted to task-conditioned decodability.
  3. [3 and 5.1] The method assumes that the 'How similar are the two numbers?' prompt is a faithful readout of internal number representation. All of the paper's controls—int()/str(), base changes, and the decision scenario—still use the word 'similar' (or 'most similar'), so they may all tap the same surface heuristic. A concrete validation would be to use a non-similarity paradigm, such as asking the model to choose the numerically nearest number or to rate 'which number is closer in value'; if the dual-distance structure persists without the word 'similar', the readout assumption is strengthened. Without such a control, the abstract's claim about 'representational spaces' should be tempered.
  4. [4.1] The statement that all integers in 0–999 are 'represented as unique tokens' in all six models is an empirical tokenizer claim, but no tokenizer analysis or evidence is provided. If some three-digit numbers are multi-token in these models, the Levenshtein-distance effect could partly reflect token-boundary artifacts rather than character-level string similarity. Please verify this claim with the actual tokenizers and report the fraction of single-token versus multi-token numbers in the 0–999 range for each model.
minor comments (5)
  1. [Figure 5 caption] The caption contains a typo: 'faction' should be 'fraction'.
  2. [4.5] The example triplet construction says the digits are sampled from {2,...,9}, but the first example is '3, 3, 1 → 331', which includes 1; the text is inconsistent and should be corrected.
  3. [5.2 and Table 2] The claim that the probes 'nicely dissociate' the integer and string subspaces is overstated: the string probe correlates 0.527 with the Log-Linear measure, and the integer probe correlates 0.393 with Levenshtein, so the dissociation is partial. The correlations are also not specified as Pearson; please state the correlation type.
  4. [Appendix A] The int()/str() manipulation changes the surface string by adding parentheses and function names, so part of the observed context effect may be a formatting effect rather than type semantics. Consider a control that wraps the numbers in neutral characters (e.g., 'Number: (NUM)') to isolate the semantic effect.
  5. [5.1 and Appendix E] The linear ℓ1-distance control is reported only in the appendix, but it is an important control; consider mentioning it more prominently in the main text, possibly with a sentence in Section 5.1 explaining that the Log-Linear metric outperforms linear magnitude.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: external benchmarks are regressed against model outputs, and probe correlations are out-of-sample supervised fits rather than by-construction equivalences.

full rationale

The derivation chain is not circular. The primary behavioral result regresses model-elicited similarity matrices (the dependent variable) on two external benchmarks, Levenshtein edit distance and Piantadosi's Log-Linear distance, both defined independently of the model outputs; the reported R2 values therefore measure fit, not a quantity that was fed into the model. The probing analysis trains a linear probe on 9,500 pairs and evaluates on a held-out 250,000-pair grid; although the probe targets are the same two distance measures used in the behavioral analysis, the reported correlations are out-of-sample decodability results rather than fitted values relabeled as predictions. The probe is trained on residuals of the last token after the 'Rating:' instruction, so it can only show that the two distances are linearly decodable from task-conditioned activations; this limits the strength of the internal-representation inference but is not a by-construction equivalence. The choice of Log-Linear over linear distance was made after comparing fits on the same data, a researcher-degree-of-freedom concern but not circularity. Self-citations (Marjieh et al., 2024a; 2024b; Bai et al., 2024) are background and related work and are not load-bearing for the main claim. The decision triplets are deliberately constructed to separate Levenshtein and Log-Linear predictions; that is diagnostic design, not circularity. No equation in the paper defines the target result in terms of itself.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on two external distance benchmarks (Levenshtein and Log-Linear) and on the assumption that prompt-level similarity ratings reflect internal representations. The model-specific regression weights are fitted per context, but the distance measures themselves are not fitted to the data, which keeps circularity low. No new entities are introduced.

free parameters (4)
  • Regression intercept alpha = varies by model and context; not listed in paper
    Fitted intercept in the linear model that maps z-scored Levenshtein and Log-Linear similarities to LLM similarity ratings.
  • Levenshtein regression coefficient beta = varies by model and context; not listed in paper
    Weight of the string edit distance predictor in the combined similarity model, fit with scikit-learn LinearRegression.
  • Log-Linear regression coefficient gamma = varies by model and context; not listed in paper
    Weight of the numerical distance predictor in the combined similarity model, fit with scikit-learn LinearRegression.
  • Epsilon regularizer in dLog = 10^-4
    Chosen by hand to handle the number 0 in the log transform; its value is small and does not materially affect the fit or the central claim.
assumptions (4)
  • domain assumption Log-Linear distance is a valid benchmark for numerical similarity
    Taken from Piantadosi (2016) rational analysis of the approximate number system; the paper assumes this is the right numerical metric and shows it fits better than linear distance.
  • domain assumption Levenshtein distance is a valid benchmark for string similarity
    Standard string edit distance; the paper assumes string-likeness of number tokens is captured by edit distance.
  • domain assumption Elicited similarity ratings from LLMs reflect their internal representations
    The core methodology assumes prompt-level similarity judgments are a faithful behavioral readout of the model's representation; partially validated by internal probes for Llama-3.1-8b only.
  • domain assumption Linear separability of distances in embeddings via probes is evidence of representation
    Probing literature assumption; the paper trains linear probes to predict distances and interprets high correlation as evidence of encoding.

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Cite this review

Pith. "Pith review of What is a Number, That a Large Language Model May Know It?." pith.science (2026). https://pith.science/paper/KBPMUJ5J

@misc{pith2026250201540,
  author       = {Pith},
  title        = {Pith review of: What is a Number, That a Large Language Model May Know It?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KBPMUJ5J}},
  note         = {Machine review of arXiv:2502.01540}
}
read the original abstract

Numbers are a basic part of how humans represent and describe the world around them. As a consequence, learning effective representations of numbers is critical for the success of large language models as they become more integrated into everyday decisions. However, these models face a challenge: depending on context, the same sequence of digit tokens, e.g., 911, can be treated as a number or as a string. What kind of representations arise from this duality, and what are its downstream implications? Using a similarity-based prompting technique from cognitive science, we show that LLMs learn representational spaces that blend string-like and numerical representations. In particular, we show that elicited similarity judgments from these models over integer pairs can be captured by a combination of Levenshtein edit distance and numerical Log-Linear distance, suggesting an entangled representation. In a series of experiments we show how this entanglement is reflected in the latent embeddings, how it can be reduced but not entirely eliminated by context, and how it can propagate into a realistic decision scenario. These results shed light on a representational tension in transformer models that must learn what a number is from text input.

Figures

Figures reproduced from arXiv: 2502.01540 by the authors.

Figure 1
Figure 1. LLM number similarity matrices (symmetrized) over all integer pairs in the range 0 − 999, along with two theoretical similarity matrices derived from a Levenshtein string edit distance and a psychological Log-Linear numerical distance (highlighted in black). resentation of numbers (Piantadosi, 2016) and a string edit distance, suggesting an entanglement. We show how this en￾tanglement can be reduced but not eliminat… view at source ↗
Figure 2
Figure 2. Context effects on LLM-elicited number similarity matrices and their decomposition. A. LLM similarity matrices under the effect of ‘type’ specification: int() vs. str() (see Appendix A for prompts). B. Coefficient of determination (R 2 ) for the different similarity matrices under the default ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The effect of other number bases on elicited similarity. A. LLM similarity matrices over all integer pairs in the range 0 − 999 represented in base 4 and 8 along with the corresponding Levenshtein distance measures (see Appendix A for prompts). B. Coefficient of determination (R 2 ) for the various similarity matrices under the different base contexts (including the base 10 results from [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Decoded string and integer subspaces from Llama-3.1-8b using linear probes (see Methodology). The decoded similarity matrices are provided as insets along with their multidimensional scaling solutions (MDS). Integers are labeled every 5 points. correlation is .917 with…
Figure 5
Figure 5. Figure 5: Probing string-bias in a naturalistic decision scenario. Bar plots indicate the faction of times an incorrect (Levenshtein-aligned) option was chosen for the 3-digit and 5-digit scenarios considered, and the two possible presentation orders (see Methodology). ‘(Rev.)’ …
Figure 6
Figure 6. Figure 6: Coefficient of determination (R 2 ) for the different similarity matrices under the default ( [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Ablations for the training the probe. The vertical axis shows the correlation between the groundtruth similarities and the probe similarities on the test set. The plot on the left features how the correlation varies with the layer the probe is trained on using the full…
Figure 8
Figure 8. Figure 8: (left) Similarity judgments extended to the 2000 × 2000 setting. (right) Original prompt where “similar” is replaced with “closer” in the 2000 × 2000 setting. Both plots are for GPT-4o. F.3. Higher Temperature In the final analysis, we re-run the temperature 0 setting …

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.