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Mathematical Reasoning in Large Language Models: Assessing Logical and Arithmetic Errors across Wide Numerical Ranges

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read LLMs make more reasoning errors as numbers grow larger

desk verdict Useful benchmark and a plausible trend, but the plausibility confound keeps the central claim from being fully clean. read the letter →

arxiv 2502.08680 v1 pith:VQ6I373V submitted 2025-02-12 cs.LG cs.AIcs.CL

Safal Shrestha , Minwu Kim , Keith Ross This is my paper · ORCID
classification cs.LGcs.AIcs.CL
keywords largelanguagemodelsmathematicalreasoningnumericalperturbationlogicalerrorsarithmeticGSM8Kout-of-distributionnumbersautomatedgrading
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 do not treat math word problems as abstract reasoning tasks: when the same problem is rewritten with bigger numbers, their logical mistakes become more frequent. The paper introduces GSM-Ranges, a generator that takes 100 GSM8K problems and rewrites the numbers across six magnitude bands, from same-digit values up to the millions. It also introduces a grading method that converts each model's answer into Python code, executes the code, and checks whether the remaining error is arithmetic (non-logical) or a genuine reasoning failure (logical). Across eight of nine tested models, logical error rates climb as numbers grow, up to 14 percentage points for the worst model, even though the reasoning steps are unchanged. The result matters because it separates “can’t compute” from “can’t reason” and shows that out-of-distribution magnitudes disproportionately impair the latter.

What carries the argument

The central apparatus is twofold. GSM-Ranges systematically rewrites the numbers in 100 GSM8K problems into six magnitude bands (same-digit, 100–1,000, 1,000–10,000, 10,000–100,000, 100,000–1,000,000, and 1,000,000–10,000,000) while keeping the wording fixed. The grading method asks GPT-4o to translate each wrong response into Python code that mirrors the response's stated logic, executes that code, and classifies the response as a non-logical error if the corrected computation matches ground truth and as a logical error otherwise. This two-step design lets the authors attribute rising error rates to reasoning rather than to computation alone.

What would settle it

Run the same GSM-Ranges perturbation protocol on problems where every intermediate arithmetic result stays below 1,000 (for instance by dividing all quantities by a common factor), and check whether logical error rates still rise with printed number size. If they stay flat, the observed trend is driven by computational load rather than by numerical magnitude as such.

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Extended reading notes

Core claim

The paper argues that perturbing only the numerical values in GSM8K word problems, while leaving the question structure and required reasoning steps intact, causes a consistent upward trend in logical error rates across nearly all evaluated models. The worst-case increase is 14 absolute percentage points (Gemma 2 2B), with several 7B models showing double-digit rises; GPT-4o is the only model with a near-zero gap. The paper also shows that arithmetic (non-logical) errors rise even more steeply with scale, that the same arithmetic operations are solved far more accurately in isolation than inside word problems, and that sampling many responses reveals the correct logic often exists in the model's distribution even when greedy decoding fails.

Load-bearing premise

The result assumes that replacing problem numbers with larger values leaves the logical difficulty of a problem unchanged, so any increase in errors must come from the numerical scale itself rather than from a change in the reasoning task.

Editorial extensions

If this is right

  • Benchmarks that score only final answers overstate a model's reasoning ability, because large numbers inflate both logical and non-logical errors.
  • Sampling many responses (for example pass@48) recovers correct logic that greedy decoding misses, so the correct reasoning is present in the model's distribution even when the greedy trace fails.
  • Reasoning models such as o3-mini keep logical error rates nearly flat as numbers scale, but they spend substantially more tokens on larger-numbered problems, indicating that the model senses the difficulty shift.
  • The varying gap between Level 1 perturbations and original GSM8K questions across models is consistent with training-data contamination and warns against using GSM8K accuracy alone to compare reasoning robustness.

Reading between the lines

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

  • Editorial inference: the logical-error gap may partly be an artifact of the GPT-4o grader translating long, error-laden responses; a grader that runs the model's arithmetic with an external calculator could separate computational load from reasoning more cleanly.
  • Editorial inference: the flattening of logical errors beyond Level 3 suggests a training-data frequency threshold; a testable prediction is that models finetuned on log-uniformly distributed numbers would show a near-flat logical-error curve.
  • Editorial inference: since the main experiments use greedy decoding, the trend could reflect decoding instability rather than the absence of correct logic; majority voting over temperature samples would test this directly.
  • Editorial inference: if out-of-distribution magnitudes are the mechanism, then numeric scaling is a cheap general probe for distributional robustness beyond arithmetic, applicable to other reasoning benchmarks that contain quantities.
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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 / 6 minor

Summary. The paper introduces GSM-Ranges, a dataset generator derived from GSM8K that perturbs numerical values in math word problems across six numerical ranges (same-digit, 100–1,000, ..., 1M–10M), and a grading methodology that uses GPT-4o to translate a model's response into executable Python code, thereby classifying errors as logical (reasoning-related) or non-logical (arithmetic and number-copy). The authors evaluate nine models and report that logical error rates increase by up to 14 absolute percentage points as perturbation level rises, while arithmetic accuracy is higher for standalone operations than for computations embedded in word problems. They also report recall-rate experiments with multiple sampling passes and a case study of o3-mini, which shows robustness across perturbation levels. The paper releases its code and dataset.

Significance. If the central claim survives scrutiny, the paper makes a useful contribution: it broadens evaluation of LLM math reasoning to out-of-distribution numerical scales and proposes a practical way to separate logical from non-logical errors, an improvement over final-answer-only grading. The public release of the dataset generator and the manual validation of the grader (98.5% on 200 responses) are positive features. The reported trend—that logical errors grow with numerical magnitude even when the question text is structurally unchanged—is important for understanding LLM robustness. However, the main empirical claim is currently threatened by a confound between numerical scale and semantic plausibility of the perturbed problems, and the statistical support for the 'consistent upward trend' is incomplete. These issues need to be addressed before the findings can be considered solid.

major comments (4)
  1. [Section 3.2, Appendix A.4, Table 1] The perturbation generator filters only for non-negative integer values and non-negative intermediate/final answers, but not for semantic plausibility; as a result, high-level perturbations produce problems such as 'Judy teaches 3,124,213 dance classes every day' (Table 1) and 'Mary is 1,922,674 years younger than Joan' (Appendix A.4). The central claim in Section 5.1.1 that 'the logical reasoning required to solve the problems remains unchanged across all perturbation levels' is therefore not directly supported, because absurd premises can cause models to disengage or adopt different reasoning strategies, producing exactly the 'question misinterpretation' and 'omitted information' errors that the paper counts as logical errors. Since the observed increase in logical errors (up to 14 percentage points in Figure 4) is the main empirical result, the paper should control for plausibility, for example by constructing a separate perturbation set where large numbers are semantically plausible (e.g., populations, distances, or currency amounts) or by analyzing error rates only on the subset of Level 4–6 problems that remain semantically coherent.
  2. [Section 4.4 and Appendix A.3] The grading methodology is validated on only 200 manually classified responses (98.5% accuracy), while the main experiments grade roughly 270,000 responses (9 models × 6 levels × 5,000 problems); the paper does not report how the 200 samples are stratified across models and perturbation levels, nor whether the three misclassifications are concentrated in a particular model or level. More importantly, because GPT-4o is both the grader and one of the evaluated models, and because the code-translation prompt explicitly corrects number-copy errors, the possibility remains that GPT-4o's translation fidelity degrades for large-number responses, which would create a spurious trend in logical errors. The authors should report per-level validation accuracy, ideally with a second annotator or a different judge model, and should test whether the grader's error rate varies systematically with perturbation level.
  3. [Section 5.1.1, Figure 4, Table 3] The paper claims a 'consistent upward trend in logical errors as the perturbation level increases, across all nine evaluated models except GPT-4o,' but the underlying rates in Table 3 are not strictly monotonic (e.g., Qwen 2.5 7B shows 9.0, 10.2, 10.1, 9.9 across levels 3–6, and Mathstral 7B shows 13.9, 15.2, 14.8 across levels 4–6), and Figure 4 reports only point estimates of the Level-6-minus-Level-1 difference without confidence intervals or a formal trend test. To support the central claim, the authors should either fit a monotone trend model (e.g., isotonic regression) or report bootstrap confidence intervals for the differences, and they should state explicitly which models exhibit a statistically significant increase.
  4. [Section 3.2 and Section 5.2.1] The perturbation levels alter not only numerical scale but also the computational difficulty of the required arithmetic (e.g., number of digits in multiplication and addition), and the paper's own results show that non-logical/arithmetic errors rise sharply with level. Because logical and arithmetic processing share limited working memory in autoregressive decoding, a model that is struggling with large-number arithmetic may drop steps, invert operators, or omit information—errors that the paper classifies as logical. The recall experiment in Section 6.1 suggests the correct logic is present in the model's distribution, but the paper does not directly separate 'scale-induced difficulty in keeping track of steps' from 'genuine logical misunderstanding.' A control using numbers of similar magnitude but simpler arithmetic (e.g., powers of ten, or problems designed to minimize carrying) would help disentangle these explanations.
minor comments (6)
  1. [Section 3 and Figure 1 caption] The dataset name 'SV AMP' should be 'SVAMP'.
  2. [Section 3.2] The last perturbation range is written as '1,000,000–10,000,00'; it should presumably be '1,000,000–10,000,000'.
  3. [Section 4.4] The sentence 'randomly sample 200 of these responses and and manually classify each one' contains a duplicated 'and'.
  4. [Table 3 heading] The model name 'Mathtral 7B v0.1' should be 'Mathstral 7B v0.1'; please check for consistent spelling throughout the appendix.
  5. [Section 5 and Section 6.1] The statement 'All inferences are done in the greedy decoding setting' conflicts with the recall experiment in Section 6.1, which uses temperature 0.8 and top_p 0.95; please clarify the exact sampling settings for each experiment.
  6. [Section 4.1] The definition of 'non-logical error' groups arithmetic errors and number-copy errors, which are arguably distinct phenomena; the paper should justify this grouping and report their relative frequencies.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the paper is an empirical measurement, and its central trend is not entailed by construction.

full rationale

The paper does not derive its main conclusion from its own definitions or fitted parameters. GSM-Ranges holds question text structure fixed while perturbing numerical values, and the subsequent rise in logical error rates is an observed, contingent outcome across model responses, not a consequence of the construction. The logical-error label is operationally defined: if correcting arithmetic and number-copy mistakes still leaves an answer that does not match ground truth, the response is counted as a logical error. This classification is checked against ground truth and executed code, so it is not circular in the sense of assuming the conclusion. Using GPT-4o to grade responses, including some of its own outputs, is a possible measurement-bias concern, but the paper mitigates it with manual validation (98.5% agreement on 200 sampled responses and 50/50 successful number-copy corrections), so the grading instrument is not assumed valid by definition. The semantic-implausibility confound noted by the skeptic (e.g., 'Mary is 1,922,674 years younger than Joan') is a genuine threat to internal validity, because implausible premises may change perceived problem difficulty, but that is a confound, not a circular reduction: the paper's invariance assumption is an empirical premise rather than a tautology. There are no load-bearing self-citations, no imported uniqueness theorems, no fitted parameters renamed as predictions, and no known result repackaged as derivation. The paper is self-contained as an evaluation study; its limitations are about validity and generalizability, not circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on assumptions about the invariance of problem logic under perturbation, the fidelity of GPT-4o as a code translator, and the representativeness of 100 sampled GSM8K questions. No free parameters are fitted to the data; the perturbation ranges and sample sizes are methodological choices.

free parameters (3)
  • Perturbation range boundaries = 100-1000, 1000-10,000, 10,000-100,000, 100,000-1,000,000, 1,000,000-10,000,000
    Chosen by hand to define six numerical scales; not fitted to outcomes, but the observed trend depends on these scales.
  • Multiplication scaling rule = Only one side of a multiplication is scaled
    Keeps final answers within manageable limits; may reduce arithmetic difficulty at high levels and could affect error rates.
  • Number of base questions and variations = 100 base questions, 50 variations per level
    Chosen for compute budget; affects confidence intervals but not the direction of findings.
assumptions (5)
  • domain assumption GSM8K problems are logically coherent and have unique solutions.
    The perturbation generator starts from these problems, and logical correctness is judged against the original ground truth.
  • domain assumption The logical reasoning required to solve a problem is invariant under numerical perturbation.
    Section 5.1.1 states this invariance to attribute the observed logical error increase to numerical scale rather than problem structure.
  • domain assumption GPT-4o-generated Python code faithfully reproduces the reasoning steps in the response.
    The grading methodology depends on this translation; manual validation on 200 samples supports it.
  • domain assumption A corrected answer matching the ground truth is a valid test of logical correctness.
    This operationalization defines 'logical error' as any error that persists after fixing computation.
  • domain assumption LLMs' training data is dominated by low-range numbers.
    Used in the conjecture that large numbers are out-of-distribution; not directly tested.

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Cite this review

Pith. "Pith review of Mathematical Reasoning in Large Language Models: Assessing Logical and Arithmetic Errors across Wide Numerical Ranges." pith.science (2026). https://pith.science/paper/VQ6I373V

@misc{pith2026250208680,
  author       = {Pith},
  title        = {Pith review of: Mathematical Reasoning in Large Language Models: Assessing Logical and Arithmetic Errors across Wide Numerical Ranges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQ6I373V}},
  note         = {Machine review of arXiv:2502.08680}
}
read the original abstract

Mathematical reasoning in Large Language Models (LLMs) is often evaluated using benchmarks with limited numerical ranges, failing to reflect real-world problem-solving across diverse scales. Furthermore, most existing evaluation methods only compare model outputs to ground-truth answers, obscuring insights into reasoning processes. To address these limitations, we introduce GSM-Ranges, a dataset generator derived from GSM8K that systematically perturbs numerical values in math problems to assess model robustness across varying numerical scales. Additionally, we propose a novel grading methodology that distinguishes between logical and non-logical errors, offering a more precise evaluation of reasoning processes beyond computational accuracy. Our experiments with various models reveal a significant increase in logical error rates-up to 14 percentage points-as numerical complexity rises, demonstrating a general weakness in reasoning with out-of-distribution numerical values. Moreover, while models demonstrate high accuracy on standalone arithmetic tasks, their performance deteriorates substantially when computations are embedded within word problems. These findings provide a comprehensive evaluation of LLMs' mathematical reasoning capabilities and inform future research directions for improving numerical generalization in language models.

Figures

Figures reproduced from arXiv: 2502.08680 by the authors.

Figure 1
Figure 1. Cumulative frequency distribution of nu [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Illustration of grading process for LLM re [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Logical & non-logical error rates across different perturbation levels. The left panel illustrates the increase [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Logical error gaps across perturbation levels. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Recall rates across perturbation levels and [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Results of o3-mini across perturbation levels. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Forward citations

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Reference graph

Works this paper leans on

25 extracted references · 21 canonical work pages · cited by 4 Pith papers

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    Avoid directly copying the results of these operations or the final answer from the response

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    Gsm-symbolic: Understanding the limitations of mathematical reasoning in large language models. Preprint, arXiv:2410.05229. OpenAI. 2024. Gpt-4 technical report. Preprint, arXiv:2303.08774. OpenAI. 2024. Gpt-4o system card. Published: August 8, 2024. Accessed: 2024-12-20. OpenAI. 2025. Introducing OpenAI o1. Accessed February 7, 2025. Arkil Patel, Satwik ...

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    Provide the output in the following format: { “extracted_answer”: “<final numerical value of the answer>”, “explain”: “<detailed explanation of the response logic>”, “python_code”: “```python\n<generated Python function>\n```” } • This is the list of numbers extracted from the question: {number_list}. • This the response: {response}. A.4 Various Types of ...

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    Compute the baby shower outfits: 8852986 × 2 = 17705972

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    Example 2: Operator Error Question: Finn watches 9 360 266tadpoles swimming in the pond

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    Since 12947038 more hide under the rock, add them as well: 16541186 + 12947038 =29488224 Error: The model makes an operator error by incorrectly adding the 12 947 038tadpoles that hid under the rock instead of subtracting them. Example 3: Contextual Value Error Question: Glori...

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