Recognition: 2 theorem links
· Lean TheoremMeasuring Mathematical Problem Solving With the MATH Dataset
Pith reviewed 2026-05-10 12:55 UTC · model grok-4.3
The pith
The MATH dataset shows that scaling up Transformer models is insufficient for strong mathematical reasoning.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We introduce MATH, a dataset of 12,500 challenging competition mathematics problems with full step-by-step solutions. Despite increasing accuracy with larger models and pretraining, accuracy remains relatively low even with enormous Transformers, and scaling trends indicate it will be impractical to achieve strong mathematical reasoning without new algorithmic changes.
What carries the argument
The MATH dataset, consisting of 12,500 competition math problems each paired with a detailed solution, used to evaluate model performance on mathematical problem solving.
If this is right
- Current scaling of model size and compute will not suffice to solve advanced math problems effectively.
- New algorithmic innovations from the research community will be necessary for progress in mathematical reasoning.
- Models trained on the auxiliary pretraining dataset can improve but still fall short on MATH.
- Step-by-step solutions in the dataset can be used to train models to generate explanations.
Where Pith is reading between the lines
- Progress on MATH may require techniques that go beyond pattern matching in large datasets, such as symbolic reasoning or verification methods.
- If scaling continues to underperform on MATH, it could indicate limitations in how Transformers process mathematical structures compared to other tasks.
- Future benchmarks might need to incorporate more diverse or harder problems to track true advances in reasoning.
Load-bearing premise
That the MATH problems are a faithful and comprehensive measure of general mathematical problem-solving ability and that the observed performance trends with model scale will continue without new algorithmic changes.
What would settle it
Demonstrating a Transformer-based model that achieves high accuracy on the MATH dataset through scaling alone, without novel algorithms, would falsify the claim that scaling is impractical for strong mathematical reasoning.
read the original abstract
Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces the MATH dataset of 12,500 competition-level mathematics problems, each with a full step-by-step solution, together with a large auxiliary pretraining corpus of mathematical text. It evaluates a range of Transformer models on MATH, reports that final-answer accuracy remains low even for the largest models tested, and concludes that continued scaling of model size and compute will be insufficient to reach strong mathematical reasoning performance if current trends persist, thereby calling for new algorithmic advances.
Significance. If the empirical measurements hold, the work supplies a demanding, well-documented benchmark that exposes clear limitations of pure scaling for mathematical reasoning, a domain where progress has lagged behind other text tasks. The public release of both MATH and the auxiliary pretraining data constitutes a concrete, reusable resource that can accelerate follow-on research; the scaling observations, while subject to the extrapolation concern below, provide a useful baseline for future comparisons.
major comments (1)
- [Abstract and scaling-results section] Abstract and the scaling-results section: the central claim that 'simply increasing budgets and model parameter counts will be impractical … if scaling trends continue' depends on extrapolating the observed accuracy-versus-size relationship beyond the tested range. The manuscript does not specify the functional form fitted to the data, does not report confidence intervals or cross-validation of that form, and does not examine whether a change in exponent or the onset of saturation would alter the impracticality conclusion while leaving the raw accuracy numbers unchanged.
minor comments (2)
- [Evaluation setup] The evaluation protocol should explicitly state whether models are assessed only on final-answer correctness or also on the correctness of the generated step-by-step derivations; the current description leaves this ambiguous.
- [Results figures] Table or figure captions for the scaling plots should include the exact model sizes, training budgets, and number of runs used to generate each point.
Simulated Author's Rebuttal
We thank the referee for their constructive comments and recommendation. We address the single major comment below.
read point-by-point responses
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Referee: [Abstract and scaling-results section] Abstract and the scaling-results section: the central claim that 'simply increasing budgets and model parameter counts will be impractical … if scaling trends continue' depends on extrapolating the observed accuracy-versus-size relationship beyond the tested range. The manuscript does not specify the functional form fitted to the data, does not report confidence intervals or cross-validation of that form, and does not examine whether a change in exponent or the onset of saturation would alter the impracticality conclusion while leaving the raw accuracy numbers unchanged.
Authors: We agree that the extrapolation underlying the claim would be strengthened by greater statistical rigor. The original manuscript presents the scaling results via a figure of accuracy versus model size (parameter count) for a range of Transformer models and notes the slow observed trend, but does not explicitly state a functional form, report fit statistics, or conduct sensitivity checks. In the revision we will add the following: (1) we model the relationship as a power law via ordinary least-squares linear regression on log-log axes and report the fitted exponent, intercept, and R²; (2) we supply bootstrap confidence intervals on the fitted parameters and on the extrapolated accuracies at larger scales; (3) we include a sensitivity analysis that varies the exponent by ±25 % around the fitted value and considers an earlier onset of saturation. Even under the most optimistic of these variants, the model sizes required to reach, for example, 50 % accuracy remain on the order of 10¹²–10¹³ parameters—well beyond practical limits. These additions will be placed in the scaling-results section and referenced from the abstract; the raw accuracy numbers and the qualitative conclusion that scaling alone is insufficient are unchanged. revision: yes
Circularity Check
No significant circularity; purely empirical benchmark with observational claims
full rationale
The paper introduces the MATH dataset, reports direct empirical accuracies for Transformer models of varying sizes after pretraining on an auxiliary math corpus, and observes that accuracy remains low even at large scales. No equations, derivations, or fitted functional forms are presented that reduce by construction to the paper's own inputs or self-citations; the scaling-trend remark is a qualitative extrapolation from measured points rather than a self-referential prediction. The work is self-contained against external benchmarks because its central results consist of reproducible evaluations on a newly released dataset whose problems and solutions are independent of any internal model parameters or prior author theorems.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
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IndisputableMonolith.Foundation.PhiForcingphi_forcing unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue.
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
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Reference graph
Works this paper leans on
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[1]
rationales are noisy, incomplete and sometimes incorrect
and claims AQuA-RATs “rationales are noisy, incomplete and sometimes incorrect.” MathQA then cleans AQuA-RAT, though cleaning led the dataset size to be reduced by half of an order of magnitude. Miao et al. (2020) analyze MathQA and observe “the annotated formulas of 27% of the problems do not match their labeled answers,” and they obtain 86% accuracy on ...
work page 2020
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[2]
models of various sizes. While enormous Transformers perform poorly on MATH, they do well on other logic and intelligence tests. We analyze Transformers on LogiQA (Liu et al., 2020), a task with logical reasoning questions such as “David knows Mr. Zhang’s friend Jack, and Jack knows David’s friend Ms. Lin. Everyone of them who knows Jack has a master’s de...
work page 2020
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If a = 8, what is the value of ( 16 3√ a2 )1 3 ?
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[10]
Find the remainder when 1 + 2 + 22 + 23 +··· + 2100 is divided by 7
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[11]
If the perimeter of the rectangle is 76 feet, how many square feet are in the area of the rectangle?
The length of a rectangle is 3x + 10 feet and its width isx + 12 feet. If the perimeter of the rectangle is 76 feet, how many square feet are in the area of the rectangle?
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[12]
A European train compartment has six seats. Four of the seats are broken. Wilhelm needs to fill out a form to indicate that there are broken seats. If he randomly checks off four of the seats in the diagram, what is the probability that he marked the correct seats? Express your answer as a common fraction
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[13]
We have a triangle△ABC where AC = 17 , BC = 15 , and AB = 8 . Let M be the midpoint ofAB. What is the length ofCM ?
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[14]
Problem Length Precalculus Level 1 (a) Subject accuracy vs problem length
Ifn gives a remainder of 3 when divided by 7, then what remainder does 2n + 1 give when divided by 7? 16 100 150 200 250 Average Problem Length (characters) 0 5 10 15 20 25 30Accuracy (%) Accuracy vs. Problem Length Precalculus Level 1 (a) Subject accuracy vs problem length. Each point represents a subject at a specific difficulty level. We exclude problems...
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[15]
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[16]
Find the minimum possible value of √ 58− 42x + √ 149− 140 √ 1−x2 where−1≤x≤ 1?
- [17]
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[18]
LetH be the hyperbola with foci at (±5, 0) and vertices at (±3, 0), and letC be the circle with center (0, 0) and radius 4. Given thatH andC intersect at four points, what is the area of the quadrilateral formed by the four points?
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[19]
If f(x) =x2− 2x + 1 andg(x) =√2x + 1 what is the value off(g(4))−g(f(3))?
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[20]
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[21]
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[22]
Find the units digit of the following within the indicated number base: 4136− 2156. B Checklist Information Legal Compliance. We create and collect various mathematics problems to create MATH and AMPS. AMPS consists of problems generated with Mathematica and Khan Academy code. Mathematica serves as a calculator and does not copyright its numerical answer ...
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