The Limits of Simulated Reasoning: AI Struggles with Advanced Mathematical Proofs | Large language models course free | Llm machine learning tutorial geeksforgeeks | Llm machine learning | Turtles AI

A recent study highlights the limitations of AI models based on simulated reasoning (SR), which, while capable of solving routine mathematical problems, are still not up to the level required to tackle complex proofs, such as those in the Mathematics Olympiad.

Key Points:

• SR models excel at simple mathematical problems, but fail at complex proofs.
• Performance on United States Mathematics Olympiad (USAMO) problems is very low, with average scores below 5%.
• The difference between solving a problem and providing a proof highlights the shortcomings in AI’s deep understanding of mathematics.
• Current AI models are not capable of abstract reasoning like humans, but future improvements could fill the gap.

A new study by a team of researchers from ETH Zurich and INSAIT Sofia University has highlighted a fundamental contradiction in AI models that use “simulated reasoning” (SR). While these models have made significant strides in many areas, including basic mathematics, they still fall short of the most complex challenges, such as those encountered in high-level mathematical competitions. The report focuses in particular on their inability to provide complete and correct mathematical proofs, such as those required in the United States Mathematics Olympiad (USAMO), one of the most prestigious events in the world for young mathematicians.

When researchers presented the SR models with problems from the USAMO 2025 program, the results were less than satisfactory. Although one of the models, Google’s Gemini 2.5 Pro, scored relatively well (10.1 out of 42 points, or about 24 percent success), most of the other models fell well short of an acceptable result. Some models, like DeepSeek R1 and Grok 3, scored just 2 points, while others, like Qwen’s QwQ and OpenAI’s o1-pro, only managed to average 1.2 points out of 42, with scores never exceeding 5%. This performance gap is significant, as the Math Olympiad requires rigorous proofs of complex theorems, not simple equation solutions.

The distinction between solving a mathematical problem and providing a complete proof is crucial. Solving a problem involves finding a correct answer, as in the case of an equation or a sum. But a proof requires justifying each step logically, constructing a line of reasoning that shows how you arrive at the conclusion unambiguously. While SR models can produce correct answers for routine mathematical problems, such as calculating the sum of two numbers, they lack the ability to articulate and justify the steps required to prove a theorem. This type of deep reasoning, which requires constructing abstract logical arguments, is beyond the capabilities of current SR models, which rely primarily on pattern recognition.

The researchers’ analysis revealed several types of common errors in models’ attempts to prove something. In many cases, the algorithms’ answers contained significant logical gaps: unproven claims, errors in intermediate steps, or the inclusion of contradictory arguments. For example, in a problem that required finding specific integers for which a mathematical formula would always produce an integer result, Qwen’s QwQ model incorrectly excluded some possibilities that were actually allowed by the problem’s formulation, leading to an incorrect conclusion. Even more problematic is that these models often provide incorrect answers in language that seems certain, without showing awareness of the errors in the reasoning process, making them less reliable.

The researchers suggest that one reason for these errors may be the way the models are trained. SR models are designed to “think” in a simulated way, using chains of reasoning that mimic, in some sense, the human process. However, these models lack a true conceptual understanding of the mathematical questions they address. When faced with new situations that require reasoning that cannot be directly replicated from the training data, the models lack the ability to adapt and correct their course. The lack of abstract reasoning capacity is a structural limitation that currently hinders the effectiveness of these models in complex contexts.

Another challenge is the so-called "pattern matching", a technique that underpins AI models such as those based on Transformer. SR models rely primarily on identifying patterns in data and adapting these patterns to new problems. While this approach is effective for tasks where the answers are relatively simple and well-defined, it falls short when more sophisticated logical reasoning is required. In other words, while SR models can solve elementary mathematical problems with precision, they are not yet able to do the same on more advanced problems that require the construction of complex logical arguments.

Despite these limitations, research suggests that SR models could improve over time. One approach that could prove useful in the future is the integration of symbolic reasoning techniques, which combine the capabilities of neural networks with more formal logical methods. Some experiments in this field, such as DeepMind’s AlphaGeometry project, are trying to develop neurosymbolic models, which could, in theory, improve AI’s ability to perform complex reasoning more reliably by avoiding the confabulation errors seen in current models. However, these advances are still in their early stages, and it remains to be seen whether they will be sufficient to bridge the gap between AI and human mathematical reasoning.

Overall, the gap between current AI models and advanced mathematical reasoning is still considerable, and it is not certain that expanding the capabilities of these models alone will overcome the challenges that require deeply abstract thinking. But it is possible that, with further developments in training technologies and architectural design, these models could bridge the gap in the future.

For now, progress continues to be slow but steady as the scientific community explores new ways of developing AI applied to mathematics.