section_path: Applications/Mathematics
section_name: Mathematics
paragraph_text: Mathematics
  
  In mathematics, special forms of formal step-by-step reasoning are used. In contrast, LLMs such as GPT-4 Turbo, Gemini Ultra, Claude Opus, LLaMa-2 or Mistral Large are working with probabilistic models, which can produce wrong answers in the form of hallucinations. Therefore, they need not only a large database of mathematical problems to learn from but also methods such as supervised fine-tuning or trained classifiers with human-annotated data to improve answers for new problems and learn from corrections. A 2024 study showed that the performance of some language models for reasoning capabilities in solving math problems not included in their training data was low, even for problems with only minor deviations from trained data.
  
  Alternatively, dedicated models for mathematic problem solving with higher precision for the outcome including proof of theorems have been developed such as Alpha Tensor, Alpha Geometry and Alpha Proof all from Google DeepMind, Llemma from eleuther or Julius.
  
  When natural language is used to describe mathematical problems, converters transform such prompts into a formal language such as Lean to define mathematic tasks.
  
  Some models have been developed to solve challenging problems and reach good results in benchmark tests, others to serve as educational tools in mathematics.