This paper investigates the possibility of approximating multiple mathematical operations in latent space for expression derivation. To this end, we introduce different multi-operational representation paradigms, modelling mathematical operations as explicit geometric transformations. By leveraging a symbolic engine, we construct a large-scale dataset comprising 1.7M derivation steps stemming from 61K premises and 6 operators, analysing the properties of each paradigm when instantiated with state-of-the-art neural encoders.Specifically, we investigate how different encoding mechanisms can approximate expression manipulation in latent space, exploring the trade-off between learning different operators and specialising within single operations, as well as the ability to support multi-step derivations and out-of-distribution generalisation. Our empirical analysis reveals that the multi-operational paradigm is crucial for disentangling different operators, while discriminating the conclusions for a single operation is achievable in the original expression encoder. Moreover, we show that architectural choices can heavily affect the training dynamics, structural organisation, and generalisation of the latent space, resulting in significant variations across paradigms and classes of encoders.

This paper proposes a methodology for generating and perturbing detailed derivations of equations at scale, aided by a symbolic engine, to evaluate the generalisability of Transformers to out-of-distribution mathematical reasoning problems. Instantiating the framework in the context of sequence classification tasks, we compare the capabilities of GPT-4, GPT-3.5, and a canon of fine-tuned BERT models, exploring the relationship between specific operators and generalisation failure via the perturbation of reasoning aspects such as symmetry and variable surface forms. Surprisingly, our empirical evaluation reveals that the average in-distribution performance of fine-tuned models surpasses GPT-3.5, and rivals GPT-4. However, perturbations to input reasoning can reduce their performance by up to 80 F1 points. Overall, the results suggest that the in-distribution performance of smaller open-source models may potentially rival GPT by incorporating appropriately structured derivation dependencies during training, and highlight a shared weakness between BERT and GPT involving a relative inability to decode indirect references to mathematical entities. We release the full codebase, constructed datasets, and fine-tuned models to encourage future progress in the field.

Automating discovery in mathematics and science will require sophisticated methods of information extraction and abstract reasoning, including models that can convincingly process relationships between mathematical elements and natural language, to produce problem solutions of real-world value. We analyze mathematical language processing methods across five strategic sub-areas (identifier-definition extraction, formula retrieval, natural language premise selection, math word problem solving, and informal theorem proving) from recent years, highlighting prevailing methodologies, existing limitations, overarching trends, and promising avenues for future research.

In order for language models to aid physics research, they must first encode representations of mathematical and natural language discourse which lead to coherent explanations, with correct ordering and relevance of statements. We present a collection of datasets developed to evaluate the performance of language models in this regard, which measure capabilities with respect to sentence ordering, position, section prediction, and discourse coherence. Analysis of the data reveals the classes of arguments and sub-disciplines which are most common in physics discourse, as well as the sentence-level frequency of equations and expressions. We present baselines that demonstrate how contemporary language models are challenged by coherence related tasks in physics, even when trained on mathematical natural language objectives.