  In this paper, we discuss how pure mathematics and theoretical physics can be
applied to the study of language models. Using set theory and analysis, we
formulate mathematically rigorous definitions of language models, and introduce
the concept of the moduli space of distributions for a language model. We
formulate a generalized distributional hypothesis using functional analysis and
topology. We define the entropy function associated with a language model and
show how it allows us to understand many interesting phenomena in languages. We
argue that the zero points of the entropy function and the points where the
entropy is close to 0 are the key obstacles for an LLM to approximate an
intelligent language model, which explains why good LLMs need billions of
parameters. Using the entropy function, we formulate a conjecture about AGI.
  Then, we show how thermodynamics gives us an immediate interpretation to
language models. In particular we will define the concepts of partition
function, internal energy and free energy for a language model, which offer
insights into how language models work. Based on these results, we introduce a
general concept of the geometrization of language models and define what is
called the Boltzmann manifold. While the current LLMs are the special cases of
the Boltzmann manifold.
