If LLMs Can Speak Mathematics, Where Does the Expert Stand?

One useful way to understand mathematical modeling is recognizing that mathematics is, in essence, a (highly compressed) language.

Ordinary language already compresses meaning. Instead of saying, “the wooden board supported by four long wooden cylinders,” we say “table.” Even words such as board and cylinder are compressed representations of more detailed concepts. We then create more specific categories: desk, dining table, conference table, and so on. A dictionary partially reverses this compression by explaining what each word represents.

Mathematics works in a similar way, but its symbols and definitions are more explicit.

Suppose we define (x_{ij}) as the amount of goods shipped from supplier (i) to customer (j). Instead of repeatedly writing “the total amount of goods shipped from all suppliers to all customers,” we can write:

[
\sum_i \sum_j x_{ij}
]

This notation is compact, but it is meaningful only because we have defined the symbols, indices, units, and boundaries of the problem. We might also require that the total shipped by supplier \(i\) not exceed its capacity: \(\sum_j x_{ij} \leq C_i\). Writing this constraint is easy. Determining what “capacity” actually means (and whether it reflects production, inventory, transportation, contracts, or time) is where domain expertise becomes essential.

Beyond Words and Notations

A mathematical model is therefore more than a collection of equations (and inequalities). It is a compressed representation of how we believe part of the world works. It captures entities, relationships, constraints, assumptions, and objectives in a precise language.

Ordinary language also contains implicit models of reality. If someone says, “I put the desk in my pocket,” we immediately notice a conflict with our understanding of size and physical possibility. We may assume the person is joking, speaking metaphorically, or using the wrong words. In that sense, people carry informal models of the world and use them to judge whether a sentence makes sense.

Mathematical modeling requires the same discipline, but at a more explicit level. A modeler must be able to:

1. Understand how the real system operates.
2. Decide which details matter and which can be omitted.
3. Represent the important elements at appropriate levels of abstraction.
4. Move between those levels without losing meaning.
5. Check whether the resulting model remains consistent with reality.

A strong mathematical modeler resembles an eloquent and perceptive speaker. Eloquence comes not merely from knowing many words, but from understanding the world those words describe (through work, travel, business, and interaction with different kinds of people). Similarly, modeling skill comes not merely from mastering mathematical techniques, but from exposure to (and understanding) real systems and the people who operate them. That experience helps a modeler recognize what matters, what can be abstracted, and which assumptions are unrealistic.

The potential role of large language models

LLMs are useful for interpreting descriptions, reorganizing information, and translating ideas from one form into another. If you understand a business process and can describe it precisely, an LLM may help identify decision variables, objectives, constraints, and assumptions. It can help translate a verbal description into the initial structure of a mathematical model.

But producing a plausible formulation is not the same as producing a valid one. The model still needs to be reviewed by someone who understands the domain, tested against real cases, and checked for missing constraints and incorrect assumptions.

AI can help with the translation. It cannot replace the understanding that makes the translation meaningful.

—Javad Seif
Pasadena, California
October 13, 2026