# Optimal Presentations of Mathematical Objects

## TL;DR

This paper explores how different symbol libraries influence the presentation of mathematical objects, especially natural numbers, revealing properties and establishing bounds on presentation complexity.

## Contribution

It introduces a framework for analyzing optimal presentations of mathematical objects using specific symbol libraries, linking presentation properties to mathematical insights.

## Key findings

- Bounds on presentation length and shape established
- Connections made between presentations and properties of natural numbers
- Insights into how symbol choices affect mathematical object representation

## Abstract

We discuss the optimal presentations of mathematical objects under well defined symbol libraries. We shall examine what light our chosen symbol libraries and syntax shed upon the objects they represent. A major part of this work will focus on discrete sets, particularly the natural numbers, with results that describe the presentation of the natural numbers under specific symbol libraries and what those presentations may reveal about the properties of the natural numbers themselves. We shall present bounds and constraints on the length and shape of presentations, connect already existing problems in other fields of mathematics to questions relevant to these presentations and otherwise illuminate why such a study can produce exciting results.