Introduction

Mathematical research has been a cornerstone of human progress for centuries, but the style, rigor, topics, and authorship of research papers have evolved significantly over time. This article delves into the differences between mathematical research papers from the 1700-1800s and modern ones, focusing on four key aspects: language style and notation, rigor, topics and subject matter, and authorship collaboration and diversity.

Language Style and Notation

The transition from the 1700-1800s to modern times saw significant changes in how mathematical research was communicated. Initially, scientific papers were primarily written in Latin, a language that was widely used across Europe. However, by the 19th century, there was a gradual shift towards using French and later English. This shift was partly driven by the increasing influence of French and English mathematical communities and the need for a more accessible and standardized language.

A notable example is Leonhard Euler, who used Latin extensively in his works, such as Methodus Generalis Summandi Progressiones. His writings were rich in prose and often lacked the formal rigidity that is characteristic of modern mathematical papers. In contrast, modern research papers are predominantly in English, with authors adhering to a strict formal structure that includes definitions, lemmas, theorems, and proofs.

Rigor in Mathematical Research

The level of rigor in mathematical research has also evolved over time. The 18th century saw less emphasis on formal rigor, with mathematicians like Euler manipulating infinite series with little concern for convergence or divergence. This lack of rigor can be attributed to the fact that mathematicians often developed techniques that worked in practice, even if they weren't theoretically sound. By the mid-19th century, however, the work of mathematicians like Dedekind and Weierstrass began to establish a more rigorous framework for mathematics, which continues to shape the field today.

The modern approach to rigor is characterized by the formal definition-lemma-theorem-proof structure. This structure ensures that mathematical arguments are logically sound and can be verified by other mathematicians. Additionally, the use of modern notation and rigorous mathematical language has made it easier for contemporary researchers to understand and build upon each other's work.

Topics and Subject Matter

Mathematical research in the 18th and 19th centuries was relatively limited in scope. For instance, the first paper on graph theory was published by Euler in 1736, but significant progress in this field did not occur until the late 19th century. Similarly, the development of topology, representation theory, and other areas of modern mathematics were still in their infancy.

Today, mathematical research encompasses a wide range of topics, including graph theory, combinatorics, topology, and more. The field has expanded to include theoretical computer science, homological algebra, and many other domains. This broad scope reflects the growing complexity and interconnectedness of modern mathematical research.

Authorship Collaboration and Diversity

Historically, mathematical research was primarily conducted by individual mathematicians, with few exceptions. In the 18th and 19th centuries, most publications were single-authored and written by European, male, and predominantly white authors. This lack of diversity was due to a lack of societal support for mathematical research in non-European regions and a cultural orthodoxy that excluded women from the field.

Modern mathematical research is a collaborative effort, with many papers being co-authored by multiple mathematicians from diverse backgrounds. This trend towards collaboration began in the late 19th century, with the first notable collaborative papers being published in the 1870s. The rise of collaborative research, particularly in the 20th century, has been a significant factor in the growth and diversity of the field.

While there has been progress in increasing diversity in mathematics, there is still a long way to go. Despite the increased representation of women and mathematicians from different cultural backgrounds, the majority of mathematicians in leading institutions remain male. However, the globalization of mathematical research has led to a broader distribution of scholarly activity, with mathematicians from countries such as Vietnam, Iran, Israel, and Brazil contributing to the field.

In conclusion, the evolution of mathematical research from the 18th and 19th centuries to the present day is marked by significant changes in language style, rigor, topics, and authorship. While the historical context provides valuable insights into the development of modern mathematics, ongoing efforts to promote collaboration and diversity will continue to shape the future of the field.