Understanding the Distinction Between Mathematicians and Mathematical Foundationalists
Mathematics as a discipline is vast and multifaceted. At its core, there are researchers who spend their time unraveling new theorems, developing novel mathematical concepts, and solving complex problems. However, there is an often-overlooked distinction between these so-called mathematicians and foundationalists who work on the very foundations of mathematics.
Foundationalists in Mathematics
To add to Jack Huizenga's answer, there is a clear distinction made between people who work on the foundations of mathematics—such as category theorists and logicians—and other mathematicians who do not need or desire to delve into these foundational areas. These foundationalists focus on subjects like category theory, logic, and the foundational aspects of probability theory and decision theory.
For instance, in the field of statistics, the foundational knowledge consists of measure-theoretic probability, decision theory, and the logic of probabilistic inference. While more than 95% of statisticians do not need to learn this material in depth, they must have a basic understanding to ensure the assumptions underlying a given method are valid in real-world scenarios.
Mathematicians in Research
A working mathematician is a mathematician who is engaged in research, working on unsolved problems in mathematics aimed at proving new theorems. They typically focus on the application and development of existing mathematical theories, contributing to the advancement of various branches of mathematics.
These working mathematicians might contribute to fields like algebra, geometry, number theory, and more, each with its own set of unsolved challenges. Unlike foundationalists, these mathematicians do not necessarily need to delve into the intricate details of set theory or other foundational aspects of mathematics, as their work often revolves around practical problem-solving and innovation.
The Opposite of a Mathematician
While the term working mathematician is commonly used, there is a parallel term that often refers to individuals who may not fully engage in the rigorous research required of a mathematician. This could include applied mathematicians, data scientists, or statisticians who focus more on the application of mathematical tools and theories to real-world problems rather than on the pure exploration and proof of mathematical concepts.
In contrast to working mathematicians, the opposite of a mathematician would be someone who may lack a deep understanding of mathematical theory and its foundational aspects. These individuals might be data analysts or practitioners in fields like engineering or economics who apply mathematical models and algorithms without necessarily understanding the theoretical underpinnings.
The Importance of Foundational Knowledge
Although not all mathematicians need to be foundationalists, there is a consensus that a working mathematician should have enough foundational knowledge to evaluate the validity of their work and ensure that their methodologies are sound. This foundational knowledge is crucial for assessing whether the assumptions of a given method are met in real-world scenarios, thus ensuring the reliability and credibility of their mathematical contributions.
Foundationalists, on the other hand, contribute to a deeper understanding of the axioms and principles that underpin mathematics. Their work is essential for ensuring that the theoretical frameworks in mathematics are robust and coherent, even if such knowledge is not always needed by the majority of mathematicians.
The Book Parallel
The term working mathematician is often seen in the context of a book by Michael Atiyah and his collaborator, publishing a work that delves into the lives and contributions of this class of scholars. The book not only celebrates the achievements of mathematicians but also highlights the critical importance of their work in shaping the landscape of mathematics.
In summary, while the term mathematician can encompass a wide range of specialties and applications within the field, the distinction between mathematicians and foundationalists is important. Understanding the roles of these different classes of scholars can provide insight into the diverse and interconnected nature of mathematics as a discipline.