Patrick Reany
15 December 2023
On 9 December 2023, I had this short chat with BingChat:
Patrick:
Serge Lang said something like 'The notation must be functorial with respect to the ideas.'
What does this mean? Does it mean that the structure inherent in the ideas must be mapped
faithfully to the structure invented in the notation or terminology?
BingChat:
Serge Lang, a prominent mathematician, emphasized that notation should be functorial with
respect to ideas. Let’s unpack this concept:
1. Functoriality: In mathematics, a functor is a mapping between categories that preserves
the structure. It takes objects and morphisms from one category to another in a consistent
way. Functors respect the relationships and properties inherent in the mathematical structures
they operate on.
2. Ideas: When mathematicians work with mathematical objects (such as groups, rings,
or spaces), they often have underlying ideas or concepts in mind. These ideas represent
the essence of what they are studying.
3. Notation: Notation refers to the symbols, expressions, and terminology used to represent
mathematical concepts. It includes variables, operators, function names, and other symbols.
4. Functorial with Respect to Ideas: Lang’s statement suggests that the notation we choose
should align with the underlying ideas or concepts. In other words:
The notation should faithfully capture the essential properties and relationships of the
mathematical objects.
When we define symbols or introduce terminology, they should reflect the inherent structure
and behavior of the mathematical entities.
For example:
• If we define a new operation on a group, the notation for that operation should
respect the group’s properties (such as associativity or inverses).
• If we introduce a term in algebraic geometry, it should capture the geometric or
algebraic essence of the concept.
In summary, functorial notation ensures that the symbols and terminology we use accurately
convey the mathematical content and align with the underlying ideas. It’s about maintaining
consistency and clarity in mathematical communication. Lang’s emphasis on this principle
reflects his commitment to precision and rigor in mathematical exposition [1,2].
1. arxiv.org
2. mattlord.net
3. en.wikipedia.org
4. math.stackexchange.com