Capturing concepts with math

2026-01-12

How do we know that the definition of a continuous curve captures “the concept” of a curve you can draw without lifting your pencil?

How do you know that your definition of surface consists exactly of what you would consider to be a surface?

You can make a definition and check against specific examples and non-examples to see if the definition is consistent with those particular examples. That is about the best you can do.

Sometimes you can show that there are a lot of objects which satisfy the definition you come up with that might not line up with the concept you originally had. Your definition captures a superset of the objects your concept captured. For example, I’d argue that everywhere non-differentiable continuous functions, were they conceivable prior to the definitions of “continuous” and “differentiable”, would probably not count as intuitively continuous. Moreover, there is generally no guarantee that all of the intuitive objects satisfy a given definition meant to capture them. There is no guarantee that your definition even captures all objects that your original conception “captures” — you can only verify finitely objects coincide in your intuitve category and also your formal definition.

We are thus left with two options after making a definition:

• only care about the referent of the mathematical definition, regardless of whether it coincides with the original concept
• believe that the original concept is fully captured by the definition and speak as though there is no distinction.

I think this is a place where a mathematician and a philosopher (possibly including natural philosophers) diverge.

This might be a reason for the increasingly abstract nature of mathematics: if there is no “real” concept that a definition is meant to capture then the contents of this post are of no concern.