I'm trying to understand the nature of mathematical objects.
As far as I understand it, mathematics studies these objects. Geometric shapes are one kind of such object, including 1D shapes, namely lines. I have two questions about them.
Firstly, is this the case that:
- The length of a line (whether evaluated or not to some number, e.g., 3) is an object in its own right, and is thus somehow "associated with" the line (as in, lengths cannot exist by themselves, they have to be lengths of something), or
- The length of a line is a property of said line, and thus is not a mathematical object, or
- Both are true, because properties are also mathematical objects in their own right?
If (3) is correct, is there some kind of relationship between properties (as objects) and the objects of which they are properties, which is not found in objects which are not properties of other objects? For example, some kind of function or relation, which basically acts as a "this object X is a property of object Y" designator.
Secondly, imagine I am trying to evaluate some length. Let's say I am a secondary school student and I am trying to evaluate the length of the hypothenuse of a right triangle using the Pythagorean theorem, and I am given sides a = 3cm, b = 4cm. Does the word "evaluate" mean "to discover the mathematical object (here, a number) represented by the symbol c in the Pythagorean theorem?"
The reason I'm asking this is because I am trying to understand the nature of mathematical notation, and what it represents. At school, I was really good at maths, but I never realised that there exists a separation between what mathematics (and its notation) is about, and the notation itself. To me, the subject was always just about following procedures to try and get some answer, and it was never as conceptual as I have come to understand it over the past few months. I've found it to be much more aesthetically beautiful since!