In my opinion, this topic is more philosophical than mathematical, but if it is not, I will ask it on another forum.
I'm talking about non-reserved symbols here. Not about 0, 1 or π.
As I understand it, in math we can use words or symbols to refer to a specific object (even if the actual referent of a symbol is unknown to us) or we can use them to formulate rules or laws. In the first case a symbol is called an unknown, and in the second case a symbol is called a variable. Examples:
- Let n be an even number, then there exists a number k such that n=2k. In this example, the symbols n and k are references to specific numbers, but I don't know which ones.
- Let some car move and its velocity increases linearly. I can write v = 5*t. This is a rule and I can use it to calculate the velocity at any given time.
In equations, we can use symbols in both senses (I wrote above n = 2k and v = 5t).
My problem is that in equations the meaning of the symbols can be interpreted in both ways, and this confuses me. In any given equation, if you don't have context, it's not clear exactly how to understand the symbols in that equation. You can perceive an equation as a statement on specific unknown numbers and as a predicate(Like in this answer). And for some reason it doesn't cause problems. This article says that unknowns and variables are generally two different names for the same thing.
Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation.
I understand that our actions with equations do not depend on the meaning of the symbols, but the result of solving the equations changes.
Let me explain with an example: Let's say I have an expression n ∈ [0, 1]. If n is a reference to a specific number, then I cannot say which one. I can only indicate the segment in which it is located. But if n is a variable, then the condition is simply written here. In the first case, we are trying to define a specific object, and in the second, we are studying the conditions. The meaning of x = 2 is different depending on the meaning of x. This is either an indication that the referent x is 2 or a condition with the only suitable object 2.
- Do you agree with my division of the use of symbols in mathematics?
- If you agree, then why are they not specifically distinguished? Why can they be perceived in both ways?