Why are undefined references and variables not specifically differentiated?

Question

In my opinion, this topic is more philosophical than mathematical, but if it is not, I will ask it on another forum.

My understanding

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:

1. 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.
2. 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 difficulties

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.

Questions

1. Do you agree with my division of the use of symbols in mathematics?
2. If you agree, then why are they not specifically distinguished? Why can they be perceived in both ways?

Answer 1

I just don't see your problem. Mathematicians do make these distinctions.

They begin work by describing f(x), eg whether x is being considered over real number values, complex plain etc. They say 'let n be -' whatever role it will have. But then they may wish to iterate what they are doing, for instance beginning only with variations in x, but then also varying something previously being treated as a fixed variable. In particular, you might begin with a multivariate equation, and use tactics like partial differentiation substitution and special conditions, to arrive at something soluble in one unknown. It's a whole process.

In grade school math, it's often assumed the domain under consideration is real numbers. Once other domains are regularly considered, and especially if not specifying could lead to irrelevant or 'unphysical' results, we need to specify. Sometimes mathematicians assume there are contextual cues, like that the number domain is obvious, or, irrelevant (perhaps not yet confirmed say.

Perhaps you could give examples of where you have been confused, or think this issue could cause problems. But I think in practice types of variable need to be specified, case by case and as a work develops it's argument.

Answer 2

Much of what you are distinguishing boils down to the use of the appropriate quantifier. In your sentence 1, the variable k is existentially quantified, while n could be existentially or universally quantified depending on what exactly you want it to say. It could be glossed as

(for any even number n)(there exists a number k) such that: n = 2k. 

The existential quantifier here tells us that for any given value of n, there is some value of k for which the equation holds.

In your sentence 2, the v and the t are implicitly universally quantified. For a car accelerating uniformly with an acceleration of a, the equation can be glossed as:

(for any velocity v)(for any acceleration a)(at any time t) the following holds: v = a * t.

The universal quantifiers here tell us that this equation holds for any value of the variables v, a, t.

The reason it appears confusing is because mathematicians often don't put quantifiers explicitly into their formulas and leave the variables free. Hence it may not be immediately obvious what quantifier is intended. If a mathematician writes a formula such as x² + 4x - 8 = 0, this is implicitly existentially quantified, i.e. it could be written:

(∃x)(x² + 4x - 8 = 0)

The existential quantifier tells us that this equation holds for a particular value of x. (Actually, two values.)

But if a mathematician writes an identity relation such as sin²x + cos²x = 1, this is implicitly universally quantified, and could be written:

(∀x)(sin²x + cos²x = 1)

In both cases, the variable is doing the same job. It can be thought of as ranging over a class of possible values.