When can variables simply be variables?

Question

This may seem a somewhat strange question, but I've been tying myself in knots about it recently. When constructing a polynomial ring, you must formally define a polynomial as an ordered ω-tuple, with the first entry being the constant value, the second being the coefficient of x, etc, and I think I understand why we need to do this...but I'm not entirely sure.

When do we need to define polynomials in this way rather than just as terms? If I try to solve a system of linear equations with a column matrix $(x,y)$ (I know this isn't a column matrix but I'm not quite sure how to make matrices in LaTeX) do I need to invoke a similar construction? If I simply am solving a single equation such as $x^2+x+3=0$ would I need to work in a polynomial ring to correctly (in a strictly formal setting) do so?

If I work with an abstract group, would I need some special construction or am I allowed to manipulate variables just as if they were"usual" objects?

I know this question may be somewhat pointless - we don't need to define these things in such a way to use them - but I'm trying to figure out how some concepts translate from ZFC to the main body of mathematics...and I'm just not sure how to deal with variables like this. When are variables....simply variables? If I have a rather large gap or error in my understanding that is causing this confusion I'd happily satisfy myself with plugging that gap, too. If that be the case I'm sorry for wasting your time, in a way.

Answer 1

I agree with the previous answer that the question is not very clear; thus, I add this "long comment".

From a "mathematical logic" perspective, a variable is a syntactical object.

It is used mainly as a "place holder; consider the expression : "$x$ is a Philosopher" (that we can "formalize" in first-order language as $Phil(x)$).

The $x$ is used as a pronoun, i.e. the above expression is like : "he is a Philosopher". In order to "understand" it, we have to assign to the pronoun a denotation (in "real life" this can happen simply in the "context" of the speech act).

In the same way, in order for the expression $Phil(x)$ to have meaning, we have to assign to the variable $x$ a reference.

If we consider simple examples like the above, we can dispense with variables; we can simply write : "$---$ is a Philosopher".

The real benefit of using variables is in expression with multiple "place holders", i.e. when we have to treat cases like "$x$ is father of $y$" (i.e. $Father(x,y)$).

But we have also another usage of variables : the quantifiers $\forall$ and $\exists$. Again, we have to specify the bound variable with the quantifier symbol (e.g. $\forall x$) in order to manage expressions with multiple quantifiers.

The usage of variable in algebra has a very ancient tradition, but is "ambiguous".

We have two type of expressions :

• identities, like : $(x-1)(x+1)=x^2 -1$

• equations, like : $x^2-4=0$.

The first one is an assertion that must be understood as universally quantified, i.e. as : $\forall x[(x-1)(x+1)=x^2 -1]$ and it is true in e.g. $\mathbb R$.

The second one is not "always true" in e.g. $\mathbb R$; when we assert it in a "normal" algebraic context we are not asserting the false sentence : $\forall x (x^2-4=0)$.

In this case, we are "asking for" the truth value of : $\exists x (x^2-4=0)$, i.e. we are "searching for" the solutions (if any) of the equation.

Answer 2

I don't quite understand your question, but note that these are really very different ways of talking. Its not a subtle difference. Consider the equation $$(*) \quad x^2-1=0.$$ If $x$ is a variable of type $\mathbb{R}$, then $(*)$ is equivalent to the condition $x = 1 \vee x = -1$. If $x$ is a formal variable, then $(*)$ is simply false (in the polynomial ring $\mathbb{R}[x]$). So if you're trying to solve equations, the former approach (where $x$ is a variable of type $\mathbb{R}$) is probably the way to go.

Actually, it is possible to speak of the "solutions" to $(*)$ from the formal polynomial rings perspective. By definition, the solutions to $(*)$ are the elements of the set $$\{a \in \mathbb{R} \mid [a/x](x^2-1) = [a/x](0)\},$$ where $[a/x]$ is notation for the unique $\mathbb{R}$-algebra homomorphism $\mathbb{R}[x] \rightarrow \mathbb{R}$ satisfying $[a/x](x) = a$. But there's an abuse of notation going on here; in the formal polynomial ring perspective, the "equation" $(*)$ is really more like an ordered pair $(x^2-1,0)$, or perhaps an ideal $(x^2-1)$. Its not being treated as a truthvalue; as I said earlier, as a truthvalue, it is simply false.