For mathematicians, = does not mean equality

Polynomials in multiple variables always have the variables ordered. Sometimes the ordering is not important or explicitly stated but in reality they are supposed to be ordered.

If we start with x y + 2 = 0 we can apply the function

f(x) = x - 2

to both sides of the equation. This gives us the equation

x y = -2

This is a different equation than the one we started with but these two equations have the same solution set. Can go a step further and transform this equation to

y = -2/x

This equation has the same solution set as the first equation. All three equations are equivalent. Solutions are ordered pairs of numbers.

Commonly in basic courses like calculus we tell students that the last form is preferable and we write solutions as (x, -2/x). We call this set the graph of the equation but really it’s the solution set of the equation.

In algebraic geometry x y + 2 = 0 is preferable. The solution set is called an algebraic variety. The solutions are ordered pairs in affine space.

The rules of algebra, as taught in low level courses, are rules that allow one to transform a given equation into simpler equation. In one variable the goal is to end up with something like x = 3 because such an equation is easy to solve. The solution is the 1-tuple 3.

Not all sets are recursively enumerate so using enumerate as you did can cause problems.

I know what a polynomial ring is. I am not questioning that the string "x+3" can be interpreted as an element of Q[x].

What I am questioning is the necessity to interpret "x+3" as an element of Q[x].

>The expression x+3 meets the definition of a polynomial.

Only if you take x=(0,1,0,...) and adopt the convention that any member q of the base field Q is assumed to represent qx^0 = (q,0,0,...).

That is to say, x+3 only meets the definition of a polynomial because you insist on interpenetrating it as such.

However, we can also handle "x+3=-1" without ever defining the notion of a polynomial.

Eg, we can say, suppose x \in Q such that "x+3=-1". From this premise, we can derive presisly what specific element of Q x must be.

In a more general setting, we might only be able to derive a set of potential values that x could have, or derive that x cannot possibly exist.

As I mentioned in my prior comment, I see no reason to intererperet the "x+3" in "x+3=-1" as a polynomial. If we were to do so, the question would be asking: find t \in Q such that (x+3)[t]=(-1)[t]. Where (g)[t] is polynomial evaluation.

Applying the definition of polynomial evaluation, we would get that the above equation implies: t+3=-1.

Are you now going to insist that "t+3" is a polynomial. Bearing in mind that we have defined t to be an element of Q, which was necessary to apply it as the second argument of polynomial evaluation; and we only got "t+3" as the output of polynomial evaluation, which is defined to result in an element of the base field.

We could modify are notion of polynomial evaltuation to instead be of the form R[x] X R[x] -> R[x], which also gives us (for free) the ability to apply polynomials to other polynomials. But if we were to do this, then when we say that the solution to "x+3=-1", is -4, we are taking "-4" itself to be a polynomial.

In practice this is fine (we identify the base field with the subring of degree 0 polynomials all the time). However, this entire approach breaks down when you start working with functions that do not fit within the framework of polynomial rings.

For instance, suppose I said that "(x+3)! = 120". Are you still going to insist that "x+3" is a polynomial?

What if I define a function id: Q -> Q. In the equation "id(x+3) = 2, are you still going to insist that "x+3" is a polynomial?

Geometry distinguishes between equivalence and value. An "angle" isn't its degrees, but the geometric figure (two rays or segments meeting at an end-point of each). It's the measure of the angle that is the degrees.

You don't say "angles are equal" - you say they are congruent. It's their measures that are "equal".

Although congruency implies measure equality, it doesn't really mean that, but that the shapes are the same (can be rotated/translated to coincide).

>The second sentence I quoted is not true.

I assume you are refering to the sentence:

> If we wanted to consider x+3 a polynomial, we would be asking for the t value such that (x+3)[t] = (-1)[t]

Bearing in mind that the example I have in mind is the equation "x+3=-1" with the solution of "2", in what sense in the above sentence not true?

>When presented with the equation x+3=-1 x+3 is a polynomial. -1 is a polynomial.

Fair enough. In that case, I assume you would consider the equation "x+3=-1" to be false, as it is clear that (3,1,0,0,...) != (-1,0,0,...). Unless of course you are asking, as I had suggested, that you are looking for the particular element at which polynomial evalutation yields an equal result on both sides. If this is the case then, as far as I can tell, you are introducing the machinery of formal polynomials for the sole purposes of overloading the "=" symbol in a confusing way.

>I gather you do not think x^2 - x + 1 = 0 is a polynomial equation.

Define "polynomial equation" If you are asking if I would consider that an equation taking place in R[x], then (absent some other context) the answer is no. Even with other context I would say that "x^2 - x + 1 = 0" is false as a polynomial equation. You might be able to get me to call equations done in the quotient ring R[x]/ polynomial equations, in which case "x^2 - x + 1 = 0" would be both a polynomial equation and true at the same time.

If you are asking if I would call "x^2 - x + 1 = 0" in an informal setting, then the answer is yes. However, I do not see how this is relevant, as the whole point of this comment chain was the formal notion of polynomials.

>x^3+4x a polynomial?

Informally, yes. Formally, it depends on context. However, absent some context, I would not consider "x^3 +4x" to be a formal polynomial.

>Is there any other reasonable interpretation using accepted mathematical conventions?

Yes, x^3 +4x is the member of the base ring corresponding to "(x * x * x) + (4 * x)", where x is some other member of the ring.

>Perhaps you don’t think 2/(x+3) is a member of R(x). What is it a member of then?

I am glad you asked. I believe my above answer regarding x^3+4x still applies. However, let me ask you: Is x^3+4x a member of R(x)?

>What is it a member of then?

Again, depending on context. Without context, I would consider 2/(x+3) to be a member of R.

Aren't we all taught to write "y = 3" as the answer to algebra questions? That's how I always thought of it, not as assignment, but as a declaration of truth.