[sykh]

Why? Suppose I don't know what R[x] or R(x) is. We certainly don't teach highschoolers what either of those are, and they seem to be able to do "algebra" just fine.

We don’t teach high schoolers what is really going on. We mask what is really going on because making it all precise is not effective or helpful at this stage of development. We teach rules to manipulate equations. We don’t use the language of algebraic geometry because that is too complicated. We don’t say to them the variety of the ideal generated by x+2 is the same as the ideal generated 4x+8 are the same so that x=-2 is equivalent to 3x+1=-x-7.

However, we are insisting that the LHS is a polynomial (again, in the formal sense). This means that there is no variable. The "x" in the LHS is literally a value. It makes no sense to ask what values of (0,1,0,0,...) make that equation true. The fact that we give (0,1,0,...) a standard name of x does not suddenly make the question sensical. Nor does the fact x is often used to represent variables.

You are clearly not an algebraist. The polynomial

(3, 1, 0, 0, 0, ....)

Induces a natural map from R to R that is generally called “evaluation”. It maps the number 5 to 8 for instance. We abuse notation and say to begininnng students, “replace x with 5”. We dumb things down. Instead of asking for the pre-image of 8 under this natural map we ask for what values of x do we get a value of 8. The shorthand way of writing this is to say solve the equation:

x+3=8

That you don’t know this is disconcerting since you’ve obscuoulsy had more than an elementary mathematical education. You are confusing the simplistic view of what is taught in basic courses with what is really going on.

Ask a million mathematicians, “is x^2+3x a polynomial” and without hesitation they’ll say yes. Because in standard uasage of that expression it is a polynomial. That’s the default interpretation.

Read the first chapter of any beginning algebraic geometry textbook. x^2+3x-1=0 is an algebraic variety. This is the standard interpretation of that equation.

[gizmo686]

>The polynomial (3, 1, 0, 0, 0, ....) Induces a natural map from R to R that is generally called “evaluation”.

I am well aware of this. If you look at my comments within this very chain, you will see that I made reference to polynomial evaluation, which I will call Ev(f, x). I Am not denying that the partial application given by f' = x -> f(f,x) is naturally induced by the polynomial f. Nor that this is so natural that it often makes sense to identify f with f' so that we would consider f=f', even though they are different types of objects.

>Ask a million mathematicians, “is x^2+3x a polynomial” and without hesitation they’ll say yes.

As will I, because the distinction between formal polynomials and expressions which can be naturally modeled as polynomials is so unimportant that it is almost never worth thinking about.

Put another way, how would you compute the following sets:

{ x \in C | x^3 - x = 8 }

{ x \in C | x^3 = x + 8 }

{ x \in C | log(x) = x^x }

{ x \in C | sin(x) = .7 }

{ x \in N | exists y \in N such that 5x + 3y = 1 }

Are you really claiming that these questions are ill-posed without stating them in terms of algebraic geometry?

[sykh]

I believe you are the only person trained in mathematics in the world that thinks x^2+3x=0 is not a polynomial equation.

You don’t understand the underlying algebraic theory. This is evidenced by your claim

The "x" in the LHS is literally a value. It makes no sense to ask what values of (0,1,0,0,...) make that equation true.

You can’t write such a statement if you really understand that “what values make x+3 the number 5” is a nicer way of conveying the question “under the natural evil map induced by x+3 what is the pre-image of 5”. The text of your I quoted is wrong.

[gizmo686]

> I believe you are the only person trained in mathematics in the world that thinks x^2+3x=0 is not a polynomial equation.

Then you are not reading my comments.

> To be clear, outside of very particular contexts I would still call x^2-x+1=0 a polynomial equation, because it is extremely useful to talk about polynomials without invoking all of the machinery of formal polynomials.

>You don’t understand the underlying algebraic theory.

My claim is that the underlying algebraic theory is not necessary to rigorously state or solve the problems under discussion.

>You can’t write such a statement if you really understand that “what values make x+3 the number 5” is a nicer way of conveying the question “under the natural evil map induced by x+3 what is the pre-image of 5”.

My claim is that "what values make x+3 the number 5" translates directly into { x \in F | x+3=5}. I further claim that the "x+3" in this interpenetration is not a polynomial in the formal sense

I am not disputing that this is set is the same set as the pre-image of 5 under the natural map induced by x+3.