What are the uses of cross-theoretic identifications within mathematics?

Question

I've been thinking about the identification of objects from different mathematical theories. For example, when you do set theoretic constructions of the natural numbers and identify, e.g., 0 with the empty set. Or, when you use Cartesian coordinate systems and identify points in n-dimensional Euclidean spaces with n-tuples of real numbers.

What is the purpose of these identifications? What advantage do mathematicians gain from studying geometry from an analytic perspective, or studying number theory from a set theoretic perspective? Do all of these cross-theoretic identifications serve a single purpose or are their different reasons to adopt them (e.g., maybe set-theoretic identifications are more important because of some foundationalist assumption that "set theory is the ultimate court of appeals in mathematics")?

Answer 1

The question runs together two different issues, it seems:

1. What is the use of identifying objects from different mathematical theories -- e.g identifying numbers with sets or identifying points in n-dimensional Euclidean spaces with n-tuples of real numbers?
2. What advantage do mathematicians gain from studying geometry from an analytic perspective, or studying number theory from a set theoretic perspective?

Now, there are (for example) plenty of examples of proofs in geometry that are easier when you go via a coordinate system. But note that these do not depend on identifying points in a Euclidean space with a tuple of reals. It is enough if we can set up isomorphisms between structures in Euclidean three-space, for example, and structures in $\mathbb{R}$. Likewise in other cases.

The required isomorphisms won't be unique. In the case of choosing a coordinate scheme, we need to choose origin, orientation, and the size of a unit along each coordinate axis. Of course, for a particular problem, some choices will be a more covenient than others! But still, the association of points with tuples will involve more-or-less arbitrary choices. No coordinate scheme can be said to reveal which tuple of numbers a given point "really" is. So, often, talk of identifying objects from the different mathematical domains strictly speaking overshoots: it is less misleading to say the tuples "represent" or "model" the space.

Answer 2

Any rotation in $\mathbb{R}^3$ is a rotation around an axis: i.e. it leaves a line through zero fixed*. But rotations in $\mathbb{R}^2$ don't have this property: they can rotate the whole space. Why should this be true?

Rotations in $\mathbb{R}^n$ can be described by certain $n \times n$ matrices. The idea of "leaving a line through zero fixed" is equivalent to saying that for our matrix $\mathbf{A}$ there exists a vector $v$ such that $\mathbf{A}v = \lambda v$ for some scalar $\lambda$. (The fixed line would be the line of all scalar multiples of $v$).

This property that $v$ has is called being an eigenvector of $\mathbf{A}$, for which $\lambda$ is the associated eigenvalue. Every eigenvector has an eigenvalue, and vice versa. (So we're going to find $v$s by finding their $\lambda$s).

Eigenvalues are solutions of a certain polynomial associated with $\mathbf{A}$ -- that is, the polynomial in the variable $x$ given by $\det(xI - \mathbf{A})$. Right now we only care about one specific property of that polynomial: if the matrix is $n \times n$, it's got degree $n$.

Okay, so: let $n = 3$. If the polynomial has any solutions (in the reals) then we've got our eigenvalue, and, hence our axis. And we're done, since our polynomial is a cubic, and every cubic has (at least) a real solution.

Why? Try drawing a cubic that doesn't cross the $x$-axis.

On the other hand it's perfectly possible to draw a quadratic that doesn't cross the $x$-axis (such as $x^2 + 1$): that's why there is no fixed line when $n = 2$.

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In order to do this, we thought of a type of function (the rotation) as a matrix, an $n \times n$ block of numbers. We made some arguments about the block of numbers to 1) conclude things we couldn't have concluded about a mere function, and 2) get a new object, our polynomial. Then, we thought of the polynomial as a curve in the plane (using analytic geometry like you mentioned), and made some arguments about curves in the plane.

So cross-theoretic identifications let us see something we never would have otherwise.

*_{This is called Euler's Theorem, but also just picture it.}