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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")?

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  • $\begingroup$ @ThomasAndrews ...did you mean to comment on a different question? $\endgroup$ – JP McCarthy Mar 12 '15 at 22:37
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    $\begingroup$ Umm, yes :) @JpMcCarthy $\endgroup$ – Thomas Andrews Mar 12 '15 at 23:04
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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.

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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.

a cubic polynomial

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 $n = 2$.


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.

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  • $\begingroup$ This, however, only depends on there being a bijection between an $n$ dimensional Euclidean space and $\mathbb{R}^n$. It doesn't require identifying points with tuples. $\endgroup$ – Peter Smith Mar 13 '15 at 7:11

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