Examples where adding complexity made a problem simpler

Question

I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples:

1. $S^n$ is never contractible, but $S^{\infty}$ is.

2. The vanishing viscosity method of PDE's.

3. Higher-dimensional topology as opposed to low-dimensional topology (in some specific cases)

4. Singular homology as opposed to simplicial homology

5. Cube complexes as opposed to 3-manifolds

etc.

What other examples are there where a more complex object is simpler to analyze than a 'simpler' object? I realize that you could say that if it is easier to analyze, then it is less complex, so let me restate it this way:

What examples are there where one object seems much more complicated than another, but in fact has a simpler structure?

I've been thinking about things like the Ising model for magnetic phase changes and also about Navier-Stokes; perhaps the simplifications used to derive them make them harder to analyze in the end.

Answer 1

The complex object $\mathbb C$ is in some respects easier to understand that the apparently simpler object $\mathbb R$. (E.g., if we consider zeroes of polynomials, or the behavior of power series.)

Answer 2

It seems to me that this topic is a main feature of modern mathematics, if not its characteristic feature, compared with classic mathematics: moving the complexity from individuals to the species. A complicated property in a simple situation becomes a simpler property in a suitable more complex setting, and the trade-off can be very convenient.

The easiest and oldest example I have in mind is the Cartesian product: instead of talking of two or more objects $a\in A$, $b\in B$ and $c\in C$ we can consider just one object $(a,b,c)$ in the possibly more complicate setting $A\times B\times C$ (so that, for instance, we can describe a system of equations in $\mathbb{R}$ as a single equation in $\mathbb{R}^n$ ). This approach is so powerful because modern mathematics exploits the axiomatic method to its further consequences: working in a more complicated setting does not necessarily entail more difficult methods. What is needed is an abstract point of view, to identify the exact hypotheses that make a method work, and to understand to what extent these hypotheses pass to the more complicated constructions.

To remain in the example of the equations above, for instance, the Contraction Principle passes from $\mathbb{R}$ to $\mathbb{R}^n$ or even more complicated structures with no formal difference in the statement and in the proof; the key point is that $\mathbb{R}^n$ is complete just as it is $\mathbb{R}$. To make one more example in the same spirit, a linear system of ODE with constant coefficients is written $\dot u = A u$ and solved as $u(t)=e^{tA} u _ 0$, exactly as the more elementary scalar equation $\dot u = a u$, and from the same archeotypal equation the whole theory of evolution equations spring off, by which it is possible to treat evolution equations and PDE's.

The acme of this method is reached in Category Theory, that expressly created a language to construct more and more complex settings, where every universal concept can find a simple formulation. Here's one last example: a rather complicated situation as it is an adjunction of functors between two categories, can be seen as an "initial object", the most simple categorical notion, in a more complicated category, which is the starting point to Freyd's existence theorem for adjoint functors (edit: for details on this example check Todd Trimble's answer to a further question).

Answer 3

One can argue that Projective Geometry is more complex than Euclidian Geometry (it is less intuitive and came after), but some behaviours are easier in the projective world (two lines always meet).

There are also lots of cases where results are proved for dimension high enough, but get harder in small dimension, like the Poincare conjecture.

Answer 4

Quasi-periodic tilings as projections of periodic ones from a higher-dimensional space

Answer 5

Much of modern algebraic geometry fits this paradigm. The introduction of schemes makes some things much more complicated -- non-uniqueness of the embedded components of the primary decomposition compared to the uniqueness of the decomposition of a variety into its irreducible components comes to mind -- but the technical advantages of scheme theory have proven themselves many times over in the 50+ years since they were introduced.

Answer 6

In Oxtoby's "Measure and [Baire] Category" it is mentioned that facts such as "Elements with property X form a set of measure zero/first category/countable" can be seen as existence results, as in the proof that there are many irrational numbers, since the set of rational numbers is "small" in all three senses, while the set of real numbers is "big".

From that point of view this three "complicated" ideas, [the Baire Category theorem, naive set theory and Lebesgue measure] introduce very simple ways of proving the existence of potentially complicated objects.

Answer 7

"The shortest path between two truths on the real line passes through the complex plane."
– Jacques Hadamard.

Answer 8

Passing from classical statistics to Bayesian statistics: Introducing a prior probability distribution on the parameter space "reduces" formally statistical inference to a problem of probability theory. If that is OK from an applied standpoint is another question!

Answer 9

Some smaller examples: sometimes a group is best understood by embedding it in a larger group. One can use the embedding of a finite group in a large enough $GL(n, \mathbb{C})$ as a very general example (representation theory is quite powerful, even just in characteristic 0), but some far more localized examples:

The reason plenty of subgroups of $S_{6}$ have two non-conjugate embeddings in $S_{6}$ is explainable by embedding $S_{6}$ in $Aut(S_{6})$. Likewise for $AGL_{3}(2)$ and $M_{12}$.

Much of the structure of $M_{22}$ and $M_{23}$ is most easily understood in terms of $M_{24}$.