nLab too simple to be simple

There is a general principle in mathematics that

A trivial object is too simple to be simple.

Quite often, classical references will define ‘simple’ (or an analogous term) in naïve way, so that a ‘trivial’ object is simple, but later it will become clear that more sophisticated theorems (especially classification theorems) work better if the definition is changed so that the trivial object is not simple. Usually this can be done by changing ‘if’ to ‘iff’ (or sometimes changing ‘or’ to ‘xor’) in the classical definition.

Examples include:

But of course one may still find definitions used that disagree.

Not that anybody would be naïve enough to believe otherwise, but perhaps the basic example is that

The point is that, in many cases, the naïve definition imposes only a uniqueness requirement (so that some set of possibilities —such as the set of proper divisors of a prime number, or the set of non-invertible elements of a field— must be a subsingleton) when it should in fact impose an existence and uniqueness requirement (so that the set of possibilities must be a singleton). With truth values, uniqueness is automatic, so existence is easier to notice.

More abstractly, the naïve definition is about $(-1)$-truncation, while the more sophisticated definition is about $(-2)$-truncation, which is more often relevant.

The general pattern is a progression of definitions (of ‘simple’) from more to less naïve:

1. Suitable for a vague summary but obviously not precisely correct: there are no foos. (Example: in a field, every element is invertible.)
2. Original naïve definition: there are no nontrivial foos. (Example: in a field, every element other than $0$ is invertible.)
3. Sophisticated definition: there are no nontrivial foos, but there is the trivial foo. (Example: in a field, every element other than $0$ is invertible, and $0$ is non-invertible.)

In many of the above examples one can obtain the sophisticated definition from the naïve definition by replacing a 2-ary function by a function of arbitrary (finite) arity. For example we would replace

• $n$ is prime if whenever $n=a.b$ we have $n=a$ or $n=b$

with

• $n$ is prime if whenever $n=\prod_{i=1}^k a_i$ we have $n=a_i$ for some $i$

Then $1$ is not a prime because it is equal to the empty product ($k=0$) but not equal to any of the $a_i$ (because there aren’t any)! Similarly we have:

This illustrates one advantage of using unbiased rather than biased definitions: if one has replaced “for all $n$” with “for both $0$ and $2$” then it is very easy to forget the “$0$” case and end up with a definition that fails for the trivial case.

In a similar vein we can define path connected by

• A space is path connected if for each (Kuratowski)-finite subset there is a path passing through every point of that subset.

Then the empty space is not path connected because it has no paths at all and hence no path through the empty subset.

Last revised on August 18, 2019 at 07:07:48. See the history of this page for a list of all contributions to it.