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:

- 1 is not a prime number.
- The trivial ring is not a field.
- The zero object is not a simple object (which is the trope-namer).
- The empty space is not a connected space.
- The improper ideal is not a maximal ideal or a prime ideal.
- The improper filter is not an ultrafilter.
- The empty function to the empty set is not a constant function.

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:

- Suitable for a vague summary but obviously not precisely correct: there are no foos. (Example: a field has no non-invertible elements.)
- Original naïve definition: there are no nontrivial foos. (Example: a field has no non-invertible elements except $0$.)
- Sophisticated definition: there are no nontrivial foos, but there is the trivial foo. (Example: a field has no non-invertible elements except $0$, but $0$ 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:

- The empty space is the empty union, and therefore not connected.
- In the trivial ring the empty product is equal to zero, and so this ring is not integral.
- False is the empty disjunction, and hence is not true.

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.

Revised on February 11, 2017 09:39:30
by Max New
(71.232.30.31)