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)(-1)-truncation, while the more sophisticated definition is about (2)(-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: a field has no non-invertible elements.)
  2. Original naïve definition: there are no nontrivial foos. (Example: a field has no non-invertible elements except 00.)
  3. Sophisticated definition: there are no nontrivial foos, but there is the trivial foo. (Example: a field has no non-invertible elements except 00, but 00 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

  • nn is prime if whenever we have n=an=a or n=bn=b


  • nn is prime if whenever n= i=1 ka in=\prod_{i=1}^k a_i we have n=a in=a_i for some ii

Then 11 is not a prime because it is equal to the empty product (k=0k=0) but not equal to any of the a ia_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 nn” with “for both 00 and 22” then it is very easy to forget the “00” case and end up with a definition that fails for the trivial case.

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