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too simple to be simple

Too simple to be simple

Too simple to be simple

Idea

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.

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

  1. Imprecise summary: there are no foos. (Example: in a field, every element is invertible.)
  2. Classical definition: there are no nontrivial foos. (Example: in a field, every element other than 00 is invertible.)
  3. Modern definition: there are no nontrivial foos, but there is the trivial foo. (Example: in a field, every element other than 00 is invertible, and 00 is non-invertible.)

Examples

Examples include:

But of course one may still find definitions used that disagree (see discussion at connected space and empty space, for example).

This should be distinguished from barring the trivial object entirely. Historically, many people wanted to say that the empty set is not a set, and some people still say that the trivial ring is not a ring or that the improper filter is not a filter. This allows one to state later definitions (as of field or ultrafilter) without having to exclude the trivial object, but this does the exclusion too early.

Perhaps the basic example is that

Nobody would be naïve enough to believe otherwise in this case, of course. The point with this example is that, in the other 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.

Relationship to biased definitions

In many of the above examples one can obtain the sophisticated definition from the naïve definition by replacing a 22-ary function by a function of arbitrary (finite) arity. For example we would replace

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

with

  • 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. (Unfortunately, it seems to be easy to forget 00 even when using unbiased definitions, but hopefully less easy.)

In a similar vein we can define path connected by

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 December 2, 2019 at 13:08:31. See the history of this page for a list of all contributions to it.