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If blah, then blah blah (Mathematical Definitions)

September 18, 2012

On Mon, 5 Sep 2011, Charles Wells <charles at abstractmath.org> wrote:

This may help:  The definition of category requires things such as “For every object in the category, there is an arrow such that [it is the identity arrow for that object]“.  This can be reworded: “If A is an object in the category, then A has…”  When the category is the empty category, this last quoted sentence is *true*.  It is *vacuously* true.  All the other requirements that the definition of category makes can be reworded this way, too.  So in the empty category, all the requirements are true, so it is a category.

This works with any definition in which all the requirements can be reworded as implications (if…then statements).  It does not work, for example, for the definition of group.  One of the requirements is that a group must have an identity element.  This is an absolute requirement.  It does not say anything like “If blah blah then the group must have an identity element.”  So the empty set is not a group.

*The way you can word a definition affects the properties of the thing defined.

*Isn’t that amazing?

Charles Wells

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10 Comments
  1. Well, this is perhaps because the ultimate axioms are axioms of distinction, spatial axioms (if you think of the controversial author George Spencer-Brown).

    From nothing, a distinction creates something (an inside and an outside).
    In reality, the implication itself, can be translated into a distinction between inside and outside.
    Where the inside is PERCEIVED AS a “cause” or “premiss”
    and the outside is PERCEIVED AS a “result”, or “effect” or “conclusion”, etc.

    But it’s difficult to extend all this to the rest of mathematics (or even predicate calculus).
    Hence the “controversy” around the (googable) “Laws of Form” and their author,
    George Spencer-Brown (also googable).

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