Previously, I studied mathematics for a couple of years, and thus walked the first few steps along this beautiful road. In the mathematical conventions, I discovered, from time to time, a lack of consistency which seem to have passed by unnoticed, or maybe not?
Here’s an example:
Sometimes, when we want to make a new definition by the use several axioms, a strange kind of “semi-inconsistency” arises.
Let us assume we want to define a new concept, C, by using a description, D (which can refer to one or more previous definitions), together with a requirement that D satisfies some axioms (let us name them A1, A2 and A3). Such a definition would often be stated like this:
Definition: C is a D that satisfies the following axioms:
i) A1
ii) A2
iii) A3
By this we mean that all the axioms (in this case three of them) must be satisfied by D. But then we could as well formulate the definition in this way:
Definition: C is a D that satisfies the following logical statement: A1 ⋀ A2 ⋀ A3
which can be written in several different ways (since the logical operator ⋀ is commutative).
An example could be:
Definition: C is a D that satisfies the following statement: A3 ⋀ A1 ⋀ A2
and thus we see that the original definition could as well be written this way:
Definition: C is a D that satisfies the following axioms:
i) A3
ii) A1
iii) A2
In short, if we want to define something by using a list of axioms (strictly speaking an unordered list, even though it is often numbered), then the order of those axioms should not have any influence on the definition. This means we need to express all the included axioms in such a way that they are independent of one another. But this is not always the case with some of the fundamental mathematical definitions, see the example below.
In several books discussing fundamental abstract algebra, the following definition of a group is given:
In this definition Axiom ii) must be presented before Axiom iii), because otherwise the symbol e (the neutral element) has not been described when it is used in Axiom iii).
You could, in principle, imagine very complicated definitions that contain a list with hundreds of axioms. Then, imagine that these hundreds of axioms depend on each other in a complex way.
I suggest a new convention for this kind of definitions:
In all definitions that use a list of axioms, we should always state the listed axioms in such a way that they do not depend on each other.
As an example, I rewrite the group definition mentioned above:
This new definition is better, since the three group axioms now “have the same status” (i.e. they are mutually independent, and commutative) – and the order in which we list the axioms is unimportant here. We would have to define the term neutral element separately, but that is no problem (actually, it is better). What is your opinion here?
