Consider the following definition of vector spaces:
Why are the listed conditions called "axioms"? My understanding of axioms is that they are base assumptions which are taken to be true. Thus, they're not really meant to be proven. Yet from this definition, it's necessary to show that the axioms are "satisfied" for a specific set in order to conclude that the set is a vector space. Is that somehow different than "proving" the axioms are true for the given set?
On
My understanding of axioms is that they are base assumptions which are taken to be true.
In some sense, that is true here. They are the base assumptions you are allowed to make when someone gives you a tuple $(V,+,0,-,\cdot)$ and tells you "this is a vector space". If you know some first order logic: you could make the language of real vector spaces, which has function symbols $+,-,0$ and for each real number $r$ a function symbol $r\cdot$. Then the above axioms are all formulatable in the this language. (Edit: and you need to include the theory of the real numbers as well, which is left implicit here.) The models of this set of axioms are vector spaces; and to prove that something is a vector space, you prove that it satisfies those axioms.
Yet from this definition, it's necessary to show that the axioms are "satisfied" for a specific set in order to conclude that the set is a vector space. Is that somehow different than "proving" the axioms are true for the given set?
Not really, except for the very subtle difference that something can be true without being provable. If someone asks you "is this tuple a vector space", your only recourse is showing that all the axioms hold, or that one of them does not.
On
The axioms are defined to be true for vector spaces.
When you are trying to show that something is a vector space, you are in fact verifying (not "proving") that the axioms hold.
It is a bit of wordplay here.
On
It might help to think of this in programming terms:
In mathematics there are certain objects which fall into certain classes; an example of a mathematical class is 'vector space'. The axioms define the interface of the class; the theorems derivable from these axioms in effect explore the kinds of computations that can be done using only the operations provided by the interface (plus some basic logic etc.). By showing that a mathematical object satisfies the axioms, you are not trying to compute the features of the interface from anything else; rather, you are trying to show that that object implements that interface. To do that, you have to identify which operations of the object (if any!) correspond to which operations of the interface (and -every- interface operation has to be matched!), and show that matching operations 'behave' the same.
Axioms in modern mathematics means the same as a set of properties, or conditions. Different things may or may not satisfy them. When you define a vector space you are basically saying that a vector space is anything which satisfies those axioms. The axioms are not intended to have any deeper meaning than that. They simply single out some things among all by listing properties. It's simply a definition. In the olden days, some philosophers and teachers insisted on attaching some vague unverifiable truth to the notion of axiom. That is long gone (or should be).