What is an "axiomatic definition"? Every definition is an axiomatic definition?

Question

In Bourbaki's Elements of the History of Mathematics, one reads the expression "axiomatic definition" ("the axiomatic definition of vector spaces"). In Roussas Introduction to Probability, one reads the "axiomatic definition of probability due to Kolmogorov". What's the difference between an axiomatic definition and other types of definitions? Is there any? I think we may define anything the way we want, but some things are compatible with the theory and others do not (does this make sense in mathematics?), for example, a square exists in Euclidean geometry, and this statement can be proved. Are the axioms of an axiomatic definition different from foundational axioms? (same word, but different meanings?) P.S. There are many other examples: like "the axiomatic definition of area" and so on... In summary: What is an "axiomatic definition"? Isn't every definition an axiomatic definition?

Answer 1

My answer to your question "isn't every definition an axiomatic definition" is no, but there are two flavours to this answer.

The first is that definitions in physics are often not axiomatic, e.g. "a vector is something with a direction and a magnitude". This definition is probably not very satisfactory for most mathematicians, because it lacks rigorous. You could say that a mathematical definition must be rigorous, and in that case you could argue that every mathematical definition is an axiomatic definition, since it could be expressed as a sentence in an axiomatic system.

However, the second flavour is that an "axiomatic definition" sometimes refers to a more subtle, less rigid concept. I'll give an example to illustrate, and though this example is a bit obscure, I think it should be able to illustrate the general concept, and hopefully someone else can give a less obscure example.

In category theory, Grothendieck came up with a particular kind of category called a topos. His definition goes something like this.

Start with a "site" $J$, and construct the category of "sheaves" on that site, $Sh(J)$. A topos is any category that is of the form $Sh(J)$ for some $J$. (Or, any category equivalent to a category of this form.)

Later on, people came up with a different but equivalent definition of a topos, which goes like this:

A topos is a category which satisfies the following properties.

• It has finite limits;
• It has power objects;
• It is cocomplete;
• It has a small generating set.

There is a qualitative difference between these two definitions. The first says that a topos is a category that comes from a certain construction; the second says that a topos is a category that satisfies certain axioms. The latter is referred to as the axiomatic definition of a topos. Perhaps there could be some disagreement, but I think most category theorists would say that the first definition is not an axiomatic definition of a topos, despite the fact that the definition could be written as a statement of first-order logic.