Paradigms
There are many alternative logical foundations for mathematics.
They differ in some cases just in detail or strength.
In other cases the differences are more fundamental, possibly representing radically divergent views on the nature of mathematics.

Dimensions
We consider five different characteristics or dimensions of logical foundation systems, sometimes clearly separated, other times not so.
The formal the semantic, the logical, the ontological and the conceptual.


Formal
The formal aspect concerns the mathematical theorems which can be proven using the foundation system.
Foundation systems are partially ordered according to their prooftheoretic strength.

Semantic
Gödel's first incompleteness theorem guarantees that truth and theoremhood do not coincide in any foundation system adequate for mathematics.
It is therefore desirable to have an account of the intended meaning of the language independent of the definition of formal derivability.


Logical
In some cases it is possible to separate out a part of the system which is concerned with logic and independent of matters ontological.

Ontological
Ontology is an important part of semantics, and differences both paradigmatic and in detail, are likely to be reflected in or caused by ontology.

Conceptual
Over and above the logical and ontological features which determine the strenght of the system there is likely to be some conceptual apparatus which provides the first stages in developing mathematics.

