philosophy

# Contents

## Idea

Pluralism is a philosophical position which states that there are multiple equally valid approaches to a particular field. This contrasts with monism, which states that there is only one valid approach to a particular field.

## Pluralism in the foundations of mathematics

In the context of the foundations of mathematics, pluralism states that rather than singling out a single approach to the foundations of mathematics, such as ZFC or ETCS, as the universal approach to the foundations of mathematics, each different approach are all regarded as valid approaches to the foundations of mathematics.

Approaches to the foundations of mathematics could be distinguished by its characterization of equality (whether as a proposition, a judgment, or as a type), equivalence (whether it is more fundamental than equality for collections), and propositions/logic (whether it is primitive or derived), which distinguishes material set theory, structural set theory, categorical set theory, allegorical set theory, higher order logic, dependent type theory, FOLDS, class theory, formal category theory, preset theory, and so forth, from each other.

There is another dimension in which approaches to the foundations of mathematics are distinguished, which is the strength of the theory; these concerns are orthogonal to the above concern regarding the characterization of equality, equivalence and propositions/logic: these distinguish predicative mathematics, strongly predicative mathematics, weakly predicative mathematics, constructive mathematics, classical mathematics, set-level foundations, higher foundations, and so forth, from each other.

To show that these two dimensions are independent of each other, we give some examples. On one hand, one could have a material set theory in which all sets have a choice operator; on the other hand, one could merely have a material set theory which is strongly predicative. In addition, we could have a categorical set theory in which all sets have a choice operator, and a categorical set theory which is strongly predicative. Formal category theory is usually considered to be higher foundations, but set-level formal category theory could be considered by adding an axiom or rule that every category is a discrete category.

There is also mathematics based in linear logic/linear dependent type theory, as well as paraconsistent mathematics, which from the pluralistic view are also true in their own ways.