### Applicability as an Adequacy Condition

This is a second post on the notion of a "foundation of mathematics". For example, Frege, Cantor, Russell and Zermelo gave foundations for parts of mathematics. I suggested five adequacy conditions for a claimed foundation $F$ for mathematics: (Austerity), (Non-Circularity), (Justification), (Interpretability Strength), (Structural Invariance).

That was a rough proposal. It occurs to me that I neglected an important condition:

In general, a theory of classes or sets is

That was a rough proposal. It occurs to me that I neglected an important condition:

*applicability*. So, here is a revised list:(Austerity) $F$ is conceptually austere.I've also reformulated the (Structural Invariance) condition. Beginning with $ZFC$, to discuss the natural numbers, one must definitionally extend in some way, defining "$x$ is a natural number". This can be done, but in many different ways, and the choice between these is

(Non-Circularity) $F$ is conceptually non-circular.

(Justification) $F$ has an intuitive epistemic justification.

(Applicability) $F$ should be applicable to non-mathematical subject matter.

(Interpretability Strength) $F$ has high interpretability strength.

(Structural Invariance) $F$ should not enforce arbitrary choices of reduction.

*arbitrary*. I believe that the*sui generis*approach sketched a while ago,*Sui Generis Mathematics*, provides the right way to resolve this problem.*Sui generis abstracta*(e.g., pairs, sequences, cardinals, equivalence types) are characterized by*abstraction principles*added to set theory as primitives.In general, a theory of classes or sets is

*applicable*because one can assert the existence of classes of*non-classes*(e.g., "set-theoretic atoms") and*non-mathematical entities*(e.g., spacetime points) using*comprehension*, as follows:$\exists X \forall x(x \in X \leftrightarrow \phi(x))$where $\phi(x)$ can be any predicate that applies to the non-classes, or to non-mathematical entities: e.g., lumps of material, regions of space, measuring rods, etc. So, one can talk about:

The class of spacetime points in the future light cone of some given event $e$.Without this, one cannot do physics. For example, one cannot do the kinetic theory of gases without assuming that one has a class $C$ of molecules in a certain spatial region $R$ (with volume $V$), with cardinality $N = |C|$, and some probability distribution of velocities over the molecules in $C$.

The class of molecules in a sample of gas or fluid.

## Comments

## Post a Comment