Friday, 19 July 2013

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: applicability. So, here is a revised list:
(Austerity) $F$ is conceptually austere.
(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.
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 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 $e$.
The class of molecules in a sample of gas or fluid. 
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 $C$.

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