# [FOM] Mathematical concepts and intensions

Aatu Koskensilta aatu.koskensilta at xortec.fi
Mon Jun 9 02:03:35 EDT 2003

```According to Carnap and later a number of intensional semanticist, the
intension of a concept is a function which maps each possible world to
the extension of the concept in that possible world. Now this squares
well with our intuitions when working with ordinary concepts.

Mathematics is supposedly necessary. This is the key reason why
mathematics can be done extensionally; the "intensions" of mathematical
concepts "are" constant functions, presumably. But how to model the
"intensions" of mathematical concepts then? Intension("square circle") =
Intension("bijection between the well-orders of the naturals and the
reals") if and only if the continuum hypothesis is false, but surely
these concepts have different intensions in any case?

The question can be posed as: how do we need to modify

Intension(A) = Intension(B) <-->df [](Extension(A) = Extension(B))

in order to allow necessarily co-extensive mathematical concepts to have
different intensions?

One solution is obviously to hold that "square circle" and "complete
axiomatisable FOL theory of arithmetic" do indeed have same intension,
but differ as concepts, giving us the implication sequence

A = B --> Intension(A) = Intension(B) --> ExtensionInWorld(M,A) =
ExtensionInWorld(M,B)

where no implication can be an equivalence in general. But this raises
the question as to how are we to understand the identicity of two
concepts if it is not identicity of their intensions?

--
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

```