(ii) Hodges' model? Ah! You mean the naïve conceptual framework that ...
... thinks it's a theory!
Apparently, the support of computers, through computer proof assistants, to help test and generate proofs in mathematics is creating confusion for mathematicians.
"When you see “2 + 2 = 4”, what does “=” mean? It turns out that’s a complicated question, because mathematicians can’t agree on the definition of what makes two things equal. ...
It was the late 19th century when things began to change, with the development of set theory, which provides the logical foundations for most modern mathematics.
To understand why, take the sets {1, 2, 3} and {a, b, c}. Clearly, the elements of each set are different, so the sets aren't equal. But there are also ways of mapping between the two sets, by identifying each letter with a number. Mathematicians call this an isomorphism. In this case, there are multiple isomorphisms because you have a choice of which number to assign to each letter, but in many cases, there is only one clear choice, called the canonical isomorphism. ...
Because a canonical isomorphism of two sets is the only possible way to link them, many mathematicians now take this to mean they are equal, even though it isn't technically the same concept of equality that most of us are used to."
Does this create an 'opening' for others? Others, whose commute also falls between the cold-logic of the hard sciences, and the fuzzy fields of the humanities? Others who may find the threshold as daunting, as it is (truly!?) tantalising.
orcid.org/0000-0002-0192-8965