Exercise 3: Why This Is a Category Error, Not a Broader Application — Possible Solution
============================================================================================

A genuinely broader, legitimate application of a mathematical result
would involve extending it to a new context that still shares the real
structural features the original proof actually depends on - the same
kind of formal system, the same relevant properties (consistency, the
ability to express arithmetic), just applied somewhere new but
relevantly similar.

Gödel's theorems are precise, technical claims about FORMAL AXIOMATIC
SYSTEMS specifically capable of expressing elementary arithmetic - they
depend on a very particular kind of structure: explicit axioms,
explicit rules of inference, and a specific kind of internal
self-reference the proof exploits. Religion, free will, and "science"
as a general practice aren't formal axiomatic systems in this specific,
technical sense at all - they don't have the precise, arithmetic-
capable formal structure the actual proof requires in order to apply.

Using the theorems to argue about these domains isn't extending the
proof to a new but structurally similar case - it's borrowing the
theorems' vague-sounding CONCLUSION ("there are limits to what can be
proven") while discarding the specific technical apparatus (formal
systems capable of arithmetic) that the actual proof depends on to
reach that conclusion in the first place. That's a category error: it
treats a precise result about one specific kind of mathematical object
as if it were a general philosophical principle applicable to entirely
different kinds of things that don't share the relevant structure at
all.

This is exactly the pattern the real scholars mentioned in the chapter
(Franzén, Sokal & Bricmont) have specifically criticized - and it's
also why Gödel's own real Platonist philosophical views are relevant:
some popular readings use his theorems to argue for a kind of
philosophical uncertainty or anti-realism that Gödel himself, as the
person who actually proved the theorems, didn't personally believe
followed from his own work at all.

ANSWER: This is a category error because Gödel's theorems depend on a
specific technical structure (formal axiomatic systems capable of
expressing arithmetic) that religion, free will, and general "science"
simply don't share - using the theorems in these contexts borrows only
the vague-sounding conclusion while discarding the precise mathematical
apparatus that actually makes the proof work, rather than genuinely
extending it to a structurally similar new case.

WHY THIS WORKS AS AN ANSWER
------------------------------
This identifies the real, specific technical dependency (formal
arithmetic-capable systems) the theorems require and explains why its
absence in these other domains makes the extension illegitimate rather
than simply broader.