Exercise 3: Why Treating Intuitionism Seriously Matters for This Capstone — Possible Solution
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If intuitionism were framed as a fringe curiosity - an eccentric,
easily-dismissed minority opinion - then Jarden's proof would read as
simply, uncontroversially settled, with a rejection from intuitionism
mentioned only as a historical footnote not worth taking seriously.
That framing would actually undercut the entire point of ending the
course this way, since it would imply the "what counts as proof"
question from Chapter 1 really was fully resolved by the end, just with
one small dissenting group left over.

But intuitionism is a real, historically serious philosophical
position, developed by a working mathematician (Brouwer) specifically
because of genuine, principled concerns about what mathematical
existence claims should actually require - not casual skepticism, but
a coherent, worked-out alternative framework with its own internal
logic and real consequences for which proofs it accepts. Taking that
seriously means the constructive/non-constructive divide isn't a
settled matter with one clearly "correct" side and one fringe holdout -
it's a genuine, live disagreement about what mathematical proof itself
is supposed to guarantee.

This framing choice directly serves the capstone's own real purpose:
rather than closing the course by declaring the Chapter 1 question
finally answered, treating intuitionism seriously lets the course end
honestly, showing that even a single, valid, classical proof (Jarden's)
sits at the center of a real, unresolved disagreement among working
mathematicians about what "proof" fundamentally requires - the same
open question the course started with, now demonstrated concretely
rather than just stated abstractly.

ANSWER: Treating intuitionism as a real, serious philosophical position
rather than a fringe view is what lets this capstone honestly show the
"what counts as proof" question from Chapter 1 remains genuinely
unresolved rather than quietly settled - if intuitionism were dismissed
as a curiosity, Jarden's proof would appear uncontroversial, undercutting
the entire point of ending the course on an open question rather than a
false sense of closure.

WHY THIS WORKS AS AN ANSWER
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This connects the specific framing choice (taking intuitionism
seriously) directly to the capstone's own stated purpose of leaving
Chapter 1's question genuinely open, rather than treating the framing
as an arbitrary stylistic choice.