Tiers of answers to halfbaked questions
[ This article is itself somewhat halfbaked. ]
There's this thing that happens on Stack Exchange sometimes. A
somewhatclueless person will show up and ask a halfbaked question
about something they are thinking about. Their question is groping
toward something sensible but won't be all the way there, and then
several people will reply saying no, that is not sensible, your idea
is silly, without ever admitting that there is anything to the idea at
all.
I have three examples of this handy, and I'm sure I could find many
more.
One recent one concerns chirality (handedness) in topology.
OP showed up to ask why a donut seems to be achiral while a coffee cup
is chiral (because the handle is on one side and not the other). Some
people told them that the coffee cup is actually achiral and some
others people told them that topology doesn't distinguish between
left and righthanded objects, because reflection is a continuous
transformation.
(“From a topological point of view, no object is distinguishable from its mirror image”.)
I've seen many similar discussions play out the same way in the past.
But nobody (other than me) told them that there is a whole branch of
topology, knot theory, where the difference between left and
righthanded objects is a major concern. Everyone else was just acting
like this was a nonissue.
This category theory example
is somewhat more obscure.
In category theory one can always turn any construction backward to
make a “dual” construction, and the “dual” construction is different
but usually no less interesting than the original. For example, there
is a categorytheoretic construction of “product objects”, which
generalizes cartesian products of sets, topological product
spaces, the direct product of groups, and so on. The dual
construction is “coproduct objects” which corresponds to the disjoint
union of sets and topological spaces, and to the free product of groups.
There is a standard notion of an “exponential object” and OP wanted to
know about the dual notion of a “coexponential object”. They gave a
proposed definition of such an object, but got their proposal a little
bit wrong, so that what they had defined was not the actual
coexponential object but instead was trivial. Two other users pointed
out in detail why their proposed construction was uninteresting.
Neither one pointed out that there is a coexponential object, and
that it is interesting, if you perform the dualization correctly.
(The exponential object concerns a certain property of a mapping !!f
:A×B\to C!!. OP asked insead about !!f : C\to A× B!!. Such a
mapping can always be factored into a product !!(f_1: C\to A)×(f_2:
C\to B)!! and then the two factors can be treated independently. The
correct dual construction concerns a property of a mapping !!f :
C\to A\sqcup B!!, where !!\sqcup!! is the coproduct. This admits no
corresponding simplification.)
A frequentlyasked question is (some halfbaked variation on)
whether there is a smallest positive real number. Often this is
motivated by the surprising fact that !!0.9999\ldots = 1!!, and in
an effort to capture their intuitive notion of the difference,
sometimes OP will suggest that there should be a number
!!0.000\ldots 1!!, with “an infinite number of zeroes before the
1”.
There is no such real number, but the question is a reasonable one
to ask and to investigate. Often people will dismiss the question
claiming that it does not make any sense at all, using some formula
like “you can't have a 1 after an infinite sequence of zeroes,
because an infinite sequence of zeroes goes on forever.”.
Mathematically, this response is complete bullshit because
mathematicians are perfectly comfortable with the idea of an
infinite sequence that has one item (or more) appended after the others.
(Such an object is said to “have order type !!\omega + 1!!”, and is
completely legitimate.) The problem isn't with the proposed object
itself, but with the results of the attempt to incorporate it into
the arithmetic of real numbers: what would you get, for example, if
you tried to multiply it by !!10!!?
Or sometimes one sees answers that go no further than “no, because
the definition of a real number is…”. But a better engagement with
the question would recognize that OP is probably interested in
alternative definitions of real numbers.
In a recent blog article I proposed
a classification of answers to certain halfbaked software questions
(“Is it possible to do X?”):
 It surely could, but nobody has done it yet
 It perhaps could, but nobody is quite sure how
 It maybe could, but what you want is not as clear as you think
 It can't, because that is impossible
 I am not able rightly to apprehend the kind of confusion of ideas
that could provoke such a question
and I said:
Often, engineers will go straight to #5, when actually the answer is
in a higher tier. Or they go to #4 without asking if maybe, once the
desiderata are clarified a bit, it will move from “impossible” to
merely “difficult”. These are bad habits.
These mathematically halfbaked questions also deserve better answers.
A similar classification of answers to “can we do this” might look like this:
 Yes, that is exactly what we do, only more formally. You can
find out more about the details in this source…
 Yes, we do something very much like that, but there are some
significant differences to address points you have not considered…
 Yes, we might like to do something along those lines, but to make
it work we need to make some major changes…
 That seems at first like a reasonable thing to try, but if you
look more deeply you find that it can't be made to work, because…
 I am not able rightly to apprehend the kind of confusion of ideas
that could provoke such a question
The category theory answer was from tier 4, but should have been from
tier 2. People asking about !!0.0000…1!! often receive answers from
tier 5, but ought to get answers from tier 4, or even tier 3, if you
wanted to get into nonstandard analysis à la Robinson.
There is a similar hierarchy for questions of the type “can we model
this concept mathematically”, ranging from “yes, all the time” through
“nobody has figured that out yet” and “it seems unlikely, because”, to
“what would that even mean?”. The topological chirality question was
of this type and the answers given were from the “no we can't and
we don't” tiers, when they could have been from a much higher tier:
“yes, it's more complicated than that but there is an entire subfield
devoted to dealing with it.”
This is a sort of refinement of
the opposition of “yes, and…” versus “no, but…”,
with the tiers something like:
 Yes, and…
 Yes, but…
 Perhaps, if…
 No, but…
 No, because…
 I am embarrassed for you
When formulating the answer to a question, aiming for the upper tiers
usually produces more helpful, more useful, and more interesting
results.
[ Addendum 20200525: Here's a typical dismissal of the !!0.\bar01!! suggestion: “This is confusing because !!0.\bar01!! seems to indicate a decimal with ‘infinite zeros and then a one at the end.’ Which, of course, is absurd.” ]
[ Addendum 20230421: Another example, concerning “almost orthogonal” unit vectors ]
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