# The Universe of Discourse

Fri, 24 Apr 2020

There's this thing that happens on Stack Exchange sometimes. A somewhat-clueless person will show up and ask a half-baked 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.

1. 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 right-handed 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 right-handed objects is a major concern. Everyone else was just acting like this was a nonissue.

2. 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 category-theoretic 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 “co-exponential 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 co-exponential 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 co-exponential 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.)

3. A frequently-asked question is (some half-baked 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 half-baked software questions (“Is it possible to do X?”):

1. It surely could, but nobody has done it yet
2. It perhaps could, but nobody is quite sure how
3. It maybe could, but what you want is not as clear as you think
4. It can't, because that is impossible
5. 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 half-baked questions also deserve better answers. A similar classification of answers to “can we do this” might look like this:

1. Yes, that is exactly what we do, only more formally. You can find out more about the details in this source…
2. Yes, we do something very much like that, but there are some significant differences to address points you have not considered…
3. Yes, we might like to do something along those lines, but to make it work we need to make some major changes…
4. 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…
5. 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:

1. Yes, and…
2. Yes, but…
3. Perhaps, if…
4. No, but…
5. No, because…
6. 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 ]