The Universe of Discourse

Sun, 19 Feb 2023

I had an unusually interesting batch of Math Stack Exchange posts recently.

I think all of my answers to these questions are worth reading in full, and if you like the math posts on my blog, you will like reading these SE posts also. Well, most of them. Maybe.

Summaries follow.

Confusion about equality: mathematical objects versus the symbols that describe them

This one is from last September but I'm really happy with it because it thoroughly addresses up a very common misconception about mathematical notation:

Based on my understanding of equality, the statement !!(1+1)+1=1!!, contains no mathematical content beyond !!1=1!!, since the group element !!(1+1)+1!! literally is the group element !!1!!. This bothers me...

It should; it's wrong.

I'm frequently surprised by how often this fallacy shows up on Math SE, often asserted as an obvious truth by people I thought would know better. So it's worth explaining in detail. I expect I'll be able to refer people to this answer when it comes up in the future.

A brief summary of my answer is:

• Mathematical expressions denote computations, not values.
• !!A=B!! means that that two computations eventually produce the same value.
• This does not, in general, mean that the computations have the same meaning.

What does italic i mean in integral calculator?

The i means the imaginary unit, that !!i^2=-1!! thing. No surprise there. But the reason was a bit interesting. OP had Wolfram α compute some horrendous double integral:

and the answer should have been a real number, so what was !!i!! doing in there?

The answer: Floating-point roundoff error. Check out Claude Leibovici's detailed explanation of where the roundoff error comes from, it's much smarter than what I said, which was to mumble something about how Wolfram α's “probably … used … some advanced technique …” which sounds wise but actually I had no idea what it might have done. Claude Leibovici actually has an explanation.

I was going to leave this out but I wanted to remind you all how much I despise floating-point arithmetic.

Can Peano's 9th axiom be expressed using a self-referential set definition?

This is one of those not-quite-baked questions where the initial answers act like it does not make sense (tier 4 or 5). But it does make sense and there is a good answer (tier 1).

The question asks if you can define the set of natural numbers by saying something like

If !!K = \{0\} \cup \{S(k)\mid k \in K\}!!, then !!K=\mathbb{N}!!.

The initial comments said no, it's self-referential. But so is:

$$n! = \begin{cases} 1, & \text{if n = 0} \\ n\cdot (n-1)!& \text{otherwise} \end{cases}$$

and nobody bats an eyelash at that. (The author of the comments later retracted his rejection.)

In fact it requires only a little bit of elaboration to make sense of such “circular” definitions. To interpret $$X = f(X)$$ you need to do two things. First, think of !!f!! as a mapping, and ask if it has any fixed points, any arguments !!x!! for which !!x=f(x)!! holds. And then, from the set of fixed points, find some unambiguous way to identify one of the fixed points as the one you want. If !!f!! is a mapping from sets to sets, it often happens that the family of fixed points is closed under intersections, and you can select the unique minimal fixed point that is a proper subset of all the others.

This was all formalized by Dana Scott in the 1960s and it continues to underlie formal treatments of programming language semantics.

Is there a scenario for when changing the order of different quantifiers in a nested quantifier retain the original meaning?

This is interesting because some of the replies make the mistake of conflating the meaning of an expression with its value, a problem I discussed above in connection with something else. Two expressions of first-order logic may be logically equivalent, but this does not imply that they have the same meaning.

The question also looks superficially like “What is the difference between !!\forall x\exists y. R(x,y)!! and !!\exists y\forall x. R(x,y)!!, which is a FAQ. But it is not that question.

The question concerned expressions of the type !!\forall x.\exists y.P(x,y)!! and was further complicated by the implicit quantifier on the !!P!!. Are we asking if !!\forall x.\exists y.P(x,y)!! always has a different meaning from !!\forall y.\exists x.P(x,y)!! for all !!P!!? Or for a particular !!P!!? There are several similar-sounding questions that could be asked here, and my thinking about the variations is still not clear to me.

English (and standard mathematical terminology) is not well-equipped to discuss this sort of thing intelligibly. Or perhaps I just don't know how to do it. I had to work hard to write something I was satisfied with.

Cantor set - is it made of !![a,b]!! intervals or exclusively of singletons?

This question is a bit confused (every set is made of singletons) and I was worried that some know-it-all would jump in and tell this person that really the Cantor set is very simple. When actually the Cantor set is really weird and this is why it is such an important counterexample to so many plausible-seeming conjectures. As Von Neumann supposedly said, in mathematics one doesn't understand things, one just gets used to them. It can be hard for people who have gotten used to the Cantor set to remember what it is like for people who are grappling with it for the first time — or to remember that they themselves may not understand as well as they imagine they do.

When I write an answer to a question like this, in which I need to say “your idea is somewhat confused”, I like to place that remark in close proximity to “… because the situation is a confusing one” so that OP doesn't feel that they are the only person in the world who is puzzled by the Cantor set.

(Sometimes they are the only person in the world who is puzzled by whatever it is, and then it's okay for them to feel that way. I wouldn't lie and say that the situation was a confusing one when I thought it wasn't. If the matter is actually simple it's better to say so, because that can be valuable information. Beginners often overthink simple issues. But the Cantor set is not one of those situations!)

A valuable pedagogical strategy is finding a simpler example. The Cantor set does not have all the same properties as !!\Bbb Q!!. But !!\Bbb Q!! does seem to share with the Cantor set the specific properties that were troubling this person. Does !!\Bbb Q!! contain any intervals? Like the Cantor set, no. Is !!\Bbb Q!! a union of singletons? It's not clear what OP meant by this, but, uh, probably? And if not we can at least find out more about what OP thought they meant, by asking about !!\Bbb Q!!. So it's a good idea to take the focus off of the Cantor set, which is weird, complicated, and unfamiliar, and put it on !!\Bbb Q!!, which is much less weird, somewhat less complicated, and much more familiar. Then with that foundation laid, you are in a better position to climb up to !!\Bbb R\setminus\Bbb Q!! (Similar to !!\Bbb Q!!, but uncountable) and then to the Cantor set itself.

Deriving that a cube has six sides via a square and combinatorics

This is probably my favorite question of the month, because it seems quite half-baked, but there is an excellent answer available. As often happens with half-baked questions, the people who don't know the answer jump to the conclusion that no answer is possible, and say dumb stuff like:

What is the definition of a "cube" in your problem?

This is going the wrong direction. The point is to find the ‘right’ definition of the cube; if OP could define “cube” in the way they wanted, they wouldn't need to ask the question.

A better way to answer this question is to understand that what OP is looking for is actually a suitable definition of “cube”. A more mathematically sophisticated person might have asked:

How can we understand the cube as a combinatorial object, developed from the square?

The word “cube” in this question does not mean some specific mathematical object, but rather the informal intuitive cube. A correct answer will explain how to approach the informal idea of the cube in a mathematical way.

There is a nice (tier 1!) answer in this case: A segment is composed of an interior !!i!! and two endpoints, so we can represent it as !!S=i+2!!. Then !!S^3!! is a cube and its analogous combinatorial description is !!(i+2)^3 =i^3+6i^2+12i+8!!. Ta daa! The answer has a more detailed explanation.

There were a couple of followup comments that annoyed me, objecting that what I had presented was not a proof. That was a feature, not a bug. The question hadn't asked for a proof, and I had not tried to provide one.

One of the comments went further, and called it “a nice coincidence”. It's not, it's just generating functions.

I think the “coincidence” person has a profound misunderstanding of how mathematics operates. I wrote several hundred words explaining why but then realized that I had finally been able to articulate an idea I've been groping around to get hold of for decades. This is too precious to me to stick in at the tail end of an anthology article; it deserves its own article. So I am saving the next five paragraphs for next week. Or next year. Whenever I can do it justice.