Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: community: general

Topic: citing unnumbered results

Tom Hirschowitz (Jun 22 2021 at 07:23):

I feel stupid, but how does one cite a particular result in a paper where results are not numbered? E.g., how do I refer to the first proposition in Beck's distributive laws?

Morgan Rogers (he/him) (Jun 22 2021 at 07:36):

Page numbering is standard, or section number, or location in reference to other numbered results (e.g. "after Theorem 1")

Tom de Jong (Jun 22 2021 at 07:37):

In the particular example you gave, I would say "the proposition on page 97".

John Baez (Jun 22 2021 at 19:19):

In the humanities, reference by page numbers is standard - so journals preventing open access to the published versions of papers is a disaster: you can read a preprint but you can't cite it adequately. (If you disagree, argue with my wife.)

John Baez (Jun 22 2021 at 19:20):

We're so lucky in math that we have theorem numbers, equation numbers and the like! We should use them.

John Baez (Jun 22 2021 at 19:22):

I agree with Morgan and Tom de Jong's answers, but this is a cautionary tale: you don't want to put future authors in Tom Hirshowitz's situation.

Mike Shulman (Jun 22 2021 at 19:26):

Now that we have (La)TeX, however, I find it's very rare to see totally un-numbered theorems in any recent paper, since the default is to number them. Un-numbered equations are more common, but I can't remember the last time I wanted to refer to an equation in someone else's paper.

Mike Shulman (Jun 22 2021 at 19:28):

At the moment I feel like the "frontier" in this regard is convincing everyone to number theorems, definitions, lemmas, etc. with the same counter, which is subordinate to the section counter. If "Definition 3" could come after "Theorem 16" in "section 5", it's almost impossible to find unless you also give a page number.

John Baez (Jun 22 2021 at 19:28):

In physics a lot of journals require that you number every equation, and it's common to refer to equations.

But this just shows: physics is about equations, while math is about isomorphisms and equivalences. :upside_down:

Mike Shulman (Jun 22 2021 at 19:29):

I expect there are some parts of mathematics that are about equations.

Mike Shulman (Jun 22 2021 at 19:30):

(And, of course, by "equation" I really meant "displaymath", which is irrelevant to whether it contains $=$ or $\cong$ or $\simeq$.)

Mike Shulman (Jun 22 2021 at 19:31):

(And, of course, under univalence, isomorphisms and equivalences are equations. Just sayin'.)

Tim Hosgood (Jun 22 2021 at 19:42):

my favourite style is the EGA type one, where every single sentence/paragraph is numbered sequentially, meaning you can refer to "Theorem (6.8.1)" or "the construction in (2.6.1.1)" etc.

Tim Hosgood (Jun 22 2021 at 19:43):

i wish this were the standard...

Jon Sterling (Jun 23 2021 at 04:25):

@Tim Hosgood I have written a latex package to support this style if you like it…

Tom Hirschowitz (Jun 23 2021 at 05:39):

Seems to be here, to save others one or two google searches.

Ulrik Buchholtz (Jun 23 2021 at 07:25):

@Jonathan Sterling this is great! Will you put it on CTAN?

Zhen Lin Low (Jun 23 2021 at 07:30):

I used a similar scheme for my thesis (though my model was PTJ's _Stone spaces_ rather than EGA). It was rather complicated to implement in my recollection...

Matteo Capucci (he/him) (Jun 23 2021 at 08:46):

Mike Shulman said:

Now that we have (La)TeX, however, I find it's very rare to see totally un-numbered theorems in any recent paper, since the default is to number them. Un-numbered equations are more common, but I can't remember the last time I wanted to refer to an equation in someone else's paper.

Indeed, I try to number all my equations in the hope people will find easy to refer to them (citing or even just discussing/reading the paper)

Matteo Capucci (he/him) (Jun 23 2021 at 08:48):

Unfortunately it doesn't seem to be standard practice (it wasn't among my coauthors so far)

John van de Wetering (Jun 23 2021 at 09:48):

A fellow pair of PhD students hacked something together for their theses so that every paragraph had a number that could be hyperlinked to in the document, and such that the page numbers were replaced by a pair of paragraph numbers that specified what was on that page. So you had something like 3.5.III--3.7.IV on the bottom of the page. Apparently hacking the index so that it displayed the correct "page number" was very bothersome, and the thesis committee members were not impressed by this wizardry, and requested regular page numbers anyway :P

David Michael Roberts (Jun 23 2021 at 11:01):

Matteo Capucci (he/him) said:

Unfortunately it doesn't seem to be standard practice (it wasn't among my coauthors so far)

Halmos discouraged it in his writing advice, perhaps that influenced people

David Michael Roberts (Jun 23 2021 at 11:01):

Or was merely a symptom

Morgan Rogers (he/him) (Jun 23 2021 at 12:24):

My personal approach is that any result that seems concrete enough to be citeable should be presented as a numbered result, but that's an approach that only works in maths papers, and it comes at the cost of needing to intersperse such results with sufficient commentary to sustain the flow of motivation through the paper.

Jon Sterling (Jun 23 2021 at 14:25):

@Ulrik Buchholtz I do plan to put it on CTAN at some point! In the meanwhile, here is the code: https://github.com/jonsterling/latex-dieudonne

Jon Sterling (Jun 23 2021 at 14:26):

I mainly created it for my thesis, but I am curious if others will decide to use it :)

Mike Shulman (Jun 23 2021 at 15:20):

I've said this to @Jonathan Sterling before, but I think it is execrable to imagine having both (1·1) and (1.1) that refer to different things. I find that even worse than having a separate Theorem 3 and Definition 3.

Jon Sterling (Jun 23 2021 at 15:23):

Indeed, this is an unfortunate trade-off; I am very open to any ideas you may have for how to address it. The reason for this 'interpunct' numbering is to avoid the (IMO much worse) problem of having a "section 1.1" and a "node 1.1" which are different; an alternative way to avoid this is to have the sections use the same counter as the nodes, but this is also terrible because your TOC will appear to skip sections.

Jon Sterling (Jun 23 2021 at 15:23):

For what it's worth, the French avoid this by simply avoiding patterns of section/subsection/node that introduce the ambiguity I refer to. Maybe that is ultimately the best approach, but it seemed to me to be somewhat stilted.

Jon Sterling (Jun 23 2021 at 15:24):

Anyway, I would love if we could come to a solution to this that avoids the numerous trade-offs.

Mike Shulman (Jun 23 2021 at 15:25):

I don't think any section-type object that's numbered at the same level as theorems should be appearing in the TOC. For instance, in an ordinary article the TOC would generally include only top-level sections like 1, 2, 3, so that subsections 1.1, 1.2 can be numbered with the same counter as theorems 1.3, 1.4 without looking odd in the TOC.

Jon Sterling (Jun 23 2021 at 15:26):

(deleted my reply because I misunderstood your message haha)

Mike Shulman (Jun 23 2021 at 15:26):

You can get this effect very simply with

\usepackage{amsthm}
\swapnumbers
\newtheorem{thm}[subsection]{Theorem}

and then use \subsection{} for the unnamed paragraphs.

Jon Sterling (Jun 23 2021 at 15:27):

That may be a reasonable approach, but I am not ready to give up having subsections in the TOC. Furthermore, I am not convinced that it is a good idea to number subsections at the same level as theorems, etc.

Mike Shulman (Jun 23 2021 at 15:29):

If you want subsections in the TOC, then number the theorems with the subsubsection counter instead.

Jon Sterling (Jun 23 2021 at 15:30):

I think your scheme only works if every theorem-level node appears within a subsection. But this is not always the case...

Mike Shulman (Jun 23 2021 at 15:30):

No it doesn't. Try it.

Jon Sterling (Jun 23 2021 at 15:31):

What is the number assigned to the theorem in the following:

section Introduction
theorem My theorem.

Mike Shulman (Jun 23 2021 at 15:31):

1.1.

Jon Sterling (Jun 23 2021 at 15:31):

OK, good. Now what about:

section Introduction.
theorem My theorem.
subsection My subsection.

Mike Shulman (Jun 23 2021 at 15:31):

theorem 1.1, subsection 1.2.

Jon Sterling (Jun 23 2021 at 15:32):

Cool. Now the TOC is:

1. Introduction
   1.2. My subsection.

(and I consider the above unacceptable.) Did you have any thoughts on how to address that?

Mike Shulman (Jun 23 2021 at 15:33):

Yes, I already said how to address that. If you want subsections in the TOC, then use the subsubsection counter for theorems instead (and subsubsections for un-named paragraphs).

Jon Sterling (Jun 23 2021 at 15:33):

OK, but I hope you understand that I can continue coming up with examples of the same form for whatever depth you choose.

Mike Shulman (Jun 23 2021 at 15:34):

But any given paper has a fixed \tocdepth, so there is always a level below that that you can use for theorems.

Jon Sterling (Jun 23 2021 at 15:34):

Let's go one level deeper, as you suggest. In that case, consider the following:

section Intro
theorem My theorem
subsectino My subsection

Now the TOC is good, but the theorem is numbered "1.0.1".

Jon Sterling (Jun 23 2021 at 15:35):

It might be that this is an OK trade-off. How do you feel about it?

Ian Coley (Jun 23 2021 at 15:35):

You can also advance the subsection counter by 1 after \section{First}

Mike Shulman (Jun 23 2021 at 15:36):

If you have subsections in the TOC, then by default I would expect every section to be divided into subsections, and if something appears before the first subsection I would expect its number to include a "0" so that I know where to find it.

Mike Shulman (Jun 23 2021 at 15:36):

So yes, I would say that is the expected behavior; I wouldn't even consider it a trade-off.

Jon Sterling (Jun 23 2021 at 15:37):

OK! That is perfectly fair. I personally do see it as a trade-off, but I am not sure it is worse than what I did. It might be better. For me, what is attractive about my approach was that it is uniform regardless of the ultimate section-depth of a given piece of work. But I do recognize and agree with the readability problems that you pointed out.

Jon Sterling (Jun 23 2021 at 15:38):

Personally I have been wondering if a better approach can be found by re-thinking section-numbers themselves; to me the section-subsection-etc. hierarchy is too rigid and seems to be causing problems. But I don't yet have any concrete ideas. I'll let you know if I come up with something :)

In the meanwhile, it would perhaps be reasonable to add a 'mode' to my package that supports the behavior you prefer.

Mike Shulman (Jun 23 2021 at 15:39):

Of course, it's not as if you have to decide in advance with my approach what the tocdepth is going to be. Just replace \subsection{} by a macro; then if the tocdepth changes you just have to change the definition of that macro and the counter used by \newtheorem.

Jon Sterling (Jun 23 2021 at 15:40):

indeed, I think that at the end of the day there can be a single configuration parameter that generates the behavior that you describe. It would even be possible in principle to automate this by means of some awful .aux-file hackery (but that's probably not worth it).

Mike Shulman (Jun 23 2021 at 16:10):

One thing I do find objectionable is to get theorems numbered 1.0.1 in a section that doesn't have any subsections. In principle I feel like either all sections should have subsections or none of them should, but sometimes it's hard to resist the temptation to do otherwise. At the moment in a paper I'm writing like this I'm experimenting with changing the numberwithin for different sections, so that if section 1 has no subsections then it contains theorem 1.1 and definition 1.2, while if section 2 has subsections then subsection 2.1 contains theorem 2.1.1 and subsection 2.2 contains definition 2.2.1.

Jon Sterling (Jun 23 2021 at 16:19):

Ah that's a good observation, yes. I would like to address that too, and I think to do so in non-manually basically would involve keeping track (in the aux file) of the depth used below a particular section, and using that to determine the appropriate indexing level for theorem nodes.

Jon Sterling (Jun 23 2021 at 16:19):

I noticed that the TAC templates cause these terrible situations to happen almost immediately LOL

Ralph Sarkis (Jun 23 2021 at 17:17):

An alternative I am convinced is better but I don't know how to optimally implement yet is to have names instead of numbers to refer to theorems (and definitions and lemmas, etc.). Moreover, if a theorem has parameters, the name should also contain the parameter so that we can instantiate the parameters at the same time as we refer to the theorem. E.g.:

Theorem $\mathsf{HOR}(\phi,\eta)$. Let $\phi: F \Rightarrow G$ and $\eta: F' \Rightarrow G'$ be natural transformations, then $\eta G \circ F'\phi = G'\phi \circ \eta F$.

Then, if there is a proof where you need to use this equality but with different transformations, you can refer to it with $\mathsf{HOR}(\lambda,\mu)$.

Ralph Sarkis (Jun 23 2021 at 17:17):

Bonus feature: When this will fit in my perfectly designed interactive web document format, clicking or hovering on the reference will show you the theorem with the substitutions done for you already.

Ralph Sarkis (Jun 23 2021 at 17:17):

Other bonus feature: You can now define higher-order theorems. Right now, I feel like it would only be useful for proving approximations and bounds, but I'd love to be proven wrong.

Jules Hedges (Jun 23 2021 at 17:20):

Heh, nice idea. An example of a "higher order theorem" is one that assumes a conjecture, it can take a proof of the conjecture as an argument

Mike Shulman (Jun 23 2021 at 17:39):

Yeah, don't even get me started on the TAC style.

Mike Shulman (Jun 23 2021 at 17:40):

My standard macro file defines a special \iftac because there are so many things that have to be done differently to work around its idiosyncracies.

Mike Shulman (Jun 23 2021 at 17:41):

(But I'll put up with it for such an otherwise-great journal.)

Mike Shulman (Jun 23 2021 at 17:44):

By the way, I personally am not a huge fan of numbering all paragraphs. It feels a bit pretentious to me, like "every word that drops from my mouth is so precious that it deserves a number." More importantly, it makes it hard to distinguish visually between "the comments and the code". In my mind, numbered blocks like Theorem and Definition are for precise mathematical statements (the things you would formalize in a proof assistant), while unnumbered paragraphs between them are for informal comments and motivation (the things that at most would be comments in a proof assistant file).

Mike Shulman (Jun 23 2021 at 17:45):

However, I respect it, and I'll go along with it when my coauthors prefer it (as long as you don't make me try to distinguish between (1·1) and (1.1)!).

Jon Sterling (Jun 23 2021 at 17:55):

To my mind, this numbering-all-paragraphs style is only applicable in certain kinds of writings... I would not use it in a research paper. I am using it in my thesis, however.

Mike Shulman (Jun 23 2021 at 18:01):

@Jonathan Sterling Is our project on strict gluing not a "research paper"?

Jon Sterling (Jun 23 2021 at 18:02):

I was planning to change the format over to something more standard ;-)

Jon Sterling (Jun 23 2021 at 18:02):

My thinking is that publication venues are typically not kind to submissions whose format is idiosyncratic, and I don't blame them.

Tom Hirschowitz (Jun 23 2021 at 19:32):

@Ralph Sarkis Nice idea, but names don't help you locate the result within the paper, do they? I understand that this can be remedied easily when reading on any kind of computer, but on paper? How about numers+name: Theorem 1.2.3.BOO ? (Only half-joking... This might in fact rectify the deficiencies alluded to above by Mike and Jon, perhaps at the cost of introducing others.)

Ralph Sarkis (Jun 23 2021 at 19:45):

You're right... unless, by some miracle, you are able to give your theorems meaningful names in alphabetical order.

Mike Shulman (Jun 23 2021 at 20:42):

I think it's already not uncommon for especially important theorems or lemmas to be given a name in addition to a number.

John Baez (Jun 23 2021 at 23:16):

Remark 0. 0.0. Every 0.1. word 0.2. that 0.3. drops 0.4. from 0.5. my 0.6. mouth 0.7 is 0.8. so 0.9. precious 0.10. that 0.11. it 0.12. deserves 0.13. a 0.14. number.

fosco (Jan 19 2022 at 16:55):

David Michael Roberts said:

Matteo Capucci (he/him) said:

Unfortunately it doesn't seem to be standard practice (it wasn't among my coauthors so far)

Halmos discouraged it in his writing advice, perhaps that influenced people

I would be curious to know the rationale behind Halmos' argument. I can't find a single reason to discourage clarity

fosco (Jan 19 2022 at 16:57):

John Baez said:

Remark 0. 0.0. Every 0.1. word 0.2. that 0.3. drops 0.4. from 0.5. my 0.6. mouth 0.7 is 0.8. so 0.9. precious 0.10. that 0.11. it 0.12. deserves 0.13. a 0.14. number.

Wittgenstein's Tractatus coff coff

Patrick Nicodemus (Jan 20 2022 at 18:35):

fosco said:

David Michael Roberts said:

Matteo Capucci (he/him) said:

Unfortunately it doesn't seem to be standard practice (it wasn't among my coauthors so far)

Halmos discouraged it in his writing advice, perhaps that influenced people

I would be curious to know the rationale behind Halmos' argument. I can't find a single reason to discourage clarity

A bit of a change of topic - Recently I learned about Halmos's bizarre role in the Pacific Journal of Mathematics affair. Interesting piece of mathematical history, I wonder how widespread this attitude on logic is today. (It is always the ideologues who intensely claim to be non-ideological.)

https://arxiv.org/pdf/1607.00149

David Michael Roberts (Jan 20 2022 at 22:41):

fosco said:

John Baez said:

Remark 0. 0.0. Every 0.1. word 0.2. that 0.3. drops 0.4. from 0.5. my 0.6. mouth 0.7 is 0.8. so 0.9. precious 0.10. that 0.11. it 0.12. deserves 0.13. a 0.14. number.

Wittgenstein's Tractatus coff coff

I think there needs to be more depth in the counters to even be compared; W goes down to sub-sub-sub-sub-sections.

David Michael Roberts (Jan 20 2022 at 22:46):

Patrick Nicodemus said:

A bit of a change of topic - Recently I learned about Halmos's bizarre role in the Pacific Journal of Mathematics affair. Interesting piece of mathematical history, I wonder how widespread this attitude on logic is today. (It is always the ideologues who intensely claim to be non-ideological.)

https://arxiv.org/pdf/1607.00149

I'm not surprised to see Katz's name on that, he's a real champion for non-standard analysis.

John Baez (Jan 20 2022 at 23:34):

I wonder how widespread this attitude on logic is today.

What is "this attitude on logic"? I'm afraid I'm too lazy to read the paper to find out.

Todd Trimble (Jan 21 2022 at 01:12):

With no pretension to scholarship (i.e., I don't actually care enough to go through the Katz et al. paper with a fine-toothed comb), I think it is fair to wonder whether ideology or partisanship is at work in that paper. It just strikes me as a tad heavy-handed.

Halmos did have strong opinions (as many people do), and he is quite candid about them in his automathography. One message that came through is that he found logic texts, at least those in the 50's, as obfuscatory and hiding the essential points. For example, at one point he had an epiphany that propositional logic was all about free Boolean algebras, something he never got from the texts, and this insight struck him so forcefully that he wanted to similarly algebraize all of logic. I am somewhat sympathetic. Lawvere would do something similar in his thesis, blurring the lines between syntax and semantics in universal algebra (the theory is a free model). Lawvere would, still later, "algebraize" predicate logic and its extensions (hyperdoctrines, etc.), and did it a whole lot better than Halmos, but I think I can see similar impulses at work.

I don't know that Halmos behaved at all dishonorably, or acted like a bully, etc. Clearly the problem settled by Robinson and Bernstein was of great personal interest to him and he wanted to understand what they had done on his own terms and in his accustomed language. (Maybe he was a bully? I'm not sure.)

Yes, he does make some disparaging remarks about category theory, but I imagine this had something to do with the natural aging process. :-)

Aside: I wrote Halmos once. In his automathography, he boasts about the wonderful course on topological groups that he taught at Chicago, and about his wonderful take-home exam ("Geniuses are expected to solve all fifteen"). I liked one of the problems -- does there exist a nontrivial connected Hausdorff topological group of exponent 2? -- and built a little category theory talk around it. Later I wrote to him with my solution. He responded approximately thus: "Your solution is in the language of category theory, and your language frightens me. My own solution was in the language of measure theory and went like this...".

Todd Trimble (Jan 21 2022 at 01:44):

There is one episode involving Halmos that leaves a bad taste, though: he gave Keisler's book Elementary Calculus: An Infinitesimal approach to Errett Bishop to review, and Bishop savaged the book, which I think might have been pretty predictable. So at least this does give one pause. (Not that Halmos was sympathetic to Bishop's constructivism -- he portrayed it as a kind of religious thing -- but let it be noted that Bishop was a Halmos student, in the area of functional analysis.)

Todd Trimble (Jan 21 2022 at 01:46):

Keisler-Bishop controversy