Etiquette regarding mathematical writing

Question

This is a question about etiquette regarding writing mathematical papers, but I feel like this venue is more appropriate than MathOverflow (though I could be mistaken). The question is as follows:

If one proves a theorem, which was known conditionally, how should one refer to the conditional proof?

For example, one may write "Theorem XX was previously proved by Y, under the assumption of Conjecture Z". Is this appropriate?

On a related note, if in the literature Author A proved Theorem S, which was previously proved conditionally by Author B, how should B be acknowledged when referring to Theorem S? Would it be appropriate to call Theorem S "A's Theorem" or "A-B's theorem"?

Answer 1

I don't think that "etiquette", per se, is a real issue. If the previous author's result was published, it was no doubt considered to be a useful contribution, even with the proviso (conditional). You don't have to remind everyone that it's OK to publish such results.

Simply state the facts: "Smith 1 provided a proof of theorem X under the assumption that Y is true. Here, we present a new proof without the necessity of relying on theorem Y"

As to the issue about how the theorem is named, there can be quite a bit of variability. Eponymous theorems or hypotheses (Pythagoras's theorem, the Riemann Hypothesis, etc.) compete with more descriptively named theorems (the prime number theorem, the convolution theorem, etc). Early proofs of the prime number theorem rely, to my knowledge, on complex analysis, whereas the proofs by Atle Selberg and Paul Erdős are "elementary" (in the way that only mathematicians can sensibly use that word) ... but the theorem is still known as the prime number theorem. It is likely that the system of referring to the result, rather than the method of proof, will still hold, whether the later simplification is doing away with an underlying conditional (as you propose), or changing the approach altogether.

Answer 2

For example, one may write "Theorem XX was previously proved by Y, under the assumption of Conjecture Z". Is this appropriate?

No, I think this is sloppy phrasing. I see two issues here. The main problem is that it's technically incorrect to refer to a mathematical assertion as a "theorem" unless it has been proved (meaning that is has been proved unconditionally on anything else).

A more minor issue of a similar nature is that I think in cases of conditional proofs, one should not say "proved" but "proved conditionally". While "under the assumption" conveys the same idea of conditionality, in your phrasing "proved" and "under the assumption" are separated by several words and a comma, creating some risk of confusion. Writing "proved conditionally" makes things clearer and is the best practice IMO. My point is that, as with "theorem", "proved" is one of the most important words in a mathematician's vocabulary, and we should be careful to use it in a way that does not dilute its meaning.

In terms of how to correct the main issue, I can think of two possibilities. The first one is to rephrase the sentence to avoid the use of the word "theorem" when referring to the conditionally proved statement. For example, if the assertion that was proved conditionally has been conjectured to hold unconditionally, then its formal status is that of a conjecture. Therefore it would be correct to say

  "Conjecture XX was previously proved conditionally by Y, on the assumption of Conjecture Z."

If the conditionally-proved claim was not a conjecture, you can still refer to it as a "claim", an "assertion", or a "hypothesis", or simply state its content as part of the sentence. E.g.:

  "Claim XX was previously proved conditionally by Y, under the assumption of Conjecture Z"

  "The indecomposability of non-Riemannian hypersquares was previously proved conditionally by Y, under the assumption of Conjecture Z,"

etc.

The second approach to correcting the issue with the use of "theorem" would be to use the word "theorem" after all but in a correct manner. The point is that Y did in fact prove a theorem, just not the theorem you were referring to but the theorem that includes the extra assumption of Conjecture Z; thus you could write simply

  "Theorem XX was previously proved by Y"

but take care to add the conditional assumption of Conjecture Z as part of the statement of Theorem XX. That is, Theorem XX should now be phrased as

  "Theorem XX. If Conjecture Z holds, then [insert claim here]."

When describing your own contribution of coming up with an unconditional proof of the same claim (if that is what you are doing), you would say something like

  "The main result of the current paper is an unconditional version of Theorem XX. We prove:

  Theorem AA. [insert claim here (same as before but no mention of Conjecture Z)]"

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On a related note, if in the literature Author A proved Theorem S, which was previously proved conditionally by Author B, how should B be acknowledged when referring to Theorem S? Would it be appropriate to call Theorem S "A's Theorem" or "A-B's theorem"?

There's no universal rule about who gets credit for theorems in cases where the proof is an accumulation of insights from different authors. This would depend on how significant the relative contributions of A and B are towards the final unconditional result. One can imagine situations in which author B's initial contribution was only a trivial step and the real work of figuring out why the claim is true unconditionally was done almost entirely by author A; and other situations where author B did come up with some very significant insight already in the conditional result, which was improved upon only in a small way by author A to arrive at the unconditional result. In the first of the two scenarios, it might make more sense to call this "A's theorem" and in the second, "A-B's theorem" (or possibly "B-A's theorem") could be more appropriate.

Even the above dichotomy is a simplification of the many nuances that could occur. Also, however you end up calling the theorem, it's possible that someone will be unhappy and think they deserve more credit.