"yields" vs "yields that" in math context

Question

I have learned that a commonly mistake in math papers is the phrase by ... we have that ... instead it would be correct to just leave the that.

Now I am wondering how to correctly use yield. In particular, when I am saying something like

Applying theorem XY yields a=b.

I had a discussion about that and I would say this is correct but some say that it should be

Applying theorem XY yields that a=b.

I found this on this website, which suggests that I don't need the that, but I am still not sure.

Another occurrence where I am not sure whether this is correct is Theorem XY yields the convergence of the sequence. Can I use yield as a synonym for imply?

How do I correctly use the verb yield?

Answer 1

I will try to argue by a parallel (sometimes a dangerous route but it may work here).

Squeezing lemons yields juice. Applying theorem XY yields a=b. Correct, if you see a=b as an entity.

Squeezing lemons yields that juice flows. Applying theorem XY yields that a=b. Correct if you see a=b as a statement that a becomes equal to b (others may disagree, and it is certainly awkward). But it adds nothing to your first version so cannot be recommended.

Similarly with "Theorem XY yields the convergence of the sequence." If you regard the convergence of the sequence as a conceptual entity to be proven valid, this seems to me correct. Otherwise "Theorem XY yields that the sequence converges".

Answer 2

I disagree with the first sentence of this question. Both "we have ..." and "we have that ..." are OK in mathematical usage if the "..." is an equation. If "..." is a verb phrase, though, "that" is required; it's incorrect to say "we have the function is continuous". Generally, you need "that" before verb phrases, and you should omit "that" before noun phrases. Equations can be (and are) regarded as both sorts of phrases, so "that" is optional before them.

(I wrote this answer from the point of view of a mathematician, because that's what I am. At least one philosopher has insisted on attaching vast significance to the presence or absence of "that", and has claimed that attention to this significance solves the semantic paradoxes. I have not understood his arguments, nor do I now remember his name.)

Answer 3

Yes, "that" is redundant there and you need only write "Applying theorem XY yields a=b".

As regards the second point, what is implied is stated indirectly and is rather readily obvious-- for instance, Pythagoras theorem implies that hypotenuse is the longest side of a triangle. In other words, even if the theorem doesn't mention this fact explicitly, you can tacitly assume it to be true, provided you know the statement/proof of the theorem.

On the other hand, we say that a certain theorem or proposition yields a particular result when we first prove it. In other words, the result isn't obvious at once. Proof of the Pythagoras theorem yields the result hyp^2=base^2+perpendicular^2 ; from the (yielded) result it could be implied that hypotenuse is the longest side.

Answer 4

Since others have addressed the use of "that," I'll address your second example, "Theorem XY yields the convergence of the sequence." Usually, "convergence" doesn't take an article except when referring to the specific convergence of some sequence. Also, "yield" arguably sounds odd here because convergence is more of a timeless feature of nature and less of a new output of a theorem. It would sound more natural to say

Theorem XY implies sequence convergence.

or

Theorem XY implies convergence of the sequence. (This is wordier.)

or

Theorem XY implies that the sequence converges. (This is livelier.)