# "There exists" vs "There is determined"

Mathematical writing tend to be very repetitive. To be clear, I do not consider this as something necessarily evil: Mathematics is a language in its own right, and a very technical one, where most of the effort is to be put on the value of the mathematical content.

On the other hand, I do often find myself trying hard (but not necessarily with success!) to make my write-ups in English look smoother from the point of the exposition. In particular, I like avoiding to repeat the very same stylistic elements in consecutive sentences whenever this is possible (that is, I happen to know how to do): For instance, by alternating sentences of the form "[...] there exists an object X such that [...]" to sentences of the form "[...] there is determined an object Y for which [...]". However, someone in my circles made me notice that my use of "there is determined" may not be correct, and this is the reason why I'm back to you again:

Is it legitimate to use "there is determined" in the way I'm doing in the above, as an alternative to "there exists"?

I know I could just use "there is", but my question is about "there is determined", which, in some situations, sounds better to my ears (except that it might be incorrect!).

• this question is pretty specific to the conventions of the field of mathematics. not sure anyone here is really going to be able to answer it -- unless someone happens to be a mathematician.
– abcd
Commented May 1, 2017 at 6:51
• I'm voting to close this question as off-topic because it is about the conventions of mathematical writing, no English. Commented May 1, 2017 at 8:34
• Beware - if you change form simply to avoid repetition, you risk distracting the reader with "elegant variation". Commented May 1, 2017 at 10:53
• If you use 'there is determined,' you open yourself to being asked who did the determining and what methods did they use. If you are the one who determined that X exists, then it exists -- done. If you want smoother, use one form, especially the shorter 'there is an object X.' Commented May 5, 2017 at 18:18

I wouldn't do it; not with determine, at least. Though I sympathize with your plight.

As pointed out, determine has a more complex meaning than simple existence.
Logically, it's a difference between an assertion of uniqueness, given a presupposition of existence

`⊦ ¬(∃y) (P(y) ⋀ x ≠ y) | (∃x) P(x)`

and just a simple assertion of existence

`⊦ (∃x) P(x)`,

Or, in logical English, between

`Assert that [there is no y such that [P(y) and x ≠ y], given an x such that P(x)]`

which asserts uniqueness and presupposes existence like determine, and just the last proposition:

`Assert that [there is an x such that P(x)]`,

which asserts only the existence of some `x`, which is not necessarily unique.

Logic, unfortunately, works only for assertions and doesn't deal well with presuppositions, so one has to be careful about lexical substitutions.

Instead, you can use exist as an intransitive verb without There-insertion

• A line exists that bisects both closed curves [from the Ham Sandwich Theorem (n=2)]

The rule of Extraposition from Noun Phrase allows one to move the relative clause that bisects both closed curves to a position after the verb exists, so you get the same right-branching effect as There-insertion, except there's no there there.

Or, if you want to use there, you can use one of the (mostly) locative predicates that also govern There-insertion, like lie

• On a vertical line `r = ½` lie all known nontrivial zeroes of the Riemann zeta function.

Or you can refer to the proof or reasoning that got you there

• It can be shown that (there is) some `x` in `A` (that) meets/must meet these criteria.
• There is nothing wrong with the grammar described in this answer, but I'm not sure it is appropriate for a mathematical paper. "In a hole in the ground there lived a Hobbit" is a good way to start a novel, but "On a vertical line lie all known zeros of the zeta function" is not a good way to write math, IMO. One of the nice things about math papers in a foreign language is that learning enough Russian, Japanese, etc., vocabulary and grammar to read them is trivial, compared with the effort required to understand the math itself. IMO the same should be true of a math paper in English. Commented May 1, 2017 at 2:36
• If the OP wants to write better math, then read some mathematicians who wrote well in English - for example Bertrand Russell, G. H. Hardy, Richard Courant & Bertrand Robbins.(No doubt there are more modern examples, but those authors' writing is timeless.) Commented May 1, 2017 at 2:44
• @alephzero: Nobody's writing is timeless. Commented May 1, 2017 at 11:33

When mathematicians use the phrase

X and Y determine Z,

they mean that, given X and Y, a unique Z exists.

Thus, if one believes that the standard rules about the passive construction apply here,

there is determined an object Z

would mean that a unique object Z exists, so it is certainly not a synonym for "there exists".

I wouldn't even use it in cases where a unique Z exists, because it sounds rather unidiomatic to me. But for those cases, I won't say that it is categorically wrong. And Googling, one finds this expression has been used in a handful of math papers.

• My guess is that the there is determined comes from translation/influence from non-English languages. I've caught myself using a similar construction with a host of verbs in a translation I've been working on the past few weeks. Commented Apr 30, 2017 at 20:59

Firstly, "there is determined" doesn't mean the same thing as "there exists." "There is determined" implies that something was discovered or deduced, whereas "there exists" merely implies that the thing is.

But a more broadly relevant point is that

## parallel construction is good style.

From The Elements of Style by Strunk & White:

Express co-ordinate ideas in similar form.

This principle, that of parallel construction, requires that expressions of similar content and function should be outwardly similar. The likeness of form enables the reader to recognize more readily the likeness of content and function. Familiar instances from the Bible are the Ten Commandments, the Beatitudes, and the petitions of the Lord's Prayer.

The unskilful writer often violates this principle, from a mistaken belief that he should constantly vary the form of his expressions. It is true that in repeating a statement in order to emphasize it he may have need to vary its form. For illustration, see the paragraph from Stevenson quoted under Rule 10. But apart from this, he should follow the principle of parallel construction.

Formerly, science was taught by the textbook method, while now the laboratory method is employed.

Formerly, science was taught by the textbook method; now it is taught by the laboratory method.

In the example above, the second is preferable, because parallel construction is easier to understand and has an element of rhetorical aesthetic.

• @PeterShor Here's a quote from Barack Obama: To give workers the power, to unionize for better wages, to update the social safety net to reflect the way we live now, and make more reforms to the tax code. He uses parallel form to the benefit of his rhetoric. But I'm more immediately curious why he didn't say "to make??" Commented Apr 30, 2017 at 16:28
• I'm not convinced that these recommendations apply to my case. The motivation provided by Peter Shor sounds much more convincing to me. Style is a personal thing, and besides that, I don't have only in mind situations where the ideas I want to convey are coordinate in the sense of this answer. Commented Apr 30, 2017 at 16:36
• @SalvoTringali then you should do what feels right to you. Style is a personal thing -- I agree with that completely, I simply wanted to offer a perspective Commented Apr 30, 2017 at 16:38
• Even Strunk and White can't get the usefulness of parallel structure wrong. It's simple programming -- once you've loaded the parsing routine for one constituent, you're primed to parse the next ones very easily if they have the same structure. it's a way of getting into the reader's software and massaging it, which is what every writer wants to do. Commented Apr 30, 2017 at 16:41
• @RaceYouAnytime Yes, and that's why my question is about the correctness or legitimacy of the use I'm making of the expression "there is determined", which should be something slightly more objective than style. Commented Apr 30, 2017 at 16:43