"There exists" vs "There is determined"

Question

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!).

Answer 1

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.

Answer 2

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.

Answer 3

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.