Learn English – “Such that” vs. “Subject to”

abbreviationsmathematics

In the attached youtube video, at 13:12, the lecturer gives a footnote on the difference between Such that and Subject to when expanding the initials mathematical abbreviation S.T. in certain mathematics problems.

He claims that some people read this as Such that and some as Subject to where the former is a common mistake. In his words: "subjugate to followed by restriction is not the same as such that, restrictions" .

Can anyone elaborate why such that, restrictions, as the lecturer states it, is wrong? In general, which is the correct form and where lies the difference between the two?

Best Answer

tl;dr : Use either one, unless you have a specific stylistic issue with one.

Here's what professor Stephen Boyd says, adjusted slightly to make the spoken version more readable.

This is a question of style, and, by the way, this is bad style.

and

Nevertheless ... it's somewhere in the yellow range ... it's not what such that means in English ... subject to are the English words that say exactly what happens when you do something subject to restriction. You don't do something such that restrictions. That's actually not ... English. You will see this, but, ah, not from me you won't.

Prof. Boyd can't dismiss the usage entirely, because he knows that other members of the mathematics community use such that, but he draws a stylistic line that urges caution ("the yellow range"), and he makes a linguistic argument roughly like the following:

  1. mathematical usage should always map back to English usage
  2. "such that restriction," as an expression, does not exist in English usage
  3. therefore "such that restriction" violates rule 1 and
  4. it should not be mathematical usage

I have two objections to this that will also answer your question. Each objection demonstrates that both usages are correct from the standpoint of language usage, even if one entails a certain risk of frustrating someone who shares Prof. Boyd's pedantry.

Such that can be used in English to indicate a logical restriction. Such that acts as a conjunction. One use is to indicate the logical result or extent of an event:

The roads were icy such that it was difficult to drive.

This usage is close to so that.

Such that is also used in English to indicate a condition or restriction. As the answer by Hellion here states, such that expresses "HOW something is to be done." A mathematical restriction pertains to one meaning of how: the solution(s) of x (the results) must fit the conditions or restrictions; that is how it has to be solved.

This usage works even outside of math. Oxford Dictionaries gives this sentence as an example for such that:

‘the linking of sentences such that they constitute a narrative’

Constituting a narrative is a condition or restriction on how sentences are linked. They explain in part how I should link sentences. I need to be aware of how the linking builds a narrative. So English does use such that to set restrictions, and therefore similar usages should be valid in math.

Such that is established mathematical usage irrespective of 1.

Such that is widely used in the mathematical sciences to introduce conditions, properties, and restrictions that must be satisfied. Wolfram weighs in:

A condition used in the definition of a mathematical object, commonly denoted : or |.

This usage was standardized by the time Bertrand Russell wrote The Principles of Mathematics (1903), a philosophical study of mathematics that required him to delve deeply into mathematical terminology. He writes a section on using such that, which begins:

This brings me to the notion of such that. The values of x which render a propositional function ϕx true are like the roots of an equation - indeed the latter are a particular case of the former - and we may consider all the values of x such that ϕx is true (p. 20).

Such that ϕx is true sets a condition or restriction on the values of x being found.

I've found similar advice in writing guides for mathematics, as a search for "such that" and math will turn up. This one, provided by authors Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling at UC Davis (2007), explains "such that":

(The such that sign) means “under the condition that”. However, it is much more common (and less ambiguous) to just abbreviate “such that” as “s.t.”.

Common collocations of "such that" include "there exists" and "there is," as seen in this sample:

For any sets A and B we can assume there is a set S such that A, B ⊂ S

Nonetheless, the expression such that does not have a necessary structure, and can be applied to results or conditions. So it is an accepted mathematical usage that can be used for s.t.

What about subject to?? It's almost synonymous. This article never differentiates between the two:

In economics, the letters "s.t." are used as an abbreviation for the phrases "subject to" or "such that" in an equation.

And the top answer on what subject to means in this question resorts to using such that to explain its meaning:

It is a way to specify constraints. To put it very simply, the problem "do 'X' subject to 'Y'" means that, you have to do "X" (whatever X is), but you have to do it such that "Y" is also satisfied in the process.

It might be possible to differentiate them more strictly through some rule where subject to always defines constraints and such that always gives results. I think that's what Prof. Boyd would prefer, and there may be an argument for precision in that. However, that's not how English works, and that's not how mathematics currently uses the two terms.