Is something 'best possible' or 'the best possible?'

Question

Which one is correct at the end of the sentence, best possible or the best possible?

In a science paper (not in a title), should one write

This constant is best possible

or

This constant is the best possible?

Example.

Theorem: every positive number is >0.

This theorem is (the) best possible. (Since you cannot write a number bigger than 0 instead of 0, that would make the theorem false.)

An example for non-mathematicians. Suppose we talk of hobbits, who are at most 4 feet tall. If Bilbo is exactly 4 feet tall, then would you say

Bilbo is the tallest possible

or

Bilbo is tallest possible?

Answer 1

As medica pointed out in comments, "Possible + countable noun can take an indefinite article (a/some) but best necessitates a definite article." So in everyday English one would only ever say "the best possible".

The mathematics link establishes a convention used by that particular author. It says:

"Best possible". "Best possible" is an adjective; it indicates sharpness. We write "This result is best possible", just as we would write "This result is sharp". Writing "This result is the best possible" says that this result is better or more valuable aesthetically than all other results in the world, which is not what is meant. The definite article should not be used here. Think of "best possible" as a technical term that is already a specific predicate adjective, so no definite article is needed.

Whether that convention is more widespread among other mathematicians, I can't say. I've never encountered it personally among numerous math/science/engineering classes.

Obviously scientific/technical fields make up technical terms all the time, but when you take words that have established meaning and use them in a completely counter-intuitive way, you're just begging to be misunderstood.

"Best possible" uses the superlative. Whether you include the definite article or not, it clearly implies that the result is the best among all possible results- which is completely the opposite of what the author supposedly wants to say.

Answer 2

The superlative form of adjectives is in fact very often used without any article at all. The time when this is most likely (please forgive me, no irony intended), is when the adjective is occurring as a predicative complement without a following noun.

• This is most convenient.
• John is tallest.

It is even more common when the particular times, events or situations in which X will be most or least likely/possible/convenient and so forth are specified afterwards:

• This is most/best/least possible when oxygen levels are very high.
• The leaves will be greenest, if the plant is well fed with some kind of plant food.

However, best possible in the example that you give seems to be a fixed phrase within the register of mathematical academia - as is pointed out in Lynn's answer. Used in this way, it seems to be the case that a best possible constant is a constant of a particular standard, quality or type, whereas the best possible constant is the most meritorious amongst a number of given constants.