2 Corinthians 10:5 mentions the ‘casting down of reasonings’ but this phrase is omitted in the Douay-Rheims Vulgate version. Why is this the case, and which translation should we trust?
The Vulgate is a Latin translation of the Greek New Testament, so I am not all that sure why the content of the Vulgate is of much concern to Christians. The Douay-Rheims Vulgate is an English translation of the Latin translation, making it even less important to most believers (except some Roman Catholics). Bear this in mind.
The Greek has the two words logismous kathairountes, which means, literally, “arguments, overthrowing” Therefore, this phrase, which you have as “casting down of reasonings” (from the KJV?) belongs in the text. I found the Vulgate of this passage. It is: et omnem altitudinem extollentem se adversus scientiam Dei, et in captivitatem redigentes omnem intellectum in obsequium Christi. The initial phrase, essentially, means casting down of all arguments, so the phrase is in the Latin Vulgate. I looked at the Douay-Rheims, and it seems to pull the phrase “demolish arguments” into 2 Cor 10:4. In other words, it is a bit of a paraphrase, and the word order has been moved around enough that the first two words of the Greek of 2 Cor 10:5 find themselves in 2 Cor 10:4. My conclusion is that there is no important controversy here, although some folks might want to find one.