How can we be sure that all nations across the world are actually able to read and interpret the same contextual intention of the source material, regardless of how cultural differences affect how each nation approach the Bible?

This might be a better question for /r/DebateAChristian or /r/AcademicBiblical then for /r/AskHistorians. If you ask there you will probably get better answers to "How can we know the Bible is Correctly translated?" Rather then descriptions of how Biblical translation works.

I am asking this because, if a Japanese bible translators sat down and translated the english bible, how can we be sure that the integrity of the english version is "correctly" jotted down into japanese?

In the Modern context, they would be translating from Greek/Hebrew, the original language, to Japanese. This results in one translation from the original to the Japanese text.

Now, I'm going to assume for a moment that you are not particularly interested in Japanese translations specifically. If you are this site will provide you with some more information about Japanese translations.

I'm mostly going to summarize Persistent Problems Confronting Bible Translators by Bruce M. Metzger. In this article, Bruce lays out several issues with Bible translation.

The First problem confronted by Bible Translators is variant manuscripts. In some manuscripts the words themselves are slightly different. You know how you sometimes don't include all the right words in a text message because you are in a hurry? It's similar to that. However, a historian can overcome this problem by analyzing several different copies of the same text and comparing them. An older document might have more weight then a younger document. A version that shows up three times in different lineages might have more weight then one that only shows up twice. This is external evidence of a reading. There is also internal evidence that can be used, such as how an author uses a word or phrase. The vast number of copies of the Bible allow us to put together a fairly accurate picture of what the original manuscript looked like.

After translators have decided which wording of the Hebrew or Greek text should be taken as the basis of the English rendering, the next problem has to do with ascertaining the precise meaning of the words. A huge tool here is context. What are they talking about? This is very similar to how we learn new words in general, what context are they being used in? We can also takeapart the words to figure out what they mean. For example, if we know what paper is and we know what a clip is we can make a guess at what a paperclip is. We can also use comparative linguistics to compare words in different languages. This is like how some words in French 'Bleu' are similar to the words that they translate to in English 'blue'

Another issue that confronts translators is punctuation. However, punctuation as we know it today was not invented in the time that the Bible was written. The position of a comma within a sentence can totally alter how it is understood. How do you break up the words? This is a question that has led to some theological dispute. The key here is to understand what the thought is that is being communicated.

Yet another problem confronting translators arises when a Hebrew or a Greek word can be either translated or transliterated. What should be done with proper names that can also be used as common nouns? For example, imagine if you had a guy named Wolf who worked with Wolves. Things could get confusing quickly. Once again, context is key.

There is also stylistic issues such as how closely do you want to mirror the text vs wanting the text to be understandable. In Korean a idiom that is translated 'playboy' technically means 'windy boy'.

That is an overview of some of the issues that can be encountered in translating the Bible.

I'll note however, that many Christians don't consider their translation to be authoritative, but the autographic text. The Chicago Statement on Biblical Inerrancy states the following in article X.

"WE AFFIRM that inspiration, strictly speaking, applies only to the autographic text of Scripture, which in the providence of God can be ascertained from available manuscripts with great accuracy. We further affirm that copies and translations of Scripture are the Word of God to the extent that they faithfully represent the original. WE DENY that any essential element of the Christian faith is affected by the absence of the autographs. We further deny that this absence renders the assertion of Biblical inerrancy invalid or irrelevant."

To complement /u/SobanSa's answer, in a sense this is no different than the broader question, "Is translation possible?" to which Translation theory basically says, "Yes, but."

Modern translations are rarely based on an English version, they almost always involve a translation project built upon critical editions of the Hebrew/Aramaic/Greek texts of the Bible, so there's no 'long-way-round' problem of translating through multiple languages.

"How can we be sure?" - the only way you can evaluate translations is if you yourself have a solid grasp of the source language and the target language, and the requisite cultural/background knowledge in both as well. Then you can say, "Yes, this is a pretty accurate translation" or "No, something is amiss here."

All translation involves betrayal - a choice to preserve one element while sacrificing another. That's not bible translation, that's all translation. So, yes, there is loss in translation (otherwise no historian would need to learn original languages for their research, would they?). But that gap is often filled by other material - in the case of the Bible, commentary, notes, etc..

How can we be sure that all nations across the world are actually able to read and interpret the same contextual intention of the source material, regardless of how cultural differences affect how each nation approach the Bible?

In a sense, you can't. But I'm not sure why anyone would think they can - you can't guarantee or control this process. All you can ask is, "what is the cultural and historical context of the source material?", "what is the cultural and historical context of the target language into which it's being translated?", "how well does a particular translation represent the meaning of the source text in the cultural and historical context of the target language?".

You probably would get further discussion on /r/AcademicBiblical but I wouldn't bother /r/DebateAChristian, since the latter is not devoted to academic discourse at all.

I'm happy to answer further questions (I have studied Hebrew, work with Ancient Greek daily, and have quite a few other languages under my belt), or provide some follow up reading and references if you like.

There are several cases in which translations have to make the choice between the text and its spirit. For example, there are Psalms that are acrostics, with one verse starting with aleph, bet, gimel, dalet, etc. Do you preserve the acrostic structure and lose the meaning of the words or translate the words and lose the acrostic?

One of my favorite successful translations involve Judges 10:4, which involves a pun in Hebrew and describes Jair, whose 30 sons owned 30 towns and rode on 30 donkeys. Threr is a delightful Jewish translation that describes Jair's sons as "owning 30 boroughs and riding 30 burros."