I'm under the impression that religious Judaism believes in absolute truths. That being the case, how does the concept of "eilu v'eilu divrei elokim chaim" (Eruvin 13b) work out? When two scholars argue, how can they both be right?
A simple parable for this idea is the idea of projection. If one looks at a cylinder, for example, it could be a square or a circle. In reality it is both, or neither.
Torah, in principle, is higher then human understanding. Therefore, as the Torah comes into human understanding it gets "filtered" through their brain (even two prophets would not use the same wording). Therefore, if someone applies the Torah approach to a problem and gets to one resolution, the result is true.
However, practically, as it comes into the world, there can be only one truth, which is known as "Halacha". Therefore, when one asks a shaila, he is expecting "what to do", not "there are two approaches" because this world, a physical world, is limited in the sense that one cannot do two opposite things at a time. Therefore, the halacha could go like either one or the other, and halachically, the other could be rejected completely, as if it wasn't there (Beis Shamai v Beis Hillel). In the source of Halacha, in the higher worlds, however, they are both true.
The short of your answer I believe is that Judaism does not believe in absolute truths.
However, it is true that many Jews do believe in absolute truths, so I'll answer the question regarding that subset of Jews.
The general explanation given for Eilu v'Eilu amongst people who believe that something either is or isn't, is that eilu v'eilu refers to eight different time periods.
Many people will try to explain 'eilu v'eilu' as refering to a difference of opinion based on which of those 8 time periods you are talking about. That is, those 8 time periods are so vastly different from eachother, that the halacha might have conflicting outcomes.
Alternatively, one might argue that even within the same time period, the circumstances of each event help determine which of the two eilus is correct at any given moment.
There are certainly absolute truths in regard to things that are known or things that are fundamental principles - it is absolutely true that HaShem created the universe and redeemed the Israelites from Egypt, for example.
When it comes to things that are really unknowable, the best we can do is base our understanding of them on sound principles handed down to us from Sinai.
One subset of unknowable things is, ironically, Halachah. What I mean by that is that we cannot know with absolute certainty (ie., we cannot know the absolute truth about) what HaShem wants us to do in any given situation.
Certain things are easier to determine than others - we can know that lighting a camp fire on Shabbath is prohibited. That is explicitly forbidden by the Torah. What about operating an electric wheelchair? That would depend on the definition of certain types of Melachah and the design of the wheelchair, and whether or not the neighborhood is in a public domain. Depending on the situation, and also depending on the analysis of the applicable laws, one Posek can reach a different conclusion than another.
The same is true for things that we would like to know outside the realm of laws. What happens to us when we die? What do certain esoteric verses mean when they describe the Heavenly Court? We cannot know these things for certain. Because of that we have to rely on the Sinaitic principles that we have, and different scholars can reach different conclusions based on the limited knowledge that we do have (verses, primarily, as well as earlier scholars' analyses of the same subject).
That is Eilu VeEilu in a nutshell.
There are many different interpretations of "Elu V'Elu". It is rather difficult to say that it literally means both sides of an argument can be true, since it is applied to disputes of facts. E.g. what were the events of a story, or what did so-and-so hold. They cannot both be literally true, since they were events that actually happened.
A more likely explanation is that "Elu V'elu" means both sides are legitimate interpretations. While only one of them may in fact be true, at the time of the dispute it is not possible to prove one side wrong. And one cannot ask God, since "Lo B'shamayim He" (The Torah is no longer in heaven, it has been given over to man to interpret). Since we cannot prove which side is right, a person can learn both sides without worrying that one of them is wrong.
Absolute truth exists, but in certain areas of dispute it is not possible to find it. So both sides are considered the word of God.
I think that what has to be discussed is your basic premise that Judaism believes in absolute truths. Judaism believes that Hashem is the absolute ruler and that he gave the Torah to Israel. There is also a concept of "Lo Bashamyim Hee" i.e. the Torah was given to Israel to "figure" out to the best of our ability based on rules passed down through tradition. Whatever is honestly derived based on these rules is given credence by Hashem that it is the truth, as exemplified by the incident of Rabbi Eliezer and the Rabanan. Hashem will judge man based on what is in the Shulchan Aruch. Absolute truths is basically irrelevant. Regarding Halacha the Torah instructs us to go after the majority for practical purposes. This never implies that the losing opinion was false.
"There are 70 faces to the Torah" (Bamidbar Rabbah 13:15)
There's a difference between an explanation of a Torah verse that is wrong, and an explanation that our sages chose not to follow.
Shammai wasn't wrong. He was a scholar equally as great as Hillel. As Avi alludes to in his answer, there is a concept that when the messiah comes, we will follow Shammai's views. Actually, we do follow at least eighteen of Shammai's rulings (see mishanyos shabbos chapter 1).
In any case, the Heavenly Voice declaring that "These and these are the words of the living G-d" was simply teaching that Shammai was a genuine Torah scholar who sincerely reached a different understanding.
This is in stark contrast to the Reform movement, who simultaneously disavows the halachic process, and perverts it for its own deceitful purposes.
"Eilu v'eilu" does NOT mean that anything anyone says on a given Torah verse is valid.
Intentionally misreading Torah verses to obliterate severe negative commandments is wrong, misleading to uneducated Jews, and has no basis in Jewish practice. Using "eilu v'eilu" to justify negation of Torah law in favor of secular morality is a total fraud.
"70 faces to the Torah" means that, even today, there are varying practices that can all be classified as proper Jewish observance.
An Ashkenazi Jew can walk into a Sefardi or Yemenite or Persian synagogue, and fulfill his prayer obligations there. That same Ashkenazi Jew can enter the home of any of the above (assuming that the family is observant) and eat in their home. A European chassidic Jew from America can observe the sabbath with a Tunisian Jew from Israel (I've seen this myself), and the two will recognize each other's actions as that of a sabbath-observant Jew - even though the specific customs, mode of dress, accent in pronouncing the blessings and prayers etc, are quite different.
Judaism believes in an absolute Truth; halacha. However, there are many valid ways to fulfill halacha, and they all look, sound, feel, smell, and taste different from each other.
In the same way the square root of 4 is 2 and negative 2. Source- R Yaakov Weinberg of Ner Yisroel
I would like to offer a different perspective which is my own conjecture. I think we need to examine the full phrase which is "eilu v'eilu divrei elokim chaim" - these and these are the words of the living God (emphasis added). Then you need to ask, "why would God say both of these things?" The answer is because God learns torah!
The discussion in Gittin 6b is enlightening
True, in that discussion there was a way to make both sides of the argument be "right". But, the principle is instructive. Eilu v'eilu doesn't necessarily mean you are both right or both things are true, it means that God is saying over both of these opinions this portion of study is reviewed in the heavenly study hall!
I would furthermore postulate that only certain select arguments are of such high caliber that God himself deems them worthy of repetition, otherwise every argument would be 'eilu v'eilu'.