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Matthew and Luke clearly have different objectives with their genealogies, so we should expect differences. Here are two possible explanations for the differences:
At first glace, the genealogies of Matthew and Luke appear to be contradictory. Matthew's genealogy traces Jesus back through David's son Solomon, and Luke's genealogy traces Jesus back through David's son Nathan. Also, Matthew states that Joseph's father is Jacob, and Luke states that Joseph's father is Heli.
- And David was the father of Solomon by the wife of Uriah...
- and Jacob the father of Joseph the husband of Mary, of whom Jesus was born, who is called Christ.
- Jesus, when he began his ministry, was about thirty years of age, being the son (as was supposed) of Joseph, the son of Heli...
- the son of Melea, the son of Menna, the son of Mattatha, the son of Nathan, the son of David...
The seeming discrepancies between Matthew and Luke's genealogies are definitely not certain contradictions. There are several good reasons to believe that there is no contradiction between the two authors.
The most significant factor to consider is that it is clear that Matthew and Luke are not trying to accomplish the same things with their genealogies. Here are some significant differences:
The fact that Matthew and Luke's genealogies clearly have different purposes and themes helps us approach the differences between them with the question, "Why are they different?", rather than with the declaration, "They are contradictory."
One possible explanation for the seeming discrepancy between the two genealogies is that Matthew's genealogy is of Joseph's, while Luke's genealogy is of Mary's. There was no Greek word for "son-in-law," so perhaps Joseph was the "son of Heli" in the sense that he was married to Mary, Heli's daughter. Then, Joseph would have descended from David's son Solomon, while Mary would have descended from David's son Nathan.
Although it would have been unusual to trace a genealogy through a mother's side (Mary's side), Mary was an extraordinary woman. Furthermore, Luke's genealogy does not occur until later in the book, after Luke has introduced Mary and told part of the story from her point of view.
Another possible explanation, which was held by the church historian Eusebius, is that Matthew's genealogy is a biological genealogy, while Luke's genealogy is legal genealogy. In Jewish tradition, if a man died without having any sons, his brother would marry the widow and have a son who could carry on the family name. Eusebius noted that Melchi (Luke 3:24
Then, perhaps, when Heli died without a son, his half-brother Jacob married Heli's widow. This would mean that Joseph is legally the son of Heli, but biologically the son of Jacob. Then, Jacob would be the descendent of Solomon, and Heli would be the descendent of Nathan.
Because of all of the factors listed above, we should assume that Matthew and Luke would not write genealogies that are so directly contradictory. It is likely that they would have been able to double-check their facts, especially because genealogies were extremely important to the Jews. Furthermore, it is likely they used similar sources, and perhaps even communicated with one another, before writing their genealogies, in which case it would be silly to believe that they would write contradictory genealogies.
Some people have trouble accepting possible explanations to alleged contradictions in the Bible, especially ones that they consider a stretch. For example, Dan Barker, president of the Freedom From Religion Foundation, says this:
I have received numerous replies from Christians who think that these contradictions are either trivial or easily explained. Yet not a single "explanation" has been convincing.
The fact is that there are plausible explanations for every alleged contradiction in the Bible. That there is a plausible explanation for an alleged contradiction does not mean that it is definitely the correct explanation for the alleged contradiction.
However, as long as a possible explanation has been suggested, then it has been objectively demonstrated that there is no necessary contradiction regarding the Bible verses and passages brought up.
When people like Dan Barker say that they don't find a particular explanation for a contradiction "convincing," then that is merely their opinion. A plausible explanation has been suggested that eliminates the necessary alleged contradiction. They simply don't like it, which is not at all a relevant argument against the explanation.
To read more about solutions to Bible contradictions and difficulties, check out Norman Geisler's The Big Book of Bible Difficulties: Clear and Concise Answers from Genesis to Revelation. While we do not agree with some of Geisler's theology, particularly concerning his view of predestination, this book is still an excellent resource. It is thorough and filled with research.
Another book to check out is Tim Chaffey's Demolishing Supposed Bible Contradictions: Exploring Forty Alleged Contraditions, which also answers many alleged contradictions in the Bible.