Lets examine two possibilities, and the evidence for each. One the one hand, there is the possibility that Paul knew nothing or next to nothing about the historical Jesus. Supporting this possibility is the almost complete lack in Pauls letters of any details of the Jesus tradition recorded in the gospels. Paul says little to nothing about the earthly Jesus.
Another possibility is that Paul knew a great deal about Jesus, but either expected his readers/hearers to be acquainted with the Jesus tradition already, or he just didnt care, or he felt that talking about the earthly Jesus (whom he didnt follow) would undermine his fragile apostleship, or some combination. Id like to go into the evidence for this possibility in some detail.
1. Prima facie evidence.
We may note first of all that Paul was in a pretty ideal place to know a great deal about Jesus. Jesus was a contemporary of his, and he is the only Christian write we can say for sure was around for a pretty long period of time while Jesus was (some estimates put their birth around the same years). We also know that he cared a great deal (and took the time to learn) about what his fellow Jews and Christians and even pagans were preaching and teaching. After all, he spent a few years prior to converting (which probably happened only a few years after Jesus was crucified) persecuting the earliest Christians, and it is not unlikely he took the time to find out at least a bit about Jesus and who he was and what his followers were preaching even then, and perhaps even while Jesus was living. We dont know, but given his active nature and the fact that he did care, both before and after converting, what other groups traditions were (even if it was only to persecute/revile them), it seems unlikely that he didnt find out a fair amount of the Jesus tradition even before converting. Once he converted, again he is the only early Christian author whom no one disputes in terms of his centrality in the earliest stage of Christianity. Personally, I have no problem taking the author of Luke/Acts at his or her word when s/he uses the first person, nor do I find it unbelievable that an actual living beloved disciple of Jesus stands behind the text (i.e. it was by a disciple of him), but while I am far from being alone here, there are plenty of those who disagree. Paul, however, was in the thick of it. He knew Jesus brother. He knew Peter. He knew James. He was an incredibly active person in the earliest stages of the Christian community which was part of a wider oral culture. We know that the earliest Christians spent a great deal of time telling the stories of Jesus. And we know that within a very surprisingly short period of time these were written down. It seems highly unlikely that Paul, in the thick of it, a guy wanting to know whats going on, who knew the eyewitnesses, in an oral culture, someone how almost completely avoided learning anything about Jesus despite the many oral traditions circulating.
2. Evidence of the tradition in Paul
As noted earlier, there is a remarkable (if Paul knew more than he wrote) lack of correlating information in Paul with the Jesus tradition. Yet there are some interesting parallels. For example, there is the Last supper recorded in 1 Cor. 11:23-26, which is found within the gospel tradition (esp. Lk. 22.19-20). That Paul received this orally (and in a fixed manner) is given not only by the fact that it is so similar to parts of the lukan version but also because of the technical words parelabon and paredoka. These both have a long tradition within oral Jewish culture as technical terms for the giving and receiving of specifically oral teachings, in both Greek and Hebrew (corresponding to מסר/maser
and קבּל/qibbel). On this see especially M.S. Jaffe (2001) Torah in the Mouth: Writing, and Oral Tradition in Palestinian Judaism 200 BE-400 CE (Oxford: Oxford University Press). The same terms are also found within early Christian texts, both in the gospels, other epistles (not just Pauls), Didache, etc. Gerhardssons analysis of both Christian oral transmission and rabbinic oral transmission is instructive here because (and we will see more evidence for this later) it suggests (as does the fixed form of this particular tradition) that someone official instructed Paul, making him memorize a fixed oral text, not at all uncommon even outside Judaism in the ancient world.
While this may be the only time Paul demonstrates so aptly that he was instructed with a fixed formula of the Jesus tradition, he does either allude or explicitly cite others. He mentions Jesus teaching on divorce. Paul also mentions the instruction of Jesus in 1 Cor. 9:14, but does not tell us anything about it (i.e. the exact formula that was passed on). He merely gives the content. Then there is 1 Thess. 5.1-4 which is found in Q (Lk. 12:39/Mt. 24:43). Paul does not recite the whole parable, but he states quite interestingly in 5.2 that ἡ ἡμέρα Κυρίου ὡς κλέπτης ἐν νυκτὶ οὕτως ἔρχεται/the day of the Lord comes in this way as a thief in the night. The parable isnt there, the content is (the return of Jesus). Also of note is the comparison to a thief which is also in Q. Also interesting is Pauls take on food laws, where he again cites Jesus as his source (and we know from Pauls teachings on divorce that he distinguishes his own teachings from those handed down from Jesus). In Rom. 14:14 Paul states ὅτι οὐδὲν κοινὸν δι᾿ ἑαυτοῦ/that there is nothing unclean by itself which is strikingly similar to the Jesus tradition in Mark and Luke (7.15 and 11.41, respectively).
These are only a few examples, and there are only a couple more. But we should keep in mind a few things: we dont have all of Pauls letters, we dont have all the Jesus tradition that was circulating in the early Christian decades, and we know that Paul liked to concentrate on the risen Jesus and distance himself from the earthly man he likely never knew and certainly didnt follow. What is striking is that Paul was clearly aware of some of Jesus teachings, and was clearly instructed in them (rather than just having heard them). Which brings me to my next point.
1. Paul and Kephas.
There are a number of important analyses of Paul and Kephas time together referred to in Galatians. Paul went up and spent 2 weeks with just Peter. The verb used to describe their conversation type, historesai, is interesting (see e.g. Kilpatrick, G. D. (1959). HISTORESAI KEPHAN in N.T. Studies in memory of T. W. Watson and Bauernfeind, O. (1956). Die Begnegnung zwischen Paulus und Kephas Gal 1.18-20 Zeitschrift für die Neutestamentliche Wissenschaft 47). It is not simply a verb of conversation, or getting to know someone, but means to get information from. In other words, Paul spent 14 days with the head of the Jerusalem church, the head of Jesus disciples, learning from him. Now, Paul doesnt talk about what he learned, or note why he went, but this is very likely due not only to the strained relationship between the two but also because Paul is trying to get credit as a legitimate apostle and he was vying for authority with Peter. Nonetheless, he mentions this extended learning session. As C. H. Dodd said, they could hardly have been talking about the weather. What information did Paul need from Peter that would take fourteen days to get? Clear instruction from the head disciple on the Jesus tradition.
Ill wrap it up for now and allow time for rebuttal. Heres what weve got so far. First, there is good reason to think that Paul knew the Jesus tradition simply because he spent a lot of time out and about talking to people and learning what was going on, and that he had several reasons not to talk about the tradition in his letters. However, weve also seen that Paul did recite or allude to or paraphrase some parts of the Jesus tradition we find in the gospels, even in Q. From the words he uses, and the fixed aspect of one part of the tradition he cites, we can see he was instructed. And, finally, we know that Paul spent an awful long time being instructed by the head disciple/apostle Peter. It seems unreasonable to conclude anything other than that Paul knew a whole lot more about the historical Jesus than he discussed.