I've answered the question on the Hermeneutics site, which can be summarized as spiritual resurrection was an oxymoron at that point in history. I won't repeat that argument, but I will make a related argument based on what the Corinthians probably expected from a messiah.
Who were the Corinthians?
Corinth was destroyed by the Romans in 146 BC and reestablished as a colony by Julius Caesar in 44 BC. During the late Republic and early Empire, colonies were established in order to provide land to pay off and settle veterans. A hundred or so years later, when Paul arrived, it was a much more cosmopolitan city with a mixture of Romans, Greeks and Jews. The city's twin ports and strategic location meant that received much Roman attention. While the original legionary colonists had died off, it seems quite likely that their decedents would have had strong affinities to the Imperial cult—especially Divus Julius.
What would they have thought of Jesus?
Most Corinthians, we can imagine, never heard of Jesus. Of those who had, probably many supposed he had been rightly executed as a Zealot. But the primary audience of Paul's letters were Christian. They must have seen Jesus as worth following, even perhaps to the death:
Why are we in danger every hour? I protest, brothers, by my pride in you, which I have in Christ Jesus our Lord, I die every day! What do I gain if, humanly speaking, I fought with beasts at Ephesus?—1st Corinthians 15:30-32a (ESV)
Perhaps they saw Jesus in the mold of a Greek demigod, like Hercules. But more likely, they thought Jesus had experienced apotheosis upon his death in the same way many believed Julius Caesar had become a god. Caesar was not resurrected as his body was cremated in the Forum; his soul was believed to have ascended into heaven in the form of a comet. So the Corinthians could have interpreted stories of Jesus' transfiguration and his ascension in the same incorporeal manner.
There seem to have been a division in the Corinthian church (see 1st Corinthians 1:10-16 and all of chapter 3, for instance) along ethnic and cultural lines. The Jewish Christians would have understood Jesus as Messiah and looked forward to a general Resurrection event, but might not have put the two concepts together. If so, they didn't believed that Jesus bodily rose from the grave either. (But more likely, they became Christians after believing that the Christ rose from the tomb on the third day.)
What was Paul's argument?
Paul's task is to bring both sides of the Corinthian church into unity. The letter tackles a huge number of issues: sexual immorality, lawsuits, idols, worship, and spiritual gifts. His driving goal is to find ways of unifying the body. So when he gets to the topic of Resurrection, he must make a series of complicated arguments. But this is his abstract:
Now if Christ is proclaimed as raised from the dead, how can some of you say that there is no resurrection of the dead? But if there is no resurrection of the dead, then not even Christ has been raised. And if Christ has not been raised, then our preaching is in vain and your faith is in vain.—1st Corinthians 15:12-14 (ESV)
Paul's argument (and it carries throughout the chapter) was that what happened to Christ is fundamentally the same thing that we can expect to happen to us in the future. He leans hard on the word "resurrection" (x2 in these verses alone) and "raised" (x3). To his Jewish audience, the words would have clearly signaled an event that would look like:
Your dead shall live; their bodies shall rise.
You who dwell in the dust, awake and sing for joy!
For your dew is a dew of light,
and the earth will give birth to the dead.
—Isaiah 26:19 (ESV)
But notice that Paul directs his argument toward "some of you [who] say that there is no resurrection of the dead". Therefore, Paul basically sides with the Jewish Christians when it comes to resurrection theology. Thus, his argument must be opposed to the Greek and Roman views that Christ was a special case—an extraordinary man who was transported into heaven. So Paul stands on the side of the group who believes our bodies can somehow be transformed:
I tell you this, brothers: flesh and blood cannot inherit the kingdom of God, nor does the perishable inherit the imperishable. Behold! I tell you a mystery. We shall not all sleep, but we shall all be changed, in a moment, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, and the dead will be raised imperishable, and we shall be changed. For this perishable body must put on the imperishable, and this mortal body must put on immortality. When the perishable puts on the imperishable, and the mortal puts on immortality, then shall come to pass the saying that is written:
“Death is swallowed up in victory.”
“O death, where is your victory?
O death, where is your sting?”
The sting of death is sin, and the power of sin is the law. But thanks be to God, who gives us the victory through our Lord Jesus Christ.—1st Corinthians 15:50-57 (ESV)
Paul's argument assumes the Corinthians already believed Jesus had experienced a "spiritual" ascendance into heaven, so it makes little sense that he would convince them, in a strident tone, that Jesus' ascendance was just spiritual. Paul clearly expects them to believe something beyond their current beliefs: namely bodily resurrection