"Now if Christ is proclaimed as raised from the dead, how can some of you say that there is no resurrection of the dead?"The Corinthians are denying that there is such a thing as the resurrection of the dead. To be a bit more specific, they probably are not denying the idea that someone who has died can live again (that being a much more modern objection to the resurrection). After all, Paul surely preached that Christ had been raised and we can gather from Paul's argument here that Christ's resurrection is something on which he and the Corinthians agree. Instead, the Corinthians are eschewing Paul's belief that everyone will eventually be bodily resurrected from the dead. This is what Paul means he speaks of the resurrection: not merely that rising from the dead is possible but that at the end of the age all of humanity will be resurrected.
This denial of the resurrection was likely rooted in the Greek philosophical assumptions that seem to have been pervasive in Corinth. Among those assumptions was the belief that the material world, including our bodies, is basically evil and that the goal of life is to allow our good spirit to escape our corrupted flesh. As a result, the Corinthians thought that what it meant to be spiritual had very little to do with with their bodies.
In fact, the Corinthians rejection of the material world in their spirituality can even be seen in the series of issues which Paul addresses throughout the letter prior to chapter 15. The Corinthians think nothing of a man sleeping with his step mother or of engaging in prostitution because they reason that what they do with their bodies has nothing to do with their spirituality. They think of spirituality as an individual competition for spiritual knowledge rather than seeing themselves as a part of the Body of Christ and they exalt the spiritual gift of speaking in tongues above simple love and care for one another. The Corinthians make a habit of denying the physical, whether it be their own bodies or those around them, in favor of what they see as being more "spiritual". So it is easy to see why they would reject Paul's very Jewish notion of the resurrection of the dead which could just as easily be translated as "the raising of the corpses" (zombies anyone?). The crude image of a bunch of decaying bodies given life is about the furthest thing from the Corinthians definition of what it means to be spiritual.
It is fitting then that Paul's words concerning the resurrection come as his closing argument in his letter to the Corinthians. This is probably not an issue about which they have written to him but one about which he has heard reports and sees as foundational to everything else he has addressed. So Paul reminds the Corinthians that Jesus' resurrection is not an isolated event within history. Instead, it is the precursor of what will happen to all of us. We will all be raised with a new kind of body. Paul calls it a "spiritual body" in v.44, thus combining eternally the very two things, spirit and body, which the Corinthians wish to separate.
The truth is that much of the Church today still struggles to hold those two things, spirit and body, together in the way that Paul did. Often our ideas and language about our physical world resembles those of the Corinthians much more than they do Paul's. We speak of our soul going to heaven when we die rather than our hope that God will raise us up and give us an entirely new body. We struggle to articulate a meaningful sexual ethic that goes beyond a simple list of scriptural imperatives because, like the Corinthians, we're not really sure what sex has to do with spirituality anyway. We become to busy with our very "spiritual" ministries and gaining our own "spiritual" knowledge to be bothered with lowly, fleshly things like actually loving and serving others. We fail to challenge greed, consumerism, and injustice because we figure its only a matter of time before God's set this world aflame.
The resurrection forever declares to us that our bodies and this physical world matter to God. God created this stuff, inhabited this stuff, raised this stuff, and one day God will renew and transform this stuff. It is not an afterthought in God's work of salvation. It is this very stuff which is being saved.