Many classical theologians and philosophers conceived of God, not only as the cause of the existence of everything, but also as that towards which everything naturally tends, the final cause of everything. To put it in more Platonic language, God is the Good, after which everything seeks.
On this scheme of things, we might understand God's providential work along the lines of attraction: everything in some sense has some kind of awareness and attraction towards the Good, as is evidenced by the fact that everything seeks to stay alive and to enjoy a good life; consequently we infer that they are attracted in some mysterious way by the Good Itself to do this.
Now of course in the case of human persons this searching after the Good takes place through the faculties of our intellect and will: ceteris paribus, we consider something to be good, and therefore we seek after it. The Good itself, too, draws us in this way, since we would seem to have some kind of natural knowledge and awareness of it. But because we may oftentimes have mistaken conceptions of what is good -- e.g., you thought that slice of pizza was going to hit the spot but actually it gave you heartburn -- we don't always achieve the Good. Still over the course of our history, we begin to learn what things are good and what aren't, what bring fulfillment and what leave us empty.
On this view of things, it is a part of our very constitution that we seek after God/the Good. The Bible affirms this when Paul says to the philosophers at Athens that God made things such that people would search for God and perhaps grope for him and find him -- though indeed he is not far from each one of us (Acts 17.27). We have an inner attraction towards this primal and implicit perception of the Good, and we seek after it in everything that we do. This is the way God has made us.
I think this is a fine way to understand the general structure of God's providence. Whereas other traditions speak of decrees and covenants and so on, on this view of things we understand God's workings to be a part of the very structure of the universe itself. The world itself is infused with God's providence, rather than God standing apart from it and imposing upon it some plan or wish. Importantly, too, God's providence leads all things to himself, a point which Christian Platonists such as Origen, Gregory Nyssen, and many others affirmed.
With this in mind, I think we can make the following point about Paul's comments in Romans 9, that most hotly contented passage in the debate surrounding Calvinism and Arminianism, predestination and free will, etc.
It is clear that God in his providence allows us to err and wander in the search for him, though his intention is always that we find him and learn through our experiences. It may well be that on occasion God even permits that some persons be hardened and ardently refuse to bow the knee to God, that through the punishment of their sin his opposition to evil be known, as well as his appreciation for righteousness when the righteous are spared. The punishment of the unrepentant wicked and the insistence that justice be served is an important part of sustaining the moral integrity and intelligibility of the world God has created, and provides realistic moral motivation (see the discussion here). We see these things happen and we learn from them.
(Some persons may point out that God at Rom 9 is said to have hardened Pharaoh; that sounds like more than mere 'permitting.' But the point is not so obvious. The sun hardens clay and melts ice. It is not that the sun specifically seeks to harden the clay while with a different act it melts the ice; rather the sun shines all the same, and given the constitution of clay, it is hardened, whereas given the constitution of the ice, it is melted. So also when God works to liberate the enslaved Hebrews in his unceasing pursuit of the good of all, Pharaoh who is inappropriately disposed to God is hardened by what happens when his idolatry is exposed as a sham, and his ego is crushed. The language of hardening does not necessarily ascribe specific intention to harden in the grammatically active agent, just as Paul annoyed Peter does not entail that Paul intended to annoy Peter. It may be that Peter was annoyed without Paul's knowing or intending it.)
It would be a grave mistake, however, ever to suppose that God's punishment of some wicked person is final, that it is the end of God's dealings for that person. That contradicts the starting premise that God attracts all beings towards himself, and desires that they seek him and know him; it contradicts God's providence by which he put in us a desire for him, the Good. Rather the punishment of some and the sparing of others is something done in the wisdom of God for the sake of all, that all, the punished and the spared alike, may ultimately find him and enjoy him.
If the Bible agreed with my point here, there would be evidence of this; and in fact there is. Both before and after the troublesome ninth chapter of Romans, there are unequivocal affirmations of the universality of God's salvation: the universality of justification by the faithfulness of Christ is affirmed at Rom 3.21-6; the universal restoration of Adam's fall is affirmed in Rom 5.12ff.; and the salvation of the totality of the Gentiles, after which the hardened Jews will likewise be restored, is affirmed in Rom 11. Indeed Paul's discussion ends with the affirmation that God has imprisoned all in disobedience that he may be merciful to all (11.32). Even later, in a discussion of the life of Christians as servants to the Lord Jesus, he says: For to this end Christ died and lived again, so that he might be Lord of both the dead and the living (14.9).
It would be an outrage to extract from the brief statements in Rom 9 a doctrine of predestination which is not even explicitly affirmed, since Paul never speaks of final destinies or eternal damnation in Rom 9 in any case, but especially so when the universally salvific plan and work of God in Christ is unequivocally affirmed on either side of the chapter within the same letter. Moreover the reading I have suggested here is in keeping with the classical doctrine of God as the Good, as the final cause of all things.