I agree that "natural" is a more accurate contrast.
Here's an explanation of the Amillennial view:
The kingdom of God is both present and future. Amillennialists do not believe that the kingdom of God is primarily a Jewish kingdom which involves the literal restoration of the throne of David. Nor do they believe that because of the unbelief of the Jews of his day Christ postponed the establishment of the kingdom to the time of his future earthly millennial reign. Amillennialists believe that the kingdom of God was founded by Christ at the time of his sojourn on earth, is operative in history now and is destined to be revealed in its fullness in the life to come.
They understand the kingdom of God to be the reign of God dynamically active in human history through Jesus Christ. Its purpose is to redeem God’s people from sin and from demonic powers, and finally to establish the new heavens and the new earth.
The kingdom of God means nothing less than the reign of God in Christ over his entire created universe.
The kingdom of God is therefore both a present reality and a future hope. Jesus clearly taught that the kingdom was already present during his earthly ministry: “But if I drive out demons by the Spirit of God, then the kingdom of God has come upon you” (Mt. 12:28, NIV). When the Pharisees asked Jesus when the kingdom of God was coming, he replied, “The kingdom of God is not coming with signs to be observed; nor will they say, `Lo, here it is!’ or `There!’ for behold, the kingdom of God is in the midst of you” (Lk. 17:20-21). But Jesus also taught that there was a sense in which the kingdom of God was still future, both in specific sayings (Mt. 7:21-23; 8:11-12) and in eschatological parables (such as those of the Marriage Feast, the Tares, the Talents, the Wise and Foolish Virgins). Paul also makes statements describing the kingdom as both present (Rom. 14:17; 1 Cor. 4:19-20; Col. 1:13-14) and future (1 Cor. 6:9; Gal. 5:21; Eph. 5:5; 2 Tim. 4:18).
The fact that the kingdom of God is present in one sense and future in another implies that we who are the subjects of that kingdom live in a kind of tension between the “already” and the “not yet.” We are already in the kingdom, and yet we look forward to the full manifestation of that kingdom; we already share its blessings, and yet we await its total victory. Because the exact time when Christ will return is not known, the church must live with a sense of urgency, realizing that the end of history may be very near. At the same time, however, the church must continue to plan and work for a future on this present earth which may still last a long time.
Meanwhile, the kingdom of God demands of us all total commitment to Christ and his cause. We must see all of life and all of reality in the light of the goal of the redemption not just of individuals but of the entire universe. This implies, as Abraham Kuyper, the renowned Dutch theologian and statesman, once said, that there is not a thumb-breadth of the universe about which Christ does not say, “It is mine.”
This total commitment further implies a Christian philosophy of history: All of history must be seen as the working out of God’s eternal purpose. This kingdom vision includes a Christian philosophy of culture: Art and science, reflecting as they do the glory of God, are to be pursued for his praise. The vision of the kingdom also includes a Christian view of vocation: All callings are from God, and all that we do in everyday life is to be done to God’s praise. ~
"Amillennialism: A Brief Sketch of Amillennial Eschatology" by Anthony Hoekema