"Astronomy" is from the Greek meaning "star formation". It is the branch of science dedicated to studying bodies in outer space. Since astronomy is a science—an investigation of God's creation—it is perfectly acceptable for Christians to study astronomy. Where we get caught up is when scientists draw conclusions about the data that contradict the Bible.
Should a Christian study astronomy? What does the Bible say?
Genesis 1:16 says, "And God made the two great lights—the greater light to rule the day and the lesser light to rule the night—and the stars." Psalm 147:4 says of God, "He determines the number of the stars; he gives to all of them their names."
But God also said, "And beware lest you raise your eyes to heaven, and when you see the sun and the moon and the stars, all the host of heaven, you be drawn away and bow down to them and serve them, things that the Lord your God has allotted to all the peoples under the whole heaven" (Deuteronomy 4:19).
In biblical times, this "astrolatry" (worship of heavenly objects) was literal. Many cultures worshiped the sun, moon, and stars as gods. In fact, the planets in our solar system are named after Roman gods. In our day, this worship is more subtle. We don't tend to think that stars, planets, quasars, and comets are deities, but many in science do value their interpretations of astronomical data higher than the historical account given in the Bible.
The problem with a Christian studying astronomy isn't in the subject matter but in the culture of those who study the subject matter. Astronomy is one of the hardest of the sciences to understand, in part because the objects of study are so far away and not subject to human interaction. Scientific theories are validated either by experimentation or by discovering phenomenon that had previously been predicted. We can't do experiments on a quasar. And considering the supposed time it takes for stars to change, it isn't easy to predict phenomenon. Sometimes predictions do turn out to be true; earlier astronomers were proved to be right when they postulated that some stars wobble because of the gravitational pull of revolving planets. But sometimes they're wrong; science would indicate that at their supposed ages, the gas giants should have very weak magnetic fields—Saturn does, but Jupiter, Uranus, and Neptune have very strong fields.
Because so many noted scientists are atheists, their work naturally does not take into consideration the existence of a creator. Every observation, then, is interpreted by the presupposition that the creation account told in the Bible is wrong. Another explanation must be true. And so things like the slow expansion of the universe and the existence of background radiation are given as proof of theories that contradict the Bible, like that the universe is billions of years old. The anthropic principle is explained away by the improvable logic exercise of the multiverse. And young-earth creationists are so marginalized and ridiculed that even strong Christians with a scientific bent feel the need to accept secular science over what the Bible clearly states.
The Bible does mention astronomy—mostly in a poetic form that may or may not be meant to declare scientific fact. Isaiah 40:22 appears to describe the expansion of the universe. Job 26:10 seems to say the earth is a globe, not a flat surface. Job 26:7 says that God "hangs the earth on nothing"—an apt description of a planet in space.
More importantly, the Bible gives the account of the formation of heavenly objects through the words of their Creator. As Christians—particularly Christians interested in science—we need to understand that despite the fact astronomy has been studied for thousands of years, we still know very little. Even the latest interpretation is based on incomplete data. It is good to study God's creation, as long as we recognize that it is His creation.
We can trust that God's truth will become more evident the more we study His work. After all, before scientists learned about the expansion of the universe, they believed the universe was eternal and had no beginning—and they mocked Christians who insisted otherwise. Once secular astronomers realized Christians were right, they came up with the idea of the Big Bang (despite the fact they cannot explain how the Big Bang happened or how it could have resulted in stars and galaxies). Study the stars, and trust that the more we learn, the more we will be pointed back to God.
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