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A word on the authority of modern science as it relates to the age of the earth...the age of the earth relates directly to its origins. God has given us special revelation as to the origins of the earth. There were no witnesses, but we have the inspired testimony of Moses; and empirical secular science (falsely so-called) is unequipped to properly tell us how long the earth has been in existence. Although the Bible does not spell out for us exactly how old the earth is, we can properly deduce reasonable estimates based on the chronologies found in Scripture. Attempts made to exceed reasonable estimates based on the authority of modern science in opposition to God's Word are simply wrong. The modern secular scientist has his presuppostions about origins and the Christian has his based on the Bible. The age of the earth is fundamentally a theological question, not a question for empirical science to answer. This is not a rejection of science, because true science, in words of Kepler, is "thinking God's thoughts after him." It is rejection of secular presuppositions upon which modern science is based in opposition to God's Word.
As J.G. Vos says in The Separated Life,
Westminster Confession, Chap. 20:
For a young earth creationist perspective on carbon dating, see this article.
As J.G. Vos says in The Separated Life,
...experience or science can never of itself be binding on the conscience of man.
Moreover, those who wish to introduce science as an additional authority always speak as if it were a very simple matter to ascertain what science has to say on any particular question. They always speak as if somewhere there were a sort of scientific pope who could utter ex cathedra the final, united, unquestionable voice of science. They seem to presuppose that the voice of science can be heard, speaking with authoritative accents, by simply consulting a few volumes in the public library. The truth is, however, that 'science' is an abstraction. There is in the world today no such thing as the voice of science; there are only the voices of a multitude of scientists, and they are anything but agreed among themselves. Now who is to decide which of these many voices is to be accepted as the authoritative voice of science? One scientist, a professor in a great university, states that years of research have failed to demonstrate that a certain practice shortens life. Another scientist, of equal scientific standing, maintains the contrary position. Who is to decide which represents the authoritative voice of 'science'? All to often those who wish to place science alongside of Scripture as a standard of faith and conduct wish at the same time to be the judges of what is science; those who hold certain views they regard as scientists; all others they reject as being prejudiced or otherwise untrustworthy. Can any pope or church assembly decide just what kinds of science "” the opinions of just which scientists "” are authoritative and therefore, along with Scripture, binding on the conscience of man? No, in matters of science every person must decide for himself. And even if certain scientific theories are believed to be true, they cannot be binding on the conscience. We must beware of the sin mentioned in the Larger Catechism, no. 105, of 'making men the lords of our faith and conscience.' All human authority, however expert or learned, is fallible, and therefore cannot bind the conscience.
Westminster Confession, Chap. 20:
II. God alone is Lord of the conscience,(k) and hath left it free from the doctrines and commandments of men, which are in any thing contrary to His Word; or beside it, if matters of faith or worship.(l) So that, to believe such doctrines, or to obey such commands, out of conscience,(m) is to betray true liberty of conscience: and the requiring of an implicit faith, and an absolute and blind obedience is to destroy liberty of conscience, and reason also.(n)
(k) Jam. 4:12; Rom. 14:4.
(l) Acts 4:19; Acts 5:29; I Cor. 7:23; Matt. 23:8, 9, 10; II Cor. 1:24; Matt. 15:9.
(m) Col. 2:20, 22, 23; Gal. 1:10; Gal. 2:4, 5; Gal. 5:1.
(n) Rom. 10:17; Rom. 14:23; Isa. 8:20; Acts 17:11; John 4:22; Hos. 5:11; Rev. 13:12, 16, 17; Jer. 8:9.
For a young earth creationist perspective on carbon dating, see this article.