Genesis 1 and 2 do NOT state that Adam was made "a fully grown human".
1) That tradition arose from a misunderstanding (primarily from the English translation) that HADAM meant "adult male" instead of "the human one" (or even "the red one" or "the read soil human"). In English, "man" can mean either an adult human male OR simply "humanity" or "mankind". So the confusion is understandable.
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You are partially correct. The Bible does not explicitly state that Adam was "a fully grown human", but it considers that fact should be known to the reader based upon the context of the text. Let us take a look at the information we can glean from the Scriptures as to whether or not Adam was a fully formed mature man, or whether it was meant to be understood as mankind.
Gen 2:20 So the man gave names to all the livestock, the birds in the sky and all the wild animals. But for Adam no suitable helper was found.
The virst verse we look at states that there was no suitable helper for Adam. Could there have been a breeding population of males only before the creation of Eve? Anyone who has had the "birds and the bees" discussion knows the answer is a resounding "no".
21 So the LORD God caused the man to fall into a deep sleep; and while he was sleeping, he took one of the mans ribs and then closed up the place with flesh.
In this passage we see that the singular possessive is used to describe where the LORD got the material to make the woman. This is further evidence that Adam was a singular individual.
22 Then the LORD God made a woman from the rib he had taken out of the man, and he brought her to the man.
When God was finished with His creation of woman, the Bible says that "He brought her to the man". I suppose one could imagine that God was now bringing a group of women to the group of men, but a much easier understanding of the passage would lead one to believe that God made a single woman from a single man for the single man.
23 The man said,
This is now bone of my bones
and flesh of my flesh;
she shall be called woman,
for she was taken out of man.
To answer the question, which came first, the man or the woman, here we have our answer. This all comes on the heels of Adam naming all the animals in the Garden, and now he names this new creation that is woman. This is something only a mature, thinking individual could do. What more do the Scriptures say concerning this Adam?
24 That is why a man leaves his father and mother and is united to his wife, and they become one flesh.
Here, the Scriptures give us the very first marriage. Would a child marry? No. Would a population marry? Only in Utah. The passage does say that the TWO shall become ONE, so we have an understanding that there are only TWO people involved in that marriage and that they must therefeore be old enough to marry.
If we back up a few verses we can find out that Adam was mature enough to understand a direct commandment from God concerning a matter of life and death.
16 And the LORD God commanded the man, You are free to eat from any tree in the garden; 17 but you must not eat from the tree of the knowledge of good and evil, for when you eat from it you will certainly die.
The above passage demonstrates that Adam was mature enough to understand consequences for his actions.
In the NT we have Jesus' lineage being traced back to Adam and Paul states that it was by one man's sin (Adam's sin) that death entered into the world.
So, while you may be correct that Genesis does not explicitly state the matter, it is easily understood by the context.
In Christ, GB