It appears to me that there are two possibilities regarding the lack of God/Creator language in the Constitution:
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It was a conscious and intentional act
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It was accidental, as it was to be a list of laws, a practical document
#1—Pro
a) The very conscious decision to not have a religous test to hold office, as it was debated at the convention.
b) Direction: If you look at the D of I, the Articles of Confederation, and the Constitution, the documents get more secular over time.
c) The Language in The Treaty of Tripoli shows that one or some founders, Washington and or Adams assumed that the US was a purely secular country
d) The US Constitution was most closely modeled after the Virginia Constitution, which was the most secular.
e) When the Founders had the opportunity to go back and amend the document by adding the Bill of Rights, they used the very first amendment to put limits on religion.
#1—Con
a) The fact that a specific law regarding religion was debated (the test to hold office), but the “decision” to make the document as a whole secular was NOT debated, runs counter to logic.
b) The country was a very religious place. Some states had established State Religions. The Founders themselves were, at the very least, Deists, who believed in a Creator and an afterlife.
c) The language of the Treaty of Tripoli can be explained in a practical light. Hostages were being held, ships were being pirated, and we knew we couldn’t go to war with the Muslim world.
d) The language of the Treaty of Tripoli might not have even been in the Treaty that Adams signed, but added later. See “NOTE REGARDING THE Barlow TRANSLATION” [URL=http://www.yale.edu/lawweb/avalon/diplomacy/barbary/bar1796n.htm#n1]
e) As Liberal has pointed out, the embrace of secularism and break with religion was not complete. There are minor instances of religiosity in the document. If it was such and intional, deliberate act, surely these would have been removed.
#2—Pro
a) The issue was not debated. In the notes taken by the seven founders who we know took notes at the Constitutional Convention in 1787, there is no evidence of the subject even being discussed.
b) The preamble, the section that the Founders historically used to invoke God is noticeably short, pointing immediatley to the body of the document, which “merely” lists laws.
c) When the Founders had the opportunity to go back and talk about the issue of religion, they simply barred the Federal Government from establishing an official religion: “Congress shall make no law regarding the establishment of religion.” Surely, they could have added more on the subject.
D) The nation was very religious. The Founders themselves were, at the very least, Deists. Such a break from their traditional beliefs surely would have merited some discussion, in their private papers talking about the decision to omit God/Religion from the Constitution the thought so important.
#2—Con
a) The fact is that the document is largely secular.
b) It was common, almost routine, for them to invoke God and Divine Providence, yet it is not in the document.
c) Many of the Founders embraced the Rationalism of the Enlightenment, which minimized the importance and role of religion.
d) The D of I and the Articles of Confederation show the direction the Founders’ thinkiing was heading: away from religion and toward secularism.
Another piece of evidence that would help us glean the intent of the founders are their general writiings. Although I am not aware of any papers or quotes (other than those seven individuals who took notes at the Convention) that point back to their intent when creating the document. There are many quotes that do not pertain to the drafting itself that might offer us the opportunity to divine what was on their minds during the time in question, but it is not proof one way or another. I resist including any quotes here, as it will simply lead to a battle of quotes, with many of them presented incomplete, incorrect, or out of context.
That said, I tried to be fair with the merits of each position. Please add to the list as you see fit. Maybe if we all worked from one master list, we’d get closer to agreement.
My opinion, based on the facts as I see them, is to now stand more strongly behind my original theory: that the document was intended to be as purely practical as possible. I think this is best supported by: a very short preamble, the legal tenor of the body of the document, and—most powerfully—the complete lack of any discussion regarding the degree to which the Constitution should be secular or religious during the debate process.
Barring the discovery of additional notes taken at the Convention or letters discussing it after the fact (by the Founders), I do not think that either position can be proven one way or another. It does seem perfectly clear, however, that the OP’s original claim that the omission of God/Creator/Religion was deliberate because the Founders “fought over it tooth and nail” has no basis in fact.