The philosophically American way of perceiving God based on the Declaration of Independence is that God is that which grounds our ability to Reason. This concept of God was known as Deism and was an outgrowth of several philosophies, including the ancient philosophical school of Stoicism, that believed that God was revealed in the patterns of Nature and the God of Nature can be understood and experienced through observing nature and thinking rationally toward the Good Society. This was the view of Thomas Jefferson and Ben Franklin who believed that reason and science was the path to knowing God as God was revealed in Nature. So "God" in this sense, is that which grounds (or is the source of) the laws of physics and the principles, patterns, and processes that leads to the Just Society and "a more perfect union."
"God" then is that which transcends all categories of thought yet is experienced as Ultimate Reality and known as the patterns producing the stable society based on innate human Rights endowed by Nature's God and enforced by Law and Order through Reason and Evidence. The Deists did not try to capture God as a concept to be imagined and dogmatized into a State Creed. For humans can only project humanoid ideas and attributes onto God, as humans can only filter human concepts through one's human nature.
The American God of Thomas Jefferson and Ben Franklin is also not an interventionist God, otherwise on the back of the Dollar Bill the image of the the All Seeing Eye representing God in the capstone of the pyramid would be capped; and the US constitution would not begin with “We the people,” but would begin with a reference to a deity’s will in theistic terms. Instead, the Constitution is secular and the Declaration of Independence speaks of breaking away from England that was a more theocratic form of government with its “divine right of kings.” Jefferson rebelled against theistic government by invoking “Nature’s God” (the God of Deism) that instills “inalienable rights” in rational human beings free to govern themselves by Reason. This appeal to Reason by Divine fiat gave Americans the right to reject the claim to divine revelation made by government leaders in England.
The American God is the “God of Nature,” a God that is “known” through observable Reality. The God of Thomas Jefferson, Ben Franklin, George Washington, John Adams, and Thomas Paine was similar to Einstein and Spinoza’s concept of God. It was the God that is manifest via natural laws and patterns available to both the non-religious and the religious equally. For Deists, God’s will, was not found in a holy book or divine revelation, but was available to all (believer and nonbeliever) through God’s laws of nature which could be known rationally and observationally. Stephen R. Covey calls it universal principles.
Deism was the nontheism of the day back in the 1700s. Unaware of Darwinian Evolution, the appearance of design in nature led many rationalists and scientists back then to posit some kind of Higher Power. However, based on logic and lack of evidence it was clear that this God was not a revelatory interventionist deity but a Divine Power operating in and through Nature and known through human Reason and observation of Natural Laws.
Ben Franklin writes how he became a Deist and his thoughts on revealed religions in the quotes below:
Some volumes against Deism fell into my hands ... they produced an effect precisely the reverse to what was intended by the writers; for the arguments of the Deists, which were cited in order to be refuted, appeared to me much more forcibly than the refutation itself; in a word, I soon became a thorough Deist.
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Scarcely was I arrived at fifteen years of age, when, after having doubted in turn of different tenets, according as I found them combated in the different books that I read, I began to doubt of Revelation itself.
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Revealed religion has no weight with me.
Source: https://www.azquotes.com. Note all of the rest of the quotes by Ben Franklin are taken from this link.
The last quote above is important, for Ben Franklin took part in forming the symbols on the back of the Dollar Bill. As noted above, the fact that the capstone is not capped, but there is a space, is clear to me at least that Franklin was saying that America was not a theocracy directed by divine revelation to holy men or a holy text, but instead God stands apart and allows for Man to rule himself as a government of the people, governed by reason, not by receiving “revelation" and an appeal to scripture. The whole point of the separation of church and state was essentially the separation of reasoning humans capable of observing Nature, from dogmatic revelation claims leading to endless religious interpretations and creedal conflicts.
Among the Founding Fathers, I think Ben Franklin might have used the most personal sounding language for the Deist God, with words like “Father.” But it is clear that his conception of God was not of a personal deity like a father that talks directly to his children verbally and gives them rules to follow through religious gurus. Instead, Franklin thought that the Deist God revealed his ways to everyone equally through Nature itself. Therefore, the clergy were not the mouthpieces of God, but Nature was. He also did not believe in the concept of "worshipping God" as form of flattery, but that to worship God was to observe the Laws of Nature and treat people fairly and justly. Franklin wrote:
I cannot conceive otherwise than that He [God], the Infinite Father, expects or requires no worship or praise from us, but that He is even infinitely above it.
Another time Franklin said:
“Here is my Creed. I believe in one God, creator of the Universe. That he governs it by his Providence. That he ought to be worshiped. That the most acceptable service we render him is doing good to his other children. That the soul of Man is immortal, and will be treated with justice in another life respecting its conduct in this.”
From these two quotes we can gather that if one wanted to "worship God" they were doing so when doing service to one’s fellow human beings. Franklin’s God-belief was pragmatic. He said once, “God helps those who help themselves.” In other words, God is not going to help us with divine revelation to alleged revelators of Gods will or by disrupting the Laws of Nature. Instead, we have to help ourselves by applying God's gift of Reason and aligning with God's Natural Laws via Right Action. We have to work to create the world we want around us. He saw very little practical value in church going and theistic-Christian worship:
I wish it (Christianity) were more productive of good works ... I mean real good works ... not holy day keeping, sermon-hearing ... or making long prayers, filled with flatteries and compliments despised by wise men, and much less capable of pleasing the Deity.
The perception of God among the Deist Founding Fathers, was similar to how the Stoic leader, Marcus Aurelius, viewed the Logos (and Gods) when he wrote:
“Live a good life. If there are gods and they are just, then they will not care how devout you have been, but will welcome you based on the virtues you have lived by. If there are gods, but unjust, then you should not want to worship them. If there are no gods, then you will be gone, but will have lived a noble life that will live on in the memories of your loved ones.”
Ben Franklin followed a similar sentiment in regards to the Bible and the divinity of Jesus, in that he did not bother following any dogma. Regarding Jesus’s divinity, toward the end of his life he said:
As to Jesus of Nazareth, ... I have, with most of the present Dissenters in England, some doubts as to his divinity; though it is a question I do not dogmatize upon, having never studied it, and think it needless to busy myself with it, when I expect soon an opportunity of knowing the Truth with less trouble.
In other words, Franklin did not bother with the dogma wars and decided he’d find out after he died if Jesus was actually a divine being. Meanwhile, he focused on this life, this world, using science to discover how lightning works for example, and trying to live a virtue-centered life.
Rather than join the ranks of liberal Christians or try to reform the Fundamentalist strains within Christianity, Franklin instead avoided dogmatizing and was more practical and chose to live in the real world. He once said:
The things of this world take up too much of my time, of which indeed I have too little left, to undertake anything like a reformation in religion.
Ben Franklin saw religions as essentially man-made and the free market would weed out religions that are dysfunctional:
When a religion is good, I conceive it will support itself; and when it does not support itself, and God does not take care to support it so that its professors are obliged to call for help of the civil power, 'tis a sign, I apprehend, of its being a bad one.
Ben Franklin said, “I have found Christian dogma unintelligible. Early in life, I absented myself from Christian assemblies.” Regarding church attendance he said, “Lighthouses are more helpful than churches,” and “The way to see by Faith is to shut the Eye of Reason.” Regarding the conflict in nature of Bible dogma, Franklin wrote:
This modesty in a sect is perhaps a singular instance in the history of mankind, every other sect supposing itself in possession of all truth, and that those who differ are so far in the wrong ; like a man traveling in foggy weather, those at some distance before him on the road he sees wrapped up in the fog, as well as those behind him, and also the people in the fields on each side, but near him all appears clear, tho' in truth he is as much in the fog as any of them.
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If we look back in history for the character of the present sects in Christianity, we shall find a few that have not in their turns been persecutors and complainers of persecution. The primitive Christians thought persecution extremely wrong in the Pagans, but practiced it on one another. The first Protestants of the Church of England blamed persecution in the Romish Church, but practiced it upon the Puritans. They found it wrong in Bishops, but fell into the practice themselves both there (England) and New England.
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Religion I found to be without any tendency to inspire, promote, or confirm morality, serves principally to divide us and make us unfriendly to one another.
So the American Way is that which sees God as synonymous with Reality itself beyond Creeds and Dogmas as instead the Power revealed in Nature and known through Reason and Science, which provides the principles and patterns toward building The Good Life. So that whatever it is that grounds the phenomena we see around us, this Great Mystery (whether personal or impersonal), has provided us humans with reason to govern ourselves and has provided evidential patterns and principles in nature that when followed or tapped into increases societal life satisfaction and fulfillment.