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I think the theist can make a strong, general appeal to beauty and aesthetics as an intrinsic part of the universe, and as necessary in their own way as breathing and brain waves. I would contend that there is a link between the beauty we perceive in the order and proportion, dramatic contrast, design, symmetry, color, etc., of creation, and the God Who lies behind all that.
In other words, it could be argued as a subset of the teleological argument for God (the argument from design). Plenty of atheists or non-Christian types have made statements along these lines. Albert Einstein (a sort of pantheist) comes to mind:
Everyone who is seriously involved in the pursuit of science becomes convinced that a spirit is manifest in the laws of the universe -- a spirit vastly superior to that of man . . . In this way the pursuit of science leads to a religious feeling of a special sort . . .I commented on these statements:
(To student Phyllis Right, who asked if scientists pray; January 24, 1936)
In view of such harmony in the cosmos which I, with my limited human mind, am able to recognize, there are yet people who say there is no God.
(to German anti-Nazi diplomat and author Hubertus zu Lowenstein around 1941)
Then there are the fanatical atheists . . . They are creatures who can't hear the music of the spheres.
(August 7, 1941)
My religiosity consists of a humble admiration of the infinitely superior spirit that reveals itself in the little that we can comprehend about the knowable world. That deeply emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe, forms my idea of God.
(To a banker in Colorado, 1927. Cited in the New York Times obituary, April 19, 1955)
Now, I ask atheists: whence comes Einstein's "deeply felt conviction"? Is it a philosophical reason or the end result of a syllogism? He simply has it. It is an intuitive or instinctive feeling or "knowledge" or "sense of wonder at the incredible, mind-boggling marvels of the universe" in those who have it. Atheists don't possess this intuition, but my point is that it is not utterly implausible or unable to be held by even the most rigorous, "non-dogmatic" intellects, such as Einstein and Hume. And the atheist has to account for that fact somehow, it seems to me.The other approach I would use is the "argument from longing." Many atheists feel this longing or deep yearning in the enjoyment of nature or a great cathedral. We can build on that and construct a persuasive argument from it that can be interesting to them and non-confrontational.
See an article about C.S. Lewis's version of the "argument from desire" and another fabulous piece on "Aesthetic Arguments for God." See also an interview with Peter Kreeft: "A Baptism of Imagination" and his treatment of the argument from desire.
Those are two possible ways to approach such subjects with atheists and agnostics (as good as any).
Dialogue With Atheists on the Evolution of the Eye, Irreducible Complexity, and Intelligent Design (Dave Armstrong vs. Steve Conifer)
Dialogue on Materialist Evolutionary Theory and Intelligent Design (including St. Augustine's and St. Thomas Aquinas' Views on Creation and Evolution) (Dave Armstrong vs. five agnostics)
The Atheist's Boundless Faith in Deo-Atomism ("The Atom-as-God")
Was Skeptical Philosopher David Hume an Atheist? [Hume was actually a deist or "minimal theist" and espoused one version of the teleological argument (theistic argument from design) ]