There exists a widely quoted story about [18th century philosopher/mathematicians] [Denis] Diderot and [Leonhard] Euler according to which Euler, in a public debate in St. Petersburg, succeeded in embarrassing the freethinking Diderot by claiming to possess an algebraic demonstration of the existence of God: "Sir, (a+b^n)/n = x; hence God exists, answer please!"
This story turns out to be (at least in detail) false, but it was likely invented and resonated because it embodies an underlying truth almost any of us in the sciences have seen: once a mathematical equation comes out, it tends to blind the naive, and even the experienced will often skip over the equations on a first reading of any complex argument. A minor error in a mathematical expression (a forgotten minus-sign being the most common example!) can completely change its meaning, and reasoning about such things requires detailed understanding, it's something that's intellectually demanding, requiring time and mental effort. Sometimes we are willing to put that time in, but more often than not we just don't have the time, or the requisite background, and just skip over the math, hoping that it makes sense to somebody else.
Of course, there is an equation that's proof of amazingly beautiful self-consistency in mathematics, that some have taken as evidence for God:
ei π + 1 = 0
but the beauty of that expression isn't something I intend to get into right now.
What brought to my mind the apocryphal story about Euler and Diderot was a pair of recent posts by Dr. Judith Curry, who I've criticized here before. The first post seemed in some ways to finally be a response to my earlier queries about the no-feedbacks question - about which more below. But in the second she oddly chose to highlight 3 comments which claimed the whole thing was ill-defined, with one of them chock-full of equations that seems to have blinded her and others to the fact it made no more sense than Euler's apocryphal equation, ending with a claim that it's all nonsense:
... it is impossible to evaluate these 2 integrals because they necessitate the knowledge of the surface temperature field which is precisely the unknown we want to identify.
The parameter dTa/dFa is a nonsense
which is the sort of language that should remind my few regular readers of our friends Gerlich and Tscheuschner...
The following is a collection and rearrangement of some of my comments on how we know about the radiative effects of greenhouse gases like carbon dioxide and how big they are, made on another blog. I'm posting these here as in working things out to my own satisfaction to try to respond to some rather egregiously wrong claims by the blogger there, I believe I clarified a few things in a way that's worth preserving.
The starting view here has to be the Kiehl-Trenberth diagram of Earth's energy flows, even though in some ways (which I'll get to below) it may be slightly misleading.