In mathematics there is a proof strategy known as "proof by exhaustion". It works like this:

First you prove that the only possible answers are, for example, A, B, C and D. It being, say, very difficult to treat case D on its own, you could still prove it is true simply by proving that A, B and C are false. This is called proof by exhaustion because at that point you've exhausted all possibilities. D simply MUST be true whether or not the detailed workings of D are immediately obvious.

My point here is that mathematical reasoning can lead to correct conclusions regarding situations about which we know nothing at all. In physics this is called "dimensional analysis" -- because all we have to do is to make the units (the dimensions) on the left side of an equation match the units on the right side of that same equation. You will see this in action below.

Note that D having been proved to be true, any assertions by others that D is false, or that a proof exists that D is false, or that D can be shown probably to be false by virtue of [whatever] -- these "facts" can be rejected out of hand, just as perpetual motion machine designs and circle-squaring proofs can be rejected out of hand.

Note also that on the forums, proof by exhaustion usually means something quite different. It means that a claim-counterclaim dispute goes on till one or the other party drops out because he/she has become exhausted, leaving the last man standing to crow his victory as signifying the truth of his position. :D

Let's call these people "A" and "B", and let's have the dispute go as follows ...

xxxxxxxxxxxxxxxxxx

Let Person A make a non-obvious but nevertheless correct assertion, such as "All theories of lift must reduce to the downward acceleration of air, since F=mA and there is no reason to suspend Newton's Laws Of Motion for purposes of aerodynamics studies".

By this he means, "At a suitable angle of attack an airfoil will generate an upward force we call "lift". (The F part of F=mA.) By Newton's Laws this upward force must be counterbalanced by the downward acceleration of air. (The mA part of F=mA.)There are no other possibilities."

The key point here is that the downward-acceleration-of-air viewpoint must be true for ALL theories of lift. The mathematics of Newton's Laws compels this. This is not a matter of opinion, or of design preference, or of wisdom of the ancients, or of forum courtesy, or of somebody's possessing all kinds of advanced degrees in aerodynamics. It is inescapable mathematical truth.

Worse still, Person A's assertion will be true regardless of whether Person A knows anything at all about airfoil theory. He could know zero about aircraft yet still arrive at the correct conclusion solely through the combination of dimensional analysis and Newton's Laws.

In other words, whatever theory of lift Person B may present, in the end it must reduce to the case just proved using dimensional analysis. This will be true regardless of whether Person A is willing -- or even able -- to show where the downward-momentum-of-air fairy is hiding in Person B's theory of lift.

(Note that this often results in Person B believing Person A to be very unfair because Person A usually will not want to take the time and trouble to track down Person B's lift-by-momentum fairy.)

xxxxxxxxxxxxxxxxxxxx

Person A may then go on to make another outrageous statement that also simply MUST be true ...

If there is any benefit to having curved airfoils, it can only be for purposes of drag reduction, i.e. for delaying the onset of turbulent airflow, hopefully to past the trailing edge of the wing.

We will prove this one not by means of dimensional analysis but instead by exhaustion. Our having earlier proved that the shape of the airfoil has nothing to do with lift, we now note that either the shape of the airfoil has to do with drag reduction, or it does not. There are no other possibilities. So ...

If the shape of the airfoil has nothing to do with drag reduction, we should prefer flat plate wings since, even though turbulent flow sets in immediately behind the leading edge, the flat plate wing is trivially easy to manufacture.

But this immediate onset of drag-creating turbulence makes the flat plate wing inefficient, so we don't prefer it.

Therefore the shape of the airfoil must in fact have to do with drag reduction, and only with drag reduction, whether or not the airfoil designer chooses to adopt this viewpoint.

xxxxxxxxxxxxxxxxxxxx

Person A may then go on to make the most outrageous statement of all, which is ...

I don't have to give you a detailed theory of lift, and I don't have to give you a detailed theory of airfoil shape drag analysis. I only have to give you Newton's Laws, and I don't have to prove those either. :D

xxxxxxxxxxxxxxxxxxxx

Exactly this happened in Outer Marker during the Winter/Spring of 2008, with me being Person A and two dozen other people being Person B. I stuck to my guns and was roundly criticized for Refusing To Play Fair, and for Refusing To Respect The Opinions Of Others, and for Arrogantly Believing That He Is Always Right, and for the comission of a half dozen other Forum High Crimes And Misdemeanors That Ought To Result In The Expulsion Of This Terrible Person.

With that background in hand, let's now summarize the forum dispute ...

xxxxxxxxxxxxxxxxxxxx

Person A presents his proof by dimensional analysis that all theories of lift must reduce to downward acceleration of air.

Person B then says "Person A cannot possibly be right because everybody knows that wings are sucked up into the sky by the Bernoulli effect. This is because wings are curved on the top and flat on the bottom, and the air pressure is lower on the top, sucking the wing up."

Person A will then point out that a flat plate wing will exhibit lift, deflecting air downwards just as required by Newton's laws, even though the plate is flat on top and flat on the bottom..

Person B may then say "Sorry, Bernoulli's equation clearly shows that wings are sucked up into the sky. This is because the air is flowing faster over the curved top of the wing than it is over the flat bottom of the wing."

Person A will then point out that symmetric airfoils are curved on top and curved on the bottom, and that they exhibit Bernoulli flow on both their upper and lower surfaces, so by Person B's reasoning these wings must be being sucked both upwards and downwards, resulting in zero lift, which clearly is not the case.

Person A might then even point out that aircraft with wings that are curved on top and flat on the bottom are capable of inverted flight which, from the Bernoulli viewpoint, ought to result in the aircraft being sucked downward to earth instead of upward to the sky.

Person B may then say "You are dead wrong, Mister A, because I've believed all my life that wings are sucked up into the sky" ...

... Which is proof only of an unwillingness to process new information.