What does Paul mean by not beating the air or shadow boxing?
Paul means he does not run or fight aimlessly, like a boxer swinging at empty air. He knows the prize, an incorruptible crown, and disciplines himself spiritually, keeping his body under control, so his race and fight have a definite, eternal purpose.
Les explains that Paul is contrasting the discipline of athletes with the spiritual race. Runners train temperately in food and drink to enhance body strength, but only to obtain a crown of leaves and vines that will die. Paul says believers run for an incorruptible crown that lasts for eternity. Paul therefore runs not uncertainly; he knows the prize. Les suggests it may be Philippians where Paul desires to know Him and the power of His resurrection, so that he might win the prize. Thus, not beating the air or shadow boxing means Paul refuses to be aimless; he trains himself spiritually to run the race of life to win the prize. In the last verse, “I keep under my body” means keeping everything under control.
I therefore so run, not as uncertainly; so fight I, not as one that beateth the air: