I suspect; pain is a physical feeling, weakness is similarly an understanding of physical problems. By pipining the pain=negative weakness, you convince yourself that (temporarily) more pain is helpful to the cause of doing more pushups, rather than previously "not helpful" as you would have been treating pain and deciding to stop.
I have heard it work before for pushups or similar exercise by danielfilan
(side note: don't push too hard, trust your body's limits - it tends to know when its going to break)
My impression is that the advice applies more generally than to just physical pain. My impression is that there's a lot of times when your body produces pain-like signals telling you to stop, when it's really in your interest to push through.
And so, to use the emotional example, when I push past fear, I imagine it as weakness leaving my body. I imagine a) becoming a stronger person for having pushed past it and being more capable of pushing...
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