They are not really interchangeable, except for the case when they describe a person. But the meaning is slightly different. Real meanings: Keen: It is has two meanings: 'he was keen for me to bring it' Enthusiasm is not quite the right word, the person really wanted me to bring it. They expressed a strong desire for a certain outcome. 'he has a keen mind for mathematics'. His mind is made for mathematics, it is strongly suited for it. Sharp: the quality of the edge of an object, suggesting that the edge will cut: 'the knife was sharp'. It is often used figuratively - 'Turn right on the road, and then sharp left.' and this is how it gets applies to people. Where do sharpness and keenness overlap? - Sharp, when used to describe the quality of a persons mind, indicates that it is is effective, quick, useful. The person is quick to understand, to see solutions. The atonym would be 'dull' or slow. A knife blade can also be dull, from over use or bad use. - keenness: when presented with a problem, the person is quick to understand it. he engages with the problem immediately, and with enthusiasm. He is hungry to be useful. his mind is good at solving the problem presented. The suffix -ness refers to the amount of this quality, not the quality itself. 'There is a certain keenness or sharpness to that person. They are sharp / keen. Their mind is sharp / keen. They demonstrate sharpness / keenness.

sharp really comes from the other definition of sharp, like something that cuts easily. a shape blade, thus someone has a "sharp" or quick brain. keenness or keen has two definitions that I know of and people use (southern americans) sometimes people are keen on someone (to be attracted romantically) such as "now boy, you aint keen on her are ya?"(ya being you, aint being are not combined) keenness I have also heard being used as to be aware of something, but im not sure that is the correct usage. I hope this helps. the main thing is to keep in mind that keen is rarely if ever used. sharp is fairly used as to have a quick wit or to be mentally quick. good question hope this helps.

sharp = adjective keenness = noun