In the first part of the paper, a metaphysics of anticipation is presented through the ordering of the different levels of anticipation in the natural world. The interconnections between those different levels are highlighted. In the second part, an attempt is first made to delineate the various mathematically expressible forms of anticipation, but then, the limits of mathematical efforts to express anticipatory phenomena are shown; these limits arise from the difficulty in quantifying the imagination, creativity, improvisation, cultural synchronicity, and autopoesis, all of which are relevant to anticipation. And so in the third part, the paper concludes with a preliminary classification of the different computational and non-computational meanings of our concept of anticipation. ©1999 American Institute of Physics.