Systematic observations were made of the mathematical knowledge of physics students from the U.S. and other countries during their first years of graduate study at Ohio University. It was found that all were deficient in general and in "modern'' mathematical concepts, and in problem-solving skills. Sizable fractions of them did not even possess adequate concepts of "derivative,'' "integration,'' and "truth.'' Nearly all were limited to some familiarity with rather elementary calculus, and with equally elementary differential and linear equations, but they showed some ability and a pronounced willingness to perform manipulations. Roughly, they regarded mathematics as mechanical method, not as constructive thinking. In view of the significantly higher levels of mathematical fluency demanded by contemporary advances in physics and in computer usage, none of these students was adequately prepared for future-oriented study, or for research and employment in physics and related areas at the close of the 20th century. It is intended to discuss the likely causes of this state of affairs elsewhere with a view toward remedial actions.