This is particularly true of applied mathematics where we are interested in it not merely as an end in itself but as a means to some other end. Means, however, always in part determine the ends for which they are ostensibly designed, and neither tools nor languages are neutral in regard to the ends which they serve. Tools help to determine the tasks we will undertake, and languages influence what we will say. These notes are directed towards examining the limitations of mathematics both as a tool and as a language, especially in regard to possible distortions of the growth of knowledge which might result from too exclusive a reliance on mathematical tools. The rounded growth of knowledge however requires both mathematics and something that is not mathematics.
Mathematics clearly has a bias on the side of delicacy and exactness.
Where the task requires delicacy, this is all to the good. If however the empirical universe which we are trying to know is not delicate, too great a reliance on mathematics may be misleading, if it is not checked by good judgment about the nature of the empirical universe itself. I do not imply, of course, that mathematics is incapable of modifying itself in the direction of the buzz-saw for there are some signs of this. The bias, however, is at present all towards the scalpel. A more exact analogy is that of a calculating machine with a set program.