From R.G. Manley’s Waveform Analysis, this observation, at the start:
At a not very remote period in the past, a university education in Natural Philosophy, together with a small amount of private reading, enabled a man to claim fairly the he knew the whole of science, so far as it was at that time revealed. In contrast, the present-day study of science is so extensive and intensive that no one can hope to acquire a thorough knowledge of more than a small portion of one of the sciences. Specialisation is forced upon the scientific worker who desires to contribute useful original work, for the successful and economical achievement of which it is essential to be well-informed of contemporary progress by other investigators in the same field of study. This unavoidable trend towards learning “more and more about less and less” necessarily involves some considerable dependence upon the results of research and development in other subjects; and perhaps the greatest disadvantage of the situation is that one has to accept without question those results, as time along prevents a thorough ab initio investigation of principles and methods from being made.
While the foregoing remarks are broadly true of science in general there is a conspicuous exception to the rule in the case of mathematics. Mathematics is, in its utilitarian aspect, the hand maid of all the sciences, in that it provides a set of processes for solving problems posed in their most general terms. Everyday life is permeated with the use of figures, and a mathematical undercurrent is observable in any study of physical phenomena. It is not surprising, therefore, that workers in very different fields make use of identical or similar mathematical processes, the the solution of problems which have a common mathematical nature, although they made differ widely in physical significance.
It’s interesting to note that he starts by referring to the sciences as “Natural Philosophy.” That was common practice for a long time; Manley wrote this in 1945. The separation of science from any kind of philosophy was, in part, a result of the specialisation that he describes in the first paragraph.
But his case that mathematics is what ties the sciences together–and ultimately what makes them work–is one that has been lost in our “believe the science” rhetoric. He is correct that mathematics allows bridging the gaps created by the expansion of knowledge, which allows scientific analysis, especially for general discussion, to not be reserved strictly to the specialists. The virtue of that is on display in what passes for public discussion on just about any scientific or engineering topic.