Dühring expresses something I’ve run into before (Heinlein pulls a slight-of-hand with it in Rocket Ship Galileo; I noticed the slight-of-hand when I was 12). It is the belief that mathematics can be produced “without making use of the experience offered us by the external world.” In other words, that mathematics is it’s own thing, fully abstract, existing only in the mind, and independent of the content of the world–that it exists in and for itself. The validity of mathematics is mathematics; there’s no connection to anything else.
Engels concedes that pure mathematics is independently valid from the particular experience of each individual, but then, so is every other fact. But, “The concepts of number and figure have not been derived from any source other than the world of reality. The ten fingers on which men learnt to count, that is,to perform the first arithmetical operation, are anything but a free creation of the mind.”
I want to take a moment here for something that is probably a side note. The subjectivists, the schematists, the empiricists–that is, generally, those who are hostile to theory–nearly always have this in common: an impatience for and a dislike of history. This seems to hold true whether the subject is social, political, philosophical, religious, or about the natural sciences. While they may be willing to talk about what is (generally in a fixed, static, lifeless way), the process by which anything became the way it is, is contemptuously set aside as unimportant. And yet everything in the world is constantly in movement, everything is part of a process. To blind one’s self to history (in this case, to ignore or disregard that mathematics came from arithmetic, which came from attempts to understand nature, from abstracting number from real world objects) can only produce distorted, or, at best, severely limited, narrow, one-sided understanding.
“Counting requires not only objects that can be counted, but also the ability to exclude all properties of the objects considered except their number — and this ability is the product of a long historical development based on experience.”
“Before one came upon the idea of deducing the form of a cylinder from the rotation of a rectangle about one of its sides, a number of real rectangles and cylinders,
however imperfect in form, must have been examined.”
“Like all other sciences, mathematics arose out of the needs of men: from the measurement of land and the content of vessels, from the computation of time and from mechanics.”
Here, too, by the way (although Engels doesn’t yet go into it) we have a dialectical relationship. We study triangles in life, and produce the abstraction called trigonometry , we then use trigonometry to both design and use sails to drive a square-rigged ship, and then we use that experience to further refine our understanding of trigonometry. We have the mathematical concepts abstracted from nature, we have those concepts used in order to further understand aspects of nature.
In the end, the theory has built upon itself to the point where the impressionist is able to argue (albeit unconvincingly) that mathematics is just about itself. The fact that we are constantly using higher and higher forms of mathematics in our understanding of nature; and in technology to change and examine nature, must be, to these people, pure coincidence. And that these discoveries, in turn, are folded into mathematics and spur it still higher, they simply prefer not to see.