At the beginning of the 19th century. 1780 to 1880 say, the Prussian Holy Roman Empire had an opportunity to change the very nature of Philosophy and its Matrix Astrology. Religious propaganda and control had separated Philosophy from Theosophy and Christology, and taken Astrology apart and put it back together again as astronomy and divination. The divination, though still secretly practiced was religiously condemned, but the other parts of Astronomy were dismembered and religiously purified through the Ousia Debate, in which the Hupostasis of reality was an undeniable aspect of God and Theologically transformed into the worship of God through the admiration of Nature. Philosophy was sanitised by concentrating it on those revered Forefathers the early religious leaders had identified as of good moral character, even if Pagan!
Mathemaics on the other hand was gradually sanitized from its Astrological beginnings and the Gematria of the Astrologers was combed through to pick out what was conformal with the needs of the Theological Hupostasis in the Ousia of God. As a consequence the Prussian Holy Roman Empire had determined that Geometry, Algebra, and Arithmetic were the fundamental subject boundaries that any Mathematiker, or Astronomer/Astrologer should know.
The challenge to this cosy Theosophy was the burgeoning success of technology and Science, which was challenging basic accepted Cosmologies and Cosmgenies, and the observable order. Most of this took place on the edges of the Prussian Empire, especially in the Romance countries and the trading and coastal countries. Land locked Empires controlled there borders and maintained internal conformity with great assiduousness.
The break up of the old political and geographical order exposed the Prussian empire to its technological and scientific weaknesses. Of great influence in this process was Newtons Principia. Prussia was a land of Philosophers, but very few innovative technologist and scientists. Many Dukes and Regents of the empire imported or bought in these innovative thinkers from the countries outside the empire. The network of connections among the Royal families maintained connections with these developments even in the most backward looking societies.
Within Prussian Empire Philosophers were moving to catch up, and revolutionary philosophies based on Ideas not creeds were beginning to be exposited and explored. German Idealism or Prussian Imperial Renaissance takes place in time of European wide turmoil, when apparently it becomes necessay to innovate or die! Thu we find several desperate movements attempting to readjust the Empire to cope with the geopolitical reality. The result of the turmoil was the break up of the empire, but not without a valiant attempt to make the old system work or survive by radical change.
The big change that was introduced by the philosophers was to change the educational system that was too classical and theological with only rudimentary science involved. The change was to make the curriculum more "modern" more scientific and technological. Humboldt and the Philosophers took an opportunity to pass this educational reform into Prussian law. However, the necessary machinery and workforce to implement this reform were non existent! The Existing universities, higher middle and lower educational functionaries had to reinvent the curriculum from scratch! They did this y a decades long collaboration which saw the whole of the educational establishment work together in the most collegiate and open way to bring about the emperors command! This provided all involved in education a duty and an opportunity to influence the minds of future generations and many seized on this opportunity to incubate the best ideas from around the world into the curriculum of the Prussian imperial education system. The opportunity was only there for those who were prepared to do he work. Every educator had to be approved, had to pass examinations, had to demonstrate career development, keeping up to date, and had to be able to provide teaching and course material for the students which could be examined and approved, that is they had to write their own text books!
The work of Justus Grassmann and his sons were in this vein, and a heroic attempt to prepare the next generation of the empire for the future. In this context, the notion of a function derives from this intense imperial background, and this new creating of text books, terminology and subject disciplines based on the philosophy of the best prussian thinkers. One would think all would contribute to this great effort, but as usual some resisted every innovation anxous of the consequences of change. Among these amazingly was C. F. Gauss!