In the last post I introduced the mathematics education of August Wilhelm Grube, who’s Leitfaden, first published in 1842 and the in six subsequent editions, was one of the most used textbooks in arithmetic in the second half of the 19th century. Even more importantly, his educational doctrine then stood at the center of the methodological discussion. In this post I will continue to use the book by Rolf Braun about Grube, here to bring out the philosophical influences and implications of Grube’s thought.

The general purpose of this post and the last one is to highlight the religious origin of modern mathematics education. The more specific purpose of this post is to exemplify the high level of philosophical sophistication of educational thinking in the 19th century, in particular in connection with elementary mathematics. I want to show here how on of the doctrines that were to lead on to those of modern mathematics education, were shaped by the heterogeneous and quite fascinating mix of philosophical ideas present in Germany in the middle of the 19th century.

Grube’s point of departure, I read in Brauns book was the following – and these are Grube’s own words:

The faith is the sense organ of reason, of the inward facing eye, whereby man becomes convinced of the existence of non-physical beings. This belief stands over the ”knowledge” of mind, in that it is an act of reason (an act of the power of the soul to sens the super-sensible); it is thus beyond any demonstration, any evidence of the understanding, in that it contains immediate certainty in itself. (p. 70, in Braun, 1979, here and henceforth)

It if quite difficult to find a good translation of the expressions that Grube uses, that sounds well in German. And, at least to me, it is difficult even to understand what Grube means. It seems in a way that Grube operates in a world quite different from ours, and that this makes itself known in the difficulty of translation. What Grube actually writes is:

Der Glaube ist der Gefühlssinn der Vernunft, das em Innern zugewandte Auge, wodurchder Mensch vom Dasein nichtmaterieller Wesen die Überzeugung erhält. Dieser Glaube steht über dem ’Wissen’ der Verstandes, denn er ist ein Akt der Vernunft (der das Übersinnliche vernehmnenden Seelenkraft); er ist damit über jede Demonstration, jeden Verstandesbeweis erhaben, da er die Gewißheit unmittelbar in sich selber trägt.

There are many entities and capacities here, that do not square so easily with what we secular moderns find present in the world. What, for instance, is the ”Gefühlssin der Vernunft”? And that his the ”eye” that looks inward? What are the ”non-material beings” the presence of which we get convinced? What is a ”soul power” that can sense the supernatural? etc. one can go on.

The point is that this is the ”world” in which Grube’s mathematics education was conceived. While we are familiar with some aspects of it, namely those that have been handed down to us by generations of school teachers and textbook authors, what we see here are some quite other aspects, that we do not even understand.

But what Grube tries to say is that ”belief”, in contrast to the kind of conviction that you can get by rational means, such as ”proof”, is the foundation of ”reason”, in the sense of the ”reasonable” or ”practical reason”. You cannot ”find out” what is reasonable be means of rationality and demonstrations – it can only be founded on belief. Developing this theme, Grube writes:

Alles Wissen wurzelt im Glauben und geht, zur höchsten Entwicklung gelangt, wieder über zum Glauben, zu dem Punkte, wo Wahrheit empfunden, die Wissenschaft zur Angelegenheit der Herzens, zur Religion wird.

All knowledge is rooted in faith and becomes, as it reaches its highest development, faith once again, to the point where truth is felt, science becomes a matter of the heart, and becomes religion.

Grube saw as the ultimate goal of all philosophy to solve the ”riddle of existence”, and this by understanding that God is the last and highest  unity of ”being and thinking” and – as should be clear from the quotations above – he thought that this goal could only be reached through faith.

I think this is a rather unexpected philosophical context for mathematics education, and it puts Grube’s method in a new light. Religion is the basis of science, and science, in its highest stage, again returns to religion and becomes identical with it. Grube writes this only a decade after August Comte had published his book on ”positive philosphy”, the origin of positivism. But while Comte wanted to replace religion with science, letting science become a ”new religion”, Grube definitely wanted to keep religion, by arguing that science needed it as a foundation.

To be continued…