Leibniz, Herbart, Riemann – The Lives of Ideas

When I looked recently at Riemann’s famous lecture On the Hypotheses which lie at the Bases of Geometry, I gave some attention to this remark:

besides some very short hints on the matter given by Privy Councillor Gauss in his second memoir on Biquadratic Residues, in the Göttingen Gelehrte Anzeige, and in his Jubilee-book, and some philosophical researches of Herbart, I could make use of no previous labours.

I’m most interested in the conviction these thinkers share that a kind of reasoning, a willingness to look more closely, can get past the appearances of things.  This is what mathematics consistently accomplishes.

Johann Friedrich Herbart was a philosopher and would likely be referred to as an early 19th century pioneer in psychology-based theories of learning. But he wrote on the nature of many things and is said to have been significantly influenced by Leibniz’s philosophical writing (which I will get back to)

In her paper, An Overview of Riemann’s Life and Work, Rossana Tazzioli describes one of the directions that Herbart’s influence on Riemann took:

Riemann agreed “almost completely” with Herbart’s psychology, which inspired both Riemann’s model of the ether (the elastic fluid which was supposed to  fill all the universe) and his principles of Naturphilosophie. According to Herbart, the “psychic act” (or “representation”) is anact of self-preservation with which the “ego” opposes the perturbations coming from the external world. A continuous flow of representations go from the ego to the conscious and back. Herbart studied the connections between different representations in mechanical terms as compositions of force.

In his “Neue mathematische Principien der Naturphilosophie” Riemann followed Herbart’s ideas and supposed that the universe is  filled with a substance (Stoff)
flowing continually through atoms and there disappearing from the material world (Korperwelt). From this obscure assumption, Riemann tried to build a mathematical model of the space surrounding two interacting particles of substance…

But Herbart’s ideas about space or, more specfically, sensory space may have motivated some other generalizations.  In a book about Hermann von Helmnoltz and nineteenth century science, David Cahan also talks about Herbart’s influence:

Herbart argues that each modality of sense is capable of a spatial representation.  Color could be represented as a triangle in terms of three primary colors, tone as a continuous line, the sense of touch as a manifold defined by muscle contractions and still other spaces as associations of hand-eye movements.  “To be exact,” he wrote, “sensory space is not originally a single space.  Rather, the eyes and the sense of touch independently from one another initiate the production of space; afterward both are melted together and further developed.  We cannot warn often enough against the prejudice that there exists only one space, namely, phenomenal space.”  Therefore, for Herbart, “space is the symbol of the possible community of things standing in a causal relationship.”  He insisted that for empirical psychology space is not something real, a single container in which things are placed.  Rather, it is a tool for representing the various modes of interaction with the world through our senses.

Herbart treated a successful visual representation as a projected expectation or “hypothesis” formed through an unconscious inference……he treated vision as an experiment constantly performed by the brain with the aid of the eye as its measuring device.

Vision as an experiment would fit nicely with some very current ideas about the visual brain.  And, while not an analysis of anything geometric, this perspective is provocative.  It encourages the abandonment of static structures in favor of more interactive ones.

In my looking I became captivated by other ideas of Herbart’s.  Some nice expressions of them are in An Introduction to the History of Psychology by B.R. Hergenhahn.  Herbart agreed with empiricists that ideas were derived from sense experience.  But they were, for him, independently active.  From the text:

Herbart’s system has been referred to as pyschic mechanics because he believed that ideas had the power to attract or repel other ideas, depending on their compatibility.

According to Herbart, all ideas struggle to gain expression in consciousness, and they compete with each other to do so.

…and all ideas attempt to become as clear as possible…all ideas seek to be part of the conscious mind.

Herbart used the term self-preservation to describe an idea’s tendency to seek and maintain conscious expression.

Herbarts position represented a major departure from that of the empiricists because the empiricists believed that ideas, like Newtons particles of matter, were passively buffeted around by forces external to them….For Herbart, an idea was like an atom with energy and a consciousness of its own – a conception very much like Leibniz’s conception of the monad.

I need to take up Leibniz’s monad in another post.  A simple statement from Philosophy Pages will suffice for now:

Leibniz concluded that the ultimate constituents of the world must be simple, indivisible, and therefore unextended, particles—dimensionless mathematical points. So the entire world of extended matter is in reality constructed from simple immaterial substances…

These are the monads. But there is life in these monads. And material does somehow come from the immaterial…

I did become particularly interested in the kind of analysis undertaken by Herbart and Leibniz.  We inevitably learn the mathematics of the 19th century, but we’re not likely to spend time thinking about monads.  It reminded me of how I felt when I read Thomas Mann’s Magic Mountain.  Hans Castorp’s innocent and imaginative response to his first glimpse of a very young medical science reminded me that ideas can take us countless places, not just here.