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Albert Lewis's article (Annals of Science, 1977) analysing the influence of Friedrich Schleiermacher on Hermann Grassmann, stimulated many different studies on the founder of n-dimensional outer algebra.
Following a brief outline of the various, sometimes diverging, analyses of Grassmann's creative thinking, new research is presented which confirms Lewis's original contribution and widens it considerably. It will be shown that:
i. Grassmann, although a self-taught mathematician, was at the centre of a hitherto understated intellectual trend, which was defining for Germany. Initiated by Pestalozzi's concept of elementary mathematical education and culminating in the modern mathematics of the late 19th Century, it was reflected in the contributions of Grassmann, Riemann, Jacobi and Eisenstein.
ii. Hermann Grassmann, his father Justus, and his brother Robert were all demonstrably influenced by Schleiermacher's dialectic; however the two brothers responded to it in very different ways.
iii. Whilst the more philosophical parts of Hermann's 1844 Extension Theory are characterised by the influence of Schleiermacher and also by the mathematical knowledge of his father, the entire development of this work is the unfolding of a single idea based on the father's interpretation of combinatorial multiplication as a 'chemical conjunction', which was developed largely dialectically by Hermann.
In honour of Seymour Papert
(2018)
Forth is nice and flexible but to a philosopher and teacher educator Logo is the more impressing language. Both are relatives of Lisp, but Forth has a reverse Polish notation where as Logo has an infix notation. Logo allows top down programming, Forth only bottom up. Logo enables recursive programming, Forth does not. Logo includes turtle graphics, Forth has nothing comparable. So what to do if you can't get Logo and have no information about its inner architecture? This should be a case of "empirical modelling": How can you model observable results of the behaviour of Logo in terms of Forth? The main steps to solve this problem are shown in the first part of the paper.
The second part of the paper discusses the problem of modelling and shows that the modelling of making and the modelling of recognition have the same mathematical structure. So "empirical modelling" can also serve for modelling desired behaviour of technical systems.
The last part of the paper will show that the heuristic potential of a problem which should be modeled is more important than the programming language. The Picasso construal shows, in a very simple way, how children of different ages can model emotional relations in human behaviour with a simple Logo system.