A Calculus of Ideas: A Mathematical Study of Human Thought by Ulf Grenander

By Ulf Grenander

This monograph stories a concept scan with a mathematical constitution meant to demonstrate the workings of a brain. It provides a mathematical thought of human concept in response to development thought with a graph-based method of pondering. the tactic illustrated and produced through broad desktop simulations is said to neural networks. dependent frequently on introspection, it's speculative instead of empirical such that it differs appreciably in angle from the traditional knowledge of present cognitive technology.

Readership: Graduate scholars in utilized arithmetic, statistics and cognitive technology in addition to postgraduates and researchers in those fields.

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4. 4. 4: In this figure, thought1 means that Jim speaks English and Henri speaks French, while thought 2 says that Jim speaks French and Henri English. If thought1, thought 2 ∈ MIND, we could form the modality M 1 = {Jim, Henri} and M 2 = {English, French} and consider thought 3 regular, thought 3 ∈ MIND. But if thought1 ∈ MIND, thought 2 ∈ / MIND, the creation of the modalities M 1, M 2 would not be legal. We would have to introduce the contrived primitive ideas speak 1 and speak2 , the first one with out-bonds (Jim, English) and the second one with (Henri, French).

We can think of this procedure as an encapsulation process. 4a means that one should love one’s neighbor. When encapsulated it becomes a new generator that could perhaps be named “CommandX”, but in the automated working of the mind we shall use more neutral notation like ideak ∈ G with a counter k. Now let us make this more precise. Say that the MIND has produced a conscious thought with the size n = size(thought), and the generators g1 , g2 , . . , gn . With some probability pcreate (n) we shall abstract thought 27 July 2, 2012 9:55 9in x 6in A Calculus of Ideas 2.

If two generators g1 and g2 are similar in the sense that there is a group element s ∈ S such that 1 See GPT, Chapter 1. 4. An Algebra of Human Thought g1 = sg2 , it is clear that this similarity induces a partition of the generator space into modalities as equivalence classes. For example, g1 = “John” and g2 = “Jim” may be equivalent but probably not g1 = “John” and g2 = “Mary”: this is an expression of the principle “arbitrariness of the sign” to quote de Saussure although he spoke of language rather than thought.

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