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

By Ulf Grenander

This monograph experiences a notion test with a mathematical constitution meant to demonstrate the workings of a brain. It offers a mathematical concept of human concept in accordance with trend thought with a graph-based method of considering. the tactic illustrated and produced by way of huge laptop simulations is expounded to neural networks. dependent in general on introspection, it truly is speculative instead of empirical such that it differs notably in perspective from the normal knowledge of present cognitive technology.

Readership: Graduate scholars in utilized arithmetic, records and cognitive technological know-how in addition to postgraduates and researchers in those fields.

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A Calculus of Ideas: A Mathematical Study of Human Thought

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Modalities can be ordered by inclusion. For example, ANIMAL ⊂ ANIMATE . Note that this ordering is different from the partial order discussed above. It is clear that M forms a lattice, a POSET. This means that the ordering of modalities produces entities on different planes of modality. We have been denoting modalities (on the first plane) by capital letters and shall use bold faced capitals for the next plane. 12 A Concept of Concept We shall make the idea of a modality clearer. A concept, a modality M , is an item that can be used as an independent unit: it can connect to primitive thoughts as well as to other modalities as long as regularity is observed.

6 Thought Patterns A subset P ⊂ MIND(R) is called a thought pattern if it is invariant with respect to the modality group S. 7 Completion Thoughts are made meaningful by the application of the COMPLETE operation that closes out-bonds. 8 Generalization Thoughts are generalized by the application of the MOD operation from a semi-group GENERALIZATION. 9 Abstraction The device of encapsulation abstracts thoughts to ideas that can be referred to as independent units; they are automatically added to the generator space G.

On the contrary, some will occur more often than others: due to external stimuli and remembered events, some are more likely than others. To formalize this we introduce an idea function Q taking positive values over the generator space, Q(g) > 0; g ∈ G. A large value of Q(g) means that the elementary idea g is likely and vice versa. The Q-values need not be normalized to probabilities, for example Q ≡ 1 is allowed and means no preference for any generator. So a person overly concerned about his wealth will have large values for Q(money), Q(stocks), Q(rich), Q(acquire), .

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