Conscious experience is awash with underlying relationships. Moreover, for various brain regions such as the visual cortex, the system is biased toward some states. Representing this bias using a probability distribution shows that the system can define expected quantities. The mathematical theory in this article links these facts using expected float entropy (efe), which is a measure of the expected amount of information needed, to specify the state of the system, beyond what is already known about the system from relationships that appear as parameters. Under the requirement that the relationship parameters minimize efe, the brain defines relationships. It is proposed that when a brain state is interpreted in the context of these relationships the brain state acquires meaning in the form of the relational content of the associated experience. For a given set, the theory represents relationships using weighted relations which assign continuous weights, from 0 to 1, to the elements of the Cartesian product of that set. The relationship parameters include weighted relations on the nodes of the system and on their set of states. Examples obtained using Monte-Carlo methods (where relationship parameters are chosen uniformly at random) suggest that efe distributions with long left tails are most important.
The theoretical base for consciousness, in particular, an explanation of how consciousness is defined by the brain, has long been sought by science. We propose a partial theory of consciousness as relations defined by typical data. The theory is based on the idea that a brain state on its own is almost meaningless but in the context of the typical brain states, defined by the brain's structure, a particular brain state is highly structured by relations. The proposed theory can be applied and tested both theoretically and experimentally. Precisely how typical data determines relations is fully established using discrete mathematics.
Abstract. A theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn's lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains, they provide a good source of examples in the theory of uniform algebras and rational approximation. In this paper we take a more direct approach when proving their theorem by using transfinite induction and cardinality. An explicit reference to a theory of allocation maps is no longer required. Instead we find that the repeated application of a single operation developed from the final step of the proof by Feinstein and Heath is enough.
Over recent decades several mathematical theories of consciousness have been put forward including Karl Friston’s Free Energy Principle and Giulio Tononi’s Integrated Information Theory. In this article we further investigate theory based on Expected Float Entropy (EFE) minimisation which has been around since 2012. EFE involves a version of Shannon Entropy parameterised by relationships. It turns out that, for systems with bias due to learning, certain choices for the relationship parameters are isolated since giving much lower EFE values than others and, hence, the system defines relationships. It is proposed that, in the context of all these relationships, a brain state acquires meaning in the form of the relational content of the associated experience. EFE minimisation is itself an association learning process and its effectiveness as such is tested in this article. The theory and results are consistent with the proposition of there being a close connection between association learning processes and the emergence of consciousness. Such a theory may explain how the brain defines the content of consciousness up to relationship isomorphism.
In this paper we prove Shapiro's 1958 Conjecture on exponential polyno-mials, assuming Schanuel's Conjecture.
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