Author, entrepreneur, and AI professor Graham Morehead joins the Local Maximum to talk about Artificial Thought and human potential working with AI in his book, “The Shape of Thought”
I am chewing on his use of the term "homeomorphic." Mathematically, homeomorphism refers to a mapping between topological spaces that keeps the topology and has an inverse. What this means is that the two objects under consideration (topological spaces) are equivalent to each other in all their topological features.
There are similar terms for other structures: homomorphism for algebraic structures, isometry for spaces with a metric. If you are dealing with sets, then equivalence means basically that two sets have the same cardinality, i.e. there is a bijection between them. At the other end of the complexity scale, a diffeomorphism preserves the differentiable structure of a manifold.
These concepts have been generalized in a field called "category theory." Category, loosely speaking, defines a collection of objects with similar structure (groups, abelian groups, topological spaces, ...). Connecting some of these objects are, abstractly, "arrows" called "morphisms." We generally think of morphisms as being functions in the usual sense, but this is not true in all categories.
The morphisms define the structure of a category and the generalization of the concepts of homeomorphism, homomorphism, isometry, etc. is defined in category theory so that it has the intuitive meaning that the two objects related to each other by two morphisms that are each the inverse of the other have, in some sense, the same structure.
So, in some sense, talking only about the number of right answers in a vast sea of possibility may, in both the philosophical and mathematical sense, be a categorical error. If there is a differential structure (i.e. a category of manifolds) on the space of interest, for example, it might be possible to use the gradient to efficiently move toward a solution much more efficiently than a random search, which would be the case if we think of our objects merely as collections of points (i.e. the category of sets).
I am chewing on his use of the term "homeomorphic." Mathematically, homeomorphism refers to a mapping between topological spaces that keeps the topology and has an inverse. What this means is that the two objects under consideration (topological spaces) are equivalent to each other in all their topological features.
There are similar terms for other structures: homomorphism for algebraic structures, isometry for spaces with a metric. If you are dealing with sets, then equivalence means basically that two sets have the same cardinality, i.e. there is a bijection between them. At the other end of the complexity scale, a diffeomorphism preserves the differentiable structure of a manifold.
These concepts have been generalized in a field called "category theory." Category, loosely speaking, defines a collection of objects with similar structure (groups, abelian groups, topological spaces, ...). Connecting some of these objects are, abstractly, "arrows" called "morphisms." We generally think of morphisms as being functions in the usual sense, but this is not true in all categories.
The morphisms define the structure of a category and the generalization of the concepts of homeomorphism, homomorphism, isometry, etc. is defined in category theory so that it has the intuitive meaning that the two objects related to each other by two morphisms that are each the inverse of the other have, in some sense, the same structure.
So, in some sense, talking only about the number of right answers in a vast sea of possibility may, in both the philosophical and mathematical sense, be a categorical error. If there is a differential structure (i.e. a category of manifolds) on the space of interest, for example, it might be possible to use the gradient to efficiently move toward a solution much more efficiently than a random search, which would be the case if we think of our objects merely as collections of points (i.e. the category of sets).