Related papers: Mental Geometry
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
A desirable property of an intelligent agent is its ability to understand its environment to quickly generalize to novel tasks and compose simpler tasks into more complex ones. If the environment has geometric or arithmetic structure, the…
While the utility of well-chosen abstractions for understanding and predicting the behaviour of complex systems is well appreciated, precisely what an abstraction $\textit{is}$ has so far has largely eluded mathematical formalization. In…
Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it…
This paper introduces abstractions that are meaningful for computers and that can be built and used according to computers' own criteria, i.e., computable abstractions. It is analyzed how abstractions can be seen to serve as the building…
There is a vast supply of prior art that study models for mental processes. Some studies in psychology and philosophy approach it from an inner perspective in terms of experiences and percepts. Others such as neurobiology or…
One of the most fundamental questions in Biology or Artificial Intelligence is how the human brain performs mathematical functions. How does a neural architecture that may organise itself mostly through statistics, know what to do? One…
A primary goal of physics is to create mathematical models that allow both predictions and explanations of physical phenomena. We weave maths extensively into our physics instruction beginning in high school, and the level and complexity of…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
This is a guided tour through some selected topics in geometric analysis. We have chosen to illustrate many of the basic ideas as they apply to the theory of minimal surfaces. This is, in part, because minimal surfaces is, if not the…
This paper presents a geometric approach to the problem of modelling the relationship between words and concepts, focusing in particular on analogical phenomena in language and cognition. Grounded in recent theories regarding geometric…
``Can number and geometric spaces be reconstructed from their symmetries?'' This question, which is at the heart of anabelian geometry, a theory built on the collaborative efforts of an international community in many variants and with the…
Numeric planning with control parameters extends the standard numeric planning model by introducing action parameters as free numeric variables that must be instantiated during planning. This results in a potentially infinite number of…
Neural operators are neural network-based surrogate models for approximating solution operators of parametric partial differential equations, enabling efficient many-query computations in science and engineering. Many applications,…
A key competence for open-ended learning is the formation of increasingly abstract representations useful for driving complex behavior. Abstract representations ignore specific details and facilitate generalization. Here we consider the…
Following the processing of individual topics of elementary school mathematics as content of empirical theories the question is adressed wether the associated conception of mathematics finds itself under established concepts, and how it can…
This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…
We illustrate the principle: rational generating series occuring in arithmetic geometry are motivic in nature.
We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…