Related papers: Mental Geometry
The language of graph theory, or network science, has proven to be an exceptional tool for addressing myriad problems in neuroscience. Yet, the use of networks is predicated on a critical simplifying assumption: that the quintessential unit…
Despite the extensive amount of scholarly work done on Indian mathematics in the last 200 years, the conditions under which it originated and evolved is still not clear. Often, one reads the ancient texts with the present concepts and…
This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel…
In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
Mathematics cannot anymore be assimilated to a linguistic game, where formal proofs are strongly differentiated with conjectural thinking, without building any category of knowledge to understand the passage (Wittgenstein's gist). Nowadays,…
Creating learning models that can exhibit sophisticated reasoning skills is one of the greatest challenges in deep learning research, and mathematics is rapidly becoming one of the target domains for assessing scientific progress in this…
Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…
We offer a view of mathematics as an experimental science where axioms play the role of foundational theories like general relativity and quantum mechanics in physics. Under this view, axioms are provisional and inferred from experience…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
Machine learning problems have an intrinsic geometric structure as central objects including a neural network's weight space and the loss function associated with a particular task can be viewed as encoding the intrinsic geometry of a given…
This work continues the development of an intensional approach to computability initiated in previous work, in which programs and computations, rather than functions, constitute the primary objects of study. In this setting, models of…
Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis…
The cognitive framework of conceptual spaces bridges the gap between symbolic and subsymbolic AI by proposing an intermediate conceptual layer where knowledge is represented geometrically. There are two main approaches for obtaining the…
The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the…
We study how large language models (LLMs) ``think'' through their representation space. We propose a novel geometric framework that models an LLM's reasoning as flows -- embedding trajectories evolving where logic goes. We disentangle…
We present a conceptual space framework for modelling abstract concepts that unfold over time, demonstrated through a chess-based proof-of-concept. Strategy concepts, such as attack or sacrifice, are represented as geometric regions across…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
Mathematics is a mountain, but students need more than descriptions of the view: they need a trail they can actually walk. This paper presents the Math Teaching Atlas, a framework for mathematical exposition built around route units (single…