19 papers · ranked by Valyu relevance
Stanislas Dehaene, Mathias Sablé-Meyer, Lorenzo Ciccione
Concepts of exact number are often thought to originate from counting and the successor function, or from a refinement of the approximate number system (ANS). We argue here for a third origin: a shared language-of-thought (LoT) for geometry and arithmetic that involves primitives of repetition, concatenation, and…
Trieu H. Trinh, Yuhuai Wu, Quoc V. Le, He He + 1 more
Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1-4, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to…
Guo, Hongyu
Tarski's first-order axiom system E 2 for Euclidean geometry is notable for its completeness and decidability. However, the Pythagorean theorem—either in its modern algebraic form a 2 + b 2 = c 2 or in Euclid's Elements—cannot be directly expressed in E2, since neither distance nor area is a primitive notion in the…
Shuang Tian, Xiaomin Mao, Dahui Wang, Xiaoying Wang + 1 more
Perception and mental imagery are widely thought to rely on overlapping neural representations, giving rise to perception-like visual experiences during imagery generation. However, such overlap, by itself, is agnostic to the nature of the underlying neural geometry for both perception and imagery. Here we addressed…
Loe Feijs
In this article, we present a fresh perspective on language, combining ideas from various sources, but mixed in a new synthesis. As in the minimalist program, the question is whether we can formulate an elegant formalism, a universal grammar or a mechanism which explains significant aspects of the human faculty of…
Christophe Godin, Frédéric Boudon
2.3.1.#### Turtle geometry Turtle geometry is a way to define complex geometric objects in 3-D as a sequence of elementary geometric instructions. Basically, a (virtual) turtle is able to move and draw in the 3-D space as it moves. For this, the turtle is given a sequence of elementary geometric instructions that it…
Xinyuan Yan, Aaditya Krishna, Katie Van Arsdel, Ivy Gautam + 19 more
The human brain has the remarkable ability to understand and express similar concepts in multiple languages. To understand how it does so, we examined responses of hippocampal neurons during passive listening, directed speaking, and spontaneous conversation, in both English and Spanish, in a small group of balanced…
John T. Baldwin
We distinguish the axiomatic study of proofs in geometry from study about geometry from general axioms for mathematics. We briefly report on an abuse of that distinction and its unfortunate effect on US high school education. We review a number of 20th century approaches to synthetic geometry. In doing so, we…
Xerxes D. Arsiwalla, Hatem Elshatlawy, Dean Rickles
How does one formalize the structure of structures necessary for the foundations of physics? This work is an attempt at conceptualizing the metaphysics of pregeometric structures, upon which new and existing notions of quantum geometry may find a foundation. We discuss the philosophy of pregeometric structures due to…
James T. Todd, Alexander A. Petrov
Shape is an interesting property of objects because it is used in ordinary discourse in ways that seem to have little connection to how it is typically defined in mathematics. The present article describes how the concept of shape can be grounded within Euclidean and non-Euclidean geometry and also to human perception.…
Oisín Parkinson-Coombs, Rafael Núñez
Philosophers of mathematics often rely on the historical progress of mathematics in support of mathematical realism. These histories typically build on formal semantic tools to evaluate the changes in mathematics, and on these bases present later mathematical concepts as refined versions of earlier concepts which are…
Samuel Debray, Stanislas Dehaene
Mathematics is an underexplored domain of human cognition. While many studies have focused on subsets of math concepts such as numbers, fractions, or geometric shapes, few have ventured beyond these elementary domains. Here, we attempted to map out the full space of math concepts and to answer two specific questions…
Maria J. Esteban
The fundamental role of mathematics as an inspiration for artists, but also as a tool for art creation, is presented in this paper following different art fields, like architecture, sculpture, painting, photography, literature and poetry, movie making and music. The historical viewpoint is completed with recent…
José Antonio Pérez-Escobar, Colin Jakob Rittberg, Deniz Sarikaya
This paper has two aims. First, we argue that Wittgenstein’s notion of petrification can be used to explain phenomena in advanced mathematics, sometimes better than more popular views on mathematics, such as formalism, even though petrification usually suffers from a diet of examples of a very basic nature (in…
Semir Zeki, Zachary F Hale, Ahmad Beyh, Samuel E Rasche
There are different definitions of axioms, but the one that seems to have general approval is that axioms are statements whose truths are universally accepted but cannot be proven; they are the foundation from which further propositional truths are derived. Previous attempts, led by David Hilbert, to show that all of…
Suniyya A. Waraich, Jonathan D. Victor
Low-level features are typically continuous (e.g., the gamut between two colors), but semantic information is often categorical (there is no corresponding gradient between dog and turtle) and hierarchical (animals live in land, water, or air). To determine the impact of these differences on cognitive representations…
Peter Neri
Sensory operators are classically modelled using small circuits involving canonical computations, such as energy extraction and gain control. Notwithstanding their utility, circuit models do not provide a unified framework encompassing the variety of effects observed experimentally. We develop a novel, alternative…
Peter Neri
Sensory operators are classically modelled using small circuits involving canonical computations, such as energy extraction and gain control. Notwithstanding their utility, circuit models do not provide a unified framework encompassing the variety of effects observed experimentally. We develop a novel, alternative…
Authors not listed
Previously we posited that a systematic and general description of stereoisomerism could be based upon the principles of the polytopal rearrangement model. The most daunting challenge to this end is to comprehensively describe all possible geometries for arbitrary n-coordinate centres, ABn, and for this we have…