Strategy Choice Theory (SCT; Siegler and Shrager, 1984; Siegler, 2000) explains important aspects of children's arithmetic learning based upon principles including learning from developmentally naturalistic data, probabilistic representation, confidence-based retrieval, and the phase-like importance of scaffolding strategies, such as finger-counting. Here we recast SCT as a ``Small Math Model'' (SMM), employing a neural-network-based architecture analogous to LLMs. The SMM extends SCT to include counting practice, symbol (number) embedding, and gated attention. Similar to earlier work, the SMM demonstrates constructive and destructive interference between counting and addition, and the ``wave-like'' use of finger-counting as sum recall improves. We plan to extend the SMM to later aspects of the decades-long SCT program, including adaptive strategy choice and eventually strategy discovery, providing a unified platform to investigate the understanding of numerical characteristics and relationships essential for mathematical reasoning -- as it can emerge in LLM-based agents.

Large Language Models (LLMs) have demonstrated remarkable emergent capabilities, yet the robustness of their numerical reasoning remains an open question. While standard benchmarks evaluate LLM reasoning on complex problem sets using aggregated metrics, they often obscure foundational weaknesses. In this work, we probe LLM mathematical numeracy by evaluating performance on problems of escalating complexity, from constituent operations to combinatorial puzzles. We test several state-of-the-art LLM-based agents on a 100-problem challenge comprising four categories: (1) basic arithmetic, (2) advanced operations, (3) primality checking, and (4) the Game of 24 number puzzle. Our results show that while the agents achieved high accuracy on the first three categories, which require deterministic algorithmic execution, they consistently failed at the number puzzle, underlining its demand for a heuristic search over a large combinatorial space to be a significant bottleneck. These findings reveal that the agents' proficiency is largely confined to recalling and executing known algorithms, rather than performing generative problem-solving. This suggests their apparent numerical reasoning is more akin to sophisticated pattern-matching than flexible, analytical thought, limiting their potential for tasks that require novel or creative numerical insights.

An essential element of human mathematical reasoning is our number sense -- an abstract understanding of numbers and their relationships -- which allows us to solve problems involving vast number spaces using limited computational resources. Mathematical reasoning of Large Language Models (LLMs) is often tested on high-level problems (such as Olympiad challenges, geometry, word problems, and puzzles), but their low-level number sense remains less explored. We introduce "Numberland," a 100-problem test to evaluate the numerical reasoning abilities of LLM-based agents. The tasks -- basic operations, advanced calculations (e.g., exponentiation, complex numbers), prime number checks, and the 24 game -- aim to test elementary skills and their integration in solving complex and uncertain problems. We evaluated five LLM-based agents: OpenAI's o1 and o1-mini, Google Gemini, Microsoft Copilot, and Anthropic Claude. They scored 74-95% on the first three tasks that allow deterministic steps to solutions. In the 24 game, which needs trial-and-error search, performance dropped to 10-73%. We tested the top 24 solver (o1 with 73% accuracy) on 25 harder problems, and its score fell to 27%, confirming search as a bottleneck. These results, along with the types of mistakes, suggest a fragile number of LLMs, which is a bit surprising given their prowess in challenging benchmarks. The limits of LLM numerical reasoning highlight the scope of simple, targeted tests to evaluate and explain LLM math skills to ensure safe use.

We here analyse the question of developing artificial consciousness from an evolutionary perspective, taking the evolution of the human brain and its relation with consciousness as a reference model. This kind of analysis reveals several structural and functional features of the human brain that appear to be key for reaching human-like complex conscious experience and that current research on Artificial Intelligence (AI) should take into account in its attempt to develop systems capable of conscious processing. We argue that, even if AI is limited in its ability to emulate human consciousness for both intrinsic (structural and architectural) and extrinsic (related to the current stage of scientific and technological knowledge) reasons, taking inspiration from those characteristics of the brain that make conscious processing possible and/or modulate it, is a potentially promising strategy towards developing conscious AI. Also, it is theoretically possible that AI research can develop partial or potentially alternative forms of consciousness that is qualitatively different from the human, and that may be either more or less sophisticated depending on the perspectives. Therefore, we recommend neuroscience-inspired caution in talking about artificial consciousness: since the use of the same word consciousness for humans and AI becomes ambiguous and potentially misleading, we propose to clearly specify what is common and what differs in AI conscious processing from full human conscious experience.

Philip K. Dick's iconic 1968 sci-fi novel, "Do Androids Dream of Electric Sheep?" posed an intriguing question in its title: would an intelligent robot dream? In the 53 years since publication, artificial intelligence research has matured significantly. And yet, despite Dick being prophetic about technology in other ways, the question posed in the title is not something AI researchers are that interested in; no one is trying to invent an android that dreams of electric sheep. Why? Mainly, it's that most artificial intelligence researchers and scientists are busy trying to design "intelligent" software programmed to do specific tasks. There is no time for daydreaming.

The exploration of geometrical patterns stimulates imagination and encourages abstract reasoning which is a distinctive feature of human intelligence. In cognitive science, Gestalt principles such as symmetry have often explained significant aspects of human perception. We present a computational technique for building artificial intelligence (AI) agents that use symmetry as the organizing principle for addressing Dehaene's test of geometric intelligence \cite{dehaene2006core}. The performance of our model is on par with extant AI models of problem solving on the Dehaene's test and seems correlated with some elements of human behavior on the same test.

"We cannot be conscious of what we are not conscious of." Unlike the director leads you to believe, the protagonist of Ex Machina, Andrew Garland's 2015 masterpiece, isn't Caleb, a young programmer tasked with evaluating machine consciousness. Like most cerebral movies, Ex Machina leaves the conclusion up to the viewer: was Ava actually conscious? In doing so, it also cleverly avoids a thorny question that has challenged most AI-centric movies to date: what is consciousness, and can machines have it? As machine intelligence barrels forward at breakneck speed--not only exceeding human performance on games such as DOTA and Go, but doing so without the need for human expertise--the question has once more entered the scientific mainstream.

As Vatican lights Christmas tree, Pope reflects on Nativity scene'A gravely critical moment': Catholic scholars call on Bishops to support the four Cardinals Listen to God for guidance to build a better world Pope's prayer for the Immaculate in Piazza di Spagna Be like Mary – say yes to God, but not halfway, Pope Francis says'A gravely critical moment': Catholic scholars call on Bishops to support the four Cardinals ROME: Artificial intelligence is "an extremely important goal that has not been achieved yet," said Stanislas Dehaene, a professor of cognitive neuroscience at the College de France, adding that "we don't want to create a system full of machines that don't share our intuitions of what should be a better world." The Vatican hosted a high-level discussion in the world of science, gathering experts to discuss the progress, benefits and limits of advances in artificial intelligence. A new conference at the Vatican drew experts in various fields of science and technology for a two-day dialogue on the "Power and Limits of Artificial Intelligence," hosted by the Pontifical Academy for Sciences. Among the scheduled speakers were several prestigious scientists, including Stephen Hawkins, a prominent British professor at the University of Cambridge and a self-proclaimed atheist, as well as a number of major tech heads such as Demis Hassabis, CEO of Google DeepMind, and Yann LeCun of Facebook. The event, which ran from Nov. 30-Dec.

Vatican City, Dec 4, 2016 / 03:03 am (CNA/EWTN News).- This week the Vatican hosted a high-level discussion in the world of science, gathering experts to discuss the progress, benefits and limits of advances in artificial intelligence. A new conference at the Vatican drew experts in various fields of science and technology for a two-day dialogue on the "Power and Limits of Artificial Intelligence," hosted by the Pontifical Academy for Sciences. Among the scheduled speakers were several prestigious scientists, including Stephen Hawkins, a prominent British professor at the University of Cambridge and a self-proclaimed atheist, as well as a number of major tech heads such as Demis Hassabis, CEO of Google DeepMind, and Yann LeCun of Facebook. The event, which ran from Nov. 30-Dec.

Consciousness is a major feature of mammalian nervous systems. Recent evidence indicates it may extend from mammals to birds and even cephalopods (Edelman, Seth 2009). Since all major biological adaptations are functional, or sequelae of biofunctions, and since brains perform computations, it would seem that consciousness must have a basic biocomputational function. Biologically, that means of course that consciousness endows nervous systems with one or more adaptive advantages leading to higher gene frequencies for those brains. Given that mammals have existed for some 200 million years, and that mammals share the thalamocortical core that supports conscious states, it is very likely that conscious brains have gathered not just one but many biocomputational functions. That certainly accords with our common sense notions of conscious (as well as unconscious) activities.