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Category: mathematics – Page 42

Gödel’s Incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic.

The first incompleteness theorem: No consistent formal system capable of modelling basic arithmetic can be used to prove all truths about arithmetic.

In other words, no matter how complex a system of mathematics is, there will always be some statements about numbers that cannot be proved or disproved within the system.

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Computing has already accelerated scientific discovery. Now scientists say a combination of advanced AI with next-generation cloud computing is turbocharging the pace of discovery to speeds unimaginable just a few years ago.

Microsoft and the Pacific Northwest National Laboratory (PNNL) in Richland, Washington, are collaborating to demonstrate how this acceleration can benefit chemistry and materials science – two scientific fields pivotal to finding energy solutions that the world needs.

Scientists at PNNL are testing a new battery material that was found in a matter of weeks, not years, as part of the collaboration with Microsoft to use to advanced AI and high-performance computing (HPC), a type of cloud-based computing that combines large numbers of computers to solve complex scientific and mathematical tasks.

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At the end of the 20th century, analog systems in computer science have been widely replaced by digital systems due to their higher computing power. Nevertheless, the question keeps being intriguing until now: is the brain analog or digital? Initially, the latter has been favored, considering it as a Turing machine that works like a digital computer. However, more recently, digital and analog processes have been combined to implant human behavior in robots, endowing them with artificial intelligence (AI). Therefore, we think it is timely to compare mathematical models with the biology of computation in the brain. To this end, digital and analog processes clearly identified in cellular and molecular interactions in the Central Nervous System are highlighted. But above that, we try to pinpoint reasons distinguishing in silico computation from salient features of biological computation. First, genuinely analog information processing has been observed in electrical synapses and through gap junctions, the latter both in neurons and astrocytes. Apparently opposed to that, neuronal action potentials (APs) or spikes represent clearly digital events, like the yes/no or 1/0 of a Turing machine. However, spikes are rarely uniform, but can vary in amplitude and widths, which has significant, differential effects on transmitter release at the presynaptic terminal, where notwithstanding the quantal (vesicular) release itself is digital. Conversely, at the dendritic site of the postsynaptic neuron, there are numerous analog events of computation. Moreover, synaptic transmission of information is not only neuronal, but heavily influenced by astrocytes tightly ensheathing the majority of synapses in brain (tripartite synapse). At least at this point, LTP and LTD modifying synaptic plasticity and believed to induce short and long-term memory processes including consolidation (equivalent to RAM and ROM in electronic devices) have to be discussed. The present knowledge of how the brain stores and retrieves memories includes a variety of options (e.g., neuronal network oscillations, engram cells, astrocytic syncytium). Also epigenetic features play crucial roles in memory formation and its consolidation, which necessarily guides to molecular events like gene transcription and translation. In conclusion, brain computation is not only digital or analog, or a combination of both, but encompasses features in parallel, and of higher orders of complexity.

Keywords: analog-digital computation; artificial and biological intelligence; bifurcations; cellular computation; engrams; learning and memory; molecular computation; network oscillations.

Copyright © 2023 Gebicke-Haerter.

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The Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge hosted a research programme on one of the most pressing problems in modern physics: to build a theory that can explain all the fundamental forces and particles of nature in one unifying mathematical framework. Such a theory of quantum gravity would combine two hugely successful frameworks on theoretical physics, which have so far eluded unification: quantum physics and Einstein’s theory of gravity.

The Black holes: bridges between number theory and holographic quantum information programme focusses on black holes, which play a hugely important part in this area, on something called the holographic principle, and on surprising connections to pure mathematics. This collection of articles explores the central concepts involved and gives you a gist of the cutting edge research covered by the INI programme.

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Powner’s team didn’t check whether its sulfur-based catalysts had a chiral bias. That’s where Donna Blackmond, an origin of life chemist at Scripps Research, and her colleagues Min Deng and Jinhan Yu grabbed the baton. They tested two of Powner’s sulfur compounds to see whether the catalysts were sensitive to chirality as they formed dipeptides. They were, but not in the way Blackmond had expected. The catalysts created about four times as many “heterochiral” dipeptides—those pairing a left-handed amino acid (L) with a right-handed (D) one—as fully chiral products. “We thought it was bad news,” Blackmond says, because it suggested that even if amino acids on early Earth started with a bias, it would have been scrambled as proteins formed.

But as Blackmond and her colleagues looked more deeply, the news got better. In a series of experiments, the Scripps researchers started with skewed proportions of L and D amino acids—for example, 60% Ls and 40% Ds. The L, D and D, L heterochiral dipeptides formed most quickly, and as they did they pulled equal numbers of L and D amino acids out of the mix. Because of the baseline bias, eventually a predominance of Ls remained in the pool of unreacted amino acids, raising the likelihood of forming fully lefthanded dipeptides. “It’s like a domino effect,” Powner says. The first heterochiral reaction eventually encourages more homochirals to form. “And it’s a general process that works with all amino acids,” Powner says. Joyce adds: “It’s just math.”

Follow-up experiments suggested a second bias that amplifies the effect. The team found that heterochiral dipeptides precipitate out of a solution more quickly than homochiral ones, speeding the way to a relative abundance of either homochiral L, L or D, D pairs, depending the starting mix. Just why this precipitation bias occurs isn’t yet clear, Blackmond says. However, Joyce says, together with the other effect, “it beautifully fits the [experimental] data.” Blackmond adds: “The wrong answer turned out to be the right answer to get us to homochirality.”

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Alex Rosenberg is professor of Philosophy at Duke University and has made several important contributions to the philosophy of science, biology, and social science.

0:00 intro.
2:53 scientism.
5:09 naturalism and the manifest image.
7:25 pragmatism.
10:40 intentionality.
12:38 objections to eliminativism and truth.
14:35 consciousness.
16:50 biological functions, purposes, and the selected effects theory.
22:28 reductionism.
28:05 causality.
31:02 multiple realizability.
35:13 math.
39:45 morality.
44:51 humanism, art, and history.

Alex Rosenberg’s website: https://alexrosenbergbooks.com/

Alex Rosenberg Books:
Reduction and Mechanism (2020)
How History Gets Things Wrong: The Neuroscience of Our Addiction to Stories (2018)
The Atheist’s Guide to Reality: Enjoying Life Without Illusions (2011)
Darwinian Reductionism (2006)
Philosophy of Science: A Contemporary Introduction (2000)

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Last year, scientists discovered a mathematically perfect star system — and now, they’re looking into whether it might contain signs of alien tech.

Dubbed HD 110067, the star system located just 100 light-years from Earth has six exoplanets that are each perfectly spaced apart in the sort of mathematical harmony rarely seen in our chaotic Universe. In a paper published in the journal Nature last November, scientists listed off the astounding attributes of the system, which unfortunately did not include any planets in the so-called “habitable zone,” or distance from the orbit-inducing star that could support life as we know it here on Earth.

All the same, scientists aren’t done looking, and as radio astronomer and alien life-seeking expert Steve Croft of the University of Berkeley told Space.com, there’s no reason that advanced civilizations may not have visited HD 110,067 and potentially left some of their technology behind.

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New theoretical work establishes an analogy between systems that are dynamically frustrated, such as glasses, and thermodynamic systems whose members have conflicting goals, such as predator–prey ecosystems.

A system is geometrically frustrated when its members cannot find a configuration that simultaneously minimizes all their interaction energies, as is the case for a two-dimensional antiferromagnet on a triangular lattice. A nonreciprocal system is one whose members have conflicting, asymmetric goals, as exemplified by an ecosystem of predators and prey. New work by Ryo Hanai of Kyoto University, Japan, has identified a powerful mathematical analogy between those two types of dynamical systems [1]. Nonreciprocity alters collective behavior, yet its technological potential is largely untapped. The new link to geometrical frustration will open new prospects for applications.

To appreciate Hanai’s feat, consider how different geometric frustration and nonreciprocity appear at first. Frustration defies the approach that physics students are taught in their introductory classes, based on looking at the world through Hamiltonian dynamics. In this approach, energy is to be minimized and states of matter characterized by their degree of order. Some of the most notable accomplishments in statistical physics have entailed describing changes between states—that is, phase transitions. Glasses challenge that framework. These are systems whose interactions are so spatially frustrated that they cannot find an equilibrium spatial order. But they can find an order that’s “frozen” in time. Even at a nonzero temperature, everything is stuck—and not just in one state. Many different configurations coexist whose energies are nearly the same.

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