Month: May 2015

In addition to describing biological interactions, evolutionary theory has also become a valuable tool to make sense of the dynamics of social norms. Social norms determine which behaviours should be regarded as positive, and how community members should act towards each other.

In a recent publication, published in PLOS Computational Biology, researchers from RIKEN, Japan, and the Max-Planck-Institute for Evolutionary Biology (MPI) describe a new class of social norms for cooperative interactions.

Social norms play an important role in people’s everyday lives. They govern how people should behave and how reputations are formed based on past behaviours.

In the last 25 years, there has been an effort to describe these dynamics of reputations more formally, using mathematical models borrowed from evolutionary game theory. These models describe how social norms evolve over time—how successful norms can spread in a society and how detrimental norms fade.

Most of these models assume that an individual’s reputation should only depend on what this person did in the past. However, everyday experience and experimental evidence suggest that additional external factors may as well influence a person’s reputation. People do not only earn a reputation for how they act, but also based on who they interact with, and how they are affected by those interactions.

For example, with a recent series of experiments, researchers from Harvard University have shown that victims of harmful actions are often regarded as more virtuous than they actually are. To explore such phenomena more formally, researchers at the MPI for Evolutionary Biology in Plön and RIKEN, Japan, have developed a new mathematical framework to describe social norms.

According to the new framework, when a person’s action affects the well-being of another community member, the reputations of both individuals may be updated. Using this general framework, the researchers explore which properties such norms ought have to support cooperative interactions. Surprisingly, some of these social norms indeed have the property observed in the earlier experiments: when one individual defects against another, the victim’s reputation should improve.

Moreover, the researchers also observe a fundamental trade-off. Norms that are particularly good in sustaining cooperation tend to be less robust with respect to noise (such as when reputations are shaped by third-party gossip).

Overall, this work is part of a bigger effort to understand key properties of social norms in a rigorous manner. These studies shed light on which ecological and social environments facilitate cooperation, and on the effects of social norms more generally.

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Credit of the article given to Max Planck Society

An attempt to settle a decade-long argument over a controversial proof by mathematician Shinichi Mochizuki has seen a war of words on both sides, with Mochizuki dubbing the latest effort as akin to a “hallucination” produced by ChatGPT,

An attempt to fix problems with a controversial mathematical proof has itself become mired in controversy, in the latest twist in a saga that has been running for over a decade and has seen mathematicians trading unusually pointed barbs.

The story began in 2012, when Shinichi Mochizuki at Kyoto University, Japan, published a 500-page proof of a problem called the ABC conjecture. The conjecture concerns prime numbers involved in solutions to the equation a + b = c, and despite its seemingly simple form, it provides deep insights into the nature of numbers. Mochizuki published a series of papers claiming to have proved ABC using new mathematical tools he collectively called Inter-universal Teichmüller (IUT) theory, but many mathematicians found the initial proof baffling and incomprehensible.

While a small number of mathematicians have since accepted that Mochizuki’s papers prove the conjecture, other researchers say there are holes in his argument and it needs further work, dividing the mathematical community in two and prompting a prize of up to $1 million for a resolution to the quandary.

Now, Kirti Joshi at the University of Arizona has published a proposed proof that he says fixes the problems with IUT and proves the ABC conjecture. But Mochizuki and his supporters, as well as mathematicians who critiqued Mochizuki’s original papers, remain unconvinced, with Mochizuki declaring that Joshi’s proposal doesn’t contain “any meaningful mathematical content whatsoever”.

Central to Joshi’s work is an apparent problem, previously identified by Peter Scholze at the University of Bonn, Germany, and Jakob Stix at Goethe University Frankfurt, Germany, with a part of Mochizuki’s proof called Conjecture 3.12. The conjecture involves comparing two mathematical objects, which Scholze and Stix say Mochizuki did incorrectly. Joshi claims to have found a more satisfactory way to make the comparison.

Joshi also says that his theory goes beyond Mochizuki’s and establishes a “new and radical way of thinking about arithmetic of number fields”. The paper, which hasn’t been peer-reviewed, is the culmination of several smaller papers on ABC that Joshi has published over several years, describing them as a “Rosetta Stone” for understanding Mochizuki’s impenetrable maths.

Neither Joshi nor Mochizuki responded to a request for comment on this article, and, indeed, the two seem reluctant to communicate directly with each other. In his paper, Joshi says Mochizuki hasn’t responded to his emails, calling the situation “truly unfortunate”. And yet, several days after the paper was posted online, Mochizuki published a 10-page response, saying that Joshi’s work was “mathematically meaningless” and that it reminded him of “hallucinations produced by artificial intelligence algorithms, such as ChatGPT”.

Mathematicians who support Mochizuki’s original proof express a similar sentiment. “There is nothing to talk about, since his [Joshi’s] proof is totally flawed,” says Ivan Fesenko at Westlake University in China. “He has no expertise in IUT whatsoever. No experts in IUT, and the number is in two digits, takes his preprints seriously,” he says. “It won’t pass peer review.”

And Mochizuki’s critics also disagree with Joshi. “Unfortunately, this paper and its predecessors does not introduce any powerful mathematical technology, and falls far short of giving a proof of ABC,” says Scholze, who has emailed Joshi to discuss the work further. For now, the saga continues.

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*Credit for article given to Alex Wilkins*