Saturday, August 31, 2019
A beautiful landscape painting, a beautiful piano sonata — art and music are almost exclusively described in terms of aesthetics, but what about math? Beyond useful or brilliant, can an abstract idea be considered beautiful?
Yes, actually — and not just by mathematicians, reports a new study in Cognition.
Coauthored by a Yale mathematician and a University of Bath psychologist, the study shows that average Americans can assess mathematical arguments for beauty just as they can pieces of art or music. The beauty they discerned about the math was not one-dimensional either: Using nine criteria for beauty — such as elegance, intricacy, universality, etc. — 300 individuals had better-than-chance agreement about the specific ways that four different proofs were beautiful.
This inquiry into the aesthetics of mathematics began when study co-author and Yale assistant professor of mathematics Stefan Steinerberger likened a proof he was teaching to a “really good Schubert sonata.”
“As it turns out, the Yale students who do math also do a statistically impressive amount of music,” said Steinerberger. “Three or four students came up to me afterwards and asked, ‘What did you mean by this?’ And I realized I had no idea what I meant, but it just sounded sort of right. So, I emailed the psych department.”
Yale professor of psychology Woo-Kyoung Ahn replied to Steinerberger and, after further discussion, gave him the name of a psychology graduate student with whom she thought he would get along.
Enter Samuel G.B. Johnson, study co-author and now an assistant professor of marketing at the University of Bath School of Management, who was still completing his Ph.D. in psychology at Yale when he connected with Steinerberger. Johnson studies reasoning and decision making. “A lot of my work is about how people evaluate different explanations and arguments for things,” he explained.
Steinerberger said Johnson understood immediately how to design an experiment to test his question of whether we share the same aesthetic sensibilities about math that we do about other modalities, i.e. art and music, and if this would hold true for an average person, not just a career mathematician like himself.
“I had some diffuse notion about this, but Sam immediately got it,” said Steinerberger. “It was a match made in heaven.”
For the study, they chose four each of mathematical arguments, landscape paintings, and piano sonatas. Because the similarities between math and music have long been noted, Johnson explained, they also wanted to test people using another aesthetic modality — art in this case — to see if there’s something more universal about the way we judge aesthetics.
Johnson divided the study into three parts. The first task required a sample of individuals to match the four math proofs to the four landscape paintings based on how aesthetically similar they found them; the second required a different sample to do the same but instead comparing the proofs to sonatas; and the third required another unique sample of people to independently rate, on a scale of zero to ten, each of the four artworks and mathematical arguments along nine different criteria plus an overall score for beauty.
They derived these criteria from “A Mathematician’s Apology,” a 1940 essay by famous mathematician G.H. Hardy, which discusses mathematical beauty. The researchers’ nine dimensions elaborated from Hardy’s six were: seriousness, universality, profundity, novelty, clarity, simplicity, elegance, intricacy, and sophistication. When Steinerberger and Johnson analyzed the ratings given by participants in part three, they found that for both the artworks and math arguments a high rating for elegance was most likely to predict a high rating for beauty.
The final step was to calculate the “similarity scores” for the participants in group three, which revealed how aesthetically similar they considered each proof and painting were to each other based on the separate beauty criteria. They then compared these scores to the results from the first group of participants, who were asked to simply match proofs with paintings based on their own intuitive sense of aesthetic similarity — much like Steinerberger’s initial analogy of the proof to a “good Schubert sonata.”
When the results came in, Steinerberger and Johnson were surprised but pleased. They were able to take the similarity scores from participants in the third task to predict how the participants would behave in the first task. Participants in the third group agreed about which arguments were elegant and which paintings were elegant while, likewise, participants in the first group tended to match the argument the third group rated as most elegant with the painting they’d rated most elegant.
Laypeople not only had similar intuitions about the beauty of math as they did about the beauty of art but also had similar intuitions about beauty as each other. In other words, there was consensus about what makes something beautiful, regardless of modality.
“I’d like to see our study done again but with different pieces of music, different proofs, different artwork,” said Steinerberger. “We demonstrated this phenomenon, but we don’t know the limits of it. Where does it stop existing? Does it have to be classical music? Do the paintings have to be of the natural world, which is highly aesthetic?”
While quick to point out that they are not education scholars, both Steinerberger and Johnson see eventual implications of this research for math education, especially at the secondary-school level.
“There might be opportunities to make the more abstract, more formal aspects of mathematics more accessible and more exciting to students at that age,” said Johnson, “And that might be useful in terms of encouraging more people to enter the field of mathematics.”
“I think if you understand what people consider beautiful in math, then it could give insight into how people understand math in the first place and how they process it,” added Steinerberger. “There’s also the human implication of the question: How are we actually thinking about things as human beings? I think we have an obligation to collaborate with psychologists on this.”
Journal article: https://www.sciencedirect.com/science/article/pii/S0010027719300927?via%3Dihub
Friday, August 30, 2019
If our eyes could see high-energy radiation called gamma rays, the Moon would appear brighter than the Sun! That’s how NASA’s Fermi Gamma-ray Space Telescope has seen our neighbor in space for the past decade.
Gamma-ray observations are not sensitive enough to clearly see the shape of the Moon’s disk or any surface features. Instead, Fermi’s Large Area Telescope (LAT) detects a prominent glow centered on the Moon’s position in the sky.
Mario Nicola Mazziotta and Francesco Loparco, both at Italy’s National Institute of Nuclear Physics in Bari, have been analyzing the Moon’s gamma-ray glow as a way of better understanding another type of radiation from space: fast-moving particles called cosmic rays.
“Cosmic rays are mostly protons accelerated by some of the most energetic phenomena in the universe, like the blast waves of exploding stars and jets produced when matter falls into black holes,” explained Mazziotta.
Because the particles are electrically charged, they’re strongly affected by magnetic fields, which the Moon lacks. As a result, even low-energy cosmic rays can reach the surface, turning the Moon into a handy space-based particle detector. When cosmic rays strike, they interact with the powdery surface of the Moon, called the regolith, to produce gamma-ray emission. The Moon absorbs most of these gamma rays, but some of them escape.
Mazziotta and Loparco analyzed Fermi LAT lunar observations to show how the view has improved during the mission. They rounded up data for gamma rays with energies above 31 million electron volts — more than 10 million times greater than the energy of visible light — and organized them over time, showing how longer exposures improve the view.
“Seen at these energies, the Moon would never go through its monthly cycle of phases and would always look full,” said Loparco.
As NASA sets its sights on sending humans to the Moon by 2024 through the Artemis program, with the eventual goal of sending astronauts to Mars, understanding various aspects of the lunar environment take on new importance. These gamma-ray observations are a reminder that astronauts on the Moon will require protection from the same cosmic rays that produce this high-energy gamma radiation.
While the Moon’s gamma-ray glow is surprising and impressive, the Sun does shine brighter in gamma rays with energies higher than 1 billion electron volts. Cosmic rays with lower energies do not reach the Sun because its powerful magnetic field screens them out. But much more energetic cosmic rays can penetrate this magnetic shield and strike the Sun’s denser atmosphere, producing gamma rays that can reach Fermi.
Although the gamma-ray Moon doesn’t show a monthly cycle of phases, its brightness does change over time. Fermi LAT data show that the Moon’s brightness varies by about 20% over the Sun’s 11-year activity cycle. Variations in the intensity of the Sun’s magnetic field during the cycle change the rate of cosmic rays reaching the Moon, altering the production of gamma rays.
Light touch plays a critical role in everyday tasks, such as picking up a glass or playing a musical instrument. The sensation is also an essential part of the body’s protective defense system, alerting us to objects in our environment that could cause us to fall or injure ourselves. In addition, it is part of the detection system that has evolved to protect us from biting insects, such as those that cause malaria and Lyme disease, by eliciting a feeling of an itch when an insect lands on your skin.
Salk researchers have discovered how neurons in the spinal cord help transmit such itch signals to the brain. Published in the journal Cell Reports on July 16, 2019, their findings help contribute to a better understanding of itch and could lead to new drugs to treat chronic itch, which occurs in such conditions as eczema, diabetes and even some cancers.
“The takeaway is that this mechanical itch sensation is distinct from other forms of touch and it has this specialized pathway within the spinal cord,” says Salk Professor Martyn Goulding, holder of the Frederick W. and Joanna J. Mitchell Chair and a senior author of the new work.
Goulding and his colleagues had previously discovered a set of inhibitory neurons in the spinal cord that act like cellular brakes, keeping the mechanical itch pathway in the spinal cord turned off most of the time. Without these neurons, which produce the neurotransmitter neuropeptide Y (NPY), the mechanical itch pathway is constantly on, causing chronic itch. What the researchers didn’t know was how the itch signal, which under normal circumstances is suppressed by the NPY neurons, is transmitted to the brain to register the itch sensation.
David Acton, a postdoctoral fellow in the Goulding lab, hypothesized that when the NPY inhibitory neurons are missing, neurons in the spinal cord that normally transmit light touch begin to act like an accelerator stuck in the “on” position. Acton then identified a candidate for these “light touch neurons,” a population of excitatory neurons in the spinal cord that express the receptor for NPY, the so-called Y1 spinal neurons.
To test whether these neurons were indeed acting like an accelerator, Acton undertook an experiment that involved selectively getting rid of both the NPY “brake” and Y1 “accelerator” neurons. Without Y1 neurons, mice didn’t scratch, even in response to light-touch stimuli that normally make them scratch. Moreover, when Acton gave the animals drugs that activated the Y1 neurons, the mice scratched spontaneously even in the absence of any touch stimuli. The Goulding team was then able to show that the NPY neurotransmitter controls the level of Y1 neuron excitability; in other words, NPY signaling acts as a kind of thermostat to control our sensitivity to light touch. Data from other labs has found that some people with psoriasis have lower than average levels of NPY. This may mean their brakes on mechanical itching are less effective than other people’s, a potential cause of their itching.
While Y1 neurons transmit the itch signal in the spinal cord, other neurons are thought to be responsible for mediating the final response in the brain but more research is needed to continue mapping out the full pathway, according to the researchers. Understanding this will help suggest targets for drugs to turn down the sensation of itch in people who are overly responsive and could lead to ways to address chronic itch.
“By working out mechanisms by which mechanical itch is signaled under normal circumstances, we might then be able to address what happens in chronic itch,” says Acton.
Journal article: https://www.cell.com/cell-reports/fulltext/S2211-1247(19)30799-5?_returnURL=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS2211124719307995%3Fshowall%3Dtrue