#numbers
The #Number #Five Is Nothing Like the Number #Four
Our #brains #process them in completely different ways.
More than 150 years ago, the economist and philosopher William Stanley Jevons discovered something curious about the number four. While musing about how the mind conceives of numbers, he tossed a handful of black beans into a cardboard box. Then, after a fleeting glance, he guessed how many there were, before counting them to record the true value. After more than 1,000 trials, he saw a clear pattern. When there were four or fewer beans in the box, he always guessed the right number. But for five beans or more, his quick estimates were often incorrect.
Jevons’s description of his self-experiment, published in Nature in 1871, set the “foundation of how we think about numbers,” Steven Piantadosi, a professor of psychology and neuroscience at UC Berkeley, says. It sparked a long-lasting and ongoing debate about why there seems to be a limit on the number of items we can accurately judge to be present in a set.
Now a new study in Nature Human Behaviour has edged closer to an answer by taking an unprecedented look at how human brain cells fire when presented with certain quantities. Its findings suggest that the brain uses a combination of two mechanisms to judge how many objects it sees. One estimates quantities. The second sharpens the accuracy of those estimates—but only for small numbers.
It’s “very exciting” that the findings connect long-contested ideas to their neural underpinnings, Piantadosi, who was not involved in the study, says. “There’s not many things in cognition where people have been able to pinpoint very plausible biological foundations.” Although the new study does not end the debate, the findings start to untangle the biological basis for how the brain judges quantities, which could inform bigger questions about memory, attention, and even mathematics.
The ability to instantly judge the number of items in a set doesn’t have anything to do with counting. Human infants have this number sense even before they learn language. And it’s not limited to humans: Monkeys, bees, fish, crows, and other animals also have it. A monkey needs to be able to quickly judge the number of apples in a tree, and also how many other monkeys it’s competing against for those apples. A lion, when confronted by other lions, has to decide whether to fight or flee. Honeybees need to know which area has the most flowers for foraging. A guppy has better chances of escaping a predator if it joins a shoal. “The bigger the shoal, the safer that little fish is,” Brian Butterworth, a cognitive neuroscientist at University College London who was not involved in the new work, says.
This innate number sense is therefore crucial to survival, increasing an animal’s chances of finding food, avoiding predators, and ultimately reproducing. “It simply pays off for the survival of an animal to be able to differentiate numeric quantities,” says Andreas Nieder, a professor of animal physiology at the University of Tübingen, in Germany, who co-led the new study. The fact that this ability is found in diverse animals, from insects to humans, suggests that it arose a long time ago, and its neural basis has interested cognitive scientists for decades.
In 2002, when Nieder was working with the neuroscientist Earl Miller at MIT as a postdoctoral fellow, they published one of the first pieces of evidence that numbers are linked to specific neurons. In a behavioral experiment using monkeys, they found that these neurons, which are located in the prefrontal cortex, where higher-level processing takes place, have preferred numbers—favorite numbers that, when perceived, make the cells light up in brain scans.
For example, some neurons are tuned to the number three. When they’re presented with three objects, they fire more. Other neurons are tuned to the number five and fire when presented with five objects, and so on. These neurons aren’t exclusively committed to their favorites: They also fire for numbers adjacent to them. (So the neuron tuned to five also fires for four and six objects.) But they don’t do it as often, and as the presented number gets further away from the preferred number, the neurons’ firing rate decreases.
Nieder was excited by the deeper questions the work presented about the development of mathematical ability. Numbers lead to counting, and then to symbolic number representations, such as Arabic numerals that stand in for quantities. Those symbolic numbers underpin arithmetic and mathematics. “For us to know how numbers are represented [in the brain] is setting the foundation for everything that’s coming later,” Nieder says.
He went on to learn as much as he could about number neurons. In 2012, his team discovered that the neurons respond to their preferred numbers when they’re estimating the quantity of a set of sounds or visual items. Then, in 2015, they showed that crows also have number neurons. In a show of “amazing crow behavior,” Nieder says, the birds could correctly peck the number of dots or Arabic numerals displayed to them.
However, no one had identified number neurons in humans. That’s because studying the human brain is notoriously difficult: Scientists usually can’t access its activity ethically in experiments while people are alive. Brain-imaging tools don’t have the resolution needed to distinguish individual neurons, and scientific curiosity alone can’t justify implanting invasive electrodes in the brain.
To peer into a living brain, Nieder needed to find patients who already had electrode implants and who would consent to being part of his research. In 2015, he contacted Florian Mormann—the head of the cognitive and clinical neurophysiology group at the University of Bonn, who is one of the few clinicians in Germany who does single-cell recordings in human patients—to see if he and his patients would join Nieder’s search for human number neurons. Mormann said yes, and their teams got to work examining the brain activity of his epilepsy patients, who had previously had electrodes implanted to improve their medical care.
Nine patients did simple calculations in their heads while researchers recorded their brain activity. Sure enough, in the data, Nieder and Mormann saw neurons firing for their preferred numbers—the first time number neurons had been identified in the human brain. They published their findings in Neuron in 2018.
Neuroscientists are of course driven to understand the mind, Nieder says, so “finding such neurons in the human brain is extremely rewarding.” To continue their quest, Nieder and Mormann launched a new study to find out how the neurons represent odd and even numbers. The researchers recruited 17 epilepsy patients and showed them flashes of dots, ranging in number from one to nine, on computer screens. The participants indicated whether they saw an odd or even number while electrodes recorded their brain activity.
Over the next few months, as Esther Kutter, a graduate student studying with Nieder, analyzed the resulting data, she saw a clear pattern emerge—right around the number four. The data, which comprised 801 recordings of single neurons firing, showed two distinct neural signatures: one for small numbers and one for large. Above the number four, the neurons’ firing for their preferred number grew progressively less precise, and they erroneously fired for numbers close to the preferred one. But for four and below, the neurons fired precisely—with the same small amount of error whether firing for one, two, three, or four objects. The misfiring in response to other numbers was largely absent.
This surprised Nieder. He hadn’t previously seen this boundary in his animal studies: Those experiments had included numbers only up to five. He hadn’t set out to probe Jevons’s observation, nor did he expect to see a neural boundary confirm what behavioral studies had found. Up until that point, he had been convinced that the brain had just one mechanism for judging numbers—a continuum that got fuzzier the higher the numbers climbed.
The new data changed that for him. “This boundary popped out in different ways,” Nieder says. The neural patterns suggested that there is an additional mechanism that suppresses smaller-number neurons from firing for the wrong numbers. Piantadosi and Serge Dumoulin, the director of the Spinoza Centre for Neuroimaging, in Amsterdam, had both previously published papers supporting the idea that only one mechanism manages the neuronal interpretation of numbers. Yet they were struck by Nieder and Mormann’s new data showing that there are, in fact, two separate mechanisms.
It’s “real validation that large and small numbers have different neural signatures,” Piantadosi says. But he cautioned that two signatures can emerge from a single process; whether it should be described as one mechanism or two is still up for debate. “This is just beautiful,” Dumoulin says. “This type of data wasn’t available, and certainly not in humans.”
However, one more major uncertainty remains. The researchers didn’t study the prefrontal or parietal cortices, where the majority of number neurons are located in monkeys. Instead, because of where the patients’ electrodes were inserted, the study focused on the medial temporal lobe, which is involved in memory. It isn’t the first place in the human brain you’d investigate to understand numbers, Nieder says. “On the other hand, the medial temporal lobe is also not the worst place to look for such neurons.”
That’s because the medial temporal lobe is linked to number sense. It’s active when children learn calculations and multiplication tables, and it’s intimately connected to regions where number neurons are thought to lie, Nieder says. It’s not clear why number neurons are present in this region, Butterworth says. “The things that we thought were specific to the parietal lobe seem to be reflected also in parts of the medial temporal lobe.”
One possibility is that these aren’t number neurons at all. Pedro Pinheiro-Chagas, an assistant professor of neurology at UC San Francisco, thinks these could instead be concept neurons, which are located in the medial temporal lobe and are each linked to specific concepts. For example, one famous study found a concept neuron that responded directly and specifically to images of the actor Jennifer Aniston. “Maybe they are not finding the mechanisms of the number sense … Maybe they’re finding concept cells that are also applied to numbers,” Pinheiro-Chagas says. “As you have the concept of Jennifer Aniston, you could have the concept of three.”
The level of analysis is “just really outstanding,” says Marinella Cappelletti, a cognitive neuroscientist at Goldsmiths, University of London. The researchers provide “compelling evidence” for dual mechanisms in the medial temporal lobe. She thinks it would be valuable, however, to see whether these mechanisms operate in other brain regions as well, if the opportunity presents itself. “I see these findings as looking into a window,” Cappelletti says. “It would be nice to open it up a bit more and tell us more about the rest of the brain.”
The new findings have clear parallels to the limitations of working memory. People can hold only a certain number of objects in their awareness, or working memory, at one time. Experiments show that number is also four. The agreement between the boundary of number sense and that of working memory is “hard to ignore,” Cappelletti says.
The mechanisms may be related. In previous studies of number sense, when a participant stopped paying attention, they lost their ability to precisely judge the true value of numbers four and below. That suggests that the small-number system, which suppresses adjacent misfirings with small numbers, might be intimately tied to attention. Nieder now hypothesizes that the small-number system turns on only when you’re paying attention to what’s in front of you. He’s hoping to test this idea in monkeys, in addition to looking for a neural boundary at four that their experiments haven’t yet captured.
The latest research “seems to be the beginning of a new leap” in our understanding of number perception, Pinheiro-Chagas says, which could have useful applications. He hopes it will be fodder for discussions in math education and even artificial intelligence, which struggles with numerosity perception. Large language models are “pretty bad at counting,” he says. “They are pretty bad at understanding quantities.”
Better characterizing number neurons can also help us understand who we are. Next to the language system, number representation is humans’ second-biggest symbol system. People use numbers frequently and in a variety of ways, and we and our ancestors have used math to describe the world for millennia. In that sense, math is a fundamental part of being human. And, as this study starts to show, this calculation prowess might all stem from a finely tuned network of neurons in the #brain.
“Mind-blowing” #Esoteric #Origins of #Numbers & The Name of #God, Hugh Evans
https://www.youtube.com/watch?v=Sh1NM0h0M8Q
#HughEvans, an Aeronautical engineer and Chartered Accountant applies his professional scrutiny to a lifelong passion for #history; offering a new perspective and rediscovering the story.
https://originofthezodiac.com/
https://www.bitchute.com/video/yEXDFizkMgHf/
#IRELAND - Daithí Ó Fallamháin speech Immigration: Let’s talk about the numbers.
https://www.youtube.com/watch?v=vl-jWD6RAYE
Germans what do you think: should we switch?
Deutschsprachige hier: Meint ihr, wir sollten auf das Zwanzig-Eins System wechseln?
#deutsch #zahlen #zahlenbitte #sprache #language #german #numbers
From the LOOK HERE not there distraction department
Find the Robbery Tag on this chart from 2018
https://www.tcworkerscenter.org/2018/09/wage-theft-vs-other-forms-of-theft-in-the-u-s/
#Focus on the little #numbers #robbery #wagetheft
The arithmetic of American insanity
Summary (percentages express percent of American adults):
Acceptance of the Big Lie: 15%
Support for candidates who repeat the Big Lie: 11%
Support for the Jan. 6th coup attempt: 7%
Support for the political use of violence: 9%, perhaps as much as 12%
Overall fraction of Americans who are MAGAts: Roughly 10%
If I had to guess, I'd say 10% is probably pretty close to the base rate of serious mental illness in any random sample of American adults. So at least we know the MAGAts probably aren't a larger fraction of the population than the otherwise-mentally-ill.
Figuring out how many ‘MAGA Republicans’ there actually are
Analysis by Philip Bump
There are two views of who President Biden was excoriating in his speech on Thursday night.
To Biden and his team, the group was clearly delineated: “MAGA Republicans,” a group of Americans who support former president Donald Trump, reject the outcome of the 2020 election and are open to political violence as a tactic. To Biden’s critics — a group that includes but isn’t limited to Trump supporters — he was speaking more broadly, using the term “MAGA Republicans” as cover for attacking the right broadly.
The Washington Post’s Aaron Blake has walked through what Biden said and the context in which he said it, both on Thursday and last week, when he used the label “semi-fascism” to describe that group’s worldview. But what if we went one step further, trying to assign an actual numeric value to the group Biden is describing? It’s not “half the country” as Sen. Josh Hawley (R-Mo.) and others have insisted. But how much of the country is it?
We’ll start with Biden’s description of the group.
“MAGA Republicans do not respect the Constitution. They do not believe the rule of law. They do not recognize the will of the people. They refuse to accept the results of a free election. And they’re working right now, as I speak, in state after state, to give power to decide elections in America to partisans and cronies, empowering election deniers to undermine democracy itself. ... They promote authoritarian leaders and they fan the flames of political violence that are a threat to our personal rights, to the pursuit of justice, to the rule of law, to the very soul of this country. They look at the mob that stormed the United States Capitol on Jan. 6, brutally attacking law enforcement, not as insurrectionists who placed a dagger at the throat of our democracy, but they look at it as patriots.”
This is a subset of what Biden said, certainly, but, with four categories, it captures the heart of what his speech was focused on: rejection of the 2020 election, embrace of candidates who similarly reject the results, approval of the Capitol riot and a willingness to consider violence as a political tool.
(That Biden later added that the Republican Party was largely beholden to this faction of its base has been used to argue that he was impugning Republicans more broadly. You may evaluate that assertion as you wish.)
Let’s begin with the first category: Those who reject the election results. Before we pull specific numbers from polling, we should set other boundaries. Biden’s comments weren’t specifically about the views of Trump supporters but, instead, of Republicans. So to evaluate how much of the country he’s describing, we’ll look at members of the GOP or, where available, independents who align with Republicans, for our analysis. Are there Republicans who deny the 2020 election results and also hate Trump? Probably! But probably not that many. [emphasis in original]
This question of how people view the 2020 election is asked regularly. Just last month, for example, YouGov asked the question on behalf of the Economist. They determined that nearly 7 in 10 Republicans believed Biden didn’t legitimately win. So how much of the country is that? [emphasis mine]
Well, about 20 percent of the country is under age 18, so we will ignore them. How many adults are Republicans? Gallup polls on this regularly. In its most recent iteration of the poll, it found that 28 percent of the country identifies as Republican while 41 percent identify as independent. Of those independents, though, more than a third lean Republican. So 45 percent of American adults are Republican or Republican-leaning independent. [emphasis mine]
Now we just do some math, applying percentages to the total population pool. The result? About 15 percent of the country (and 19 percent of U.S. adults) are Republicans who think Biden didn’t legitimately win in 2020. About 50 million MAGA Republicans, per Biden. [emphasis mine]
Except that Biden described MAGA Republicans as holding a variety of positions, not just this one. Is that 15 percent really included in Biden’s descriptor? Or is it only the subset that also believes in the potential use of political violence, etc.? This is hard to measure, since we can’t compare subgroups across polls. So we’ll simply estimate the scale of each of the differentiators Biden listed.
We move on to support for candidates who reject the election results. This is fairly easy to determine, thanks to polling produced this summer by Pew Research Center. Pew asked Americans how they felt about leaders who said that Trump was the legitimate winner in 2020 — and whether they liked such leaders a little or a lot. A third of Republicans said they liked such leaders a lot; another 19 percent said they liked them a little. That’s about 11 percent of the country, then, that likes such leaders. [emphasis in original]
Looking at the figure for Republicans and Republican-leaning independents, the total is similar: 8 percent of the country (and 11 percent of adults) like such leaders at least a little. [emphasis mine]
Then we consider Biden’s assessment of how Jan. 6 rioters were viewed. In the YouGov poll mentioned above, respondents were asked if they approved of “Trump supporters taking over the Capitol building” on that say — an admittedly generous way of phrasing it. But more than a quarter of Republicans said they approved at least somewhat, some 6 percent of the population. [first emphasis in original; second mine]
The percentage of Republicans holding that position has hovered around 25 percent since the riot occurred.
Now we get into the trickier question: support for the political use of violence. One report from Bright Line Watch in November found limited support for the specific question at hand. Would Republicans endorse the commission of violent felonies to accomplish their political goals? Very few agreed. Asked if they supported political violence if Democrats won in 2024, though, about 10 percent of strong Republicans said they supported the use of violence. [emphasis in original]
In March, The Washington Post and our partners at ABC News asked Americans the extent to which they viewed violence against the government as potentially justified. About 4 in 10 Republicans (and the same percentage of Republicans and leaning independents) believed that it was. That’s about 9 percent of the population (or 15 percent in the case of GOP/leaners). [emphasis mine]
In July, a group of researchers from the Violence Prevention Research Program at the University of California at Davis released data including a question that’s been posed in a number of recent polls, asking respondents whether they agreed that because “the traditional American way of life is disappearing so fast ... we may have to use force to save it.” Asking the question in this way is fraught, since doing so can result in “acquiescence bias” — a tendency for people to overstate the extent to which they agree. In this case, more than half of Republicans said they did — or about 12 percent of the population. [emphasis mine]
Again, we can’t assume that these percentages all overlap. But we get a consistent picture. Over and over, about 10 percent of the population (plus or minus a few percentage points) expresses the sort of view that Biden articulated: Republican or Republican-leaning and in favor of the positions he associated with “MAGA.” [emphasis mine]
If one agrees with Biden that this group poses a threat to American democracy, it is reassuring that it constitutes a tenth of the public — and not, as Biden’s detractors had it — half.
https://www.washingtonpost.com/politics/2022/09/02/trump-republicans-biden-maga/
Start your day with #word and #maths #puzzles and #games
#Wordle, which everyone knows by now. https://www.powerlanguage.co.uk/wordle/
#Dordle, which is guessing two words at once. https://zaratustra.itch.io/dordle
#Nerdle, which is a mathematical version - guessing #numbers and arithmetic. https://nerdlegame.com/
#Queerdle, full of #queer #lgbt words. https://queerdle.com/
For an extra challenge, I like to play en #français https://wordlegame.org/wordle-in-french
And sometimes גם #עברית https://meduyeket.net/
Oh look, it's lunchtime already!
