Posts Tagged ‘Numbers’
“Things that are so far removed from our daily experience… are inherently hard to understand”*…
That’s certainly true of numbers. And as the numbers grow, the cognitive challenges grow with them. (Indeed, by way of example: 1 million seconds, is roughly 11.5 days; 1 billion seconds is almost 32 years.)
We’ve looked before at the mysterious extremes of math: zero and infinity [and here]. But as Dan Falk reminds us, the numbers in between can seem pretty strange as well– especially the extremely large ones. In a review of Richard Elwes‘ Huge Numbers: A Story of Counting Ambitiously, From 4½ to Fish 7, Falk spotlights some of the largest numbers humans have ever contemplated…
… Aficionados of huge numbers are called “googologists,” a reference to the number 10100, known as a googol. Such numbers have a peculiar sort of existence. For the vast majority of us, they’re of limited everyday value. Calculations at the supermarket checkout, or at tax time in April, typically involve far more modest figures. Perhaps we’ve read that the U.S. national debt is in excess of $38 trillion — a mind-numbing figure, to be sure, but it’s not as though any one individual needs to count it up in stacks of $20 bills.
And yet, much larger numbers await those who seek them out. Consider the kinds of numbers that crop up in problems involving combinations and permutations. For example, in how many distinct ways can one shuffle a deck of cards? Elwes takes us through the calculation, and we end up with a figure of about 8×1067. Compared to that number, the odds of getting a royal flush when dealt a five-card poker hand seem pretty decent, sitting at a mere 1 in 649,740 (still rare enough that many poker players have never held such a hand). Or consider that famous 1980s cultural touchstone, the Rubik’s cube. In how many ways can one scramble the cube? It turns out that the figure is about 43 quintillion, or 4.3×1019 — but in spite of that ridiculously large figure, people do routinely solve the puzzle, and champions can do it in mere seconds. In fact, as Elwes explains, no Rubik’s cube arrangement is more than 20 moves away from any other arrangement.
Or consider the age of the universe, estimated to be about 13.8 billion years. This may seem like a lengthy span of time, but our cosmic future is where the really big numbers come up. Elwes examines the so-called heat death of the universe, in which all matter has broken down into subatomic particles. We may reach this point in [10 raised to the 10th power, raised again to the 120th power] years — this dizzying figure is 10 raised to the power of 10120 — at which point, Elwes says, the universe will have ballooned up to a diameter of 10 to the power of 10 to the power of 10120 light years. (Yes, that’s [10 raised to the 10th power, again to the 10th power, then to the 120th power] light years.) Elwes adds a footnote: “At this point, the choice of units hardly matters; the distance is so immense that whether we choose to measure it in Planck lengths or giga-light years makes little difference.” Let that sink in!
As mind numbing as such figures are, the highest numbers contemplated by humans come not from physics but from pure mathematics and computer science. Like “Graham’s number” — an immense figure put forward as the upper-bound for solutions to a problem in a branch of mathematics known as Ramsey theory. Some readers may find the ensuing discussion of multi-dimensional hypercubes a bit challenging, but one can enjoy the payoff regardless: We end up with a number that can’t even be expressed in conventional notation, and which earned a mention in the 1980 edition of the “Guinness Book of World Records” as “the highest number ever used in a mathematical proof.”
Reading this book is a little bit like sitting in the back row of an auction house where a rare Picasso (let’s say) is up for grabs: How high is this thing going to go? And indeed, Elwes keeps going. We eventually meet the so-called busy beaver numbers, a set of numbers that crop up in theoretical computer science, when one tries to deduce whether a particular computer program will eventually stop, or keep going forever — a conundrum known as the “halting problem.” As Elwes explains, it’s not at all straightforward to distinguish the two types of programs (and if it was, it would help mathematicians tackle some of the most vexing problems in their field).
The fifth busy beaver number, known as BB(5) — associated with a computer program that can access five internal states — works out to 47,176,870. And that’s as far as we’ve gotten, Elwes explains. No one has worked out the value of BB(6), but he assures us that it’s beyond the range of any physical computer; and BB(16) leaves even Graham’s number in the dust.
But wait, there’s more! “Rayo’s number,” concocted by Agustín Rayo — a dean and professor at MIT — using set theory, is bigger still (here’s a fun video about it); and “Fish 7,” mentioned in the book’s subtitle, named for a Japanese googologist who goes by the pseudonym “Fish,” builds on Rayo’s number, and … well, the details are not easily digested, but the mind-melting nature of these numbers comes across as a feature, not a bug, of Elwes’s story… the narrative is enlivened by explorations of the peculiarities of math history…
… Archimedes tried to estimate how many grains of sand would be needed to fill up the known universe, back in the third century B.C. Did he simply have too much time on his hands? Not at all, insists Elwes: The Greek thinker was articulating an important idea — that no matter how unfathomably large a quantity may be, we can describe it with precision, thanks to mathematics. “Archimedes,” he writes, “was penning a manifesto for the expressive power of large numbers.”…
… [Elwes focuses] on numbers that are ridiculously large and yet finite. In the end, perhaps this is the most mind-boggling fact of all: that these enormous numbers, from Graham’s number to Fish 7 and beyond, fall as far short of infinity as does the humble number 1…
The mysteries of the massive: “The Mind-Boggling Science of Enormous Numbers,” @danfalk.bsky.social on @richardelwes.bsky.social in @undark.org.
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As we enumerate enormity, we might spare a thought for a seminal mathematician, Alan Turing; he died on this date in 1954. He was a foundational computer science pioneer (inventor of the Turing Machine (an influential model for the general-purpose computer), creator of the “Turing Test” (only too relevant in these AI-infected times), inspiration for “The Turing Award” (the “Nobel Prize of computing“), and cryptographer (leading member of the team that cracked the Enigma code during WWII).
“Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang.”*…
… and like infinity, zero can be a cognitive challenge. Yasemin Saplakoglu explains…
Around 2,500 years ago, Babylonian traders in Mesopotamia impressed two slanted wedges into clay tablets. The shapes represented a placeholder digit, squeezed between others, to distinguish numbers such as 50, 505 and 5,005. An elementary version of the concept of zero was born.
Hundreds of years later, in seventh-century India, zero took on a new identity. No longer a placeholder, the digit acquired a value and found its place on the number line, before 1. Its invention went on to spark historic advances in science and technology. From zero sprang the laws of the universe, number theory and modern mathematics.
“Zero is, by many mathematicians, definitely considered one of the greatest — or maybe the greatest — achievement of mankind,” said the neuroscientist Andreas Nieder, who studies animal and human intelligence at the University of Tübingen in Germany. “It took an eternity until mathematicians finally invented zero as a number.”
Perhaps that’s no surprise given that the concept can be difficult for the brain to grasp. It takes children longer to understand and use zero than other numbers, and it takes adults longer to read it than other small numbers. That’s because to understand zero, our mind must create something out of nothing. It must recognize absence as a mathematical object.
“It’s like an extra level of abstraction away from the world around you,” said Benjy Barnett, who is completing graduate work on consciousness at University College London. Nonzero numbers map onto countable objects in the environment: three chairs, each with four legs, at one table. With zero, he said, “we have to go one step further and say, ‘OK, there wasn’t anything there. Therefore, there must be zero of them.’”
In recent years, research started to uncover how the human brain represents numbers, but no one examined how it handles zero. Now two independent studies, led by Nieder and Barnett, respectively, have shown that the brain codes for zero much as it does for other numbers, on a mental number line. But, one of the studies found, zero also holds a special status in the brain…
Read on to find out the ways in which new studies are uncovering how the mind creates something out of nothing: “How the Human Brain Contends With the Strangeness of Zero,” from @QuantaMagazine.
Pair with Percival Everett’s provocative (and gloriously entertaining) Dr. No.
* Charles Seife, Zero: The Biography of a Dangerous Idea
Scheduling note: your correspondent is sailing again into uncommonly busy waters. So, with apologies for the hiatus, (R)D will resume on Friday the 25th…
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As we noodle on noodling on nothing, we might send carefully-calculated birthday greetings to Erasmus Reinhold; he was born on this date in 1511. A professor of Higher Mathematics (at the University of Wittenberg, where he was ultimately Rector), Reinhold worked at a time when “mathematics” included applied mathematics, especially astronomy– to which he made many contributions and of which he was considered the most influential pedagogue of his generation.
Reinhold’s Prutenicae Tabulae (1551, 1562, 1571, and 1585) or Prussian Tables were astronomical tables that helped to disseminate calculation methods of Copernicus throughout the Empire. That said, Reinhold (like other astronomers before Kepler and Galileo) translated Copernicus’ mathematical methods back into a geocentric system, rejecting heliocentric cosmology on physical and theological grounds. Both Reinhold’s Prutenic Tables and Copernicus’ studies were the foundation for the Calendar Reform by Pope Gregory XIII in 1582… and both made copious use of zeros.
“The control of large numbers is possible, and like unto that of small numbers, if we subdivide them”*…
It’s always been intuitively obvious that we handle small numbers more easily than large ones. But the discovery that the brain has different systems for representing small and large numbers provokes new questions about memory, attention, and mathematics…
More than 150 years ago, the economist and philosopher William Stanley Jevons discovered something curious about the number 4. 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 estimations were often incorrect.
Jevons’ description of his self-experiment, published in Nature in 1871, set the “foundation of how we think about numbers,” said Steven Piantadosi, a professor of psychology and neuroscience at the University of California, Berkeley. 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…
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…
One, two, three, four… and more: “Why the Human Brain Perceives Small Numbers Better,” from @QuantaMagazine.
* Sun Tzu
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As we stew over scale, we might spare a thought for a man untroubled by larger (and more complicated) numbers, Émile Picard; he died on this date in 1941. A mathematician whose theories did much to advance research into analysis, algebraic geometry, and mechanics, he made his most important contributions in the field of analysis and analytic geometry. He used methods of successive approximation to show the existence of solutions of ordinary differential equations. Picard also applied analysis to the study of elasticity, heat, and electricity. He and Henri Poincaré have been described as the most distinguished French mathematicians in their time.
Indeed, Picard was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
“Many of the things you can count, don’t count. Many of the things you can’t count, really count”*…
Still, we count… and have, as Keith Houston explains, for much, if not most of human history…
Figuring out when humans began to count systematically, with purpose, is not easy. Our first real clues are a handful of curious, carved bones dating from the final few millennia of the three-million-year expanse of the Old Stone Age, or Paleolithic era. Those bones are humanity’s first pocket calculators: For the prehistoric humans who carved them, they were mathematical notebooks and counting aids rolled into one. For the anthropologists who unearthed them thousands of years later, they were proof that our ability to count had manifested itself no later than 40,000 years ago.
…
Counting, fundamentally, is the act of assigning distinct labels to each member of a group of similar things to convey either the size of that group or the position of individual items within it. The first type of counting yields cardinal numbers such as “one,” “two,” and “three”; the second gives ordinals such as “first,” “second,” and “third.”
At first, our hominid ancestors probably did not count very high. Many body parts present themselves in pairs—arms, hands, eyes, ears, and so on—thereby leading to an innate familiarity with the concept of a pair and, by extension, the numbers 1 and 2. But when those hominids regarded the wider world, they did not yet find a need to count much higher. One wolf is manageable; two wolves are a challenge; any more than that and time spent counting wolves is better spent making oneself scarce. The result is that the very smallest whole numbers have a special place in human culture, and especially in language. English, for instance, has a host of specialized terms centered around twoness: a brace of pheasants; a team of horses; a yoke of oxen; a pair of, well, anything. An ancient Greek could employ specific plurals to distinguish between groups of one, two, and many friends (ho philos, to philo, and hoi philoi). In Latin, the numbers 1 to 4 get special treatment, much as “one” and “two” correspond to “first” and “second,” while “three” and “four” correspond directly with “third” and “fourth.” The Romans extended that special treatment into their day-to-day lives: after their first four sons, a Roman family would typically name the rest by number (Quintus, Sextus, Septimus, and so forth), and only the first four months of the early Roman calendar had proper names. Even tally marks, the age-old “five-barred gate” used to score card games or track rounds of drinks, speaks of a deep-seated need to keep things simple.
Counting in the prehistoric world would have been intimately bound to the actual, not the abstract. Some languages still bear traces of this: a speaker of Fijian may say doko to mean “one hundred mulberry bushes,” but also koro to mean “one hundred coconuts.” Germans will talk about a Faden, meaning a length of thread about the same width as an adult’s outstretched arms. The Japanese count different kinds of things in different ways: there are separate sequences of cardinal numbers for books; for other bundles of paper such as magazines and newspapers; for cars, appliances, bicycles, and similar machines; for animals and demons; for long, thin objects such as pencils or rivers; for small, round objects; for people; and more.
Gradually, as our day-to-day lives took on more structure and sophistication, so, too, did our ability to count. When farming a herd of livestock, for example, keeping track of the number of one’s sheep or goats was of paramount importance, and as humans divided themselves more rigidly into groups of friends and foes, those who could count allies and enemies had an advantage over those who could not. Number words graduated from being labels for physical objects into abstract concepts that floated around in the mental ether until they were assigned to actual things.
Even so, we still have no real idea how early humans started to count in the first place. Did they gesture? Speak? Gather pebbles in the correct amount? To form an educated guess, anthropologists have turned to those tribes and peoples isolated from the greater body of humanity, whether by accident of geography or deliberate seclusion. The conclusion they reached is simple. We learned to count with our fingers…
From an excerpt from Houston’s new book, Empire of the Sum: The Rise and Reign of the Pocket Calculator: “The Early History of Counting,” @OrkneyDullard in @laphamsquart.
* Albert Einstein
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As we tally, we might send carefully calculated birthday greetings to Stephen Wolfram; he was born on this date in 1959. A computer scientist, mathematician, physicist, and businessman, he has made contributions to all of these fields. But he is probably best known for his creation of the software system Mathematica (a kind of “idea processor” that allows scientists and technologists to work fluidly in equations, code, and text), which is linked to WolframAlpha (an online answer engine that provides additional data, some of which is kept updated in real time).
“Why, sometimes I’ve believed as many as six impossible things before breakfast”*…
Imaginary numbers were long dismissed as mathematical “bookkeeping.” But now, as Karmela Padavic-Callaghan explains, physicists are proving that they describe the hidden shape of nature…
Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.
In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.
Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity…
Learn more at “Imaginary numbers are real,” from @Ironmely in @aeonmag.
* The Red Queen, in Lewis Carroll’s Through the Looking Glass
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As we get real, we might spare a thought for two great mathematicians…
Georg Friedrich Bernhard Riemann died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.
Andrey (Andrei) Andreyevich Markov died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological, physical, and social sciences, and is widely used in Monte Carlo simulations and Bayesian analyses.
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