## Posts Tagged ‘**Numbers**’

## “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, andhoi 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

dokoto mean “one hundred mulberry bushes,” but alsokoroto mean “one hundred coconuts.” Germans will talk about aFaden, 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.

## “No part of mathematics is ever, in the long run, ‘useless’.”*…

The number 1 can be written as a sum of distinct unit fractions, such as 1/2 + 1/3 + 1/12 + 1/18 + 1/36…

Number theorists are always looking for hidden structure. And when confronted by a numerical pattern that seems unavoidable, they test its mettle, trying hard — and often failing — to devise situations in which a given pattern cannot appear.

One of the latest results to demonstrate the resilience of such patterns, by Thomas Bloom of the University of Oxford, answers a question with roots that extend all the way back to ancient Egypt.

“It might be the oldest problem ever,” said Carl Pomerance of Dartmouth College.

The question involves fractions that feature a 1 in their numerator, like 1/2, 1/7 or 1/122. These “unit fractions” were especially important to the ancient Egyptians because they were the only types of fractions their number system contained: With the exception of a single symbol for 23, they could only express more complicated fractions (like 3/4) as sums of unit fractions (1/2 + 1/4).

The modern-day interest in such sums got a boost in the 1970s, when Paul Erdős and Ronald Graham asked how hard it might be to engineer sets of whole numbers that don’t contain a subset whose reciprocals add to 1. For instance, the set {2, 3, 6, 9, 13} fails this test: It contains the subset {2, 3, 6}, whose reciprocals are the unit fractions 1/2, 1/3 and 1/6 — which sum to 1.

More exactly, Erdős and Graham conjectured that any set that samples some sufficiently large, positive proportion of the whole numbers — it could be 20% or 1% or 0.001% — must contain a subset whose reciprocals add to 1. If the initial set satisfies that simple condition of sampling enough whole numbers (known as having “positive density”), then even if its members were deliberately chosen to make it difficult to find that subset, the subset would nonetheless have to exist.

“I just thought this was an impossible question that no one in their right mind could possibly ever do,” said Andrew Granville of the University of Montreal. “I didn’t see any obvious tool that could attack it.”…

Bloom, building on work by Ernie Croot, found that tool. The ubiquity of ways to represent whole numbers as sums of fractions: “Math’s ‘Oldest Problem Ever’ Gets a New Answer,” by Jordana Cepelewicz (@jordanacep) in @QuantaMagazine.

* “No part of mathematics is ever, in the long run, ‘useless.’ Most of number theory has very few ‘practical’ applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role.” – C. Stanley Ogilvy, *Excursions in Number Theory*

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**As we recombine,** we might send carefully-calculated birthday greetings to Ulugh Beg (or, officially, Mīrzā Muhammad Tāraghay bin Shāhrukh); he was born on this date in 1394. A Timurid sultan with a hearty interest in science and the arts, he is better remembered as an astronomer and mathematician.

The most important observational astronomer of the 15th century, he built the great Ulugh Beg Observatory in Samarkand between 1424 and 1429– considered by scholars to have been one of the finest observatories in the Islamic world at the time and the largest in Central Asia. In his observations he discovered a number of errors in the computations of the 2nd-century Alexandrian astronomer Ptolemy, whose figures were still being used. His star map of 994 stars was the first new one since Hipparchus. Among his contributions to mathematics were trigonometric tables of sine and tangent values correct to at least eight decimal places.

## “If the doors of perception were cleansed everything would appear to man as it is, infinite”*…

For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise…

Infinity comes in many sizes. In 1873, the German mathematician Georg Cantor shook math to the core when he discovered that the “real” numbers that fill the number line — most with never-ending digits, like 3.14159… — outnumber “natural” numbers like 1, 2 and 3, even though there are infinitely many of both.

Infinite sets of numbers mess with our intuition about size, so as a warmup, compare the natural numbers {1, 2, 3, …} with the odd numbers {1, 3, 5, …}. You might think the first set is bigger, since only half its elements appear in the second set. Cantor realized, though, that the elements of the two sets can be put in a one-to-one correspondence. You can pair off the first elements of each set (1 and 1), then pair off their second elements (2 and 3), then their third (3 and 5), and so on forever, covering all elements of both sets. In this sense, the two infinite sets have the same size, or what Cantor called “cardinality.” He designated their size with the cardinal number

ℵ0 (“aleph-zero”).But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.

Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.

Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from all the different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality

ℵ1, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely

ℵ1 real numbers. In other words, the cardinality of the continuum immediately followℵ0, the cardinality of the natural numbers, with no sizes of infinity in between.But to Cantor’s immense distress, he couldn’t prove it.

In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.

To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.

The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove. As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.

These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.

In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.

In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.

They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking — not truth itself.

Ever since Cohen, set theorists have sought to shore up the foundations of infinite math by adding at least one new axiom to ZFC. This axiom should illuminate the structure of infinite sets, engender natural and beautiful theorems, avoid fatal contradictions, and, of course, settle Cantor’s question…

Two rival axioms emerged that do just that. For decades, they were suspected of being logically incompatible.

…

In October 2018, David Asperó was on holiday in Italy, gazing out a car window as his girlfriend drove them to their bed-and-breakfast, when it came to him: the missing step of what’s now a landmark new proof about the sizes of infinity. “It was this flash experience,” he said.

Asperó, a mathematician at the University of East Anglia in the United Kingdom, contacted the collaborator with whom he’d long pursued the proof, Ralf Schindler of the University of Münster in Germany, and described his insight. “It was completely incomprehensible to me,” Schindler said. But eventually, the duo turned the phantasm into solid logic.

Their proof, which appeared in May in the

Annals of Mathematics, unites two rival axioms that have been posited as competing foundations for infinite mathematics. Asperó and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true…

There are an infinite number of infinities. Which one corresponds to the real numbers? “How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.”

[TotH to MK]

* William Blake

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**As we contemplate counting,** we might spare a thought for Georg Friedrich Bernhard Riemann; he 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.

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