Posts Tagged ‘Mathematics’
“If someone separated the art of counting and measuring and weighing from all the other arts, what was left of each (of the others) would be, so to speak, insignificant”*…
Mathematics, Bo Malmberg and Hannes Malmberg argue, was the cornerstone of the Industrial Revolution. A new paradigm of measurement and calculation, more than scientific discovery, built industry, modernity, and the world we inhabit today…
In school, you might have heard that the Industrial Revolution was preceded by the Scientific Revolution, when Newton uncovered the mechanical laws underlying motion and Galileo learned the true shape of the cosmos. Armed with this newfound knowledge and the scientific method, the inventors of the Industrial Revolution created machines – from watches to steam engines – that would change everything.
But was science really the key? Most of the significant inventions of the Industrial Revolution were not undergirded by a deep scientific understanding, and their inventors were not scientists.
The standard chronology ignores many of the important events of the previous 500 years. Widespread trade expanded throughout Europe. Artists began using linear perspective and mathematicians learned to use derivatives. Financiers started joint stock corporations and ships navigated the open seas. Fiscally powerful states were conducting warfare on a global scale.
There is an intellectual thread that runs through all of these advances: measurement and calculation. Geometric calculations led to breakthroughs in painting, astronomy, cartography, surveying, and physics. The introduction of mathematics in human affairs led to advancements in accounting, finance, fiscal affairs, demography, and economics – a kind of social mathematics. All reflect an underlying ‘calculating paradigm’ – the idea that measurement, calculation, and mathematics can be successfully applied to virtually every domain. This paradigm spread across Europe through education, which we can observe by the proliferation of mathematics textbooks and schools. It was this paradigm, more than science itself, that drove progress. It was this mathematical revolution that created modernity…
The fascinating story: “How mathematics built the modern world,” from @bomalmb and @HannesMalmberg1 in @WorksInProgMag.
* Plato
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As we muse on measurement, we might recall that it was on this date in 1790, early in the French Revolution, that the French Assembly, acting on the urging of Bishop Charles Maurice de Talleyrand, moved to create a new system of weights and measures based on natural units– what we now know as the metric system.
“Topology is precisely the mathematical discipline that allows the passage from local to global”*…
Jordana Cepelewicz on two new topographical results that bring some order to the confoundingly difficult study of four-dimensional shapes…
The central objects of study in topology are spaces called manifolds, which look flat when you zoom in on them. The surface of a sphere, for instance, is a two-dimensional manifold. Topologists understand such two-dimensional manifolds very well. And they have developed tools that let them make sense of three-dimensional manifolds and those with five or more dimensions.
But in four dimensions, “everything goes a bit crazy,” said Sam Hughes, a postdoctoral researcher at the University of Oxford. Tools stop working; exotic behavior emerges. As Tom Mrowka of the Massachusetts Institute of Technology explained, “There’s just enough room to have interesting phenomena, but not so much room that they fall apart.”
In the early 1990s, Mrowka and Peter Kronheimer of Harvard University were studying how two-dimensional surfaces can be embedded within four-dimensional manifolds. They developed new techniques to characterize these surfaces, allowing them to gain crucial insights into the otherwise inaccessible structure of four-dimensional manifolds. Their findings suggested that the members of a broad class of surfaces all slice through their parent manifold in a relatively simple way, leaving a fundamental property unchanged. But nobody could prove this was always true.
In February, together with Daniel Ruberman of Brandeis University, Hughes constructed a sequence of counterexamples — “crazy” two-dimensional surfaces that dissect their parent manifolds in ways that mathematicians had believed to be impossible. The counterexamples show that four-dimensional manifolds are even more remarkably diverse than mathematicians in earlier decades had realized. “It’s really a beautiful paper,” Mrowka said. “I just keep looking at it. There’s lots of delicious little things there.”
Late last year, Ruberman helped organize a conference that created a new list of the most significant open problems in low-dimensional topology. In preparing for it, he looked at a previous list of important unsolved topological problems from 1997. It included a question that Kronheimer had posed based on his work with Mrowka. “It was in there, and I think it was a little bit forgotten,” Ruberman said. Now he thought he could answer it…
Read on for the details: “Mathematicians Marvel at ‘Crazy’ Cuts Through Four Dimensions,” from @jordanacep in @QuantaMagazine.
* Rene Thom
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As we savor surprising shapes, we might send carefully-modeled birthday greetings to William Bowie; he was born on this date in 1872. A geodetic engineer who joined the United States Coast and Geodetic Survey in 1895, he investigated isostasy (a principle that dense crustal rocks to tend cause topographic depressions and light crustal rocks cause topographic elevations).
Bowie was the first President of the American Geophysical Union from 1920 to 1922 and served as president a second time from 1929 to 1932. The William Bowie Medal, the highest honor of the AGU, is named in his honor.
“Men have become the tools of their tools”*…
Visionary philosopher Bernard Stiegler argued that it’s not our technology that makes humans special; rather, it’s our relationship with that technology. Bryan Norton explains…
It has become almost impossible to separate the effects of digital technologies from our everyday experiences. Reality is parsed through glowing screens, unending data feeds, biometric feedback loops, digital protheses and expanding networks that link our virtual selves to satellite arrays in geostationary orbit. Wristwatches interpret our physical condition by counting steps and heartbeats. Phones track how we spend our time online, map the geographic location of the places we visit and record our histories in digital archives. Social media platforms forge alliances and create new political possibilities. And vast wireless networks – connecting satellites, drones and ‘smart’ weapons – determine how the wars of our era are being waged. Our experiences of the world are soaked with digital technologies.
But for the French philosopher Bernard Stiegler, one of the earliest and foremost theorists of our digital age, understanding the world requires us to move beyond the standard view of technology. Stiegler believed that technology is not just about the effects of digital tools and the ways that they impact our lives. It is not just about how devices are created and wielded by powerful organisations, nation-states or individuals. Our relationship with technology is about something deeper and more fundamental. It is about technics.
According to Stiegler, technics – the making and use of technology, in the broadest sense – is what makes us human. Our unique way of existing in the world, as distinct from other species, is defined by the experiences and knowledge our tools make possible, whether that is a state-of-the-art brain-computer interface such as Neuralink, or a prehistoric flint axe used to clear a forest. But don’t be mistaken: ‘technics’ is not simply another word for ‘technology’. As Martin Heidegger wrote in his essay ‘The Question Concerning Technology’ (1954), which used the German term Technik instead of Technologie in the original title: the ‘essence of technology is by no means anything technological.’ This aligns with the history of the word: the etymology of ‘technics’ leads us back to something like the ancient Greek term for art – technē. The essence of technology, then, is not found in a device, such as the one you are using to read this essay. It is an open-ended creative process, a relationship with our tools and the world.
This is Stiegler’s legacy. Throughout his life, he took this idea of technics, first explored while he was imprisoned for armed robbery, further than anyone else. But his ideas have often been overlooked and misunderstood, even before he died in 2020. Today, they are more necessary than ever. How else can we learn to disentangle the effects of digital technologies from our everyday experiences? How else can we begin to grasp the history of our strange reality?…
[Norton unspools Stiegler’s remarkable life and the development of his thought…]
… Technology, for better or worse, affects every aspect of our lives. Our very sense of who we are is shaped and reshaped by the tools we have at our disposal. The problem, for Stiegler, is that when we pay too much attention to our tools, rather than how they are developed and deployed, we fail to understand our reality. We become trapped, merely describing the technological world on its own terms and making it even harder to untangle the effects of digital technologies and our everyday experiences. By encouraging us to pay closer attention to this world-making capacity, with its potential to harm and heal, Stiegler is showing us what else is possible. There are other ways of living, of being, of evolving. It is technics, not technology, that will give the future its new face…
Eminently worth reading in full: “Our tools shape our selves,” from @br_norton in @aeonmag.
Compare and contrast: Kevin Kelly‘s What Technology Wants
* Henry David Thoreau
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As we own up, we might send phenomenological birthday greetings to Immanuel Kant; he was born on this date in 1724. One of the central figures of modern philosophy, Kant is remembered primarily for his efforts to unite reason with experience (e.g., Critique of Pure Reason [Kritik der reinen Vernunft], 1781), and for his work on ethics (e.g., Metaphysics of Morals [Die Metaphysik der Sitten], 1797) and aesthetics (e.g., Critique of Judgment [Kritik der Urteilskraft], 1790).
But Kant made important contributions to mathematics and astronomy. For example: his argument that mathematical truths are a form of synthetic a priori knowledge was cited by Einstein as an important early influence on his work. And his description of the Milky Way as a lens-shaped collection of stars that represented only one of many “island universes,” was later shown to be accurate by Herschel.
Act so as to treat humanity, whether in your own person or in that of another, at all times also as an end, and not only as a means.
“The greatest value of a picture is when it forces us to notice what we never expected to see”*…
The breath-takingly broadly talented Joesph Preistley left us much– not least, Alyson Foster explains, a then-new way of understanding history…
It’s a testament to the wide-ranging and unconventional nature of Joseph Priestley’s mind that no one has settled on a term to sum up exactly what he was. The eighteenth-century British polymath has been described as, among other things, a historian, a chemist, an educator, a philosopher, a theologian, and a political radical who became, for a period of time, the most despised person in England. Priestley’s many contradictions—as a rationalist Unitarian millenarian, as a mild-mannered controversialist, as a thinker who was both ahead of his time and behind it—have provided endless fodder for the historians who have debated the precise nature of his legacy and his place among his fellow Enlightenment intellectuals. But his contributions—however they are categorized—have continued to live on in subtle and surprisingly enduring ways, more than two hundred years after his death, at the age of seventy, in rural Pennsylvania.
Take, for example, A Chart of Biography, which is considered to be the first modern timeline. This unusual, and unusually beautiful, pedagogical tool, which was published by Priestley in 1765, while he was in his thirties and working as a tutor at an academy in Warrington, England, tends to get lost in the shuffle of Priestley’s more notable achievements—his seminal 1761 textbook on language, The Rudiments of English Grammar, say, or his discovery of nine gases, including oxygen, 13 years later. But the chart, along with its companion, A New Chart of History, which Priestley published four years later, has become a curious subject of interest among data visualization aficionados who have analyzed its revolutionary design in academic papers and added it to Internet lists of notable infographics. Recently, both charts have become the focus of an NEH-supported digital humanities project, Chronographics: The Time Charts of Joseph Priestley, produced by scholars at the University of Oregon.
Even those of us ignorant of (or uninterested in) infographics can look at the painstakingly detailed Chart of Biography for a moment or two and appreciate how it has become a source of fascination. The two-foot-by-three-foot, pastel-striped paper scroll—which contains the meticulously inscribed names of approximately 2,000 poets, artists, statesmen, and other famous historical figures dating back three millennia—is visually striking, combining a formal, somewhat ornate eighteenth-century aesthetic with the precise organization of a schematic. Every single one of the chart’s subjects is grouped vertically into one of six occupational categories, then plotted out chronologically along a horizonal line divided into ten-year increments. Despite the huge quantity of information it contains, it is extremely user-friendly. Any one of Priestley’s history students could run his eye across the chart and immediately gain a sense of the temporal lay of the land. Who came first: Copernicus or Newton? How many centuries separate Genghis Khan from Joan of Arc? Which artists were working during the reign of Henry VIII? The chart was a masterful blend of form and function…
The most significant design feature of Priestley’s chart—as historians point out—was the way in which he linked units of time to units of distance on the page, similar to the way a cartographer uses scale when creating a map. (The artist Pietro Lorenzetti lived two hundred years before Titian and thus is situated twice as far from Titian as Jan van Eyck, who predated Titian by about a century.) If this innovation is hard for contemporary viewers to fully appreciate, it’s probably because Priestley’s representation of time has become a convention that’s used everywhere in visual design and seems so obvious it’s now taken for granted.
To Priestley’s contemporaries, though, who were accustomed to cumbersome Eusebian-style [see here] chronological tables or the visually striking but often obscure “stream charts” created by the era’s chronographers, Priestley’s method of capturing time on the page revealed something revelatory and new—a way of seeing historical patterns and connections that would have otherwise remained hidden. “To many readers,” wrote Daniel Rosenberg and Anthony Grafton in their book, Cartographies of Time, Priestley’s Chart of Biography offered a never-before-seen “picture of time itself.”
It was no wonder, then, that eighteenth-century readers found themselves drawn to it. A Chart of Biography sold well in both England and the United States, accruing many fans along the way. Along with the New Chart of History, it would go on to be printed in at least 19 editions and spawn numerous imitations, including one by Priestley’s future friend Thomas Jefferson, who developed his own “time chart” of market seasons in Washington, and the historian David Ramsay, who acknowledged Priestley’s influence in his Historical and Biographical Chart of the United States. The time charts marked Priestley’s first major commercial success and played a key role in establishing his reputation as a serious intellectual, earning him an honorary degree from the University of Edinburgh, and helping him secure a fellowship nomination to the Royal Society of London.
As much as anything he published, and he published a staggering amount—somewhere between 150 and 200 books, articles, papers, and pamphlets—Priestley’s time charts encapsulate his uniqueness as a thinker. Of his many intellectual gifts, his gift for synthesis—for knitting together the seemingly disparate things that caught his attention—might have been his greatest…
Read on for how Priestley went on to become the most controversial man in England: “Joseph Priestley Created Revolutionary ‘Maps’ of Time,” by @alysonafoster in @humanitiesmag from @NEHgov.
More info on the Chart– and magnified views– here.
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As we celebrate constructive charts, we might spare a thought for Edward Lorenz, a mathematician and meteorologist, best remembered as a pioneer of Chaos Theory; he died on this date in 2008. Having noticed that his computer weather simulation gave wildly different results from even tiny changes in the input data, he began investigating a phenomenon that he famously outlined in a 1963 paper— and that came to be known as the “butterfly effect,” that the flap of a butterfly’s wings could ultimately determine the weather thousands of miles away and days later… generalized in Chaos Theory to state that “slightly differing initial states can evolve into considerably different [later] states.”
“Everything we care about lies somewhere in the middle, where pattern and randomness interlace”*…
… A French mathematician has just won the Abel Prize for his decades of work developing a set of tools now widely used for taming random processes…
Random processes take place all around us. It rains one day but not the next; stocks and bonds gain and lose value; traffic jams coalesce and disappear. Because they’re governed by numerous factors that interact with one another in complicated ways, it’s impossible to predict the exact behavior of such systems. Instead, we think about them in terms of probabilities, characterizing outcomes as likely or rare…
… the French probability theorist Michel Talagrand was awarded the Abel Prize, one of the highest honors in mathematics, for developing a deep and sophisticated understanding of such processes. The prize, presented by the king of Norway, is modeled on the Nobel and comes with 7.5 million Norwegian kroner (about $700,000). When he was told he had won, “my mind went blank,” Talagrand said. “The type of mathematics I do was not fashionable at all when I started. It was considered inferior mathematics. The fact that I was given this award is absolute proof this is not the case.”
Other mathematicians agree. Talagrand’s work “changed the way I view the world,” said Assaf Naor of Princeton University. Today, added Helge Holden, the chair of the Abel prize committee, “it is becoming very popular to describe and model real-world events by random processes. Talagrand’s toolbox comes up immediately.”
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A random process is a collection of events whose outcomes vary according to chance in a way that can be modeled — like a sequence of coin flips, or the trajectories of atoms in a gas, or daily rainfall totals. Mathematicians want to understand the relationship between individual outcomes and aggregate behavior. How many times do you have to flip a coin to figure out whether it’s fair? Will a river overflow its banks?
Talagrand focused on processes whose outcomes are distributed according to a bell-shaped curve called a Gaussian. Such distributions are common in nature and have a number of desirable mathematical properties. He wanted to know what can be said with certainty about extreme outcomes in these situations. So he proved a set of inequalities that put tight upper and lower bounds on possible outcomes. “To obtain a good inequality is a piece of art,” Holden said. That art is useful: Talagrand’s methods can give an optimal estimate of, say, the highest level a river might rise to in the next 10 years, or the magnitude of the strongest potential earthquake…
Say you want to assess the risk of a river flooding — which will depend on factors like rainfall, wind and temperature. You can model the river’s height as a random process. Talagrand spent 15 years developing a technique called generic chaining that allowed him to create a high-dimensional geometric space related to such a random process. His method “gives you a way to read the maximum from the geometry,” Naor said.
The technique is very general and therefore widely applicable. Say you want to analyze a massive, high-dimensional data set that depends on thousands of parameters. To draw a meaningful conclusion, you want to preserve the data set’s most important features while characterizing it in terms of just a few parameters. (For example, this is one way to analyze and compare the complicated structures of different proteins.) Many state-of-the-art methods achieve this simplification by applying a random operation that maps the high-dimensional data to a lower-dimensional space. Mathematicians can use Talagrand’s generic chaining method to determine the maximal amount of error that this process introduces — allowing them to determine the chances that some important feature isn’t preserved in the simplified data set.
Talagrand’s work wasn’t just limited to analyzing the best and worst possible outcomes of a random process. He also studied what happens in the average case.
In many processes, random individual events can, in aggregate, lead to highly deterministic outcomes. If measurements are independent, then the totals become very predictable, even if each individual event is impossible to predict. For instance, flip a fair coin. You can’t say anything in advance about what will happen. Flip it 10 times, and you’ll get four, five or six heads — close to the expected value of five heads — about 66% of the time. But flip the coin 1,000 times, and you’ll get between 450 and 550 heads 99.7% of the time, a result that’s even more concentrated around the expected value of 500. “It is exceptionally sharp around the mean,” Holden said.
“Even though something has so much randomness, the randomness cancels itself out,” Naor said. “What initially seemed like a horrible mess is actually organized.”…
“Michel Talagrand Wins Abel Prize for Work Wrangling Randomness,” from @QuantaMagazine.
* James Gleick, The Information
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As we comprehend the constructs in chance, we might spare a thought for Caspar Wessel; he died on this date in 1818. A mathematician, he the first person to describe the geometrical interpretation of complex numbers as points in the complex plane and vectors.
Not coincidentally, Wessel was also a surveyor and cartographer, who contributed to the Royal Danish Academy of Sciences and Letters‘ topographical survey of Denmark.
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