Posts Tagged ‘Mathematics’
“There are things done today in electrical science which would have been deemed unholy by the very men who discovered electricity, who would themselves not so long before have been burned as wizards”*…
Climate change continues. There is broad evidence (and consensus) that our environment, thus our ways of life, our livelihoods— indeed, our lives— are threatened. On the heels of a call from Trump to world leaders to abandon the climate fight, followed by a disappointing COP30 conference, it’s easy to be discouraged. But that, of course, is no answer.
Rather, we have to find ways to mitigate the damage that we’ve already locked in, even as we acclerate a transition to clean energy… which begins by (re-)framing and (re-)focusing the challenge. Ember, a clean energy think tank, suggests a candidate that, while it speaks to the moral obligations addressed by one of the models it means to augment/replace, has a more positive orientation…
Humanity is graduating from burning fossil commodities to harnessing manufactured technologies—from hunting scarce fossils to farming the inexhaustible sun, from consuming Earth’s resources to
merely borrowing them.This isn’t a marginal climate substitution. It’s an energy revolution.
The magnetic centre is the electron: we are revolutionising how we generate, use, and connect
electrons. Solar and wind are conquering electricity supply. EVs, heat pumps, and AI are electrifying major new uses. Batteries and digitalisation are connecting supply and demand.Three reinforcing shifts. One energy revolution. The electrotech revolution.
At its core, this revolution is driven by physics, economics, and geopolitics. After all, the arc of energy
history bends towards solutions that are leaner, cheaper and more secure.Short-terms setbacks matter, but fundamentals matter more. And the fundamentals are stacked in electrotech’s favour.
Physics. Electrotech makes a mockery of setting fossils on fire and losing two-thirds of the energy to heat. Electrotech is three times as efficient.
Economics. Technologies get cheaper with scale. Commodities get more expensive the deeper you dig.
Geopolitics. Three quarters of the world is dependent on fossil imports. 92% of countries have renewables potential over 10x their current demand.
Electrotech has grown exponentially for decades. The difference today is that it’s too cheap to contain and too big to ignore. If current exponentials hold for five more years, global fossil demand will fall off its plateau.
Welcome to the Age of Electrotech…
A long and meaty presentation: “The Electrotech Revolution- the shape of things to come,” from @ember-energy.org.
One notes that the electrification that Ember pushes has other advocates, many of whom have been vocal for years; c.f., e.g., Saul Griffin. Still, another voice in the chorus is welcome.
* Bram Stoker
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As we plug in, we might send charged birthday greetings to Franz Aepinus; he was born on this date in 1724. A mathematician, scientist, and natural philosopher, he is best known for his research, both theoretical and experimental, into electricity and magnetism. Aepinus’ Tentamen theoriae electricitatis et magnetismi (1759; “An Attempt at a Theory of Electricity and Magnetism”) was the first work to apply mathematics to the theory of electricity and magnetism. And his experiments led to the design of the parallel-plate capacitor, a device used to store energy in an electric field.
“I think it’s much more interesting to live not knowing than to have answers which might be wrong… when we know that we actually do live in uncertainty, then we ought to admit it; it is of great value to realize that we do not know the answers to different questions.”*…
The immense complexity of the climate makes it impossible to model accurately. Instead, David Stainforth argues, we must use uncertainty to our advantage…
Today’s complex climate models aren’t equivalent to reality. In fact, computer models of Earth are very different to reality – particularly on regional, national and local scales. They don’t represent many aspects of the physical processes that we know are important for climate change, which means we can’t rely on them to provide detailed local predictions. This is a concern because human-induced climate change is all about our understanding of the future. This understanding empowers us. It enables us to make informed decisions by telling us about the consequences of our actions. It helps us consider what the future will be like if we act strongly to reduce greenhouse gas emissions, if we act only half-heartedly, or if we take no action at all. Such information enables us to assess the level of investment that we believe is worthwhile as individuals, communities and nations. It enables us to balance action on climate change against other demands on our finances such as health, education, security and culture.
For many of us, these issues are approached through the lens of personal experience and personal cares: we want to know what changes to expect where we live, in the places we know, and in the regions where we have our roots. We want local climate predictions – predictions conditioned on the choices that our societies make.
So, where do we get them? Well, nowadays most of these predictions originate from complicated computer models of the climate system – so-called Earth System Models (ESMs). These models are ubiquitous in climate change science. And for good reason. The increasing greenhouse gases in the atmosphere are driving the climate system into a never-before-seen state. That means the past cannot be a good guide to the future, and predictions based simply on historic observations can’t be reliable: the information isn’t in the observational data, so no amount of processing can extract it. Climate prediction is therefore about our understanding of the physical processes of climate, not about data-processing. And since there are so many physical processes involved – everything from the movement of heat and moisture around the atmosphere to the interaction of oceans with ice-sheets – this naturally leads to the use of computer models.
But there’s a problem: models aren’t equivalent to reality.
So, what can we do? One option is to make the models better. Make them more detailed and more complicated. That, though, raises an important question: when is a model sufficiently realistic to predict something as complex as climate change? When will the models be good enough? We don’t have an answer to this question. Indeed, scientists have hardly begun to study this problem, and some argue that these models might never be sufficiently accurate to make multi-decadal, local climate predictions.
Nevertheless, changing the way we use ESMs could provide a different and better way to generate the local climate information we seek. Doing so involves embracing uncertainty as a key part of our knowledge about climate change. It involves stepping back and accepting that what we want is not precise predictions but robust predictions, even if robustness involves accepting large uncertainties in what we can know about the future…
[Stainforth explains the current state of modeling, efforts to make them better, and the problems those efforts encounter…]
… focusing on high-resolution modelling is dangerous not only because we have no answer to the question of when a model is sufficiently realistic. Investing in this approach also means we don’t have the capacity to explore the uncertainties, which inevitably encourages overconfidence in the predictions that models make. This is a particular concern because Earth System Models are increasingly being used to guide decisions and investments across our societies. Overconfidence in model-based predictions therefore risks encouraging bad decisions: decisions that are optimised for the futures in our models rather than what we understand about the range of possible futures for reality.
By contrast, perturbed physics ensembles and storyline approaches focus on exploring and describing our uncertainties. Placing uncertainty front and centre is important. When we make an investment or a gamble, we don’t just base it on what we think is the most likely result. We consider the range of outcomes that we think are possible – ideally these are characterised by probabilities, although this isn’t always achievable. It’s the same with climate change. We should not only make plans based solely on our best estimate of what might happen. We should also consider the range of plausible outcomes we foresee. Our knowledge of uncertainty is also part of what we know about climate change. We should embrace this knowledge, expand it and use it.
If we understand the uncertainties well, we can bring our values to bear on the risks we are willing to take. Uncertainty therefore needs to be at the core of adaptation planning while also being the lens through which we judge the value of climate policy and the energy transition. In my view, climate researchers and modellers wanting to support society should focus on understanding, characterising and quantifying uncertainty, and avoid the trap of seeking climate models that make reliable predictions. They may well never exist…
A more practical approach to preparing for climate change: “The model of catastrophe,” from @aeon.co
* Richard Feynman
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As we preference plausibility (over predictability), we might send never-ending birthday greetings to August Möbius; he was born on this date in 1790. An astronomer and mathematician, he studied under mathematician Carl Friedrich Gauss while Gauss was the director of the Göttingen Observatory. From there, he went on to study with Carl Gauss’s instructor, Johann Pfaff, at the University of Halle, where he completed his doctoral thesis The occultation of fixed stars in 1815. In 1816, he became Extraordinary Professor in the “chair of astronomy and higher mechanics” at the University of Leipzig, where he remained for the rest of his career. Möbius made many contributions to both astronomy and the math that underlay it: he was among the first to conceive the possibility of geometry in more than three dimensions; he introduced homogeneous coordinates into projective geometry; and he pioneered the barycentric coordinate system… all parts of the intellectual foundation of the complex system modeling described above.
But while he was an influential scholar and professor, he is best remembered for his creation of the “Möbius strip.”
“If geometry is dressed in a suit coat, topology dons jeans and a T-shirt”*…
Paulina Rowińska on how, in the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics…
Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.
The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.
This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.
Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”
So what are manifolds, and what kind of vocabulary do they provide?…
[Rowińska explains manifolds and the history of the development of our understanding of them, concentrating on the pivotal role of Riemann…]
… Manifolds are crucial to our understanding of the universe… In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.
Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”
Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.
Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.
Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”…
“What Is a Manifold?” from @quantamagazine.bsky.social.
Apposite: Rowińska in conversation with Ira Flatow on Science Friday: “How Math Helps Us Map The World.”
* David S. Richeson, Euler’s Gem: The Polyhedron Formula and the Birth of Topology (Riemann’s work was an advance on the foundation that Euler laid in his 1736 paper on the Seven Bridges of Königsberg, which led to his polyhedron formula)
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As we get down with geometry, we might spare a thought for John Wallis; he died on this date in 1703. A clergyman and mathematician, he served as chief cryptographer for Parliament (decoding Royalist messages during the Civil War) and, later (as Savilian Chair of geometry at Oxford after the hostilities), for the the royal court. Wallis is credited with introducing the symbol ∞ to represent the concept of infinity, and used 1/∞ for an infinitesimal… which earned him (along with his contemporaries Isaac Newton and Gottfried Wilhelm Leibniz) a share of the credit for the development of infinitesimal calculus. He was a founding member of the Royal Society and one of its first Fellows.
“For the simplicity on this side of complexity, I wouldn’t give you a fig. But for the simplicity on the other side of complexity, for that I would give you anything I have.”*…
Indeed. And as Jordana Cepelewicz reports, mathematicians are on the case…
Repetition doesn’t always have to be humdrum. In mathematics, it is a powerful force, capable of generating bewildering complexity.
Even after decades of study, mathematicians find themselves unable to answer questions about the repeated execution of very simple rules — the most basic “dynamical systems.” But in trying to do so, they have uncovered deep connections between those rules and other seemingly distant areas of math.
For example, the Mandelbrot set, which I wrote about last month [see also the almanac entry here], is a map of how a family of functions — described by the equation f(x) = x2 + c — behaves as the value of c ranges over the so-called complex plane. (Unlike real numbers, which can be placed on a line, complex numbers have two components, which can be plotted on the x- and y-axes of a two-dimensional plane.)
No matter how much you zoom in on the Mandelbrot set, novel patterns always arise, without limit. “It’s completely mind-blowing to me, even now, that this very complex structure emerges from such simple rules,” said Matthew Baker of the Georgia Institute of Technology. “It’s one of the really surprising discoveries of the 20th century.”
The complexity of the Mandelbrot set emerges in part because it is defined in terms of numbers that are themselves, well, complex. But, perhaps surprisingly, that isn’t the whole story. Even when c is a straightforward real number like, say, –3/2, all sorts of strange phenomena can occur. Nobody knows what happens when you repeatedly apply the equation f(x) = x2 – 3/2, using each output as the next input in a process known as iteration. If you start iterating from x = 0 (the “critical point” of a quadratic equation), it’s unclear whether you will produce a sequence that eventually converges toward a repeating cycle of values, or one that continues to endlessly bounce around in a chaotic pattern…
[Cepelewicz runs through mathemeticians’ efforts to understand– and find explanation, if not order– in the complexity, concluding with the “entropy bagel”…]
… Galois conjugates [see here] also paved the way to the discovery of a mysterious object dubbed the “entropy bagel,” a glowing fractal ring in the complex plane. Entropy is a measure of randomness; in this context, it measures how difficult it is to predict the sequence of numbers generated by iterating x2 + c. In the last paper he wrote before he died in 2012, the renowned topologist William Thurston graphed the set of entropy values corresponding to almost a billion different real values of c — together with the Galois conjugates of those entropy values, which can be complex. The notion of entropy “is just on the real line, but somehow you can still see this shadow of the complex world,” Tiozzo said.
“You see that this is organizing itself into this incredible lacy fractal structure,” Koch said. “It’s so cool.” The entropy bagel is only one very complicated pattern that emerges from the iteration of real quadratic equations. “We’re still learning all these magical statements — little gems — about real quadratic polynomials,” she added. “You can always go back and be surprised by this thing you thought you knew extremely well.”…
Simple rules in simple settings continue to puzzle mathematicians, even as they devise intricate tools to analyze them: “‘Entropy Bagels’ and Other Complex Structures Emerge From Simple Rules,” from @jordanacep.bsky.social in @quantamagazine.bsky.social.
* Oliver Wendell Holmes
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As we untangle, we might spare a thought for cherished creator of chaos, Milton Supman (better known by his stage name, Soupy Sales); he died on this date in 2009. A comedian, actor, radio-television personality, and jazz aficionado, he is best remembered for his local and network children’s television series, Lunch with Soupy Sales (later titled The Soupy Sales Show), which ran from 1953–1966, a collection of comedy sketches frequently ending with Sales receiving a pie in the face, which became his trademark.
“Where all think alike there is little danger of innovation”*…
Last week, Northwestern Professor Joel Mokyr was awarded a half-share in The Nobel Prize in Economic Sciences (AKA The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) “for having identified the prerequisites for sustained growth through technological progress.” Anton Howes explains why this is noteworthy…
Among today’s winners of the Nobel prize in Economics is Joel Mokyr, the professor at Northwestern whose name is indelibly associated with the primacy of innovation to modern economic growth – the gradual, sustained, and unprecedented improvement in living standards that first Britain, and then country after country, have enjoyed over the past few hundred years. It was reading Mokyr’s The Enlightened Economy that first opened my eyes to the importance of studying the history of invention to explaining the causes of the Industrial Revolution, which I have since made my career.
What makes this Nobel win so remarkable, and so pleasantly surprising, is that Mokyr’s work is not the kind that is often published by economics journals, or even many economic history journals anymore. Over the past few decades, journal editors and peer-reviewers have increasingly insisted that papers must present large datasets that have been treated using complex statistical methods in order to make even the mildest claims about what caused what. Although Mokyr is a master of such methods – he was one of the early pioneers of economic history’s quantitative turn – the work for which he has won the prize is firmly and necessarily qualitative.
Mokyr’s is the economic history that gets written up in books – his classics are The Lever of Riches, The Gifts of Athena, The Enlightened Economy, and A Culture of Growth – and in readable papers shorn of unnecessary formulae. His is history accessible to the layman, though rigorously applying the insights of economics. The prize is a clear signal from the economics profession that it doesn’t just value the application of fancy statistical methods; its highest prize can go to works of history.
Whereas most of the public, and even many historians, think of the causes of modern economic growth – the beginnings of the Industrial Revolution – as being rooted in material factors, like conquest, colonialism, or coal, Mokyr tirelessly argued that it was rooted in ideas, in the intellectual entrepreneurship of figures like Francis Bacon and Isaac Newton, and in the uniquely precocious accumulation in eighteenth-century Britain of useful, often mechanically actionable knowledge. Britain, he argued, through its scientific and literary societies, and its penchant for publications and sharing ideas, was the site of a world-changing Industrial Enlightenment – the place where progress was thoughtpossible, and then became real.
One of Mokyr’s big early insights, first appearing in Lever of Riches, was that many inventions could not be predicted by economic factors. Society could enjoy remarkable productivity improvements from simply increasing the size of the market, leading to division of labour and specialization – what he labelled ‘micro-inventions’ – in the vein popularised by Adam Smith. But this could not explain an invention that appeared out of the blue, like Montgolfier’s hot air balloon in the 1780s – what he called a ‘macro-invention’, not for the magnitude of its impact, but for its novelty. Macro-inventions often required further development to make them important, but the original breakthrough could not be predicted by looking at changes in prices or the availability of resources. It ultimately came down to advances in our understanding of the world. Mokyr put the Scientific Revolution – and the factors that contributed to it – on the economist’s map.
Mokyr also looked at the relationship between different kinds of knowledge. A scientist might know, through observation, that the air has a weight. A craftsman might know, through long training and experience with glass, how to make a long glass tube. Each could not get far alone. But combining them, by creating means to ensure that scientists and craftsmen talked with one another and collaborated – through connecting their propositional and prescriptive knowledge, their heads and hands – very quickly led to the invention of thermometers, barometers, and much more besides, in an ever expanding field of knowledge. What Mokyr taught economists is that it’s not knowledge per se that makes the difference, but the way it is organized. Much of his later work has shown just how deep a pool Britain’s scientists could draw on, of skilled artisans.
In a way, Mokyr himself has practised what he preached. As editor of Princeton University Press’s book series on the Economic History of the Western World, Mokyr has for decades provided an all-important space for economists and historians to write the kinds of research that would never have been publishable in economics journals – including of explanations of the Industrial Revolution that are the polar opposite to his own. He helped keep the connection between history and economics alive.
Mokyr’s case for the primacy of knowledge and ideas was not an easy one to make to economists. They are naturally drawn to data that can be counted, and not to narrative, often no matter how well evidenced. But it appears that Mokyr’s persistence, elevated by his infectious, irrepressible sprightliness, has paid off. His prize is a long overdue recognition of the historyin economic history, and a remarkable testament to the power of ideas to persuade…
A triumph for history and the importance of ideas: “Joel Mokyr’s Nobel,” from @antonhowes.bsky.social.
See also: “Why Joel Mokyr deserves his Nobel prize,” gift article from The Economist.
* Edward Abbey, Desert Solitaire
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As we ponder the process of progress, we might send creative birthday greetings to one of the subjects Mokyr’s study, Sir Christopher Wren; he born on this date in 1632. A mathematician and astronomer (who co-founded and later served as president of the Royal Society), he is better remembered as one of the most highly acclaimed English architects in history; he was given responsibility for rebuilding 52 churches in the City of London after the Great Fire in 1666, including what is regarded as his masterpiece, St. Paul’s Cathedral, on Ludgate Hill.
Wren, whose scientific work ranged broadly– e.g., he invented a “weather clock” similar to a modern barometer, new engraving methods, and helped develop a blood transfusion technique– was admired by Isaac Newton, as Newton noted in the Principia.
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