Posts Tagged ‘Mathematics’
“I love to talk about nothing. It’s the only thing I know anything about.”*…
Try as they might, scientists can’t truly rid a space or an object of its energy. But as George Musser reports, what “zero-point energy” really means is up for interpretation…
Suppose you want to empty a box. Really, truly empty it. You remove all its visible contents, pump out any gases, and — applying some science-fiction technology — evacuate any unseeable material such as dark matter. According to quantum mechanics, what’s left inside?
It sounds like a trick question. And in quantum mechanics, you know to expect a trick answer. Not only is the box still filled with energy, but all your efforts to empty it have barely put a dent in the amount.
This unavoidable residue is known as ground-state energy, or zero-point energy. It comes in two basic forms: The one in the box is associated with fields, such as the electromagnetic field, and the other is associated with discrete objects, such as atoms and molecules. You may dampen a field’s vibrations, but you cannot eliminate every trace of its presence. And atoms and molecules retain energy even if they’re cooled arbitrarily close to absolute zero. In both cases, the underlying physics is the same.
Zero-point energy is characteristic of any material structure or object that is at least partly confined, such as an atom held by electric fields in a molecule. The situation is like that of a ball that has settled at the bottom of a valley. The total energy of the ball consists of its potential energy (related to position) plus its kinetic energy (related to motion). To zero out both components, you would have to give a precise value to both the object’s position and its velocity, something forbidden by the Heisenberg uncertainty principle.
What the existence of zero-point energy tells you at a deeper level depends ultimately on which interpretation of quantum mechanics you adopt. The only noncontentious thing you can say is that, if you situate a bunch of particles in their lowest energy state and measure their positions or velocities, you will observe a spread of values. Despite being drained of energy, the particles will look as if they’ve been jiggling. In some interpretations of quantum mechanics, they really have been. But in others, the appearance of motion is a misleading holdover from classical physics, and there is no intuitive way to picture what’s happening…
More on the development of our understanding of “zero-point energy” and on the questions that remain: “In Quantum Mechanics, Nothingness Is the Potential To Be Anything,” from @georgemusser.com in @quantamagazine.bsky.social.
For the most amusing of musings on nothing, see Percival Everett‘s Dr. No.
* Oscar Wilde
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As we noodle on nought, we might spare a thought for Kurt Gödel; he died on this date in 1978. A mathematician, logician, and author of Gödel’s proof. He is best known for his proof of Gödel’s Incompleteness Theorems (in 1931). He proved fundamental that in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular, the consistency of the axioms cannot be proved… thus ending a hundred years of attempts to establish axioms to put the whole of mathematics on an axiomatic basis. [See here for a consideration of what his finding might mean for moral philosophy…]
“Curiosity is, in great and generous minds, the first passion and the last”*…
From the arcane through the mysterious to the perplexing, a glorious collecton of obscure– but fascinating– knowledge…
Freakpages is a community-curated directory of esoteric articles across the internet, primarily from Wikipedia. Here, we encourage you to learn about interesting topics you have never heard of…
… divided into categories (Society, History, Technology, Psychology, Physics, Biology, Chemistry, Finance, Philosphy), with continuously refreshed selections from both the curators and the community.
A few examples: Egregore, Operation Northwoods, Matrioshka Brain, Zeigarnik Effect, Retrocausality, Horizontal Gene Transfer, Strange Matter Seeding, Keynesian Beauty Contest, Chinese Room…
So many more at: Freakpages
[Image above: source]
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As we explore, we might spare a thought for a man driven by an endles spirit of inquiry, William Thomson, 1st Baron Kelvin; he died on this date in 1907. A mathematician, mathematical physicist, and engineer considered by many “the Newton of his era,” Lord Kelvin was instrumental in the formulation of the first and second laws of thermodynamics, and contributed significantly to unifying physics, which was then in its infancy of development as an emerging academic discipline. He received the Royal Society’s Copley Medal in 1883 and served as its president from 1890 to 1895. In 1892 he became the first scientist to be elevated to the House of Lords. Absolute temperatures are stated in units of kelvin in his honor.
“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.
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