(Roughly) Daily

Posts Tagged ‘Copernicus

“Mathematics has not a foot to stand on which is not purely metaphysical”*…

Battle of Maida 1806, part of the the invasion and occupation of Naples by Napoleon’s French Empire (source)

Lest we forget…

A forgotten episode in French-occupied Naples in the years around 1800—just after the French Revolution—illustrates why it makes sense to see mathematics and politics as entangled. The protagonists of this story were gravely concerned about how mainstream mathematical methods were transforming their world—somewhat akin to our current-day concerns about how digital algorithms are transforming ours. But a key difference was their straightforward moral and political reading of those mathematical methods. By contrast, in our own era we seem to think that mathematics offers entirely neutral tools for ordering and reordering the world—we have, in other words, forgotten something that was obvious to them.

In this essay, I’ll use the case of revolutionary Naples to argue that the rise of a new and allegedly neutral mathematics—characterized by rigor and voluntary restriction—was a mathematical response to pressing political problems. Specifically, it was a response to the question of how to stabilize social order after the turbulence of the French Revolution. Mathematics, I argue, provided the logical infrastructure for the return to order. This episode, then, shows how and why mathematical concepts and methods are anything but timeless or neutral; they define what “reason” is, and what it is not, and thus the concrete possibilities of political action. The technical and political are two sides of the same coin—and changes in notions like mathematical rigor, provability, and necessity simultaneously constitute changes in our political imagination…

Massimo Mazzotti with an adaptation from his new book, Reactionary Mathematics: A Genealogy of Purity: “Foundational Anxieties, Modern Mathematics, and the Political Imagination,” @maxmazzotti in @LAReviewofBooks.

* Thomas De Quincey


As we count on it, we might send carefully-calculated birthday greetings to Regiomontanus (or Johannes Müller von Königsberg, as he was christened); he was born on this date in 1436. A mathematician, astrologer, and astronomer of the German Renaissance, he and his work were instrumental in the development of Copernican heliocentrism during his lifetime and in the decades following his death.


“It is well to remember that the entire universe, with one trifling exception, is composed of others”*…

This artist’s impression shows the temperate planet Ross 128 b, with its red dwarf parent star in the background. Credit: ESO/M. Kornmesser

For centuries, scientific discoveries have suggested humanity occupies no privileged place in the universe. But as Mario Livio argues, studies of worlds beyond our solar system could place meaningful new limits on our existential mediocrity…

When the Polish polymath Nicolaus Copernicus proposed in 1543 that the sun, rather than the Earth, was the center of our solar system, he did more than resurrect the “heliocentric” model that had been devised (and largely forgotten) some 18 centuries earlier by the Greek astronomer Aristarchus of Samos. Copernicus—or, rather, the “Copernican principle” that bears his name—tells us that we humans are nothing special. Or, at least, that the planet on which we live is not central to anything outside of us; instead, it’s just another ordinary world revolving around a star.

Our apparent mediocrity has only ascended in the centuries that have passed since Copernicus’s suggestion. In the middle of the 19th century Charles Darwin realized that rather than being the “crown of creation,” humans are simply a natural product of evolution by means of natural selection. Early in the 20th century, astronomer Harlow Shapley deepened our Copernican cosmic demotion, showing that not only the Earth but the whole solar system lacks centrality, residing in the Milky Way’s sleepy outer suburbs rather than the comparatively bustling galactic center. A few years later, astronomer Edwin Hubble showed that galaxies other than the Milky Way exist, and current estimates put the total number of galaxies in the observable universe at a staggering trillion or more.

Since 1995 we have discovered that even within our own Milky Way roughly one of every five sunlike or smaller stars harbors an Earth-size world orbiting in a “Goldilocks” region (neither too hot nor too cold) where liquid water may persist on a rocky planetary surface. This suggests there are at least a few hundred million planets in the Milky Way alone that may in principle be habitable. In roughly the same span of time, observations of the big bang’s afterglow—the cosmic microwave background—have shown that even the ordinary atomic matter that forms planets and people alike constitutes no more than 5 percent of the cosmic mass and energy budget. With each advance in our knowledge, our entire existence retreats from any possible pinnacle, seemingly reduced to flotsam adrift at the universe’s margins.

Believe it or not, the Copernican principle doesn’t even end there. In recent years increasing numbers of physicists and cosmologists have begun to suspect—often against their most fervent hopes—that our entire universe may be but one member of a mind-numbingly huge ensemble of universes: a multiverse.

Interestingly though, if a multiverse truly exists, it also suggests that Copernican cosmic humility can only be taken so far.

The implications of the Copernican principle may sound depressing to anyone who prefers a view of the world regarding humankind as the central or most important element of existence, but notice that every step along the way in extending the Copernican principle represented a major human discovery. That is, each decrease in the sense of our own physical significance was the result of a huge expansion in our knowledge. The Copernican principle teaches us humility, yes, but it also reminds us to keep our curiosity and passion for exploration alive and vibrant…

Fascinating: “How Far Should We Take Our Cosmic Humility?“, from @Mario_Livio in @sciam.

* John Holmes (the poet)


As we ponder our place, we might send carefully-observed birthday greetings to Arno Penzias; he was born on this date in 1933. A physicist and radio astronomer, he and Robert Wilson, a collegue at Bell Labs, discovered the cosmic microwave background radiation, which helped establish the Big Bang theory of cosmology– work for which they shared the 1978 Nobel Prize in Physics.

MB radiation is something that anyone old enough to have watched broadcast (that’s to say, pre-cable/streaming) television) has seen:

The way a television works is relatively simple. A powerful electromagnetic wave is transmitted by a tower, where it can be received by a properly sized antenna oriented in the correct direction. That wave has additional signals superimposed atop it, corresponding to audio and visual information that had been encoded. By receiving that information and translating it into the proper format (speakers for producing sound and cathode rays for producing light), we were able to receive and enjoy broadcast programming right in the comfort of our own homes for the first time. Different channels broadcasted at different wavelengths, giving viewers multiple options simply by turning a dial.

Unless, that is, you turned the dial to channel 03.

Channel 03 was — and if you can dig up an old television set, still is — simply a signal that appears to us as “static” or “snow.” That “snow” you see on your television comes from a combination of all sorts of sources:

– human-made radio transmissions,

– the Sun,

– black holes,

– and all sorts of other directional astrophysical phenomena like pulsars, cosmic rays and more.

But if you were able to either block all of those other signals out, or simply took them into account and subtracted them out, a signal would still remain. It would only by about 1% of the total “snow” signal that you see, but there would be no way of removing it. When you watch channel 03, 1% of what you’re watching comes from the Big Bang’s leftover glow. You are literally watching the cosmic microwave background…

This Is How Your Old Television Set Can Prove The Big Bang

“Look up at the stars and not down at your feet. Try to make sense of what you see, and wonder about what makes the universe exist.”*…

Our environment…

This map shows a slice of our Universe. It was created from astronomical data taken night after night over a period of 15 years using a telescope in New Mexico, USA. We are located at the bottom. At the top is the actual edge of the observable Universe. In between, we see about 200,000 galaxies.

Each tiny dot is a galaxy. About 200,000 are shown with their actual position and color. Each galaxy contains billions of stars and planets. We are located at the bottom. Our galaxy, the Milky Way, is just a dot. Looking up, we see that space is filled with galaxies forming a global filamentary structure. Far away from us (higher up in the map), the filaments become harder to see…

For a (much crisper, more vivid, and interactive) version: “The Map of the Observable Universe.”

* Stephen Hawking


As we explore, we might recall that it was on this date in 1616 that The Minutes of the Roman Inquisition recorded the conclusion that Galileo’s writings in support of Copernicus’ heliocentric view of the solar system were “foolish and absurd in philosophy, and formally heretical since it explicitly contradicts in many places the sense of Holy Scripture.” The next day, Galileo was called before Cardinal Bellarmine, who (on Pope Paul V’s instruction) ordered Galileo to abandon the teaching. Shortly thereafter, Copernicus’s De Revolutionibus and other heliocentric works were banned (entered onto the Index Librorum Prohibitorum) “until correction.”

Sixteen years later, Galileo “published” Dialogue Concerning the Two Chief World Systems (Dialogo sopra i due massimi sistemi del mondo)– that’s to say, he presented the first copy to his patron, Ferdinando II de’ Medici, Grand Duke of Tuscany.  Dialogue, which compared the heliocentric Copernican and the traditional geo-centric Ptolemaic systems, was an immediate best-seller.

While there was no copyright available to Galileo, his book was printed and distributed under a license from the Inquisition.  Still, the following year it was deemed heretical– and he joined Copernicus on the Index Librorum Prohibitorum: the publication of anything else Galileo had written or ever might write was also banned… a ban that remained in effect until 1835.

Domenico Tintoretto‘s portrait of Galileo (source)

Written by (Roughly) Daily

February 25, 2023 at 1:00 am

“Werner Heisenberg once proclaimed that all the quandaries of quantum mechanics would shrivel up when 137 was finally explained”*…

One number to rule them all?

Does the Universe around us have a fundamental structure that can be glimpsed through special numbers?

The brilliant physicist Richard Feynman (1918-1988) famously thought so, saying there is a number that all theoretical physicists of worth should “worry about”. He called it “one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man.”

That magic number, called the fine structure constant, is a fundamental constant, with a value which nearly equals 1/137. Or 1/137.03599913, to be precise. It is denoted by the Greek letter alpha – α.

What’s special about alpha is that it’s regarded as the best example of a pure number, one that doesn’t need units. It actually combines three of nature’s fundamental constants – the speed of light, the electric charge carried by one electron, and the Planck’s constant, as explains physicist and astrobiologist Paul Davies to Cosmos magazine. Appearing at the intersection of such key areas of physics as relativity, electromagnetism and quantum mechanics is what gives 1/137 its allure…

The fine structure constant has mystified scientists since the 1800s– and might hold clues to the Grand Unified Theory: “Why the number 137 is one of the greatest mysteries in physics,” from Paul Ratner (@paulratnercodex) in @bigthink.

* Leon M. Lederman, The God Particle: If the Universe Is the Answer, What Is the Question?


As we ruminate on relationships, we might spare a thought for Georg von Peuerbach; he died on this date in 1461. A mathematician, astronomer, and instrument maker, he is probably best remembered for his streamlined presentation of Ptolemaic astronomy in the Theoricae Novae Planetarum (which was an important text for many later-influential astronomers including Nicolaus Copernicus and Johannes Kepler).

But perhaps as impactful was his promotion of the use of Arabic numerals (introduced 250 years earlier in place of Roman numerals), especially in a table of sines he calculated with unprecedented accuracy.

Georg von Peuerbach: Theoricarum novarum planetarum testus, Paris 1515 [source]
Page from Peurbach’s sine table [source]

“Appearances are a glimpse of the unseen”

A shower of comets rains down on Earth while violent volcanic eruptions billow up from below. Both events may follow our planet’s passage through dark matter concentrated in the Milky Way’s plane and help to trigger extinction events.

Are we on the verge of understanding the upheavals that have shaped the earth?

Do geologists dream of a final theory? Most people would say that plate tectonics already serves as geology’s overarching idea. The discovery of plate tectonics 50 years ago was one of the great scientific achievements of the 20th century, but is the theory complete? I think not. Plate tectonics describes Earth’s present geology in terms of the geometry and interactions of its surface plates. Geologists can extrapolate plate motions both back in time and into the future, but they cannot yet derive the behavior and history of plate tectonics from first principles.

Although scientists can interpret the history through the lens of what they see today, an important question remains: Why did geologic events — such as hot-spot volcanism, the breakup of continents, fluctuations in seafloor spreading, tectonic episodes, and sea-level oscillations — occur exactly when and where they did? Are they random, or do they follow some sort of a pattern in time or space?

A complete theory of Earth should explain geologic activity in the spatial domain, as plate tectonics does quite well for the present (once you incorporate hot spots), but also in the time and frequency domains. Recent findings suggest to me that geology may be on the threshold of a new theory that seeks to explain Earth’s geologic activity in time and space in the context of its astronomical surroundings.

The solar system oscillates with respect to the midplane of the disk-shaped Milky Way Galaxy with a period of about 60 million years. The Sun’s family passes through this plane twice each period, or once every 30 million years or so. The solar system behaves like a horse on a carousel — as we go around the disk-shaped galaxy, we bob up and down through the disk, passing through its densest part roughly every 30 million years.

Surely, it is too much of a coincidence that the cycle found in mass extinctions and impact craters should turn out to be one of the fundamental periods of our galaxy. The idea seemed almost too pretty to be wrong. But people searching for cycles have been fooled before, and we still had to answer the question: How does this cycle of movement lead to periodic perturbations of the Oort Cloud comets?

The idea of a roughly 30 million-year rhythm in geologic events has a long history in the geological literature. In the early 20th century, W.A. Grabau, an expert on sedimentary strata, proposed that tectonic activity and mountain building drove periodic fluctuations in sea level with an approximately 30 million-year cycle. In the 1920s, noted British geologist Arthur Holmes, armed with a few age determinations from radioactive decay, saw a similar 30 million-year cycle in Earth’s geologic activity…

If the cycles are real, what could be driving these long-term changes in volcanism, tectonics, sea level, and climate at such regular, if widely spaced, intervals? At first, I thought that the periodic energetic impacts might somehow be affecting deep-seated geological processes. I suggested in a short note in the journal Nature that large impacts might so deeply excavate and fracture the crust — to depths in excess of 10 miles (16 km) — that the sudden release of pressure in the upper mantle would result in large-scale melting. This would lead to the production of massive flood-basalt lavas, which would cover the crater and possibly create a mantle hot spot at the site of the impact. Hot spots could lead to continental breakup, which can cause increased tectonics and changes in ocean-floor spreading rates, and in turn cause global sea levels to fluctuate. Unfortunately, no known terrestrial impact structure has a clear association with volcanism, although some volcanic outpourings on Mars seem to be located along radial and concentric fractures related to large impacts.

The potential key to resolving this geological conundrum may come from outer space. Remember that Randall and Reece suggested that Earth passes through a thin disk of dark matter concentrated along the Milky Way’s midplane every 30 million years or so. Astrophysicist Lawrence Krauss and Nobel Prize-winning physicist Frank Wilczek of Harvard University, and independently Katherine Freese, an astrophysicist at the Harvard-Smithsonian Center for Astrophysics, proposed that Earth could capture dark matter particles that would accumulate in the planet’s core. The number of dark matter particles could grow large enough so that they would undergo mutual annihilation, producing prodigious amounts of heat in Earth’s interior.

A 1998 paper in the journal Astroparticle Physics (which I am sure few geologists ever read) provided a potential missing link. Indian astrophysicists Asfar Abbas and Samar Abbas (father and son, respectively) at Utkal University also were interested in dark matter and its interactions with our planet. They calculated the amount of energy released by the annihilation of dark matter captured by Earth during its passage through a dense clump of this material. They found that mutual destruction among the particles could produce an amount of heat 500 times greater than Earth’s normal heat flow, and much greater than the estimated power required in Earth’s core to generate the planet’s magnetic field. Putting together the predicted 30 million-year periodicity in encounters with dark matter with the effects of Earth capturing this unstable matter produces a plausible hypothesis for the origin of regular pulses of geologic activity.

Excess heat from the planet’s core can raise the temperature at the base of the mantle. Such a pulse of heat might create a mantle plume, a rising column of hot mantle rock with a broad head and narrow tail. When these rising plumes penetrate Earth’s crust, they create hot spots, initiate flood-basalt eruptions, and commonly lead to continental fracturing and the beginning of a new episode of seafloor spreading. The new source of periodic heating by dark matter in our planet’s interior could lead to periodic outbreaks of mantle-plume activity and changes in convection patterns in Earth’s core and mantle, which could affect global tectonics, volcanism, geomagnetic field reversals, and climate, such as our planet has experienced in the past.

These geologic events could lead to environmental changes that might be enough to cause extinction events on their own. A correlation of some extinctions with times of massive volcanic outpourings of lava supports this view. This new hypothesis links geologic events on Earth with the structure and dynamics of the Milky Way Galaxy.

It is still too early to tell if the ingredients of this hypothesis will withstand further examination and testing. Of course, correlations among geologic events can occur even if they are not part of a periodic pattern, and long-term geological cycles may exist apart from any external cosmic connections. The virtue of the galactic explanation for terrestrial periodicity lies in its universality — because all stars in the galaxy’s disk, many of which harbor planets, undergo a similar oscillation about the galactic midplane — and in its linkage of biological and geological evolution on Earth, and perhaps in other solar systems, to the great cycles of our galaxy.

Dark matter’s shadowy effect on Earth“: Earth’s periodic passage through the galaxy’s disk could initiate a series of events that ultimately lead to geological cataclysms and mass extinctions. From Michael Rapino (@mrr1_michael)

For very different angle on the evolution of the earth, the wonderful Walter Murch: “Why Birds Can Fly Over Mount Everest.”

* Anaxagoras


As we dig deep, we might spare a thought for Harlow Shapley; he died on this date in 1972. An astronomer known as “the Modern Copernicus,” he did important work first at the Mt. Wilson Observatory, and then as head of the Harvard College Observatory. He boldly and correctly proclaimed that the globulars outline the Galaxy, and that the Galaxy is far larger than was generally believed and centered thousands of light years away in the direction of Sagittarius: he discovered the center of our Galaxy, and our position within it.


%d bloggers like this: