Posts Tagged ‘Mathematics’
“Faith is a fine invention / When gentlemen can see, / But microscopes are prudent / In an emergency”*…
Microscopy has been around for centuries; it began to emerge as a field of scientific investigation with the emergence of compound microscopes in Europe around 1620. Antonie van Leeuwenhoek developed a very high magnification simple microscope in the 1670s and is often considered to be the first acknowledged microscopist and microbiologist. His fellow pioneer, Robert Hooke (author of what many to believe to have been van Leeuwenhoek’s inspiration, the ground-breaking Micrographia, published in in 1665), wrote “By the help of microscopes, there is nothing so small, as to escape our inquiry; hence there is a new visible world discovered to the understanding.”
Optical microscopes remain central tools in science, and have been joined by optical, electron, and scanning probe microscopes (along with the emerging field of X-ray microscopy). But, as Joao Inacio Silva illustrates, they are also fascinating objects, things of beauty…
Antique microscopes are amazing scientific instruments, from times when craftsmanship was as important as functionality and performance. The beauty of these instruments is manifested in countless ways, including the history of their makers and their technological developments, and their contribution to the development of microbiology and other fields of science, and all combine to inspire a feeling of admiration that a microscope can be so beautiful, elegant, and functional after so many years. An antique microscope is a work of art as well as science.
This [site] describes a collection of microscopes, which started as a hobby some years ago, and is always being updated with interesting new instruments….
For example:
Gustave Moreau (1805 – 1880) was a manufacturer of binoculars operating in Paris from 1830. The business of Moreau was merged with other opticians in 1849, forming the Deraisme house (167 Rue Saint-Maur, Paris), which specialised in binoculars and spotting telescopes, particularly for military use. Moreau is best known for the creation of the famous ‘Monkey Microscope’. [Pictured at the top] Microscope 199 is a drum-like microscope and is engraved with ‘Moreau’ in its inside base… The instrument should be dated to the mid-19th century.
Moritz (M.) Pillischer emigrated from Hungary to London, England, in 1845. He opened an independent shop that produced microscopes and other scientific and mathematical instruments in about 1849. Moritz’s nephew, Jacob (who adopted the name “James”), moved to London around 1860 to work for his uncle. Jacob later became Moritz’s son-in-law, after marrying one of his daughters. Pillischer did not make his own lenses until 1854, but instead provided French-made objectives with his instruments. Moritz Pillischer was elected as a Fellow of the Royal Microscopical Society in 1855 and joined the Quekett Microscopical Club in 1869. By 1881, Moritz had moved to Hove, Sussex, although he retained ownership of the Pillischer optical business. He handed over ownership of the business to Jacob in 1887 and passed away in his Sussex home in 1893. Jacob joined the Quekett Microscopical in 1895, and the Royal Microscopical Society in 1898. After Jacobs’ death in 1930, the company was inherited by Jacob’s three children, Edward, Leopold, and Bertha, and the business was liquidated in 1947. Microscope 17 is a version of Pillischer’s Student microscope from c. 1860, with the serial number 1011 (Figure 1). The microscope is finished in lacquered brass and has an extendable eyepiece tube, original Pillisher lenses, rack and pinion main focus and fine focus. It has a square stage with manually adjustable slide rest. Below the stage is a mirror and a revolving wheel to control the level of light. Pillischer introduced this version of student microscope in the late 1854, and the basic form of this microscope was then used in other models over the next several decades, including the Saint Thomas Hospital (introduced in 1873) and the International (introduced in 1876) models (Figure 1). The microscope came with its original wooden box and several accessories, including a live box used for the observation of wet or dry animals. Early models of live boxes were constructed of ivory or brass and would often fit into the hole in the stage. Later, they were fitted onto a rectangular brass slide above the stage.
Many, many more delights at the “Microscope Museum“, a glorious collection of antique microscopes and other scientific instruments.
* Emily Dickinson
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As we look closely, we might spare a thought for Christain Goldbach; he died on this date in 1764. A mathematician, lawyer, and historian who studied infinite sums, the theory of curves, and the theory of equations, he is best remembered for his correspondence with Leibniz, Euler, and Bernoulli, especially his 1742 letter to Euler containing what is now known as “Goldbach’s conjecture.”
In that letter he outlined his famous proposition:
Every even natural number greater than 2 is equal to the sum of two prime numbers.
It has been checked by computer for vast numbers– to more than a trillion trillion– but remains unproved.
(Goldbach made another conjecture that every odd number is the sum of three primes; it has been checked by computer for vast numbers, but also remains unproved.)
Goldbach’s letter to Euler (source, and larger view)
“The ultimate hidden truth of the world is that it is something we make and could just as easily make differently”*…
As a new collection of his writing is published, Rebecca Solnit remembers her friend David Graeber, the late activist and anarchist who believed ordinary people have the power to change the world…
David Graeber was a joyful, celebratory person. An enthusiast, voluble, on fire with the possibilities in the ideas and ideologies he wrestled with. Every time we met – from New Haven in the early 00s to London a few years before his death in 2020 – he was essentially the same: beaming, rumpled, with a restless energy that seemed to echo the constant motion of his mind, words tumbling out as though they were, in their unstoppable abundance, overflowing. But he was also much respected in activist circles for being a good listener, and his radical egalitarianism was borne out in how he related to the people around him.
He was always an anthropologist. After doing fieldwork among traditional peoples in Madagascar, he just never stopped, but he turned his focus to his own society. Essays such as Dead Zones of the Imagination: On Violence, Bureaucracy, and ‘Interpretive Labor’ and his book Bullshit Jobs came from using the equipment of an anthropologist on stuff usually regarded as boring, or not regarded at all – the function and impact of bureaucracy. His 2011 bestseller on debt reminded us that money and finance are among the social arrangements that could be rearranged for the better.
He insisted, again and again, that industrialised Euro-American civilisation was, like other societies past and present, only one way of doing things among countless options. He cited times when societies rejected agriculture or technology or social hierarchy, when social groups chose what has often been dismissed as primitive because it was more free. And he rejected all the linear narratives that present contemporary human beings as declining from primordial innocence or ascending from primitive barbarism. He offered, in place of a single narrative, many versions and variations; a vision of societies as ongoing experiments, and human beings as endlessly creative. That variety was a source of hope for him, a basis for his recurrent insistence that it doesn’t have to be this way.
As Marcus Rediker wrote in his review of David’s posthumous book Pirate Enlightenment, “Everything Graeber wrote was simultaneously a genealogy of the present and an account of what a just society might look like.” He was concerned about inequality of all kinds, including gender inequality in this society and others, and the violence that enforces inequality and unfreedom, as well as how they might be delegitimised and where and when societies might have escaped them. He focused, in short, on freedom and its impediments…
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… The way that, as he wrote, “The ultimate hidden truth of the world is that it is something we make and could just as easily make differently.” If you truly believe that, if you perceive a world that is constructed according to certain assumptions and values, then you see that it can be changed, not least by changing those assumptions and values.
We have to recognise that ideas are tools that we wield – and with them, some power. David wanted to put these tools in everyone’s hands, or remind them that they are already there. Which is part of why he worked hard at – and succeeded in – writing in a style that wasn’t always simple but was always as clear and accessible as possible, given the material. Egalitarianism is a prose style, too. Our mutual friend the writer, film-maker, and debt abolitionist Astra Taylor texted him: “Re-reading Debt. You are such a damn good writer. A rare skill among lefties.” He texted back that August, a month before his demise: “Why thanks! Well at least I take care to do so – I call it ‘being nice to the reader,’ which is an extension of the politics, in a sense.”
In order to believe that people can govern themselves in the absence of coercive institutions and hierarchies, anarchists must have great faith in ordinary people, and David did. A sentence Lyndsey Stonebridge wrote about Hannah Arendt could apply equally well to him: “To fixate on her exceptional mind is to miss something that is important about her lessons in thinking: thinking is ordinary, she teaches; that is its secret power.”…
An edited extract from Solnit’s foreword to The Ultimate Hidden Truth of the World by David Graeber: “‘It does not have to be this way’- the radical optimism of David Graeber,” from @RebeccaSolnit in @guardian.
* David Graeber
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As we promote possibility (and remember that on this date in 1973 then-President Richard Nixon averred in a speech that “I am not a crook”), 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.
While he was an influential professor, he is best remembered for his creation of the “Möbius strip.”
“Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang.”*…
… and like infinity, zero can be a cognitive challenge. Yasemin Saplakoglu explains…
Around 2,500 years ago, Babylonian traders in Mesopotamia impressed two slanted wedges into clay tablets. The shapes represented a placeholder digit, squeezed between others, to distinguish numbers such as 50, 505 and 5,005. An elementary version of the concept of zero was born.
Hundreds of years later, in seventh-century India, zero took on a new identity. No longer a placeholder, the digit acquired a value and found its place on the number line, before 1. Its invention went on to spark historic advances in science and technology. From zero sprang the laws of the universe, number theory and modern mathematics.
“Zero is, by many mathematicians, definitely considered one of the greatest — or maybe the greatest — achievement of mankind,” said the neuroscientist Andreas Nieder, who studies animal and human intelligence at the University of Tübingen in Germany. “It took an eternity until mathematicians finally invented zero as a number.”
Perhaps that’s no surprise given that the concept can be difficult for the brain to grasp. It takes children longer to understand and use zero than other numbers, and it takes adults longer to read it than other small numbers. That’s because to understand zero, our mind must create something out of nothing. It must recognize absence as a mathematical object.
“It’s like an extra level of abstraction away from the world around you,” said Benjy Barnett, who is completing graduate work on consciousness at University College London. Nonzero numbers map onto countable objects in the environment: three chairs, each with four legs, at one table. With zero, he said, “we have to go one step further and say, ‘OK, there wasn’t anything there. Therefore, there must be zero of them.’”
In recent years, research started to uncover how the human brain represents numbers, but no one examined how it handles zero. Now two independent studies, led by Nieder and Barnett, respectively, have shown that the brain codes for zero much as it does for other numbers, on a mental number line. But, one of the studies found, zero also holds a special status in the brain…
Read on to find out the ways in which new studies are uncovering how the mind creates something out of nothing: “How the Human Brain Contends With the Strangeness of Zero,” from @QuantaMagazine.
Pair with Percival Everett’s provocative (and gloriously entertaining) Dr. No.
* Charles Seife, Zero: The Biography of a Dangerous Idea
Scheduling note: your correspondent is sailing again into uncommonly busy waters. So, with apologies for the hiatus, (R)D will resume on Friday the 25th…
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As we noodle on noodling on nothing, we might send carefully-calculated birthday greetings to Erasmus Reinhold; he was born on this date in 1511. A professor of Higher Mathematics (at the University of Wittenberg, where he was ultimately Rector), Reinhold worked at a time when “mathematics” included applied mathematics, especially astronomy– to which he made many contributions and of which he was considered the most influential pedagogue of his generation.
Reinhold’s Prutenicae Tabulae (1551, 1562, 1571, and 1585) or Prussian Tables were astronomical tables that helped to disseminate calculation methods of Copernicus throughout the Empire. That said, Reinhold (like other astronomers before Kepler and Galileo) translated Copernicus’ mathematical methods back into a geocentric system, rejecting heliocentric cosmology on physical and theological grounds. Both Reinhold’s Prutenic Tables and Copernicus’ studies were the foundation for the Calendar Reform by Pope Gregory XIII in 1582… and both made copious use of zeros.
“In mathematics, the art of proposing a question must be held of higher value than solving it”*…
Matteo Wong talks with mathematician Terence Tao about the advent of AI in mathematical research and finds that Tao has some very big questions indeed…
Terence Tao, a mathematics professor at UCLA, is a real-life superintelligence. The “Mozart of Math,” as he is sometimes called, is widely considered the world’s greatest living mathematician. He has won numerous awards, including the equivalent of a Nobel Prize for mathematics, for his advances and proofs. Right now, AI is nowhere close to his level.
But technology companies are trying to get it there. Recent, attention-grabbing generations of AI—even the almighty ChatGPT—were not built to handle mathematical reasoning. They were instead focused on language: When you asked such a program to answer a basic question, it did not understand and execute an equation or formulate a proof, but instead presented an answer based on which words were likely to appear in sequence. For instance, the original ChatGPT can’t add or multiply, but has seen enough examples of algebra to solve x + 2 = 4: “To solve the equation x + 2 = 4, subtract 2 from both sides …” Now, however, OpenAI is explicitly marketing a new line of “reasoning models,” known collectively as the o1 series, for their ability to problem-solve “much like a person” and work through complex mathematical and scientific tasks and queries. If these models are successful, they could represent a sea change for the slow, lonely work that Tao and his peers do.
After I saw Tao post his impressions of o1 online—he compared it to a “mediocre, but not completely incompetent” graduate student—I wanted to understand more about his views on the technology’s potential. In a Zoom call last week, he described a kind of AI-enabled, “industrial-scale mathematics” that has never been possible before: one in which AI, at least in the near future, is not a creative collaborator in its own right so much as a lubricant for mathematicians’ hypotheses and approaches. This new sort of math, which could unlock terra incognitae of knowledge, will remain human at its core, embracing how people and machines have very different strengths that should be thought of as complementary rather than competing…
A sample of what follows…
The classic idea of math is that you pick some really hard problem, and then you have one or two people locked away in the attic for seven years just banging away at it. The types of problems you want to attack with AI are the opposite. The naive way you would use AI is to feed it the most difficult problem that we have in mathematics. I don’t think that’s going to be super successful, and also, we already have humans that are working on those problems.
… Tao: The type of math that I’m most interested in is math that doesn’t really exist. The project that I launched just a few days ago is about an area of math called universal algebra, which is about whether certain mathematical statements or equations imply that other statements are true. The way people have studied this in the past is that they pick one or two equations and they study them to death, like how a craftsperson used to make one toy at a time, then work on the next one. Now we have factories; we can produce thousands of toys at a time. In my project, there’s a collection of about 4,000 equations, and the task is to find connections between them. Each is relatively easy, but there’s a million implications. There’s like 10 points of light, 10 equations among these thousands that have been studied reasonably well, and then there’s this whole terra incognita.
There are other fields where this transition has happened, like in genetics. It used to be that if you wanted to sequence a genome of an organism, this was an entire Ph.D. thesis. Now we have these gene-sequencing machines, and so geneticists are sequencing entire populations. You can do different types of genetics that way. Instead of narrow, deep mathematics, where an expert human works very hard on a narrow scope of problems, you could have broad, crowdsourced problems with lots of AI assistance that are maybe shallower, but at a much larger scale. And it could be a very complementary way of gaining mathematical insight.
Wong: It reminds me of how an AI program made by Google Deepmind, called AlphaFold, figured out how to predict the three-dimensional structure of proteins, which was for a long time something that had to be done one protein at a time.
Tao: Right, but that doesn’t mean protein science is obsolete. You have to change the problems you study. A hundred and fifty years ago, mathematicians’ primary usefulness was in solving partial differential equations. There are computer packages that do this automatically now. Six hundred years ago, mathematicians were building tables of sines and cosines, which were needed for navigation, but these can now be generated by computers in seconds.
I’m not super interested in duplicating the things that humans are already good at. It seems inefficient. I think at the frontier, we will always need humans and AI. They have complementary strengths. AI is very good at converting billions of pieces of data into one good answer. Humans are good at taking 10 observations and making really inspired guesses…
Terence Tao, the world’s greatest living mathematician, has a vision for AI: “We’re Entering Uncharted Territory for Math,” from @matteo_wong in @TheAtlantic.
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As we go figure, we might think recursively about Benoit Mandelbrot; he died on this date in 2010. A mathematician (and polymath), his interest in “the art of roughness” of physical phenomena and “the uncontrolled element in life” led to work (which included coining the word “fractal”, as well as developing a theory of “self-similarity” in nature) for which he is known as “the father of fractal geometry.”
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”*…
Mark Frauenfelder at Boing Boing with a glorious memory…
This cover from the July 1965 issue of Scientific American illustrates the “Four Bugs Problem” featured in Martin Gardner’s “Mathematical Games” column about op art [see here].
The setup: Four bugs are placed at the corners of a square. They start crawling clockwise (or counterclockwise) at a constant rate, with each bug moving directly toward its neighbor. As the bugs move, they always form the corners of a square that both diminishes in size and rotates. Each bug’s path forms a logarithmic spiral.
Gardner said this can be generalized to any number of bugs starting at the corners of a regular polygon with n sides. In these cases, the bugs will always form the corners of a similar polygon that shrinks and rotates as they move.
Here’s an animated version of the Four Bugs Problem you can try out. If you want to try it with a different number of bugs, go here.
Your correspondent still has his copy of that issue. “The beautiful ‘Four Bugs Problem’” from @Frauenfelder in @BoingBoing.
* Bertrand Russell, A History of Western Philosophy
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As we marvel, we might send carefully-calculated birthday greetings to Ian Stewart; he was born on this date in 1945. As a teenager, he was an avid reader of Gardner’s “Mathematical Games,” from which he developed a love of the subject that led him to become a mathematician who has gone on to make important contributions to the field, especially in catastrophe theory.
But Stewart is more widely known as a popularizer of math– who credits Gardner with modeling the skills needed to be an entertaining communicator. Indeed, from 1991 to 2001 Stewart took over the Scientific American column (which had been renamed “Mathematical Recreations”).
For a list of his (remarkable) books on math and science, see here.
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