Posts Tagged ‘Math’
“A proof tells us where to concentrate our doubts”*…
Number theorist Andrew Granville on what mathematics really is, on why objectivity is never quite within reach, and on the role that AI might play…
… What is a mathematical proof? We tend to think of it as a revelation of some eternal truth, but perhaps it is better understood as something of a social construct.
Andrew Granville, a mathematician at the University of Montreal, has been thinking about that a lot recently. After being contacted by a philosopher about some of his writing, “I got to thinking about how we arrive at our truths,” he said. “And once you start pushing at that door, you find it’s a vast subject.”
“How mathematicians go about research isn’t generally portrayed well in popular media. People tend to see mathematics as this pure quest, where we just arrive at great truths by pure thought alone. But mathematics is about guesses — often wrong guesses. It’s an experimental process. We learn in stages…
Quanta spoke with Granville about the nature of mathematical proof — from how proofs work in practice to popular misconceptions about them, to how proof-writing might evolve in the age of artificial intelligence…
[excerpts for that interview follow…]
How mathematicians go about research isn’t generally portrayed well in popular media. People tend to see mathematics as this pure quest, where we just arrive at great truths by pure thought alone. But mathematics is about guesses — often wrong guesses. It’s an experimental process. We learn in stages…
The culture of mathematics is all about proof. We sit around and think, and 95% of what we do is proof. A lot of the understanding we gain is from struggling with proofs and interpreting the issues that come up when we struggle with them…
The main point of a proof is to persuade the reader of the truth of an assertion. That means verification is key. The best verification system we have in mathematics is that lots of people look at a proof from different perspectives, and it fits well in a context that they know and believe. In some sense, we’re not saying we know it’s true. We’re saying we hope it’s correct, because lots of people have tried it from different perspectives. Proofs are accepted by these community standards.
Then there’s this notion of objectivity — of being sure that what is claimed is right, of feeling like you have an ultimate truth. But how can we know we’re being objective? It’s hard to take yourself out of the context in which you’ve made a statement — to have a perspective outside of the paradigm that has been put in place by society. This is just as true for scientific ideas as it is for anything else…
[Granville runs through a history of the proof, from Aristotle, through Euclid, to Hilbert, then Russel and Whitehead, ending with Gödel…]
To discuss mathematics, you need a language, and a set of rules to follow in that language. In the 1930s, Gödel proved that no matter how you select your language, there are always statements in that language that are true but that can’t be proved from your starting axioms. It’s actually more complicated than that, but still, you have this philosophical dilemma immediately: What is a true statement if you can’t justify it? It’s crazy.
So there’s a big mess. We are limited in what we can do.
Professional mathematicians largely ignore this. We focus on what’s doable. As Peter Sarnak likes to say, “We’re working people.” We get on and try to prove what we can…
[Granville then turns to computers…]
We’ve moved to a different place, where computers can do some wild things. Now people say, oh, we’ve got this computer, it can do things people can’t. But can it? Can it actually do things people can’t? Back in the 1950s, Alan Turing said that a computer is designed to do what humans can do, just faster. Not much has changed.
For decades, mathematicians have been using computers — to make calculations that can help guide their understanding, for instance. What AI can do that’s new is to verify what we believe to be true. Some terrific developments have happened with proof verification. Like [the proof assistant] Lean, which has allowed mathematicians to verify many proofs, while also helping the authors better understand their own work, because they have to break down some of their ideas into simpler steps to feed into Lean for verification.
But is this foolproof? Is a proof a proof just because Lean agrees it’s one? In some ways, it’s as good as the people who convert the proof into inputs for Lean. Which sounds very much like how we do traditional mathematics. So I’m not saying that I believe something like Lean is going to make a lot of errors. I’m just not sure it’s any more secure than most things done by humans…
Perhaps it could assist in creating a proof. Maybe in five years’ time, I’ll be saying to an AI model like ChatGPT, “I’m pretty sure I’ve seen this somewhere. Would you check it out?” And it’ll come back with a similar statement that’s correct.
And then once it gets very, very good at that, perhaps you could go one step further and say, “I don’t know how to do this, but is there anybody who’s done something like this?” Perhaps eventually an AI model could find skilled ways to search the literature to bring tools to bear that have been used elsewhere — in a way that a mathematician might not foresee.
However, I don’t understand how ChatGPT can go beyond a certain level to do proofs in a way that outstrips us. ChatGPT and other machine learning programs are not thinking. They are using word associations based on many examples. So it seems unlikely that they will transcend their training data. But if that were to happen, what will mathematicians do? So much of what we do is proof. If you take proofs away from us, I’m not sure who we become…
Eminently worth reading in full: “Why Mathematical Proof Is a Social Compact,” in @QuantaMagazine.
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As we add it up, we might send carefully calculated birthday greetings to Edward G. Begle; he was born on this date in 1914. A mathematician who was an accomplished topologist, he is best remembered for his role as the director of the School Mathematics Study Group (SMSG), the primary group credited for developing what came to be known as The New Math (a pedagogical response to Sputnik, taught in American grade schools from the late 1950s through the 1970s)… which will be well-known to (if not necessarily fondly recalled by) readers of a certain age.
“To understand anything, you just need to understand the little bits”*…
Oscar Schwartz begs to differ. Here, excerpts from his provocative critique of TED Talks…
Bill Gates wheels a hefty metal barrel out onto a stage. He carefully places it down and then faces the audience, which sits silent in a darkened theater. “When I was a kid, the disaster we worried about most was a nuclear war,” he begins. Gates is speaking at TED’s flagship conference, held in Vancouver in 2015. He wears a salmon pink sweater, and his hair is combed down over his forehead, Caesar-style. “That’s why we had a barrel like this down in our basement, filled with cans of food and water,” he says. “When the nuclear attack came, we were supposed to go downstairs, hunker down, and eat out of that barrel.”
Now that he is an adult, Gates continues, it is no longer nuclear apocalypse that scares him, but pestilence. A year ago, Ebola killed over ten thousand people in West Africa. If the virus had been airborne or spread to a large city center, things would have been far worse. It might’ve snowballed into a pandemic and killed tens of millions of people. Gates tells the TED attendees that humanity is not ready for this scenario — that a pandemic would trigger a global catastrophe at an unimaginable scale. We have no basement to retreat to and no metal barrel filled with supplies to rely on.
But, Gates adds, the future might turn out okay. He has an idea. Back when he was a kid, the U.S. military had sufficient funding to mobilize for war at any minute. Gates says that we must prepare for a pandemic with the same fearful intensity. We need to build a medical reserve corps. We need to play germ games like generals play war games. We need to make alliances with other virus-fighting nations. We need to build an arsenal of biomedical weapons to attack any non-human entity that might attack our bodies. “If we start now, we can be ready for the next epidemic,” Gates concludes, to a round of applause.
Of course, Gates’s popular and well-shared TED talk — viewed millions of times — didn’t alter the course of history. Neither did any of the other “ideas worth spreading” (the organization’s tagline) presented at the TED conference that year — including Monica Lewinsky’s massively viral speech about how to stop online bullying through compassion and empathy, or a Google engineer’s talk about how driverless cars would make roads smarter and safer in the near future. In fact, seven years after TED 2015, it feels like we are living in a reality that is the exact opposite of the future envisioned that year. A president took office in part because of his talent for online bullying. Driverless cars are nowhere near as widespread as predicted, and those that do share our roads keep crashing. Covid has killed five million people and counting.
At the start of the pandemic, I noticed people sharing Gates’s 2015 talk. The general sentiment was one of remorse and lamentation: the tech-prophet had predicted the future for us! If only we had heeded his warning! I wasn’t so sure. It seems to me that Gates’s prediction and proposed solution are at least part of what landed us here. I don’t mean to suggest that Gates’s TED talk is somehow directly responsible for the lack of global preparedness for Covid. But it embodies a certain story about “the future” that TED talks have been telling for the past two decades — one that has contributed to our unending present crisis.
The story goes like this: there are problems in the world that make the future a scary prospect. Fortunately, though, there are solutions to each of these problems, and the solutions have been formulated by extremely smart, tech-adjacent people. For their ideas to become realities, they merely need to be articulated and spread as widely as possible. And the best way to spread ideas is through stories — hence Gates’s opening anecdote about the barrel. In other words, in the TED episteme, the function of a story isn’t to transform via metaphor or indirection, but to actually manifest a new world. Stories about the future create the future. Or as Chris Anderson, TED’s longtime curator, puts it, “We live in an era where the best way to make a dent on the world… may be simply to stand up and say something.” And yet, TED’s archive is a graveyard of ideas. It is a seemingly endless index of stories about the future — the future of science, the future of the environment, the future of work, the future of love and sex, the future of what it means to be human — that never materialized. By this measure alone, TED, and its attendant ways of thinking, should have been abandoned…
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… TED talks began to take on a distinct rhetorical style, later laid out in Anderson’s book TED Talks: The Official TED Guide to Public Speaking. In it, Anderson insists anyone is capable of giving a TED-esque talk. You just need an interesting topic and then you need to attach that topic to an inspirational story. Robots are interesting. Using them to eat trash in Nairobi is inspiring. Put the two together, and you have a TED talk.
I like to call this fusion “the inspiresting.” Stylistically, the inspiresting is earnest and contrived. It is smart but not quite intellectual, personal but not sincere, jokey but not funny. It is an aesthetic of populist elitism. Politically, the inspiresting performs a certain kind of progressivism, as it is concerned with making the world a better place, however vaguely…
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Perhaps the most incisive critique came, ironically, at a 2013 TEDx conference. In “What’s Wrong with TED Talks?” media theorist Benjamin Bratton told a story about a friend of his, an astrophysicist, who gave a complex presentation on his research before a donor, hoping to secure funding. When he was finished, the donor decided to pass on the project. “I’m just not inspired,” he told the astrophysicist. “You should be more like Malcolm Gladwell.” Bratton was outraged. He felt that the rhetorical style TED helped popularize was “middlebrow megachurch infotainment,” and had begun to directly influence the type of intellectual work that could be undertaken. If the research wasn’t entertaining or moving, it was seen as somehow less valuable. TED’s influence on intellectual culture was “taking something with value and substance and coring it out so that it can be swallowed without chewing,” Bratton said. “This is not the solution to our most frightening problems — rather, this is one of our most frightening problems.” (Online, his talk proved to be one of many ideas worth spreading. “This is by far the most interesting and challenging thing I’ve heard on TED,” one commenter posted. “Very glad to come across it!”)…
Some thoughts on the “inspiresting”: “What Was the TED Talk?” from @scarschwartz in @thedrift_mag.
* Chris Anderson, proprietor and curator of TED
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As we unchain our curiosity, we might send ruthless curious (and immensely entertaining) birthday greetings to Martin Gardner; he was born on this date in 1914. Though not an academic, nor ever a formal student of math or science, he wrote widely and prolifically on both subjects in such popular books as The Ambidextrous Universe and The Relativity Explosion and as the “Mathematical Games” columnist for Scientific American. Indeed, his elegant– and understandable– puzzles delighted professional and amateur readers alike, and helped inspire a generation of young mathematicians.
Gardner’s interests were wide; in addition to the math and science that were his power alley, he studied and wrote on topics that included magic, philosophy, religion, and literature (c.f., especially his work on Lewis Carroll– including the delightful Annotated Alice— and on G.K. Chesterton). And he was a fierce debunker of pseudoscience: a founding member of CSICOP, and contributor of a monthly column (“Notes of a Fringe Watcher,” from 1983 to 2002) in Skeptical Inquirer, that organization’s monthly magazine.
Gardner died in 2010, having never given a TED Talk.
“Life is a Zen koan, that is, an unsolvable riddle. But the contemplation of that riddle – even though it cannot be solved – is, in itself, transformative.”*…
How hard is it to prove that problems are hard to solve? Meta-complexity theorists have been asking questions like this for decades. And as Ben Brubaker explains, a string of recent results has started to deliver answers…
… Even seasoned researchers find understanding in short supply when they confront the central open question in theoretical computer science, known as the P versus NP problem. In essence, that question asks whether many computational problems long considered extremely difficult can actually be solved easily (via a secret shortcut we haven’t discovered yet), or whether, as most researchers suspect, they truly are hard. At stake is nothing less than the nature of what’s knowable.
Despite decades of effort by researchers in the field of computational complexity theory — the study of such questions about the intrinsic difficulty of different problems — a resolution to the P versus NP question has remained elusive. And it’s not even clear where a would-be proof should start.
“There’s no road map,” said Michael Sipser, a veteran complexity theorist at the Massachusetts Institute of Technology who spent years grappling with the problem in the 1980s. “It’s like you’re going into the wilderness.”
It seems that proving that computational problems are hard to solve is itself a hard task. But why is it so hard? And just how hard is it? Carmosino and other researchers in the subfield of meta-complexity reformulate questions like this as computational problems, propelling the field forward by turning the lens of complexity theory back on itself.
“You might think, ‘OK, that’s kind of cool. Maybe the complexity theorists have gone crazy,’” said Rahul Ilango, a graduate student at MIT who has produced some of the most exciting recent results in the field.
By studying these inward-looking questions, researchers have learned that the hardness of proving computational hardness is intimately tied to fundamental questions that may at first seem unrelated. How hard is it to spot hidden patterns in apparently random data? And if truly hard problems do exist, how often are they hard?
“It’s become clear that meta-complexity is close to the heart of things,” said Scott Aaronson, a complexity theorist at the University of Texas, Austin.
This is the story of the long and winding trail that led researchers from the P versus NP problem to meta-complexity. It hasn’t been an easy journey — the path is littered with false turns and roadblocks, and it loops back on itself again and again. Yet for meta-complexity researchers, that journey into an uncharted landscape is its own reward. Start asking seemingly simple questions, said Valentine Kabanets, a complexity theorist at Simon Fraser University in Canada, and “you have no idea where you’re going to go.”…
Complexity theorists are confronting their most puzzling problem yet– complexity theory itself: “Complexity Theory’s 50-Year Journey to the Limits of Knowledge,” from @benbenbrubaker in @QuantaMagazine.
* Tom Robbins
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As we limn limits, we might send thoroughly cooked birthday greetings to Denis Papin; he was born on this date in 1647. A mathematician and physicist who worked with Christiaan Huygens and Gottfried Leibniz, Papin is better remembered as the inventor of the steam digester, the forerunner of the pressure cooker and of the steam engine.
“A hole can itself have as much shape-meaning as a solid mass”*…
Holes. Caity Weaver wonders about them:
What is a hole?
A hole is a portion of something where something is not. Beyond that, holes are slippery. (As a concept — only some in reality.) Is a hole necessarily empty on both sides, like the gaps in a slice of Swiss cheese? Or need it only be empty on one side, like a pit dug into the earth? Is a hole with a bottom less of a hole than one without one? Can a slit be a hole, or must a hole be vaguely round? Does a straw have two holes, as one Reddit user pondered, or just one — a single thick hole, if you will?…
[She then proceeds to explore the concept etymologically…]
Wait — What Is a Hole?
The Stanford Encyclopedia of Philosophy goes right for the, well… philosophical:
Holes are an interesting case study for ontologists and epistemologists. Naive, untutored descriptions of the world treat holes as objects of reference, on a par with ordinary material objects. (‘There are as many holes in the cheese as there are cookies in the tin.’) And we often appeal to holes to account for causal interactions, or to explain the occurrence of certain events. (‘The water ran out because the bucket has a hole.’) Hence there is prima facie evidence for the existence of such entities. Yet it might be argued that reference to holes is just a façon de parler, that holes are mere entia representationis, as-if entities, fictions.
[There follows a fascinating account of the theories of holes…]
Holes
A whole lot about nothing…
*Henry Moore
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As we hit ’em where they ain’t, we might spare a thought for mathematician Henri Cartan; he died on this date in 2008. A founding member (n 1934) of and active participant in the Bourbaki group, Cartan made contributions to math across algebra, geometry, and analysis, with a special focus on topology (that branch of math that plays with holes in toruses, Klein bottles, and other other-worldly shapes).
“The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.”*…
As Tom Siegfried explains, the “music of the spheres” was born from the effort to use numbers to explain the universe…
If you’ve ever heard the phrase “the music of the spheres,” your first thought probably wasn’t about mathematics.
But in its historical origin, the music of the spheres actually was all about math. In fact, that phrase represents a watershed in the history of math’s relationship with science.
In its earliest forms, as practiced in ancient Egypt and Mesopotamia, math was mainly a practical tool for facilitating human interactions. Math was important for calculating the area of a farmer’s field, for keeping track of workers’ wages, for specifying the right amount of ingredients when making bread or beer. Nobody used math to investigate the nature of physical reality.
Not until ancient Greek philosophers began to seek scientific explanations for natural phenomena (without recourse to myths) did anybody bother to wonder how math would help. And the first of those Greeks to seriously put math to use for that purpose was the mysterious religious cult leader Pythagoras of Samos.
It was Pythagoras who turned math from a mere tool for practical purposes into the key to unlocking the mysteries of the universe. As the historian Geoffrey Lloyd noted, “The Pythagoreans were … the first theorists to have attempted deliberately to give the knowledge of nature a quantitative, mathematical foundation.”…
More at: “How Pythagoras turned math into a tool for understanding reality,” from @tom_siegfried in @ScienceNews.
Apposite: Walter Murch’s ideas on “planetary harmony” (and Lawrence Weschler’s book on him and them)
* Henri Poincare
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As we seek beauty, we might recall that it was on this date in 1595 that Johann Kepler (and here) published Mysterium cosmographicum (Mystery of the Cosmos), in which he described an invisible underlying structure determining the six known planets in their orbits. Kepler thought as a mathematician, devising a structure based on only five convex regular solids; the path of each planet lay on a sphere separated from its neighbors by touching an inscribed polyhedron.
It was a beautiful, an elegant model– and one that fit the orbital data available at the time. It was of course, nonetheless, wrong.
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