Posts Tagged ‘fractals’
“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.
###
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.”
“When the graphs were finished, the relations were obvious at once”*…
We can only understand what we can “see”…
… this long-forgotten, hand-drawn infographic from the 1840s… known as a “life table,” was created by William Farr, a doctor and statistician who, for most of the Victorian era, oversaw the collection of public health statistics in England and Wales… it’s a triptych documenting the death rates by age in three key population groups: metropolitan London, industrial Liverpool, and rural Surrey.
With these visualizations, Farr was making a definitive contribution to an urgent debate from the period: were these new industrial cities causing people to die at a higher rate? In some ways, with hindsight, you can think of this as one of the most crucial questions for the entire world at that moment. The Victorians didn’t realize it at the time, but the globe was about to go from less than five percent of its population living in cities to more than fifty percent in just about a century and a half. If these new cities were going to be killing machines, we probably needed to figure that out.
It’s hard to imagine just how confusing it was to live through the transition to industrial urbanism as it was happening for the first time. Nobody really had a full handle on the magnitude of the shift and its vast unintended consequences. This was particularly true of public health. There was an intuitive feeling that people were dying at higher rates than they had in the countryside, but it was very hard even for the experts to determine the magnitude of the threat. Everyone was living under the spell of anecdote and availability bias. Seeing the situation from the birds-eye view of public health data was almost impossible…
The images Farr created told a terrifying and unequivocal story: density kills. In Surrey, the increase of mortality after birth is a gentle slope upward, a dune rising out of the waterline. The spike in Liverpool, by comparison, looks more like the cliffs of Dover. That steep ascent condensed thousands of individual tragedies into one vivid and scandalous image: in industrial Liverpool, more than half of all children born were dead before their fifteenth birthday.
The mean age of death was just as shocking: the countryfolk were enjoying life expectancies close to fifty, likely making them some of the longest-lived people on the planet in 1840. The national average was forty-one. London was thirty-five. But Liverpool—a city that had undergone staggering explosions in population density, thanks to industrialization—was the true shocker. The average Liverpudlian died at the age of twenty-five, one of the lowest life expectancies ever recorded in that large a human population.
There’s a natural inclination to think about innovation in human health as a procession of material objects: vaccines, antibiotics, pacemakers. But Farr’s life tables are a reminder that new ways of perceiving the problems we face, new ways of seeing the underlying data, are the foundations on which we build those other, more tangible interventions. Today cities reliably see life expectancies higher than rural areas—a development that would have seemed miraculous to William Farr, tabulating the data in the early 1840s. In a real sense, Farr laid the groundwork for that historic reversal: you couldn’t start to tackle the problem of how to make industrial cities safer until you had first determined that the threat was real.
Why the most important health innovations sometimes come from new ways of seeing: “The Obscure Hand-Drawn Infographic That Changed The Way We Think About Cities,” from Steven Johnson (@stevenbjohnson). More in his book, Extra Life, and in episode 3 of the PBS series based on it.
###
As we investigate infographics, we might send carefully calculated birthday greetings to Lewis Fry Richardson; he was born on this date in 1881. A mathematician, physicist, and psychologist, he is best remembered for pioneering the modern mathematical techniques of weather forecasting. Richardson’s interest in weather led him to propose a scheme for forecasting using differential equations, the method used today, though when he published Weather Prediction by Numerical Process in 1922, suitably fast computing was unavailable. Indeed, his proof-of-concept– a retrospective “forecast” of the weather on May 20, 1910– took three months to complete by hand. (in fairness, Richardson did the analysis in his free time while serving as an ambulance driver in World War I.) With the advent of modern computing in the 1950’s, his ideas took hold. Still the ENIAC (the first real modern computer) took 24 hours to compute a daily forecast. But as computing got speedier, forecasting became more practical.
Richardson also yoked his forecasting techniques to his pacifist principles, developing a method of “predicting” war. He is considered (with folks like Quincy Wright and Kenneth Boulding) a father of the scientific analysis of conflict.
And Richardson helped lay the foundations for other fields and innovations: his work on coastlines and borders was influential on Mandelbrot’s development of fractal geometry; and his method for the detection of icebergs anticipated the development of sonar.
“Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry”*…
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
—Benoit Mandelbrot, The Fractal Geometry of Nature
Benoit Mandelbrot, Sterling Professor of Mathematics at Yale and the father of fractal geometry, died last Thursday at age 85. As Heinz-Otto Peitgen, professor of mathematics and biomedical sciences at the University of Bremen, observed, “if we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last 50 years.”
“I decided to go into fields where mathematicians would never go because the problems were badly stated,” Dr. Mandelbrot once said. “I have played a strange role…” Indeed, one hopes that Mandelbrot had the consolation of his own fascination as he contemplated the diffusion pattern of the pancreatic cancer that killed him.
At TED2010, mathematics legend Benoit Mandelbrot develops a theme he first discussed at TED in 1984 — the extreme complexity of roughness, and the way that fractal math can find order within patterns that seem unknowably complicated.
* Richard Feynman
In other sad news, Barbara Billingsley, the avatar of American motherhood in her role as Mrs. Cleaver on Leave it to Beaver, passed away on Saturday.
As we marvel at patterns nested within themselves, we might recall that it was on this date in in 1962 that In 1962, Dr. James D. Watson, Dr. Francis Crick, and Dr. Maurice Wilkins won the Nobel Prize for Medicine and Physiology for their work in determining the double-helix molecular structure of DNA (deoxyribonucleic acid).
You must be logged in to post a comment.