Posts Tagged ‘Sputnik’
“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.
###
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.
“By preventing dangerous asteroid strikes, we can save millions of people, or even our entire species”*…
The probability of an major asteroid strike on earth at any given moment is low, but the consequences could be catastrophic… and the odds of it happening at some point grow frighteningly large. Happily, the B612 Foundation and Asteroid Institute has developed a way of identifying potentially dangerous asteroids so that they can be deflected by NASA…
Protecting the planet: The Asteroid Institute, @b612foundation.
* Rusty Schweickart, astronaut and co-founder of B612
###
As we dodge disaster, we might recall that it was on this date in 1957 that the space age– and the space race– began in earnest: Sputnik 1 was launched by the Soviet Union into earth orbit.
“Whenever a toddler sees a pile of blocks, he wants to tear it down”*…
Jonathan M. Guberman made a set of alphabet blocks featuring iconic images of the things he and his wife were looking forward to sharing with their newborn son. Though Guberman started before his son was born, it took until his son was nearly a year old to finish these 36 1.5″ blocks. Here’s the full set of blocks on Flickr and the full list of references included on the blocks.
Quoth Jonathan:
There are 36 blocks — the English alphabet and ten digits — showing 134 images of people, animals, monsters, robots, vehicles, organizations, devices, tools, and objects from some of our favourite movies, TV shows, books, comics, video games, poems, and sculptures, as well as a few from the real world for good measure (and a couple not-so-favourites for comic relief/alphabetical exigency; I’m looking at you, Zardoz). The only real rule I followed in choosing subjects was trying to maintain an even gender balance.
Read the full story, and examine the blocks, at the ever-enlightening Laughing Squid.
* J.J. Abrams
###
As we get in touch with our inner nerd, we might recall that it was on this date in 1958 that, as part of its participation in the International Geophysical Year, the U.S. launched its first satellite, the Explorer I— following the launch the prior year of the first two satellites, the Soviet Union’s Sputnik I and II, and beginning the space race.
A Symbolic Achievement…
source: PaperBlog
Like the ampersand, the “@” symbol is not strictly a mark of punctuation; rather, it is a logogram or grammalogue, a shorthand for the word “at.” Even so, it is as much a staple of modern communication as the semicolon or exclamation mark, punctuating email addresses and announcing Twitter usernames. Unlike the ampersand, though, whose journey to the top took two millennia of steady perseverance, the at symbol’s current fame is quite accidental. It can, in fact, be traced to the single stroke of a key made almost exactly four decades ago*…
The whole story is @ Shady Characters (“The Secret Life of Punctuation”).
* Before it became the domain address marker– and the overall symbol– for email, “@” was used to denote the unit price (or weight) of an item: 10 books @ $1.00 would total $10.00… the symbol is believed to have originated with medieval scribes who used the symbol to eliminate the two extra pen strokes that would have been necessary to write “at.”
As we check our spam filters, we might recall that it was on this date in 1958, in reaction to the Soviet’s Sputnik success the prior year, that Congress passed the legislation establishing the National Aeronautics and Space Administration (NASA)… the Space Race was on. (ARPA [now DARPA]– the sponsor of the work that spawned the internet and birthed the “@” in the form in which we all now use it– was born the same year out of the same concern over Soviet scientific progress.)
You must be logged in to post a comment.