Posts Tagged ‘deduction’
“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.
“A prudent question is one-half of wisdom”*…
The death of Queen Elizabeth I created a career opportunity for philosopher and statesman Francis Bacon– one that, as Susan Wise Bauer explains– led him to found empiricism, to pioneer inductive reasoning, and in so doing, to advance the scientific method…
In 1603, Francis Bacon, London born, was forty-three years old: a trained lawyer and amateur philosopher, happily married, politically ambitious, perpetually in debt.
He had served Elizabeth I of England loyally at court, without a great deal of recognition in return. But now Elizabeth was dead at the age of sixty-nine, and her crown would go to her first cousin twice removed: James VI of Scotland, James I of England.
Francis Bacon hoped for better things from the new king, but at the moment he had no particular ‘in’ at the English court. Forced to be patient, he began working on a philosophical project he’d had in mind for some years–a study of human knowledge that he intended to call Of the Proficience and Advancement of Learning, Divine and Human.
Like most of Bacon’s undertakings, the project was ridiculously ambitious. He set out to classify all learning into the proper branches and lay out all of the possible impediments to understanding. Part I condemned what he called the three ‘distempers’ of learning, which included ‘vain imaginations,’ pursuits such as astrology and alchemy that had no basis in actual fact; Part II divided all knowledge into three branches and suggested that natural philosophy should occupy the prime spot. Science, the project of understanding the universe, was the most important pursuit man could undertake. The study of history (‘everything that has happened’) and poesy (imaginative writings) took definite second and third places.
For a time, Bacon didn’t expand on these ideas. The Advancement of Learning opened with a fulsome dedication to James I (‘I have been touched–yea, and possessed–with an extreme wonder at those your virtues and faculties . . . the largeness of your capacity, the faithfulness of your memory, the swiftness of your apprehension, the penetration of your judgment, and the facility and order of your elocution …. There hath not been since Christ’s time any king or temporal monarch which hath been so learned in all literature and erudition, divine and human’), and this groveling soon yielded fruit. In 1607 Bacon was appointed as solicitor general, a position he had coveted for years, and over the next decade or so he poured his energies into his government responsibilities.
He did not return to natural philosophy until after his appointment to the even higher post of chancellor in 1618. Now that he had battled his way to the top of the political dirt pile, he announced his intentions to write a work with even greater scope–a new, complete system of philosophy that would shape the minds of men and guide them into new truths. He called this masterwork the Great Instauration: the Great Establishment, a whole new way of thinking, laid out in six parts.
Part I, a survey of the existing ‘ancient arts’ of the mind, repeated the arguments of the Advancement of Learning. But Part II, published in 1620 as a stand-alone work, was something entirely different. It was a wholesale challenge to Aristotelian methods, a brand-new ‘doctrine of a more perfect use of reason.’
Aristotelian thinking relies, heavily, on deductive reasoning for ancient logicians and philosophers, the highest and best road to the truth. Deductive reasoning moves from general statements (premises) to specific conclusions.
MAJOR PREMISE: All heavy matter falls toward the center of the universe. MINOR PREMISE: The earth is made of heavy matter. MINOR PREMISE: The earth is not falling. CONCLUSION: The earth must already be at the center of the universe.
But Bacon had come to believe that deductive reasoning was a dead end that distorted evidence: ‘Having first determined the question according to his will,’ he objected, ‘man then resorts to experience, and bending her to conformity with his placets [expressions of assent], leads her about like a captive in a procession.’ Instead, he argued, the careful thinker must reason the other way around: starting from specifics and building toward general conclusions, beginning with particular pieces of evidence and working, inductively, toward broader assertions.
This new way of thinking–inductive reasoning–had three steps to it. The ‘true method’ Bacon explained,
‘first lights the candle, and then by means of the candle shows the way; commencing as it does with experience duly ordered and digested, not bungling or erratic, and from it deducing axioms, and from established axioms again new experiments.’
In other words, the natural philosopher must first come up with an idea about how the world works: ‘lighting the candle.’ Second, he must test the idea against physical reality, against ‘experience duly ordered’–both observations of the world around him and carefully designed experiments. Only then, as a last step, should he ‘deduce axioms,’ coming up with a theory that could be claimed to carry truth.
Hypothesis, experiment, conclusion: Bacon had just traced the outlines of the scientific method…
Francis Bacon and the Scientific Method
An excerpt from The Story of Western Science by @SusanWiseBauer, via the invaluable @delanceyplace.
* Francis Bacon
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As we embrace empiricism, we might send carefully-transmitted birthday greetings to Augusto Righi; he was born on this date in 1850. A physicist and a pioneer in the study of electromagnetism, he showed that showed that radio waves displayed characteristics of light wave behavior (reflection, refraction, polarization, and interference), with which they shared the electromagnetic spectrum. In 1894 Righi was the first person to generate microwaves.
Righi influenced the young Guglielmo Marconi, the inventor of radio, who visited him at his lab. Indeed, Marconi invented the first practical wireless telegraphy radio transmitters and receivers in 1894 using Righi’s four ball spark oscillator (from Righi’s microwave work) in his transmitters.
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