Alec Nevala-Lee

Thoughts on art, creativity, and the writing life.

Posts Tagged ‘Gian-Carlo Rota

The mathematician’s tricks

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A long time ago an older and well-known number theorist made some disparaging remarks about Paul Erdős’s work. You admire Erdős’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdős’s work could be “reduced” to a few tricks which Erdős repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert’s beautiful results have been completely forgotten. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!

Gian-Carlo Rota, “Ten Lessons I Wish I Had Been Taught”

Written by nevalalee

February 4, 2018 at 7:30 am

The physical minimum

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Vitaly Ginzburg

When you’re entering a new field, or even after you’ve been there for a while, you eventually need to decide how much to specialize. We’re at a moment in history in which it’s impossible for any one person to know everything about his or her profession, and the most meaningful work tends to occur when we drill down deeply at one particular point. Yet somehow we need to remain generalists, too, if we’re going to have the insight and perspective to use what we find. In his Nobel Prize lecture, the physicist Vitaly Ginzburg summed up this predicament:

In the recent past it was possible to be guided by the requirement “to know something about everything and to know everything about something”…but now, it seems to me, this is no longer possible. At the same time, I am startled and dispirited when young physicists (and sometimes not so young ones) restrict themselves to the knowledge in “their” area and are not informed, if only in a general way, about the state of physics as a whole and its “hottest” areas…It is possible, on the basis of theoretical physics studied in one’s student days, to understand all modern physics, or, more precisely, to understand how matters stand everywhere in physics and be aware of the situation. Every physicist…should simultaneously know, apart from theoretical physics, a wealth of facts from different branches of physics and be familiar with the newest notable accomplishments.

So how do we keep ourselves properly informed about a field that is too complex to grasp in its entirety? We perform a kind of triage, as Ginzburg advises, and focus on the “hottest” areas—the places where important work all but begs to be done in our lifetimes. More specifically, we can make a list. As Ginzburg notes:

At the same time, we in Russia like to quote a certain Kozma Prutkov, a fictitious character, who said pompously, in particular, that “there is no way of comprehending the incomprehensible.” So one has to choose something. And so I took this path: I have made a “list” of the top problems of the day. Any such “list” is admittedly subjective. It is also clear that the “list” should vary with time. Lastly, it is clear that subjects not included in the “list” can in no way be regarded as unimportant or uninteresting…I only suggest some enumeration of the questions that, in my view, every physicist should have at least a superficial idea of. Supposedly less trivial is the statement that this is not as difficult as it might seem at first glance. The time to be spent for this purpose is, I believe, no longer than the time a good student spends preparing for an examination, say, on electrodynamics. Acquaintance with all subjects included in this “list” is what I call the “physical minimum.”

Ginzburg goes on to provide an annotated list of thirty subjects in physics, from “controlled nuclear fusion” to “neutrino physics and astronomy.” (Note that this is a list of problems, not of topics for basic education. For the latter kind of list, Gerard ’t Hooft, another Nobel laureate whom I quoted recently on the subject of how to become a bad theoretical physicist, provides a useful one here.)

Richard Feynman

And this strategy is worth following no matter what your field happens to be. (As Ginzburg says: “Naturally, this equally applies to other specialties, but I restrict myself to physicists for definitiveness.”) We can’t know everything about it, but we can prioritize, putting together a list of active problems that might benefit from new approaches, and making a point of learning enough about them to recognize any useful ideas on which we happen to stumble. Even the act of writing up the list itself has a way of directing your attention onto what actually matters. When you’re preoccupied solely with what is in front of you, it’s easy to forget about the big issues that your discipline as a whole is confronting. And even if you’re mostly aware of the top ten unsolved problems in your profession, it can be enlightening to extend the list to thirty, as Ginzburg does: there may be something to which you can contribute two-thirds of the way down, which is still pretty high. Obviously, this technique can also be applied on a smaller scale—you can list the problems that present themselves in your current project, your job, or your personal life, and make sure that they’re constantly before your eyes. But it also makes sense to aim as high as possible. There’s a huge incentive in every field to turn ourselves into what Hilaire Belloc memorably called “masters of the earthworm,” in which we spend a lifetime focusing on the one tiny corner that we can claim for our own. And it’s probably necessary. But an awareness of the larger problems is what allows us to select the most promising slice of territory.

Best of all, it enables what the scientist W.I.B. Beveridge called the transfer method, in which ideas from one area are applied to seemingly unrelated problems. It’s perhaps the most fertile source of innovation we have, but it doesn’t happen by accident. It occurs, in fact, when smart people make a list of important problems and keep them continuously in mind. As the physicist Gian-Carlo Rota says, in one of my favorite pieces of advice of any kind:

Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: “How did he do it? He must be a genius!”

We can’t all be like Feynman, but we can at least position ourselves to make whatever contributions we can. This means remaining attuned to the meaningful problems that remain unresolved; picking specialties that are likely to matter, rather than counting the spots on a sea urchin’s egg; and being ready to pivot whenever our area of expertise seems useful. In the end, we may all need to be masters of the earthworm. But even a worm can turn.

Written by nevalalee

November 16, 2016 at 8:42 am

The transfer method

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Richard Feynman

A few weeks ago, I picked up a worn paperback copy of The Art of Scientific Investigation by W.I.B. Beveridge, which I expect will join the short list of books on creativity that I’ll never get tired of reading. It was first published in 1950, but it’s still in print, and it isn’t hard to understand why. Beveridge’s book is essentially a collection of recipes or approaches for coming up with ideas, with meaty chapters devoted to the roles of reason, intuition, chance, and imagination, and it’s loaded with concrete, practical advice. Take the section on what Beveridge calls the transfer method:

Sometimes the central idea on which an investigation hinges is provided by the application or transfer of a new principle or technique which has been discovered in another field. The method of making advances in this way will be referred to as the “transfer” method in research. This is probably the most fruitful and the easiest method in research, and the one most employed in applied research. It is, however, not to be in any way despised. Scientific advances are so hard to achieve that every useful stratagem must be used.

The italics are mine. Success or failure in resolving any problem often boils down to a knowledge of the available tools, and this often requires familiarity with advances in apparently unrelated fields. One of my favorite recent examples comes from the field of adaptive optics. When astronomers are viewing an object through the earth’s atmosphere, which distorts light, they’ll shine a laser in the same direction. When the light from this artificial “guide star” returns, they can measure the distortion, then use that data to adjust their telescope to cancel out the aberrations, which gives them a much more accurate view of the object under observation. The physicist Eric Betzig took the idea of a guide star and applied it to microscopy, which also has to deal with optical information being warped by an intervening medium, which in this case is organic tissue. Taking a cue from astronomy, the technique creates a guide star by focusing light from the microscope on a fluorescent object in the sample, like an embedded bead. After using a wavefront sensor to determine how the light was warped, it can make the appropriate corrections. And because tissue causes more complex distortions than the atmosphere does, it employs yet another strategy—derived from ophthalmology, which uses it to correct images of a patient’s retina—to average out the error. The result won Betzig a Nobel Prize.

Eric Betzig

And it isn’t hard to see why Betzig paid close attention to astronomy and ophthalmology. These fields may study different classes of objects, but they’re all ultimately about dealing with properties of light as it passes from the observed to the observer, which has clear implications for microscopy. Betzig and his collaborators were shrewd enough to frame their work in the most general possible terms: it wasn’t about microscopes, but about light, and everything that dealt with similar problems was potentially interesting. Being able to correctly define your field—which has more to do with the concrete problems you’re addressing than with labels imposed from the outside—is the first step in identifying useful combinations. And even trained scientists have trouble doing this. As Beveridge notes:

It might be thought that as soon as a discovery is announced, all its possible applications in other fields follow almost immediately and automatically, but this is seldom so. Scientists sometimes fail to realize the significance which a new discovery in another field may have for their own work, or if they do realize it they may not succeed in discovering the necessary modifications.

Of course, it isn’t possible to read or absorb everything, so you need to be smart about how you filter the universe of available material, which can be done from either end. You can start with a solution and then look for interesting problems: Beveridge cites several examples of techniques, such as partition chromatography, in which researchers systematically cast about for fields in which it could be put to use. Alternatively, you can keep a handful of problems perpetually before you, and use it as a kind of sieve to isolate useful ideas, as Gian-Carlo Rota describes:

Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: “How did he do it? He must be a genius!”

This is essentially what novelists do. When you have the basic premise of a story in mind, suddenly everything you see becomes relevant—which is a good argument for coming up with at least a general outline as early as possible. But you don’t need to be a novelist, or a scientist, to find a guide star of your own.

Written by nevalalee

September 2, 2015 at 9:32 am

Quote of the Day

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Gian-Carlo Rota

A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.

Gian-Carlo Rota

Written by nevalalee

April 21, 2015 at 7:30 am

Posted in Quote of the Day

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Quote of the Day

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Richard Feynman

Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: “How did he do it? He must be a genius!”

Gian-Carlo Rota, “Ten Lessons I Wish I Had Been Taught”

Written by nevalalee

December 11, 2013 at 7:30 am

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