… applied science, purposeful and determined, and pure science, playful and freely curious, continuously support and stimulate each other. The great nation of the future will be the one which protects the freedom of pure science as much as it encourages applied science.

Writing novels is hard, and requires vast, unbroken slabs of time. Four quiet hours is a resource I can put to good use. Two slabs of time, each two hours long, might add up to the same four hours, but are not nearly as productive as an unbroken four. … Likewise, several consecutive days with four-hour time-slabs in them give me a stretch of time in which I can write a decent book chapter, but the same number of hours spread out across a few weeks, with interruptions in between them, are nearly useless.

I haven’t written a novel, and probably never will, but Stephenson’s remarks describe my experience doing math and especially developing software. I can do simple, routine work in short blocks of time, but I need larger blocks of time to work on complex projects or to be more creative.

Related post: Four hours of concentration

I imagine that sort of thing happens in the real world, though I can’t recall an example. On the contrary, I can think of examples where people were thrilled by trivial solutions such as a two-line Perl script or a pencil-and-paper calculation that eliminated the need for a program.

The difference is whether the goal is to solve a problem or to produce an impressive solution.

For until this moment he had lived in a state of pure possibility, not knowing what sort of man he was or what he must do, and supposing therefore that he must be all men and do everything. But after this morning’s incident his life took a turn in a particular direction. Thereafter he came to see that he was not destined to do everything but only one or two things. Lucky is the man who does not secretly believe that every possibility is open to him.

As Lawler summarizes,

Without some such closure — without knowing somehow that you’re “not destined to do everything but only one or two things” — you never get around to living.

It’s taken me a long time to understand that deliberately closing off some options can open more interesting options.

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Since then I’ve signed NDAs (non-disclosure agreements) for numerous companies afraid that someone might steal their ideas. Maybe they’re doing the right thing to be cautious, but I doubt it’s necessary.

I think Howard Aiken got it right:

Don’t worry about people stealing your ideas. If your ideas are any good, you’ll have to ram them down people’s throats.

One thing I’ve learned from developing software is that it’s very difficult to transfer ideas. A lot of software projects never completely transition from the original author because no one else really understands what’s going on.

It’s more likely that someone will come up with your idea independently than that someone would steal it. If the time is ripe for an idea, and all the pieces are there waiting for someone to put them together, it may be discovered multiple times. But unless someone is close to making the discovery for himself, he won’t get it even if you explain it to him.

And when other people do have your idea, they still have to implement it. That’s the hard part. We all have more ideas than we can carry out. The chance that someone else will have your idea and have the determination to execute it is tiny.

The first example is finding remainders. What is the remainder when 5,000,070,004 is divided by 9? At first it may seem that you need to divide 5,000,070,004 by 9, but you don’t. You weren’t asked the quotient, only the remainder, which in this case you can do directly. By casting out nines, you can quickly see the remainder is 7.

The second example is definite integrals. The usual procedure for computing definite integrals is to first find an indefinite integral (i.e. anti-derivative) and take the difference of its values at the two end points. But sometimes it’s possible to find the definite integral directly, even when you couldn’t first find the indefinite integral. Maybe you can evaluate the definite integral by symmetry, or a probability argument, or by contour integration, or some other trick.

Contour integration is an interesting example because you don’t do what you might think you need to — i.e. find an indefinite integral — but you do have to do something you might never imagine doing before you’ve seen the trick, i.e. convert an integral over the real line to an integral in the complex plane to make it simpler!

What are some more examples, mathematical or not, of solving a problem without doing something that at first seems necessary?

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Usefulness comes not from pursuing it, but from patiently gathering enough of a reservoir of material so that one has the quirky bit of knowledge … that turns out to be the key to unlocking the problem which someone offers.

Holmes was speaking specifically of theology. I edited out some of the particulars of his quote to emphasize that his idea applies more generally.

Obviously usefulness can come from pursuing it. But there’s a special pleasure in applying some “quirky bit of knowledge” that you acquired for its own sake. It can feel like simply walking up to a gate and unlocking it after unsuccessful attempts to storm the gate by force.

… the National Space Agency didn’t suit up an astronaut. Instead their first goal was to hit the moon — literally. And just over three years later, NASA successfully smashed Ranger 7 into the moon … It took fifteen ever-evolving iterations before the July 16, 1969, gentle moon landing …

Great scientists, creative thinkers, and problem solvers do not solve hard problems head-on. When they are faced with a daunting question, they immediately and prudently admit defeat. They realize there is no sense in wasting energy vainly grappling with complexity when, instead, they can productively grapple with smaller cases that will teach them how to deal with the complexity to come.

From The 5 Elements of Effective Thinking.

Some may wonder whether this contradicts my earlier post about how quickly people give up thinking about problems. Doesn’t the quote above say we should “prudently admit defeat”? There’s no contradiction. The quote advocates retreat, not surrender. One way to be able to think about a hard problem for a long time is to find simpler versions of the problem that you can solve. Or first, to find simpler problems that you cannot solve. As George Polya said

If you can’t solve a problem, then there is an easier problem that you can’t solve; find it.

Bracket the original problem between the simplest version of the problem you cannot solve and the fullest version of the problem you can solve. Then try to move your brackets.

A good undergraduate math student can think about a problem for a few minutes before getting nervous. A grad student may be able to think about a problem for an hour at a time. Before Andrew Wiles proved Fermat’s Last Theorem, he thought about the problem for seven years.

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… the type of complexity we are discussing in this paper is that which makes large systems hard to understand. It is this that causes us to expend huge resources in creating and maintaining such systems. This type of complexity has nothing to do with complexity theory — the branch of computer science which studies the resources consumed by a machine executing a program. The two are completely unrelated — it is a straightforward matter to write a small program in a few lines which is incredibly simple (in our sense) and yet is of the highest complexity class (in the complexity theory sense).

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Poincaré … worked regularly from 10 to 12 in the morning and from 5 till 7 in the late afternoon. He found that working longer seldom achieved anything …

Poincaré made tremendous contributions to math and physics. His two-hour work sessions must have been sprints, working with an intensity that could not be sustained much longer.

I expect most of us would accomplish more if we worked harder when we worked, rested more, and cut out half-work.

Calculus may hold a world’s record for how far an idea can be pushed. Leibniz published the first article on calculus in 1684, an essay that was a mere 6 pages long. Newton and Leibniz would surely be astounded to learn that today’s introductory calculus textbook contains over 1,300 pages. A calculus textbook introduces two fundamental ideas, and the remaining 1,294 pages consists of examples, variations, and applications — all arising from following the consequences of just two fundamental idea.

Link to video. (If you know why the embedded video doesn’t appear in some RSS readers and how to fix it, please let me know.)

I’ve seen analogous problems with statistical methods. People will not necessarily adopt a new statistical method just because it is better. And if they do use it, they may use it wrongly, just like medical devices.

(“Better” in the previous paragraph is a loaded term. Statistical methods are evaluated by many criteria: power, robustness, bias, etc. When someone says his new method is better, he means better by the criteria he cares most about. But even when there is agreement on statistical criteria, a superior statistical method may be rejected for non-statistical reasons.)

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How do you decide whether something is worth learning ahead of time, or whether it is best to learn if and when you need it? This is a common dilemma, especially in technology. There’s no easy answer. You have to decide what is best in your circumstances. But here’s a suggestion: Learn real-time skills and bicycle skills in advance.

A real-time skill is something you need for live performance. If you’re going to speak French, you have to memorize a large number words before you need them in conversation. Looking up every word in a English-French dictionary as needed might work in the privacy of your study, but it would be infuriatingly slow in a face-to-face conversation. Some skills that we don’t think of as being real-time become real-time when you have to use them while interacting with other people.

More subtle than real-time skills are what I’m calling bicycle skills. Suppose you own a bicycle but haven’t learned to ride it. Each day you need to go to a store half a mile away. Each day you face the decision whether to walk or learn to ride the bicycle. It takes less time to just walk to the store than to learn to ride the bicycle and ride to the store. If you always do what is fastest that day, you’ll walk every day. I’m thinking of a bicycle skill as anything that doesn’t take too long to learn, quickly repays time invested, but will never happen without deliberate effort.

When you’re under pressure, you don’t learn bicycle skills. You don’t make long-term investments, even if the “long-term” is 30 minutes away. I’ll just walk, thank you.

What are bicycle skills you need to learn, things that would save time in the long run but haven’t been worthwhile in the short term?

So one of the very first things we teach people in economics and statistics and business and psychology is how, in a statistically valid way, do we eliminate the weirdos. How do we eliminate the outliers so we can find the line of best fit? Which is fantastic if I’m trying to find out how many Advil the average person should be taking — two. But if I’m interested in potential, if I’m interested in your potential, or for happiness or productivity or energy or creativity, what we’re doing is we’re creating the cult of the average with science. … If we study what is merely average, we will remain merely average.

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What bothers me most about tl;dr is the mindset it implies, scanning everything but reading nothing. I find myself slipping into that mode sometimes. Skimming is a vital skill, but it can become so habitual that it crowds out reflective reading.

When I realize everything I’m reading is short and new, when my patience has atrophied to the point that I get annoyed at long tweets, I’ll read something long and old to restore my concentration and perspective.

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I have two desks in my office — one is “analog” and one is “digital.” The analog desk has nothing but markers, pens, pencils, paper, index cards, and newspaper. Nothing electronic is allowed on that desk. That’s where most of my work is born … The digital desk has my laptop, my monitor, my scanner, and my drawing tablet. This is where I edit and publish my work.

From Steal Like an Artist.

The context of this quote is a discussion of how we think differently depending on the tools we use. I wrote something along these lines a while back: Create offline, analyze online.

Johnny Carson tried to be Jack Benny but ended up Johnny Carson. David Letterman tried to copy Johnny Carson but ended up David Letterman. And Conan O’Brien tried to be David Letterman but ended up Conan O’Brien. In O’Brien’s words, “It is our failure to become our perceived ideal that ultimately defines us and makes us unique.”

Stolen from Steal Like an Artist

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Although “you aren’t going to need it” is a good principle for writing code, the analogous principle “you aren’t going to need to understand it” doesn’t seem to hold. When I’ve try to avoid understanding things in detail, I end up needing to understand the details anyway and wished I’d done so sooner. This has been my experience in math research, software development, and project management.

Obviously you can’t understand everything about everything, and some things you clearly need to know well. In the fuzzy area in between, especially when I’ve said to myself “I hope I don’t have to dig into this,” I’ve often regretted postponing the dig.

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… when you study the evidence, it’s clear that you’re not likely to encounter real interesting opportunities in your life until after you’re really good at something.

If you avoid focus because you want to keep your options open, you’re likely accomplishing the opposite. Getting good is a prerequisite to encountering options worth pursuing.

From Closing your interests opens more interesting opportunities.

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Now listen to the rule of the last inch. The realm of the last inch. The job is almost finished, the goal almost attained, everything possible seems to have been achieved, every difficulty overcome — and yet the quality is just not there. The work needs more finish, perhaps further research. In that moment of weariness and self-satisfaction, the temptation is greatest to give up, not to strive for the peak of quality. That’s the realm of the last inch — here, the work is very, very complex, but it’s also particularly valuable because it’s done with the most perfect means. The rule of the last inch is simply this — not to leave it undone. And not to put it off — because otherwise your mind loses touch with that realm. And not to mind how much time you spend on it, because the aim is not to finish the job quickly, but to reach perfection.

Via Still I Am One

It can be hard to know when something deserves the kind of polish Solzhenitsyn talks about. Sometimes you’re in the realm of rapidly diminishing return and it’s time to move on. Other times, the finishing touches are everything.

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I find for myself that my first thought is never my best thought. My first thought is always someone else’s; it’s always what I’ve already heard about the subject, always the conventional wisdom. It’s only by concentrating, sticking to the question, being patient, letting all the parts of my mind come into play, that I arrive at an original idea. By giving my brain a chance to make associations, draw connections, take me by surprise. And often even that idea doesn’t turn out to be very good. I need time to think about it, too, to make mistakes and recognize them, to make false starts and correct them, to outlast my impulses, to defeat my desire to declare the job done and move on to the next thing.

Conventional wisdom summarizes the experience of many people. As a result, it’s often a good starting point. But like a blurred photo, it has gone through a sort of averaging process, loosing resolution along the way. It takes hard work to decide how, or even whether, conventional wisdom applies to your particular circumstances.

Bureaucracies are infuriating because they cannot deliberate on particulars the way Deresiewicz recommends. In order to scale up, they develop procedures that work well under common scenarios.

The context of Deresiewicz’s advice is a speech he gave at West Point. His audience will spend their careers in one of the largest and most bureaucratic organizations in the world. Deresiewicz is aware of this irony and gives advice for how to be a deep thinker while working within a bureaucracy.

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The desire to know it all is pernicious. The only way to accomplish it is to shrink your world. That may be OK for a while to focus your attention. The danger is that you can succeed and forget that you started by drawing arbitrary boundaries.

When I was very young, I thought that if I read every volume of the World Book Encyclopedia, I’d know everything. Of course I wouldn’t know everything, only what the editors of the encyclopedia chose to include.

If you want to learn English by first learning all the vocabulary, you’ll never speak English. Even if you learn every word in a particular dictionary, you still haven’t learned every word in the language.

Computer languages are orders of magnitude simpler than human languages, but they’re still too complex to learn exhaustively. If you want to learn every nuance of C++ before writing programs, you’ll never write a program. And if you think this is because C++ is a large language, it’s hardly possible to understand C exhaustively either if you take into account all the subtleties of how features are actually implemented on various platforms.

A common problem in math is how to select a finite sample from an infinite space. Maybe you have a function defined at an infinite number of points and you want to approximate it by sampling it at a carefully chosen finite set of points. This is a good metaphor for life.

Even when things are finite, it’s often very practical to think of them as being infinite. (See Infinite is easier than big.) Many aspects of life are effectively infinite.

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Evaluate people at their best or at their worst?

I think the avoidance of technology in these cases is really an avoidance of distraction. These same fellows would probably not keep a parrot in their office if it screeched every couple of minutes, regardless of their affinity for birds.

I believe he’s right. My intention was to write more broadly about how tools influence our thinking, but the examples I gave were only about one kind of influence: distraction.

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I find it interesting that Justice Souter would have such independence of mind that he chooses not to use much of the technology that our world takes for granted. He made it to the top of his profession and had a job for life, so he could afford to be eccentric. But he wasn’t born on the Supreme Court. I would like to know whether his low-tech work habits developed before or after his legal success.

I imagine most readers of this blog could more easily relate to Donald Knuth than David Souter. Knuth obviously doesn’t reject technology, but he is selective in how he uses it.

I had the opportunity to see Knuth speak while I was in college. Much to my surprise, his slides were handwritten. The author of TeX didn’t see the need to use TeX for his slides. While he cares about the fine details of how math looks in print, he apparently didn’t feel it was worth the effort to typeset his notes for an informal presentation.

In 1990 Knuth decided to stop using email.

I’d used email since about 1975, and it seems to me that 15 years of email is plenty for one lifetime. Email is a wonderful thing for people whose role in life is to be on top of things. But not for me; my role is to be on the bottom of things.

I believe I’ve read that Knuth does most of his work on a Linux box with no network connection. He also has a Mac for creating graphics and using the Internet. He has a secretary to handle his correspondence, including email.

If you’re reading legal briefs by sunlight, your thoughts will not be exactly the same as they would be if you were reading by fluorescent light. If you’re writing a presentation by hand, you’re not going to think the same way you would if you were pecking on a computer keyboard. And if you do use a computer, your thinking is subtlety different depending on what program you use. Technology affects the way you think. The effect is not uniformly better or worse, but it is certainly real.

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It is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.

Sometimes you don’t just have a hammer, you have an array of hammers. You have rock hammers, claw hammers, and sledge hammers, all in numerous sizes. You have a variety of wooden and rubber mallets too. You’ve even got a gavel. Because you have such an impressive collection of specialized hammers, you think you’re broad in your problem solving, but your basic instinct is still only to beat on things.

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Build a better mousetrap, and cats will beat a path to your door to picket.

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Kenny G’s philosophy is as shallow as his music.

I just play for myself, the way I want to play, and it comes out sounding like me.

Coltrane’s philosophy, like his music, is more ambitious.

Overall, I think the main thing a musician would like to do is give a picture to the listener of the many wonderful things he knows and senses in the universe. That’s what music is to me — it’s just another way of saying this is a big, wonderful universe we live in, that’s been given to us, and here’s an example of just how magnificent and encompassing it is. That’s what I would like to do. I think that’s one of the greatest things you can do in life, and we all try to do it in some way. The musician’s is through his music.

As Groothuis comments, Kenny G only spoke of expressing himself, while Coltrane “expressed a yearning to represent objective realities musically.”

The work you don’t do is a sort of negative space, like the shape formed by the empty space in a painting or the silence in a piece of music. It’s hard to appreciate what’s not there. It’s hard for a business to reward the unnecessary work that someone avoids doing.

Venkatesh Rao has a different take on what makes some people far more effective than others. In his post Thrust, Drag, and the 10x Effect, he says that the people who are 10x more productive are the those who allocate large, uninterrupted blocks of time to work on difficult creative tasks.

Rao’s observation would also help explain why super programmers do not earn super wages, and it ties into the idea of negative space. People who fracture their time putting out fires seem more productive, or at least more responsive, than the people who block out time to think. It’s harder to notice someone not being frantic. Thinkers don’t fare well in environments that reward activity more than accomplishment.

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Negative space in operating systems
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But how do you handicap intelligence? With interruptions. In Vonnegut’s story, those deemed too intelligent are required to wear a device in their ear that regularly interrupts their thoughts with a loud noise.

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