“I got the easy ones wrong”

This morning my daughter told me that she did well on a spelling test, but she got the easiest words wrong. Of course that’s not exactly true. The words that are hardest for her to spell are the ones she in fact did not spell correctly. She probably meant that she missed the words she felt should have been easy. Maybe they were short words. Children can be intimidated by long words, even though long words tend to be more regular and thus easier to spell.

Our perceptions of what is easy are often upside-down. We feel that some things should be easy even though our experience tells us otherwise.

Sometimes the trickiest parts of a subject come first, but we think that because they come first they should be easy. For example, force-body diagrams come at the beginning of an introductory physics class, but they can be hard to get right. Newton didn’t always get them right. More advanced physics, say celestial mechanics, is in some ways easier, or at least less error-prone.

“Elementary” and “easy” are not the same. Sometimes they’re opposites. Getting off the ground, so to speak, may be a lot harder than flying.

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Remove noise, remove signal

Whenever you remove noise, you also remove at least some signal. Ideally you can remove a large portion of the noise and a small portion of the signal, but there’s always a trade-off between the two. Averaging things makes them more average.

Statistics has the related idea of bias-variance trade-off. An unfiltered signal has low bias but high variance. Filtering reduces the variance but introduces bias.

If you have a crackly recording, you want to remove the crackling and leave the music. If you do it well, you can remove most of the crackling effect and reveal the music, but the music signal will be slightly diminished. If you filter too aggressively, you’ll get rid of more noise, but create a dull version of the music. In the extreme, you get a single hum that’s the average of the entire recording.

This is a metaphor for life. If you only value your own opinion, you’re an idiot in the oldest sense of the word, someone in his or her own world. Your work may have a strong signal, but it also has a lot of noise. Getting even one outside opinion greatly cuts down on the noise. But it also cuts down on the signal to some extent. If you get too many opinions, the noise may be gone and the signal with it. Trying to please too many people leads to work that is offensively bland.

Related post: The cult of average

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The difference between machines and tools

From “The Inheritance of Tools” by Scott Russell Sanders:

I had botched a great many pieces of wood before I mastered the right angle with a saw, botched even more before I learned to miter a joint. The knowledge of these things resides in my hands and eyes and the webwork of muscles, not in the tools. There are machines for sale—powered miter boxes and radial arm saws, for instance—that will enable any casual soul to cut proper angles in boards. The skill is invested in the gadget instead of the person who uses it, and this is what distinguishes a machine from a tool.

Related post: Software exoskeletons

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The Jericho-Masada approach to mathematics

Pierre Cartier describing Alexander Grothendieck’s approach to mathematics:

Grothendieck’s favorite method is not unlike Joshua’s method for conquering Jericho. The thing was to patiently encircle the solid walls without actually doing anything: at a certain point, the walls fall flat without a fight. This was also the method used by the Romans when they conquered the natural desert fortress Masada, the last stronghold of the Jewish revolt, after spending months patiently building a ramp. Grothendieck was convinced that if one has a sufficiently unifying vision of mathematics, if one can sufficiently penetrate the essence of mathematics and the strategies of its concepts, then particular problems are nothing but a test; they do not need to be solved for their own sake.

Source

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Playful and purposeful, pure and applied

From Edwin Land, inventor of the Polaroid camera:

… 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.

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Slabs of time

From Some Remarks: Essays and Other Writing by Neal Stephenson:

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

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Efficiency could land you in jail

A German postman recently faced criminal charges for coming up with using more efficient routes to deliver the mail. His supervisor had informally tolerated his initiative, but could not officially sanction it since his violated procedure. He got into trouble when his suspicious peers reported him. Fortunately he was not fired, only reprimanded for not following rules.

The source I saw (thanks Tim) doesn’t give much more detail. Maybe the charges against him were not as ridiculous as they seem. Maybe he violated reasonable safety regulations, for example. But I find it quite plausible that he simply got into trouble for using his brain. Even if the incident were completely made up, it would make a good story. It’s symbolic of bureaucratic punishment of efficiency. It’s easy to find analogous examples.

If this mailman were working for a small courier company, the company might reward him and ask him for recommendations for improving other routes. Of course a small company might also fire him. But large organizations, public and private, are more likely to punish initiative. And I understand why: large organizations have to maintain consistency. The clever postman must be reprimanded for the good of the system, but it’s maddening when you’re the postman.

Related posts:

Traveling salesman art
Hierarchical exfoliation

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Beethoven, Beatles, and Beyoncé: more on the Lindy effect

This post is a set of footnotes to my previous post on the Lindy effect. This effect says that creative artifacts have lifetimes that follow a power law distribution, and hence the things that have been around the longest have the longest expected future.

Works of art

The previous post looked at technologies, but the Lindy effect would apply, for example, to books, music, or movies. This suggests the future will be something like a mirror of the present. People have listened to Beethoven for two centuries, the Beatles for about four decades, and Beyoncé for about a decade. So we might expect Beyoncé to fade into obscurity a decade from now, the Beatles four decades from now, and Beethoven a couple centuries from now.

Disclaimer

Lindy effect estimates are crude, only considering current survival time and no other information. And they’re probability statements. They shouldn’t be taken too seriously, but they’re still interesting.

Programming languages

Yesterday was the 25th birthday of the Perl programming language. The Go language was announced three years ago. The Lindy effect suggests there’s a good chance Perl will be around in 2037 and that Go will not. This goes against your intuition if you compare languages to mechanical or living things. If you look at a 25 year-old car and a 3 year-old car, you expect the latter to be around longer. The same is true for a 25 year-old accountant and a 3 year-old toddler.

Life expectancy

Someone commented on the original post that for a British female, life expectancy is 81 years at birth, 82 years at age 20, and 85 years at age 65. Your life expectancy goes up as you age. But your expected additional years of life does not. By contrast, imagine a pop song that has a life expectancy of 1 year when it comes out. If it’s still popular a year later, we could expect it to be popular for another couple years. And if people are still listening to it 30 years after it came out, we might expect it to have another 30 years of popularity.

Mathematical details

In my original post I looked at a simplified version of the Pareto density:

f(t) = c/tc+1

starting at t = 1. The more general Pareto density is

f(t) = cac/tc+1

and starts at t = a. This says that if a random variable X has a Pareto distribution with exponent c and starting time a, then the conditional distribution on X given that X is at least b is another Pareto distribution, now with the same exponent but starting time b. The expected value of X a priori is ac/(c-1), but conditional on having survived to time b, the expected value is now bc/(c-1). That is, the expected value has gone up in proportion to the ratio of starting times, b/a.

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Pure possibility

Peter Lawler wrote a blog post yesterday commenting on a quote from Walter Percy’s novel The Last Gentleman:

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.

Related posts:

Small, local, old, and particular
Don’t try to be God, try to be Shakespeare

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Nobody's going to steal your idea

When I was working on my dissertation, I thought someone might scoop my research and I’d have to start over. Looking back, that was ridiculous. For one thing, my research was too arcane for many others to care about. And even if someone had proven one of my theorems, there would still be something original in my work.

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.

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Maybe you don't need to

One life-lesson from math is that sometimes you can solve a problem without doing what the problem at first seems to require. I’ll give an elementary example and a more advanced example.

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?

Related posts:

The power of parity
How has math changed your view of the world?
Maybe you only need it because you have it

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Being useful

Chuck Bearden posted this quote from Steve Holmes on his blog the other day:

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.

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Avoiding difficult problems

The day after President Kennedy challenged America to land a man on the moon,

… 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.

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How long can you think about a problem?

The main difficulty I’ve seen in tutoring math is that many students panic if they don’t see what to do within five seconds of reading a problem, maybe two seconds for some. A good high school math student may be able to stare at a problem for fifteen seconds without panicking. I suppose students have been trained implicitly to expect to see the next step immediately. Years of rote drill will do that to you.

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.

Related posts:

Demonstrating persistence
Take chances, make mistakes, and get messy

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Pushing an idea

From The 5 Elements of Effective Thinking:

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.

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