Math Jokes

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The AMS has a 2005 paper “Foolproof: A Sampling of Mathematical Folk Humor” which is — delightfully — filled with math jokes. Excerpts:

Q: What’s sour, yellow, and equivalent to the Axiom of Choice?
A: Zorn’s Lemon.

Q: What is a topologist?
A: Someone who cannot distinguish between a doughnut and a coffee cup.

Theorem. All positive integers are interesting.
Proof. Assume the contrary. Then there is a lowest noninteresting positive integer. But, hey, that’s pretty interesting! A contradiction.

Q: How many light bulbs does it take to change a light bulb?
A: Just one, if it knows its Gödel number.

One day a farmer called up an engineer, a physicist, and a mathematician and asked them to fence in the largest possible area with the least amount of fence. The engineer made the fence in a circle and proclaimed that he had the most efficient design. The physicist made a long, straight line and proclaimed “We can assume the length is infinite…” and pointed out that fencing off half of the Earth was certainly a more efficient way to do it. The mathematician just laughed at them. He built a tiny fence around himself and said, “I declare myself to be on the outside.”

One day the Wiener family was scheduled to move into a new house. Mrs. Wiener, mindful of her husband’s propensity for forgetting, wrote the new address on a slip of paper and handed it to him. He scoffed, saying, “I wouldn’t forget such an important thing,” but he took the slip of paper and put it in his pocket. Later that same day at the university a colleague came by his office with an interesting problem. Wiener searched for a piece of paper and took the slip from his pocket to use to write some mathematical equations. When he finished, he crumpled up the slip of paper and threw it away. That evening, he remembered there was something about a new house but he couldn’t find the slip of paper with the address on it. Without any alternative course of action, he returned to his old home, where he spotted a little girl on the sidewalk. “Say, little girl,” he said, “Do you know where the Wieners live?” The girl replied, “That’s o.k., Daddy, Mommy sent me to get you.”

The full paper is here. For still more jokes, see this MathOverflow question and this page.

Hal Abelson on Math for Programmers

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Seibel: So that explains why the book is the way it is. But in general for programmers, how much math and what kinds of math are important for working programmers to know?

Abelson: I don’t even know anymore. We have these arguments at MIT all the time. People say, well, there’s math. Other people say, well, what they really need to know are algebraic structures so you understand abstract data types, how you think about axiomatizing them. And then people say what you really need to know is what a mathematical proof is so you can think rigorously. And I don’t know what to say. These arguments have been going on for thirty or forty years.

Seibel: And have you ever had a position?

Abelson: Well, the last time I had a position I tended to say that they ought to be able to like a lot of math. I’ve never been partial to the thing that says they need to be able to do proofs, but that’s just taste. And again, these days the main thing you try to understand is abstraction. Maybe that’s related to proofs. You need to be able to say look, here’s this thing and here’s the rules for how it works. And then I don’t need to look below that layer most of the time in order to get started. And there are people who really get hung up on that.

Hal is known for, among other things, co-authoring perhaps the most influential programming book of all time, Structure and Interpretation of Computer Programs. The entire interview is here.

It Probably Won A Prize

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Architect Sam Sloan coordinated a project in which employees … were able to select their own office furniture and plan office layout … Since both the Seattle and Los Angeles branches of the FAA were scheduled to move into new buildings at about the same time, the client for the project, the General Services Administration, agreed with architect Sloan’s proposal to involve employees in the design process in Seattle, while leaving the Los Angeles office as a control condition where traditional methods of space planning would be followed.

Several months following the move into the new buildings, surveys by the research team were made in Los Angeles and Seattle. The Seattle workers were more satisfied with their building and work areas than were the Los Angeles employees… [T]he Los Angeles building has been given repeated awards by the American Institute of Architects while the Seattle building received no recognition. One member of the AIA jury justified his denial of an award to the Seattle building on the basis of its ‘residential quality’ and ‘lack of discipline and control of the interiors,’ which was what the employees liked most about it. … Employees in both locations rated their satisfaction with their job performance before and after the move into the new building. There was no change in the Los Angeles office and a 7 percent improvement in rated job performance in the Seattle office.
—from The Design of Everyday Things

Why Dogs Bark At Night

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A reformed thief, telling of his success, put it this way, “I’m telling you, if I had a hundred dollars for every time I heard a dog owner tell their dog to ‘shut up and go lie down’ while I was right outside their window, I’d be a millionaire.”
<span id=”quote-attribute”>—from <em>The Design of Everyday Things</em></span>

Presumably, if he were any good, he’d already be a millionaire.

Response to BasicBookReader

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How can e-readers be improved? This is a response to Austin G. Walters BasicBookReader project.

Features For Authors

Better feedback for authors. When seeking feedback on a draft of something, authors want analytics. Where does someone stop reading? If someone puts down your book at a certain page, that’s a page that ought to be rewritten. You’ve bored them enough that they’ve decided to stop reading.

After all, an author — at least of fiction — is aiming at creating an addicting product. You want to write something that people can’t put down, the sort of book that people stay awake late into the night reading. Notice that this is what popular websites (Facebook, Reddit) have in common: they’re addicting.

I expect, though, for an author to get really useful feedback, they’re going to need quite a few “test” readers. Otherwise, the signal will get drowned in noise — maybe someone places a book down because the phone rings, etc. (Heh, noisy data caused by literal noise.) But with lots and lots of test readers, the law of large numbers should kick in, and you’ll be able to objectively identify boring passages.

Extending online analytics

The current generation of cutting edge analytics are being applied to the web. These could, in principle, be extended to books read on a screen. Take AB testing for instance. Maybe some readers get a version of the original book. Others get a version with the “stopping points” rewritten. Which version are people more likely to finish?

With enough such trials, a mediocre book can be pounded and shaped, like a blacksmith would iron, into something great.

I suspect there are other techniques that could similarly be lifted from website analytics.

Features for Readers

Export highlights as text

One of the more powerful features of Skim is the ability to export highlights from PDFs as text. I then save these annotations and can search through them via grep whenever I’m looking for a piece of evidence that I vaguely recall.

grep-through-notes

Autoscroll

I’ve spoken with people who say that they’re able to read more if the page automatically scrolls — this forces them to maintain attention of the book in front of them. I’ve been unable to duplicate this success, but it seems plausible.

Bookmarks

A useful feature in academic texts is the ability to save one’s current position in order to look up a citation or footnote. You might need to skip ahead 100 or so pages or whatever, so you want to be able to quickly jump back and forth.

Allowing a “split screen” where you can view two pages at once is similarly useful.

Analytics for readers

Are there any useful analytics for readers? It’s hard to think of any. You might correlate time of day with pages per minute, or something. These analytics could help identify when one’s at an intellectual peak. Words per minute might be similarly useful.

It’s hard to think of any other information someone might find useful. The Kindle has a feature that automatically highlights popular segments of a text purchased through them. It’s entertaining, but I’m not sure how useful.

Graphs of pages per day, total pages, and books read are, at the very least, inspiring.

Beauty

Intersecting that readability.com could build a business around taking ugly websites and making them beautiful, eh? This seems like a place where a newcomer could win over established players. When given text or ePub, the reader could lay it out is some pleasing form.

The styling could be tweaked in some form of AB testing, figuring out what’s most conducive to long reading sessions.

Saving book locations

If the app crashes or my computer does, I’d like to be able to restore exactly where I left off reading. It’s also useful to have a “farthest page read” marker in case something goes wrong. I find I’ll often press the wrong key on the Kindle and lose my place.

Fast dictionary look up

A means to quickly figure out what a word means would be useful.

Table of contents sidebar

The largest advantage print books still have over electronic ones is that it’s easy to figure out the structure of a print book at any time. You keep one finger stuck in the page you’re on and flip to the table of contents, or to the next chapter.

As far as a I know, no software has successfully replicated this yet. Skim contains a sidebar, like so:

skim-sidebar

Unfortunately, this feature breaks down when a PDF doesn’t ship with table of contents metadata.

 

Too Smart To Understand

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Here is a meme I would very much like to see die forever. I’ll be reading book reviews and come across people gushing about how great the book was — and they know it’s great because they couldn’t understand any of it.

In “Greatly Exaggerated” he is so fucking smart that I couldn’t even read the essay, because I am not, and never will be, his intellectual equal.
<span id=”quote-attribute”>—from a review of <em>A Supposedly Fun Thing I’ll Never Do Again: Essays and Arguments</em></span>

So many 5 star reviews describe a book as incomprehensible.

I want to shake them. You’ve been tricked! Intelligence isn’t about hiding your ideas in an impenetrable shroud! It’s about laying insights bare so that anyone can understand them. Great writing is a combination of interdisciplinary mastery and clarity.

Where do people get this notion of intelligence-as-obfuscation from, anyways?

What Is The Purpose of Science? Algorithm Discovery

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Consider the trial of Amanda Knox. What’s the purpose of the legal process here?

Well, let’s think about. Here’s how a trial works (at least on television): the prosecution and the defense get up in front of the jury. They present evidence — it could be DNA, surveillance videos, witness testimony, or even a tic-tac-toe playing chicken. Closing arguments follow. Then the jury deliberates and returns a verdict.

Now, the purpose of all this evidence is ostensibly to get at the truth. To figure out what it is that really happened. Did Amanda Knox kill Meredith Kercher? Or not?

We can visualize the jury, then, as a sort of machine. It takes in evidence and then applies that evidence to update two competing hypotheses: guilty or not guilty. At the end of the trial, the jury spits out its verdict.

jury-inference

Science works in the same manner.

What’s a hypothesis?

Okay, I haven’t been entirely honest. A jury doesn’t have just two hypotheses floating around in its collective head. There are a bunch of different possible explanations. When they consider the most likely explanation (“someone else did it”), they decide guilty or not guilty based on that. So in that box above with the G and NG for guilty or not guilty, it really ought to contain all possible explanations.

What are these explanations, really? They’re scenarios which could have produced the evidence. Amanda Knox murdering Meredith Kercher is one possible scenario. Rudy Guédé murdering her is another, or maybe Raffaele Sollecito did it. Or maybe it was aliens or a government conspiracy.

But what’s a scenario here, really? Consider the plight of physicists. They’re trying to uncover the underlying laws of the universe. They look at the world as it is — the evidence — and ask, “What underlying structure produced this?” Much like a paleontologist who carefully brushes away dirt to reveal a fossil.

Now, what’s a structure that produces data, evidence? An algorithm! Physicists are seeking not the laws of the universe, but the algorithm of the universe — what produced it all.

We can think, then, of science as the process of collecting evidence and then updating the likelihood of possible algorithms that might have produced it. Science is the process of algorithm discovery.

updating-hypotheses

Here the colored circles are algorithms (hypotheses) and their size is their likelihood.

Further Reading

Proofs In Math: What’s The Point?

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Tyler Cowen pointed me to an article on automated theorem proving. Namely, a computer “has solved the longstanding Erdős discrepancy problem!” This would not be such a big deal, except the 13 gigabyte proof is too complicated for anyone to understand.

So, of course, the Boeotians are in rare form as the clack of keyboards fills the metaphorical air in an attempt to sate the internet’s endless appetite for stupidity. Or: people are saying some really dumb stuff.

The discussions tend to anchor around trust. Can we trust the computer? What if there was a bug in the software? Gil Kalai, a math professor at the Hebrew University of Jerusalem, said, “I’m not concerned by the fact that no human mathematician can check this, because we can check it with other computer approaches.”

The mistake here is to assume that mathematics is about proof. It’s not. Proof is a means to producing insight, and a proof that no one can understand is close to worthless.

But don’t listen to me. Take it from Fields medalist Bill Thurston. He wrote a paper “On Proof and Progress in Mathematics.” It has profoundly influenced my ideas about what intellectual enterprise and mathematics particularly are about.

We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

…

In not too many years, I expect that we will have interactive computer programs that can help people compile significant chunks of formally complete and correct mathematics (based on a few perhaps shaky but at least explicit assumptions), and that they will become part of the standard mathematician’s working environment. However, we should recognize that the humanly understandable and humanly checkable proofs that we actually do are what is most important to us, and that they are quite different from formal proofs.

…

When I started working on foliations, I had the conception that what people wanted was to know the answers. I thought that what they sought was a collection of powerful proven theorems that might be applied to answer further mathematical questions. But that’s only one part of the story. More than the knowledge, people want personal understanding.

Or as Richard Hamming put it, “The purpose of computing is insight, not numbers.”

Herbert Simon’s Ant

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Here’s a metaphor that comes to me by way of Nobel laureate and Turing award recipient Herbert Simon.

Imagine watching an ant on the beach. Its path looks complicated. It zigs and zags to avoid rocks and twigs. Very reminiscent of complex behavior — what an intelligent ant!

Except an ant is just a simple machine. It wants to return to its nest, so it starts moving in a straight line. When it encounters an object, it zigs to avoid it. Repeat until the destination is reached.

Trying to simulate the path itself would be difficult, but simulating the ant is easy. It’s maybe a half-dozen rules.

The point of this parable is to illustrate the interaction between the environment and perceived complexity. Lots of complex looking things are really the result of the territory, the shape of the beach, and not the agent, in this case, an ant.

But, of course, with this metaphor, I’m not really talking about ants. I’m talking about people. How much of the complexity of human behavior is really the product of the environment?

Consider yesterday’s post. Zach Weinersmith wrote this about writer’s block:

If you can’t write, read more. In my experience, writer’s block is not a condition, but a result. Lots of people seem to think they can play video games 12 hours a day, then one day happen upon a great idea. It doesn’t work that way. You’ve got to put in time on input if you want good output.

Now, what’s so interesting about this? Well, it’s a lot like that ant. Humans can’t just sit and intuit something complicated — we have to go and engage with the complexity of our environment.

Zach Weinersmith On The Importance Of Reading Books

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Zach Weinersmith writes Saturday Morning Breakfast Cereal, my current favorite webcomic. I wanted to know: Where’s this creativity spring from?

Turns out, he reads. A lot.

I try to read 3-5 books a week in many different subjects. Whenever I stop that, I run out of ideas reaaallll fast.

When asked about his writing process:

Usually I just read a lot (at LEAST 4-6 hours a day) then sit myself in front of a blank google doc and try to write.

Or how to overcome writer’s block:

If you can’t write, read more. In my experience, writer’s block is not a condition, but a result. Lots of people seem to think they can play video games 12 hours a day, then one day happen upon a great idea. It doesn’t work that way. You’ve got to put in time on input if you want good output.

When someone asked how he manages to publish everyday:

I try very hard to read a lot and write a lot.

And when asked about his inspiration:

I really just try to read a lot, think a lot, then write for an hour a day. I also schedule my life a lot. So far that seems to be working all right.

On what to read:

It’s very liberating to read a book on a subject you think is boring. You might end up surprised.

I read pretttty much anything.

His broader philosophy here:

Most people tend to like what they like. I try to like things I dislike.

Why does he read so much?

Also, in general, if you’re an entertainer, you have precisely one job – be more interesting than the people you are entertaining. Otherwise why should they listen to you. That’s why if you want to improve at your work, self-cultivation is the best route.

Remember, you get paid to be more insightful than people who don’t write. That means you have to read more and think more.

He’s not the first to push the virtues of reading. Warren Buffet says his secret is “reading 500 pages a day.” Alan Kay reportedly attempts to read a book a day. Who knows how much Chomsky reads? Stephen King says, “If you want to be a writer, you must do two things above all others: read a lot and write a lot.” And so on.