These posts have been the most popular this year:
- Hacking debt (about not enough time spent hacking, not about financial debt)
- Subscribing to a Twitter account via RSS
- The weight of code
- Extreme syntax
- Which Unicode characters can you depend on?
These posts have been the most popular this year:
I’ve started a new blog called Symbolism. Each post will take a symbol and say a few words about it.
I’ve also started a Twitter account to go with the new blog, @DailySymbol.
The second chapter of Out of the Silent Planet opens by describing a room as “a strange mixture of luxury and squalor.” It gives examples such as the room as having fine armchairs but no carpets or curtains, strewn with debris. The room has “empty champagne-bottles” and “teacups a quarter full of tea and cigarette-ends.” The room belongs to a scientist and an investor who have the resources to live in beauty and comfort, but instead have a few luxurious items in a pigsty. The scene is a metaphor for science and business detached from humane uses, one of the themes of the book.
“Much as beavers, who as a species hate the sound of running water, plaster a creek with mud and sticks until alas that cursed tinkle stops, so do category theorists derive elaborate and obscure definitions in an attempt to capture a concept that to most of us seemed perfectly clear before they got to it. But at least sometimes this works admirably …”
When does a rational portion of a circle have a rational cosine?
If r is a rational number, cos(2πr) rational if and only if the denominator of r is 1, 2, 3, 4, or 6.
This means that the special values of cosine you learn in a trig class, with a simple argument and simple value, are the only ones possible. (Here simple argument means an angle with an integer number of degrees and simple value means a rational number.) And if you see a result such as cos(π/7) = 837/929, you know it can’t be exactly correct, though in this case it’s very close.
“Progress in science depends on new techniques, new discoveries, and new ideas, probably in that order.” — Sidney Brenner
I’m not sure whether I agree with Brenner’s quote, but I find it interesting. You could argue that techniques are most important because they have the most leverage. A new technique may lead to many new discoveries and new ideas.
The other day Mark Jones sent me this photo of Space Shuttle Endeavour.
Sometimes a quick answer to a question is priceless. It can even be valuable to know that you could get a quick answer to a question, even if you never ask. For example, if your company is considering doing something new, knowing that there’s someone to help could make the difference in the decision to go forward.
Next year I’ll be offering this sort of micro-consulting and mentoring. For a monthly retainer, I will be available to answer questions and give advice. This would be for questions I could answer on the spot or with minimal research; anything more involved would have to be a separate consulting project. You would be guaranteed my availability for a certain amount of time per month and a quick turn-round on correspondence. (My response might be “I don’t know,” but I’d get back to you promptly.)
I’ve done some of this kind of consulting, and clients have found it very valuable. I’d like to do more of this next year as a way to fill some of the interstitial time between larger projects. I also expect it will lead to larger projects, e.g. “We like your idea of what we should do. Could you do it for us?”
If this sounds interesting to you, please contact me.
Seth Godin tells the following joke in The Icarus Deception:
Heisenberg looks around the bar and says, “Because there are three of us and because this is a bar, it must be a joke. But the question remains, is it funny or not?”
And Gödel thinks for a moment and says, “Well, because we’re inside the joke, we can’t tell whether it is funny. We’d have to be outside looking at it.”
And Chomsky looks at both of them and says, “Of course it’s funny. You’re just telling it wrong.”
Yesterday Jack Kennedy told me about a graph he’d made as part of a project he’s working on and I asked if I could post it here.
The Galois group of a quadratic polynomial x2 + bx + c is either A2 or S2. If b2 – 4c is a perfect square, the polynomial has rational roots and the Galois group is the trivial group A2. Otherwise there are distinct irrational roots and the Galois group is the two-element group S2.
As b and c range over integers, color a pixel yellow if the group is A2 and black otherwise. This produces the image below.
Note that what appear to be the crossed lines y = ±x intersecting at 0 are actually the lines y = ±(x+1) intersecting at (-1,0).
Today’s a prime day. Whether you write the date in American (MMDDYY), European (DDMMYY), or ISO (YYYYMMDD) format, you get a prime. That is, 112913 and 291113 and 20131129 are all prime numbers.
We’ll call a date an American prime date if MMDDYY is prime, a European prime date if DDMMYY is prime, and an ISO prime date if YYYYMMDD is prime. (Single-digit days and months are padded with a zero.) If a date is prime by all three criteria, we’ll call it an international prime date. Today is an international prime date, and there won’t be another one until August 11, 2019.
If a date is an American prime date and a European prime date, we’ll call it a transatlantic prime date. After today, the next transatlantic prime dates are December 4 and 13 this year. There will be no transatlantic prime dates in 2014, 2015, and 2016 since these dates correspond to numbers that are divisible by either 2 or 5. The first transatlantic prime date of 2017 will be January 16.
When I was in grad school, I had a course in Banach spaces with Haskell Rosenthal. One day he said “We got the definition wrong.” It took a while to understand what he meant.
There’s nothing logically inconsistent about the definition of Banach spaces. What I believe he meant is that the definition is too broad to permit nice classification theorems.
I had intended to specialize in functional analysis in grad school, but my impression after taking that course was that researchers in the field, at least locally, were only interested in questions of the form “Does every Banach space have the property …” In my mind, this translated to “Can you construct a space so pathological that it lacks a property enjoyed by every space that anyone cares about?” This was not for me.
I ended up studying differential equations. I found it more interesting to use Banach spaces to prove theorems about PDEs than to study them for their own sake. From my perspective there was nothing wrong with their definition.
Related post: Remembering Ted Odell
The phrase necessary but not sufficient refers to something that you’ve got to have, but it isn’t enough. For example, being divisible by 2 is a necessary but not sufficient condition for being divisible by 6. Odd numbers are not divisible by 6, so being even is necessary. But evenness is not sufficient because, for example, 8 is an even number not divisible by 6.
Wrongly believing that nice theoretical properties are sufficient for a good model is known as a reification error. I don’t know of a name for wrongly believing theoretical properties are necessary. Believing theoretical criteria are sufficient when they’re not is a sophomoric error. Believing theoretical criteria are necessary when they’re not is a more subtle error.
Maybe it would be helpful to use a phrase like “beneficial but not sufficient” to indicate that some property increases our confidence in a model, though it may not be necessary.
Many people have drawn Venn diagrams to locate machine learning and related ideas in the intellectual landscape. Drew Conway’s diagram may have been the first. It has at least been frequently referenced.
By this classification, Hector Cuesta’s new book Practical Data Anaysis is located toward the “hacking skills” corner of the diagram. No single book can cover everything, and this one emphasizes practical software knowledge more than mathematical theory or details of a particular problem domain.
The biggest strength of the book may be that it brings together in one place information on tools that are used together but whose documentation is scattered. The book is great source for sample code. The source code is available on GitHub, though it’s more understandable in the context of the book.
Much of the book uses Python and related modules and tools including:
It also uses D3.js (with JSON, CSS, HTML, …), MongoDB (with MapReduce, Mongo Shell, PyMongo, …), and miscellaneous other tools and APIs.
There’s a lot of material here in 360 pages, making it a useful reference.
When I was in college, my advisor and I published a paper in a journal called “Applicable Analysis.” At the time, I thought that was a good name for a journal. It suggested research that was toward the applied end of the spectrum but not tied to a specific application.
Now when I hear “applicable analysis” I wonder what inapplicable analysis or inapplicable math in general would be. I’d hesitate to call any area of math inapplicable. Certainly some areas of math are applied more frequently and more directly than others, but I’ve been repeatedly surprised by useful applications of areas of math not traditionally classified as “applied.”