What do you mean by can’t?

You can’t subtract 4 from 3 (and stay inside the natural numbers, but you can inside the integers).

You can’t divide 3 by 4 (inside the ring of integers, but you can inside the rational numbers).

You can’t take the square root of a negative number (in the real numbers, but in the complex numbers you can, once you pick a branch of the square root function).

You can’t divide by zero (in the field of real numbers, but you may be able to do something that could informally be referred to as dividing by zero, depending on the context, by reformulating your statement, often in terms of limits).

When people say a thing cannot be done, they may mean it cannot be done in some assumed context. They may mean that the thing is difficult, and assume that the listener is sophisticated enough to interpret their answer as hyperbole. Maybe they mean that they don’t know how to do it and presume it can’t be done.

When you hear that something can’t be done, it’s worth pressing to find out in what sense it can’t be done.

Related post: How to differentiate a non-differentiable function

Read More

Optimism can be discouraging

Here’s an internal dialog I’ve had several times.

“What will happen when you’re done with this project?”

“I don’t know. Maybe not much. Maybe great things.”

“How great? What’s the best outcome you could reasonably expect?”

“Hmm …  Not that great. Maybe I should be doing something else.”

It’s a little paradoxical to think that asking an optimistic question — What’s the best thing that could happen? — could discourage us from continuing to work on a project, but it’s not too hard to see why this is so. As long as the outcome is unexamined, we can implicitly exaggerate the upside potential. When we look closer, reality may come shining through.

 Related posts:

How much does typing speed matter?
Wouldn’t trade places

Read More

Titles better than their books

What got you here won’t get you there. I’ve been thinking about that title lately. Some things that used to be the best use of my time no longer are.

I bought Marshall Goldsmith’s book by that title shortly after it came out in 2007. As much as I liked the title, I was disappointed by the content and didn’t finish it. I don’t remember much about it, only that it wasn’t what I expected. Maybe it’s a good book — I’ve heard people say they like it — but it wasn’t a good book for me at the time.


I’ve written before about The Medici Effect, a promising title that didn’t live up to expectations.


“Standardized Minds” is a great book title. I haven’t read the book; I just caught a glimpse of the cover somewhere. Maybe it lives up to its title, but the title says so much.

There is a book by Peter Sacks Standardized Minds: The High Price Of America’s Testing Culture And What We Can Do To Change It. Maybe that’s the book I saw, though it’s possible that someone else wrote a book by the same title. I can’t say whether I recommend the book or not since I haven’t read it, but I like the title.


I started to look for more examples of books that didn’t live up to their titles by browsing my bookshelves. But I quickly gave up on that when I realized these are exactly the kinds of books I get rid of.

What are some books with great titles but disappointing content?

Read More

Public reaction to Ebola

Ebola elicits two kinds of reactions in the US. Some think we are in imminent danger of an Ebola epidemic. Others think Ebola poses absolutely zero danger and that those who think otherwise are kooks.

Nothing can be discussed rationally. Even narrow scientific questions lead to emotionally-charged political arguments. Those who have a different opinion must be maligned.

The big question is whether the Ebola virus can spread by air. Experts say “probably not” but some are cautious. For example, Ebola researcher C. J. Peters says “We just don’t have the data to exclude it.” But people who know absolutely nothing about virology are firmly convinced one way or the other.



Read More

John Napier

Julian Havil has written a new book John Napier: Life, Logarithms, and Legacy.

I haven’t read more than the introduction yet — a review copy arrived just yesterday — but I imagine it’s good judging by who wrote it. Havil’s book Gamma is my favorite popular math book. (Maybe I should say “semi-popular.” Havil’s books have more mathematical substance than most popular books, but they’re still aimed at a wide audience. I think he strikes a nice balance.) His latest book is a scientific biography, a biography with an unusual number of equations and diagrams.

Napier is best known for his discovery of logarithms. (People debate endlessly whether mathematics is discovered or invented. Logarithms are so natural — pardon the pun — that I say they were discovered. I might describe other mathematical objects, such as Grothendieck’s schemes, as inventions.) He is also known for his work with spherical trigonometry, such as Napier’s mnemonic. Maybe Napier should be known for other things I won’t know about until I finish reading Havil’s book.

Read More

Wouldn’t trade places

Last week at the Heidelberg Laureate Forum, I was surrounded by the most successful researchers in math and computer science. The laureates had all won the Fields Medal, Abel Prize, Nevanlinna Prize, or Turing Award. Some had even won two of these awards.

I thought about my short academic career [1]. If I had been wildly successful, the most I could hope for would be to be one of these laureates. And yet I wouldn’t trade places with any of them. I’d rather do what I’m doing now than have an endowed chair at some university. Consulting suits me very well. I could see teaching again someday, maybe in semi-retirement, but I hope to never see another grant proposal.


[1] I either left academia once or twice, depending on whether you count my stint at MD Anderson as academic. I’d call my position there, and even the institution as a whole, quasi-academic. I did research and some teaching there, but I also did software development and project management. The institution is a hospital, a university, a business, and a state agency; it can be confusing to navigate.

Read More

Proof maintenance

Leslie Lamport coined the phrase “proof maintenance” to describe the process of producing variations of a proof over time.

It’s well known that software needs to be maintained; most of the work on a program occurs after it is “finished.” Proof maintenance is common as well, but it is usually very informal.

Proofs of any significant length have an implicit hierarchical structure of sub-proofs and sub-sub-proofs etc. Sub-proofs may be labeled as lemmas, but that’s usually the extent of the organization. Also, the requirements of a lemma may not be precisely stated, and the propositions used to prove the lemma may not be explicitly referenced. Lamport recommends making the hierarchical structure more formal and fine-grained, extending the sub-divisions of the proof down to propositions that take only two or three lines to prove. See his paper How to write a 21st century proof.

When proofs have this structure, you can see which parts of a proof need to be modified in order to produce a proof of a new related theorem. Software could help you identify these parts, just as software tools can show you the impact of changing one part of a large program.

Read More

Love locks

If you walk across the Seine in Paris on the Pont des Arts you’ll see thousands and thousands of love locks. I saw this morning that Heidelberg has its own modest collection of love locks on the Old Bridge across the Neckar.

love locks on Old Bridge across Neckar

These may be new. If they were here last year, I didn’t notice them.

There are several other points along the Old Bridge that have locks but nowhere are there very many.

love locks on Pont des Arts across Seine

Photo credit: Disdero via Wikimedia Commons

Read More

Steep learning curves you wish you’d climbed sooner

I asked on Twitter today “What steep learning curves do you wish you’d climbed sooner?” Here’s a summary of the replies:

  • R
  • Version control
  • Linear algebra
  • Advanced math
  • Bayesian statistics
  • Category theory
  • Foreign languages
  • How to not waste time
  • Women

IgorCarron‘s response didn’t fit into the list above. He said “I wish I had known that sensing all the way to machine learning is about approximating the identity” and gave a link to this post.

Read More

Radiation equipment

John Tukey said that the best thing about being a statistician is that you get to play in everyone’s backyard. This morning I got to play in IsoTherapeutics‘ backyard. The most photogenic thing on the tour they gave me was their box for working with highly radioactive material with robotic arms. (There was nothing hot inside at the time.)


Read More

Twitter news

Starting next week, @MedVocab will post two tweets a day, once in the morning and once in the afternoon (CDT).

I’ve stopped posting to @DailySymbol. It was a fun experiment, but it was time to wrap it up.

My most popular account, @CompSciFact, now has over 100,000 followers. It’s interesting how some Twitter accounts take off and some don’t. CompSciFact has done quite well but I’ve shut down several other accounts that never gained much of a following.

You can find a list of my accounts here with a very brief description of each. Some of the accounts are a little broader than the name implies.


Read More