Quoted in New York magazine: The Chalk for Math Professors

The Chalk for Math Professors: Hagoromo Fulltouch, by Alex Ronan for the current issue of New York magazine, part of the Status Survey on various items for professionals.

The smooth texture flows so easily across the chalkboard like a fountain pen … One puts up mathematics on the chalkboard as if tracing out an idea in the air.

Meanwhile, I happened to be at a conference at the Research Institute for Mathematical Sciences in Kyoto last week, and I gave my talk using the chalkboard and the plentiful supply of Hagoromo chalk provided there.

Chalk1 chalk2









Related MathOverflow post.

Moved to a new server, new WordPress installation

I’ve recently moved my site to a new server and apologize for the brief disruption in service. Many thanks to Sam Coskey and Peter Krautzberger for all the help with the move, and for all their help over the years with getting me started on WordPress at Boolesrings.  They really opened my eyes to the possibilities.

I am currently still fixing various technical issues with the new installation, but please post a comment if you see any issue that needs attention. My intention is that all links to jdh.hamkins.org will still work as before.

– Joel David Hamkins



Contemplating large cardinals

Joel David Hamkins with cardinals


Modified image courtesy of Jerome Tauber.


Doubled, squared, cubed: a math game for kids or anyone

The number that must not be named

Doubled, squared, cubed is a great math game to play with kids or anyone interested in math.  It is a talking game, requiring no pieces or physical objects, played by a group of two or more people at almost any level of mathematical difficulty, while sitting, walking, boating or whatever.  We play it in our family (two kids, ages 7 and 11) when we are sitting around a table or when walking somewhere or when traveling by train.  I fondly recall playing the game with my brothers and sisters in my own childhood.

The game proceeds by first agreeing on an allowed number range.  For youngsters, perhaps one wants to allow the integers from 0 to 100, inclusive, but one will want to have negative numbers soon enough, and of course much more sophisticated play is possible. Eventually, one lessens or even abandons the restriction altogether. The first player offers a number, and each subsequent player in turn offers a mathematical operation, which is to be applied to the current number, which must not be mentioned explicitly.  The resulting number must be in the allowed number range.

The goal of the game is successfully to keep track of the number as it changes, and to offer an operation that makes sense with that number, while staying within the range of allowed numbers.  The point is to have some style, to offer an operation that proves that you know what the number is, without stating the number explicitly.  Perhaps your operation makes the new number a nice round number, or perhaps your operation can seldom be legally applied, and so applying it indicates that you know it is allowed to do so.  You must offer only operations that you yourself can compute, and which do not rely on hidden information (for example, “times the number of grapes I ate at breakfast” is not really permissible).

A losing move is one that doesn’t make sense or that results in a number outside the allowed range. In this case, the game can continue without that person, and the last person left wins.  It is not allowed to offer an operation that can always be applied, such as “times zero” or “minus itself“, or which can always be applied immediately after the previous operation, such as saying “times two” right after someone said, “cut in half”.  But in truth, the main point is to have some fun, rather than to win. Part of the game is surely simply to talk about new mathematical operations, and we usually take time out to discuss or explain any mathematical issue that may come up.  So this is an enjoyable way for the kids to encounter new mathematical ideas.

Let me simply illustrate a typical progression of the game, as it might be played in my family:

Hypatia: one

Barbara: doubled

Horatio: squared

Joel: cubed

Hypatia: plus 36

Barbara: square root

Horatio: divided by 5

Joel: times 50

Hypatia: minus 100

Barbara: times 6 billion

Horatio: plus 99

Joel: divided by 11

Hypatia: plus 1

Barbara: to the power of two

Horatio: minus 99

Joel: times itself 6 billion times

Hypatia: minus one

Barbara: divided by ten thousand

Horatio: plus 50

Joel: plus half of itself

Hypatia: plus 25

Barbara: minus 99

Horatio: cube root

Joel: next prime number above

Hypatia: ten’s complement

Barbara: second square number above

Horatio: reverse the digits

Joel: plus 3 more than six squared

Hypatia: minus 100

and so on!

As the kids get older, we gradually incorporate more sophisticated elements into the game, and take a little time out to explain to young Hypatia, for example, what it means to cube a number, to take a number to the power two, or what a prime number is.  I remember playing the game with my math-savvy siblings when I was a kid, and the running number was sometimes something like $\sqrt{29}$ or $2+3i$, and a correspondingly full range of numbers and operations. It is fine to let the youngest drop out after a while, and continue with the older kids with more sophisticated operations; the youngsters will rejoin in the next round.  In my childhood, we had a “challenge” rule, used when someone suspects that someone else doesn’t know the number: when challenged, the person should say the number; if incorrect, they are out, and otherwise the challenger is out.

Last weekend, I played the game with Horatio and Hypatia as we walked through Central Park to the Natural History Museum, and they conspired in whispering tones to mess me up, until finally I lost track of the number and they won…

A brief history of set theory…

A few months ago, Peter Doyle sent me a cryptic email, containing only the following photo and a subject line containing the title of this post.

A brief history of set theory, by François Dorais (photo by Peter Doyle)

I was mystified, until François Dorais subsequently explained that he had given a short presentation on recent progress in foundations for prospective graduate students at Dartmouth.

I’m glad to know that the upcoming generation will have an accurate historical perspective on these things!  🙂

Quoted in New Scientist magazine, June 2013

I was quoted briefly in Mathematicians think like machines for perfect proofs, New Scientist, by Jacob Aron, June 26, 2013.  (Actually, my quote there is a little out of context, as my remark there was referring only to research in set theory, where anyone would view the switch to another foundation as a distraction.)

The Myth of Just Do It

Barbara’s piece this week in the Philosopher’s Stone in New York Times:

OPINION   | June 09, 2013
The Myth of ‘Just Do It’

The idea that thinking interferes with doing is often taken for granted. But the realities at the highest levels of athletic and artistic performance are more complex.

→ go to The Myth of ‘Just Do It’

How well do you think this applies to expert action in mathematics? Go and post comments at the NYT…


Barbara was interviewed by Christie Nicholson at CBS News, Smart Planet, in the Pure Genius series:

CBS News, Smart Planet | June 28, 2013
Innovation / Pure Genius
Q&A: Barbara Montero, philosopher,
on the myth of ‘Just Do It’

Christie Nicholson interviews Barbara Gail Montero

There is a widely held view that thinking about one’s performance while performing ruins our ability to perform well. Many athletes say that once you’ve mastered the skill, one ought to let go of thinking and well, to quote Nike’s tag line, “Just do it.” Professional golfer Dave Hill said, “Golf is like sex. You can’t be thinking about the mechanics of the act while you are performing.”

I first heard that quote from Barbara Montero, associate professor of Philosophy at College of Staten Island and Graduate Center, City University of New York and author of a forthcoming book, Mind, Body, Movement: The Relevance of Consciousness to Expert Performance. (This is a working title, to be published by Oxford University Press.) Montero holds that thinking is not detrimental to successful expert performance. She describes the kind of thinking that might interfere but also the type of thinking that is actually necessary for an expert to improve upon his or her top performance.

SmartPlanet spoke with Montero, to hear more about the ‘just do it’ philosophy and why she feels it is misguided.

→ go to the interview

Quoted in Simons Foundation Science News Feature

I was quoted in The Global Math Commons, Simons Foundation, Science News Features article by Erica Klarreich, May 18, 2013, concerning MathOverflow.

In memory, Clark John Hamkins (1930- 2013)

Clark John Hamkins, Hoyerswort 2005

Clark John Hamkins at            Hoyerswort, 2005

My father, Clark John Hamkins (1930 – 2013) passed away May 11, 2013 at his home in Brunswick, Maine. He was a good man, witty, kind, intelligent, honest, hard-working, caring, curious, patient, philosophical, mathematical, practical, fair.  The world has just lost a great human being. He was a loving family man, married to my mother, Monica, for 55 years (their wedding anniversary was yesterday), with six children and thirteen grandchildren.

US Patent 3756058

US Patent 3756058

He was an engineer’s engineer, one who could take apart any machine and put it back together and tell you all about the ways in which the design was flawed or how it was clever.  I think of his work as a kind of meta-engineering, for he designed the machines for manufacturing a product, bending pipes and twisting wire, rather than the product itself.   He held a number of patents for his inventions, including some for his design of a manufacturing apparatus for winding semi-toroidal transformers. When I was a kid, he designed and built in his wood shop a hand-crank centrifugal honey-extractor, which we used to harvest the crop from our bee hives.  He taught all his kids their way around a wood shop, how to cut, saw, nail, drill, screw, sand, bore, file, buff, solder, join, plane, dowel, glue, measure, how to use all manner of hand tools and the jig saw, the band saw, the table saw, the mitre saw, the drill press, the belt sander. He was beyond serious, with a lathe in his shop.  “Use the right tool for the job,” I can still hear him say, and “measure twice, cut once; measure once, cut twice,” a warning to those who would rush their work.  He made all manner of toys for his children and then for his grandchildren.  He made cabinets, tools, furniture, items of all kinds, large and small, fine and plain.  Dad's dulcimerHe kept and used a wood shop his entire adult life, insisting even in his final days that he go down into it.  One of his final projects was to design and build a beautiful, melodious whale-motif dulcimer.

He was an artist, and as a young man produced oil paintings, which move me to this day.  Long ago, before having kids, he made architectural drawings for the family home he had planned, and it is clear in retrospect that he hadn’t planned at that time on having such a large family.  (The joke in our family is that Dad wanted four kids and Mom wanted two, and they both got their wish!)  Later, he would inevitably win our family games of pictionary, where one is given the task to draw the meaning of a word selected randomly from the dictionary. His drawings always started with a clear simple line, capturing the essence of the concept, which was then fleshed out in fuller artistic flair until his partner said the word.  I found an old math book of his, from his own school days, with a chapter on Polar Coordinates in which he had drawn a wonderful Eskimo, carrying a harpoon and fish, and an igloo.

He was a teacher at heart.  He explained the slide rule to me when I was young, and gave a lesson on logarithms and log tables and the accompanying explanation of linear interpolation.  He explained to me as a child what $x^2$ and $x^3$ meant, and then, with a twinkle in his eye, as I listened wide-eyed and dumbstruck, about what $x^{2.3}$ meant.  Some of my fondest young memories with him include viewings of the moon and planets through a telescope, as I shivered in my pajamas in the cool night air, pondering the craters and drinking hot cocoa. He would discuss the various historical astronomical theories, comparing Copernicus with Ptolemy and the role of Tycho Brahe.  He loved a good scientific controversy, and liked even more to figure things out himself.  He liked to put a scientific explanation in a historical context.

He was a programmer, programming computers in the earliest days. I brought his cast-off computer cards, punched paper tapes and so on into my third-grade classroom for show-and-tell.  I remember him poring over expansive flowcharts spread across the kitchen table in the evenings. He made sure that his kids were learning how to program.

He was a voracious reader, consuming books in science, philosophy, literature, history, archeology, biology, physics, mathematics. You name it; he read it.

He was a skeptic.

He was a rebel physicist, refusing to accept the conclusion of the Michelson-Morley experiment on relativistic length foreshortening and time contraction.  And he backed up his beliefs by writing an account of this part of physics, re-developing the theory from an alternative perspective of his own invention, involving a notion of time translation, which avoided the need for those paradoxical conclusions, but ended up deriving the same fundamental equations.

I will miss him.

My younger brother Jon’s eulogy

Joining the Doctoral Faculty in Philosophy

I am recently informed that I shall be joining the Doctoral Faculty of the Philosophy Program at the CUNY Graduate Center, in addition to my current appointments in Mathematics and in Computer Science.  This means I shall now be able to teach graduate courses in philosophy at the Graduate Center and also to supervise Ph.D. dissertations in philosophy there.  I am pleased to become a part of the GC Philosophy Program, which is so highly ranked in the area of mathematical logic, and I look forward to making a positive contribution to the program.


Interviewed by Richard Marshall at 3:AM Magazine

I was recently interviewed by Richard Marshall at 3:AM Magazine, which was a lot of fun. You can see that his piece starts out, however, rather over-the-top…


playing infinite chess

Joel David Hamkins interviewed by Richard Marshall.

Joel David Hamkins is a maths/logic hipster, melting the logic/maths hive mind with ideas that stalk the same wild territory as Frege, Tarski, Godel, Turing and Cantor. He thinks we all can go there and that we all should. He gives tips about the Moebius strip to six year olds and plays around with his sons homework. He has discovered all sorts of wonders involving supertasks, infinite-time Turing machines, black-hole computations, the mathematics of the uncountable, the lost melody phenomenon of infintary computability (which really should be the name of a band), set theory and multiverses, infinite utilitarianism, and infinite chess. He’s also thinking about whether we really have an absolute notion of the finite and doubts if any of this is brain melting, which is just a testimony to his modesty. He also thinks that although maths is open to all he thinks mathematicians could use more metaphors and silly terminology to get their ideas across better than they do. All in all, this is the grooviest of the hard core maths/logic groovsters. Bodacious!

→ continue to the rest of the interview

The interview is now available at Marshall’s site 3:16, along with the full collection of his interviews.

The use and value of mathoverflow

François Dorais has created a discussion on the MathOverflow discussion site, How is mathoverflow useful for me? in which he is soliciting response from MO users.  Here is what I wrote there:

The principal draw of mathoverflow for me is the unending supply of extremely interesting mathematics, an eternal fountain of fascinating questions and answers. The mathematics here is simply compelling.

I feel that mathoverflow has enlarged me as a mathematician. I have learned a huge amount here in the past few years, particularly concerning how my subject relates to other parts of mathematics. I’ve read some really great answers that opened up new perspectives for me. But just as importantly, I’ve learned a lot when coming up with my own answers. It often happens that someone asks a question in another part of mathematics that I can see at bottom has to do with how something I know about relates to their area, and so in order to answer, I must learn enough about this other subject in order to see the connection through. How fulfilling it is when a question that is originally opaque to me, because I hadn’t known enough about this other topic, becomes clear enough for me to have an answer. Meanwhile, mathoverflow has also helped me to solidify my knowledge of my own research area, often through the exercise of writing up a clear summary account of a familiar mathematical issue or by thinking about issues arising in a question concerning confusing or difficult aspects of a familiar tool or method.

Mathoverflow has also taught me a lot about good mathematical exposition, both by the example of other’s high quality writing and by the immediate feedback we all get on our posts. This feedback reveals what kind of mathematical explanation is valued by the general mathematical community, in a direct way that one does not usually get so well when writing a paper or giving a conference talk. This kind of knowledge has helped me to improve my mathematical writing in general.

So, thanks very much mathoverflow! I am grateful.


The Logic Bike

In 2004 I was a Mercator Gastprofessor at Universität MünHoratio on the Logic Bikester, Institut für mathematische Logik  und Grundlagenforschung, where I was involved with interesting mathematics, particularly with Ralf Schindler and Gunter Fuchs, who is now at CUNY.

At that time, I had bought a bicycle, and celebrated Münster’s incredible bicycle culture, a city where the number of registered bicycles significantly exceeds the number of inhabitants.  I have long thought that Münster gets something fundamentally right about how to live in a city with bicycles, and the rest of the world should take note.  I am pleased to say that in recent years, New York City is becoming far more bicycle-friendly, although we don’t hold a candle to Münster.

At the end of my position, I donated the bicycle to the Logic Institute, where it has now become known as the Logic Bike, and where I have recently learned that over the years it has now been ridden by a large number of prominent set theorists; it must be one of the few bicycles in the world to have its own web page!

Quoted in Science News

I was quoted briefly in Infinite Wisdom: A new approach to one of mathematics’ most notorious problems, Science News, by Erica Klarrreich, August 30, 2003, in an article about Woodin’s attempted solution of the continuum hypothesis.