I met Kaplansky in 1999, when he was 82 and still actively
engaged in mathematical research. At that time, he was Director Emeritus of the
Mathematical Sciences Research Institute
(MSRI) in Berkeley, Calif., where I was spending the summer as Journalist in Residence.
Kaplansky spent much of his time then in the MSRI library,
poking into various nooks and crannies of mathematical history. Tidying up
loose ends and filling in gaps in the mathematical literature, he patiently
worked through mathematical arguments, proved theorems, and prepared papers for
publication. His remarkably wide-ranging efforts belied the oft-repeated notion
that mathematicians are most productive when they are young.
A distinguished mathematician who made major contributions
to algebra and other fields, Kaplansky was born in Toronto, Ontario, several
years after his parents had emigrated from Poland. In the beginning, his
parents thought that he was going to become a concert pianist. By the time he
was 5 years old, he was taking piano lessons. That lasted for about 11 years,
until he finally realized that he was never going to be a pianist of
distinction.
Nonetheless, Kaplansky loved playing the piano, and music remained
a lifelong hobby. "I sometimes say that God intended me to be the perfect
accompanist—the perfect rehearsal pianist might be a better way of saying
it," he said. "I play loud, I play in time, but I don't play very
well."
While in high school, Kaplansky started to play in dance
bands. During his graduate studies at Harvard, he was a member of a small combo
that performed in local night clubs. For a while, he hosted a regular radio
program, where he played imitations of popular artists of the day and commented
on their music.
A little later, when Kaplansky became a math instructor at
Harvard, one of his students was Tom Lehrer, later to become
famous for his witty ditties about science and math (see "Tom
Lehrer’s Derivative Ditties" for several examples).
In 1945, Kaplansky moved to the University of Chicago, where
he remained until 1984, when he retired, then became MSRI director.
Songs had always interested him, particularly those of the
period from 1920 to 1950. These songs tended to have a particular structure:
the form AABA, where the A theme is repeated, followed by a contrasting B
theme, then a return to the original A theme.
Early on, Kaplansky noticed that certain songs have a more
subtle, complex structure. This alternative form can be described as
AA'BAA'(B/2)A", where A is a four-bar phrase, A' and A" are variants,
and B is a contrasting eight-bar phrase. "I don't think anyone had noticed
that before," he said.
Kaplansky's discovery is noted in a book
about the American musical by the late Chicago film scholar Gerald
Mast.
Kaplansky argued that the second structure is really a
superior form for songs. To demonstrate his point, he once used it to turn an
unpromising source of thematic material—the first 14 decimal digits of pi—into
a passable tune. In essence, each note of the song's chorus corresponds to a
particular decimal digit.
When Chicago colleague Enid
Rieser heard the melody at Kaplansky's debut lecture on the subject in
1971, she was inspired to write lyrics for the chorus.
A SONG ABOUT PI
Through all the bygone ages,
Philosophers and sages
Have meditated on the circle's mysteries.
From Euclid to Pythagoras,
From Gauss to Anaxag'ras,
Their thoughts have filled the libr'ies bulging histories.
And yet there was elation
Throughout the whole Greek nation
When Archimedes did his mighty computation!
He said:
Philosophers and sages
Have meditated on the circle's mysteries.
From Euclid to Pythagoras,
From Gauss to Anaxag'ras,
Their thoughts have filled the libr'ies bulging histories.
And yet there was elation
Throughout the whole Greek nation
When Archimedes did his mighty computation!
He said:
CHORUS
3 1 41 Oh (5) my (9), here's (2) a (6) song (5) to (3) sing
(5) about (8,9) pi (7).
Not a sigma or mu but a well-known Greek letter too.
You can have your alphas and your great phi-bates, and omegas for a friend,
But that's just what a circle doesn't have—the beginning or an end.
3 1 4 1 5 9 is a ratio we don't define;
Two pi times radii gives circumf'rence you can rely;
If you square the radius times the pi, you will get the circle's space.
Here's my song about pi, fit for a mathematician's embrace.
Not a sigma or mu but a well-known Greek letter too.
You can have your alphas and your great phi-bates, and omegas for a friend,
But that's just what a circle doesn't have—the beginning or an end.
3 1 4 1 5 9 is a ratio we don't define;
Two pi times radii gives circumf'rence you can rely;
If you square the radius times the pi, you will get the circle's space.
Here's my song about pi, fit for a mathematician's embrace.
The chorus is in the key of C major, and the musical note C
corresponds to 1, D to 2, and so on, in the decimal digits of pi.
You can hear a performance of the song
by singer-songwriter Lucy Kaplansky
(Irving Kaplansky's daughter) on YouTube. A club headliner, recording artist,
and former psychologist, Lucy Kaplansky has her own distinctive style but
doesn't mind occasionally showcasing her father's old-fashioned tunemanship.
In 1993, Irving Kaplansky wrote new lyrics for the venerable
song "That's Entertainment!" to celebrate his enthusiasm for
mathematics. He dedicated the verses to Tom Lehrer.
THAT'S MATHEMATICS
The fun when two parallels meet
Or a group with an action discrete
Or the thrill when some decimals repeat,
That's mathematics.
A nova, incredibly bright,
Or the speed of a photon of light,
Andrew Wiles, proving Fermat was right,
That's mathematics.
The odds of a bet when you're rolling two dice,
The marvelous fact that four colors suffice,
Slick software setting a price,
And the square on the hypotenuse
Will bring us a lot o' news.
In genes a double helix we see
And we cheer when an algebra's free
And in fact life's a big PDE.
We'll be on the go
When we learn to grow with mathematics.
Or a group with an action discrete
Or the thrill when some decimals repeat,
That's mathematics.
A nova, incredibly bright,
Or the speed of a photon of light,
Andrew Wiles, proving Fermat was right,
That's mathematics.
The odds of a bet when you're rolling two dice,
The marvelous fact that four colors suffice,
Slick software setting a price,
And the square on the hypotenuse
Will bring us a lot o' news.
In genes a double helix we see
And we cheer when an algebra's free
And in fact life's a big PDE.
We'll be on the go
When we learn to grow with mathematics.
With Lagrange everyone of us swears
That all things are the sums of four squares,
Like as not, three will do but who cares.
That's mathematics.
Sporadic groups are the ultimate bricks,
Finding them took some devilish tricks,
Now we know--there are just 26.
That's mathematics.
The function of Riemann is looking just fine,
It may have its zeros on one special line.
This thought is yours and it's mine.
We may soon learn about it
But somehow I doubt it.
Don't waste time asking whether or why
A good theorem is worth a real try,
Go ahead--prove transcendence of pi;
Of science the queen
We're all of us keen on mathematics.
That all things are the sums of four squares,
Like as not, three will do but who cares.
That's mathematics.
Sporadic groups are the ultimate bricks,
Finding them took some devilish tricks,
Now we know--there are just 26.
That's mathematics.
The function of Riemann is looking just fine,
It may have its zeros on one special line.
This thought is yours and it's mine.
We may soon learn about it
But somehow I doubt it.
Don't waste time asking whether or why
A good theorem is worth a real try,
Go ahead--prove transcendence of pi;
Of science the queen
We're all of us keen on mathematics.
Original version
posted July 12, 1999
References:
Albers, D.J., G.L. Alexanderson, and C. Reid. 1990. Irving Kaplansky. In More Mathematical People: ContemporaryConversations. Academic Press.
Mast, G. 1987. Can't
Help Singin': The American Musical on Stage and Screen. Overlook
Press.
Kaplansky, I. 1992. The deep young man. Mathematical Intelligencer 14(No.
4):62.
An online video of Irving's Kaplansky's lecture on "Fun
with mathematics: Some thoughts from seven decades" is available at http://www.msri.org/realvideo/ln/msri/1999/misc/kaplansky/1/index.html.
In the interest of full disclosure, I should note that I
went to the same high school (Harbord Collegiate in Toronto) as Kaplansky and
also attended the University of Toronto, though my schooling occurred a
generation later.
