We all have
moments in our childhood that stick in our memory. For me, one of those
is the time I bought an old math textbook at a library book sale. I can
remember the crisp, bright sunny morning, the rows of books on tables
in the parking lot. Among them, I found a browned small book with brittle
pages that were covered with geometric diagrams, a math textbook from
the 1920's that was utterly different from anything I'd been given in
school. I loved all the pictures, the geometry so precisely laid out
and easily understood. Why weren't our textbooks like that, I wondered?
I guessed it was the "New Math" that my mother, a teacher at
the time, was always talking about. But how could math be new or old?
Didn't we learn that it existed sort of out of time? That the axioms
had always been there, invisible, hovering in the universe waiting to
be memorized by ages of children sitting at school desks?

"Come unto me," wrote Charles Fort in 'Wild Talents', "and
maybe I'll make you stylish. It is quite possible to touch up beliefs
that
are now considered dowdy, and restore them to fashionableness. I conceive
of nothing, in religion, science, or philosophy, that is more than the
proper thing to wear, for a while." For those of us in the New Math
generation, when axioms came off the hook and into our lives, the influence
of one man, Nicolas Bourbaki, reigned supreme. Gone were the illustrated
texts filled with gorgeously rendered geometric figures, replaced by
a ruthlessly logical set of assertions that built into an understanding,
a perception of math as the ultimate science. Beneath everything, at
the core of the universe, only numbers.

I was too young to ever know or later hear about the influence of Nicolas
Bourbaki, so 'The Artist and the Mathematician' came as something of
a revelation for me. Aczel's book is the complex story of how disenfranchised
French mathematicians managed to, for a not-so-brief moment, take over
the math world with a colossal student prank writ large. Aczel has a
lot of untangling to do, but he's up to the task, and his history offers
a fascinating glimpse at how our perceptions of the universe itself can
be influenced by the fashions of the moment. We like to think of academia
and the sciences as exempt from the tidal pull of momentary trends, but
that's not the case. 'The Artist and the Mathematician' delves into the
murky origins of what was for me universal truth, but for those who created
it a statement of artistic as much as academic expression. And a gotcha
aimed at those whom they felt had ignored them.

Two men were the opposing poles of the group who perpetrated the Bourbaki
hoax. Alexandre Grothendieck was the dirt-poor soon of anarchist parents
born shortly after the turn of the twentieth century. He was also a natural
mathematician, a genius who re-invented theories simply because he had
never gone to a school so informed as to teach said knowledge. André Weill
was the more serene son of wealthy parents, whose climactic moment at
college involved a prank and a huge pile of trash. Together, with others,
these men decided to change the way math would be taught, as it happened,
around the world. They joined with other disenfranchised French mathematicians
and vowed to strip away everything but the most rigorous proofs and axioms
and re-write math textbooks from the first assertion forwards, in the
manner of the Greeks. But rather than publish their work under the variety
of names, they decided to create a fake person, Nicolas Bourbaki. Bourbaki,
who did not exist, would go on to publish many papers and eventually
end up in the faces of American schoolchildren in the form of the New
Math.

Aczel has a very difficult task to undertake here, and he pulls it off
with some élan. With a large number of personalities to set up
and a complicated story, he unfolds a portrait of academia and math history
that is revelatory and every bit as messy as the process itself. Grothendieck
and Weil are fascinating characters, particularly the former. His involvement
in the Bourbaki group, which mingled art and math and social concerns,
lead him down the path to politics...and thence to the French Pyrenees,
where he still lives in a sort of self-imposed exile. His genius for
math did not, alas extend to other realms of thought. Aczel's story of
his heroic rise from the interment camps and his tragic fall when he
decided to lecture on politics instead of polynomials is frightening,
funny and poignant.

The history here is not simple and does not lend itself to an easy this-then-that
summary. Aczel does a great job of conveying the murky, tangled web of
academic progress, of the discipline of mathematics derailing itself
due to pride and boneheaded stubbornness. But we also see the birth of
Structuralism, the ties between abstract art and abstract math. And there's
always that prank looming like a huge teetering pile of trash in the
background. In later years, of course, the identity of Nicolas Bourbaki
became widely known, and the Bourbaki group eventually succumbed to internecine
battles as the allure of the New Math, experienced by three generations
of students gave way to Newer Math. A new fashion came into town and
left the old one looking dowdy. Behind the academic runways, madmen,
failures and geniuses mingled; single men transformed from one to the
other and a group of men, writing as a single voice transformed the world.
It is perhaps true that at the root of the universe, somewhere beneath
the atoms, the electrons, the quanta, the particle-of-the-week, there
lies nothing more complex than a single zero. But somebody's got to get
that zero dressed up and off to school. It's a simple beginning, but
humans have a way of making even the most simple concepts as complex
as humanity itself.