leaping into math from a jewel-encrusted parapet: a review of David Berlinski’s A Tour of the Calculus

Maybe any class after such a long exile from formal studies would have seemed miraculous.  (Though I admit, I’m having a hard time getting into the stats textbook I’m trying to pre-read this summer – no wings to soar with here, no fragrant dripping leaves to stand under.  Statistics may actually be as boring as you would expect.)0679747885

Or maybe it’s because I fed an enthusiasm for mathematics last term by regularly reading from David Berlinski’s A Tour of the Calculus – who begins with his hopes that this writing will be effective in

. . . enabling anyone who has read what I have written to experience that hot flush that accompanies any act of understanding, saying as he or she puts down the book, Yes, that’s it, now I understand.

Berlinkski’s book kept playing the music whenever I lost the tune.  When I began to feel bogged down by the class itself – the teacher a pleasant fellow but given to explanation by definition, by bare assertion, repeated a little louder but in the same words – I re-opened Berlinkski’s book.

“I have written this book for men and women who wish to understand the calculus as an achievement in human thought. It will not make them mathematicians, but I suspect that what they want is simply a little more light shed on a dark subject. And that is something we all could use: a little more light.”

Because, the Class Itself was sometimes irritating, sometimes not particularly enlightening.  Exciting only because it was fast-moving.

My teacher, with his honest homely John-Boy face, his surprisingly sweet grin, had so much data to input into our minds.  And we were so often uncomprehendingly stubborn in our ignorance.

How steep a learning curve it was for everyone I don’t know, but I had jumped into the second semester of the class.  And I had forgotten so much.

Did you know 8! and 23! are not just happy numbers?  That there is actually more of  a difference between logarithm and algorithm than the scrambling of their first four letters?

During one of the first lessons when I had gasped aloud in recognition, grasping suddenly a connection, my teacher had embarrassingly reprimanded the entire class against making “aha-noises.”  I would have thought (though I had not meant to be audible) that an “aha-noise” would be the least offensive of noises to a teacher’s ear.  I vowed (muttering silently now) I would not ask any more questions, but figure things out on my own.

However after the first test (which, thank you very much, I totally aced), the teacher stopped sighing whenever I raised my hand.  In fact, I had stopped raising my hand, but my face has always been too transparent.

“MJ?” my teacher would stop, mid-sentence, when I furrowed my brow, “Do you have a question?”

“No, I just have to think about this some more.”

And then he would repeat himself verbatim in the way that so many of the mathish that I have known and loved will do.  To them, there is just one way to understand something.  And it is the way they understand it.

By the midterm, the teacher had taken me aside after class, “I have to ask you.  What exactly are you majoring in?”

“I’m not right now majoring in anything.  Just working through prerequisites before applying for a master’s in urban planning at PSU.”

“Well, your undergraduate then.”

“Well, I have a master’s in English.”

“And your undergraduate, too?  Whoa, I’ll have to watch my grammar.”

“I don’t do that kind of English,” which is the thing I say – not exactly true, but not false either – because I do adore misspellings and grammatical errors for the stories they tell, for the real-time human pronunciations and false etymologies I think these errors reveal.  At the same time, I love the correct grammatical forms for the roots of long-ago language they preserve.  But this thing I say is at any rate apparently effective at putting others back at their ease.

“You should . . . ,” hesitates my teacher, “I’m surprised you haven’t considered majoring in math.  You show marked ability.”

That was nice.


I must say I think another reader starting the study of the maths anew might find even more useful elucidation in a slimmer book by the same David Infinite Ascent: A Short History of Mathematics (Modern Library Chronicles)Berlinksi:  Infinite Ascent: a short history of mathematics.   I am reading this now – it is also enjoyable, seems more straightforward, less manic, less playful, and not nearly so purple.

But for myself, I like the purple and portentous playfulness of Tour of the Calculus very much:

Now imagine a gorgeous tower, its parapet jewel-encrusted, the creamy Perugian hills in the background, lacy clouds above.  And on this tower an Italian dandy, dressed in silks puffed at the wrists and at his thighs, is fingering a large and lavish rose and rufous stone, a fabulous ruby or garnet, something luscious and lustrous.  He dangles an elegant forearm over the parapet, holding the ruby in his upturned palm, and then slowly and with vast sensual deliberation rotates his wrist so that the precious stone, its cut facets catching the golden Tuscan light, slides from his polished palm and winking colored fire slips off into space.

I will never think of Galileo’s law of falling bodies the same.

What if I had had a teacher who taught math this way?

Reading this book I realized that what I’ve fought against in the way math was taught me is the colorlessness.  An unnecessary colorlessness.

What I loved about Berlinski’s Tour of the Calculus is the color and passion and personalities – the inescapable human history.  My reading all the more enriched as my copy is a used book once owned by Khanna(?) and much notated in mysterious and graceful-looking Hindi.

Throughout the term I’ve been asking myself – how would I teach math to a class of people like the self I was when I first failed at math?

Because it seems to me math is often taught falsely – force-fed to non-consenting students as if it were something fallen full-formed from heaven in the purest black and white, simplified to its barest essence to be easily digested – a kind of manna.  Which was also, come to think of it, incomprehensible to the children of Israel.  And they named it from their incomprehension – manna? which is Hebrew for – what is it?

What I loved about the Tour of the Calculus is the place it made for creative unknowing, a place for play in the ideas of mathematics, for the process of discovery that is what mathematics must be to living, breathing mathematicians:

The great Hindu and Arabic mathematicians of the Middle Ages took quite another tack.  Whatever incommensurable magnitudes might be, they treated such things as if they were really numbers – irrational numbers, the irrational a nice inadvertent touch signifying the madness loitering about the very notion – and learned many tricks by which such numbers might be manipulated.  In the twelfth century, for example, Bhaskara demonstrated correctly that √3 + √12 = 3√3, an achievement, I might add, utterly beyond the collective intellectual power, say, of the English department at Duke University.  (It is pleasant to imagine members of the department sitting together in a long lecture hall, Marxists to one side, deconstructionists to the other, abusing one another roundly as they grapple with the problem.)  But neither Bhaskara nor anyone else ever made clear what items such as √3 were. The symbols resisted, as symbols so often do, any attempt to invest them with meaning.  Sitting in their perfumed gardens, those thousand and one Arabian mathematicians carried out their calculations with a charming and insouciant assurance that all that gibberish actually made sense.

And this unknowingness, this poetic willingness to suspend disbelief for the purposes of momentary flight – I think that is what I loved most about studying math most recently.  That there may be just one right answer, but there isn’t just one way of getting there.












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