June 2006

Letter to a calculus student

Dear Calculus Student,

Let me begin with a quotation from the great philosopher Bertrand Russell. He wrote, in Mysticism and Logic (1918): "Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."

Beauty is one of the last things you are likely to associate with the calculus. Power, yes. Utility, that too. Hopefully also ingenuity on the part of Netwon and Leibniz who invented the stuff. But not beauty. Most likely, you see the subject as a collection of techniques for solving problems to do with continuous change or the computation of areas and volumes. Those techniques are so different from anything you have previously encountered in mathematics, that it will take you every bit of effort and concentration simply to learn and follow the rules. Understanding those rules and knowing why they hold can come only later, if at all. Appreciation of the inner beauty of the subject comes later still. Again, if at all.

I fear, then, that at this stage in your career there is little chance that you will be able to truly see the beauty in the subject. Beauty -- true, deep beauty, not superficial gloss -- comes only with experience and familiarity. To see and appreciate true beauty in music we have to listen to a lot of music -- even better we learn to play an instrument. To see the deep underlying beauty in art we must first look at a great many paintings, and ideally try our own hands at putting paint onto canvas. It is only by consuming a great deal of wine -- over many years I should stress -- that we acquire the taste to discern a great wine. And it is only after we have watched many hours of football or baseball, or any other sport, that we can truly appreciate the great artistry of its master practitioners. Reading descriptions about the beauty in the activities or creations of experts can never do more than hint at what the writer is trying to convey.

My hope then is not that you will read my words and say, "Yes, I get it. Boy this guy Devlin is right. Calculus is beautiful. Awesome!" What I do hope is that I can at least convince you that I (and my fellow mathematicians) can see the great beauty in our subject (including calculus). And maybe one day, many years from now, if you continue to study and use mathematics, you will remember reading these words, and at that stage you will nod your head knowingly and think, "Yes, now I can see what he was getting at. Now I too can see the beauty."

The first step toward seeing the beauty in calculus -- or in any other part of mathematics -- is to go beyond the techniques and the symbolic manipulations and see the subject for what it is. Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, the true beauty of calculus can only be fully appreciated by digging deep enough.

The beauty of calculus is primarily one of ideas. And there is no more beautiful idea in calculus than the formula for the definition of the derivative: $f'(x) = \lim_{h \rightarrow 0}\frac{f(x+h) - f(x)}{h} \hspace{1in}(*)$ For this to make sense, it is important that $h$ is not equal to zero. For if you allow $h$ to be zero, then the quotient in the above formula becomes $\frac{f(x+0) - f(x)}{0} = \frac{f(x) - f(x)}{0} = \frac{0}{0}$ and $\frac{0}{0}$ is undefined. Yet, if you take any nonzero value of $h$, no matter how small, the quotient $\frac{f(x+h) - f(x)}{h}$ will not (in general) be the derivative.

So what exactly is $h$? The answer is, it's not a number, nor is it a symbol used to denote some unknown number. It's a variable.

What's that you say? "Isn't a variable just a symbol used to denote an unknown number?" The answer is "No." Sir Isaac Newton and Gottfried Leibniz, the two inventors of calculus, knew the difference, but as great a mind as the famous 18th Century philosopher and theologian (Bishop) George Berkeley seemed not to. In his tract The analyst: or a discourse addressed to an infidel mathematician, Berkeley argued that, although calculus led to true results, its foundations were insecure. He wrote of derivatives (which Newton called fluxions):

"And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?"
The "evanescent increments" he was referring to are those $h$'s in formula (*). Berkeley's problem -- and he was by no means alone -- was that he failed to see the subtlety in the formula. Like any great work of art, this formula simultaneously provides you with different ways of looking at the same thing. If you look at it just one way, you will miss its true meaning. It also asks you, nay like all great works of art it challenges you, to use your imagination -- to go beyond the experience of your senses and step into an idealized world created by the human mind.

The expression to the right of the equal sign in (*) represents the result of a process. Not an actual process that you can carry out step-by-step, but an idealized, abstract process, one that exists only in the mind. It's the process of computing the ratio $\frac{f(x+h) - f(x)}{h}$ for increasingly smaller nonzero values of $h$ and then identifying the unique number that those quotient values approach, in the sense that the difference between those quotients and that number can be made as small as you please by taking values of $h$ sufficiently small. (Part of the mathematical theory of the derivative is to decide when there is such a number, and to show that if it exists it is unique.) The reason you can't actually carry out this procedure is that it is infinite: it asks you to imagine taking smaller and smaller values of $h$ ad infinitum.

The subtlety that appears to have eluded Bishop Berkeley is that, although we initially think of $h$ as denoting smaller and smaller numbers, the "lim" term in formula (*) asks us to take a leap (and it's a massive one) to imagine not just calculating quotients infinitely many times, but regarding that entire process as a single entity. It's actually a breathtaking leap.

In Auguries of Innocence, the poet William Blake wrote:

To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour
That's what formula (*) asks you to do: to hold infinity in the palm of your hand. To see an infinite (and hence unending) process as a single, completed thing. Did any work of art, any other piece of human creativity, ever demand more of the observer? And to such enormous consequence for Humankind? If ever any painting, novel, poem, or statue can be thought of as having a beauty that goes beneath the surface, then the definition of the derivative may justly claim to have more beauty by far.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition.

Devlin's newest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs) was published recently by Thunder's Mouth Press.