Here we have an interesting article by mathematician Keith Devlin followed by a wit response from a reader. Food for thought.


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 -> 0}[f(x+h) - f(x)]/h

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

[f(x+0) - f(x)]/0 = [f(x) - f(x)]/0 = 0/0

and 0/0 is undefined. Yet, if you take any nonzero value of h, no matter how small, the quotient

[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

[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.


Hi Dr. Devlin,

My name is Murray Pendergrass. I am a math student at Western Washington University, a small public liberal arts college in the Pacific Northwest where I am pursuing a BS in Mathematics.

Sometime around 2006 you authored a post on Devlin's Angle titled "Letter to a calculus student" and I suppose someone in the math department at my school enjoyed it because it has been tacked to a bulletin board on the math floor for quite sometime. I would have only been going into the 8th grade when it was originally posted, with absolutely no idea that I would ever become interested in mathematics. I did take a calculus course my junior year of high school, but I don't think I could even briefly explain what a derivative was by the time the course was over (time well spent, obviously).

I must have first seen your article either my sophomore or junior year of college, 2014 most likely. I would have either been in precalculus or calculus I (differential calculus), and still completely unaware that I would end up declaring a math major. At that time I would have still been a member of the business school. I was probably waiting outside a professor's office for office hours when the title caught my eye,

" 'Letter to a calculus student' … Hmm, maybe I should read this."

However being the impatient person that I am, I believe I started in and thought "ok this is boring, I'll check the next page and see if it gets better,

"Nope, second page is boring too. Oh well."

And I have to admit, it was not until last night that I actually read the whole thing through for the first time.

But not long after that first initial and brief encounter with the letter my passion for mathematics truly began to develop and I realized that you can actually major in math without being a child prodigy (yes I actually thought this for quite sometime). It would have been shortly after this time, less than a year ago, that I realized I wanted to major in math. Since changing majors, very few hours have been spent not working on math.

I was studying at school late last evening when I decided to take a break and cruise up and down the hallway when for the second time in my life I noticed the letter tacked to the bulletin board. I must walk past it every single day but it was not until last night that it caught my eye again and I thought "I've seen this before! Oh wow I should give it a shot now that I am passionate about math."

In the very first sentence you open with a quote by Bertrand Russell, someone I have taken great interest in over the last year since mathematical logic has become a particular interest of mine. I immediately knew this was going to be a whole different experience reading this letter, and I was right.

What provoked me to feel the need to write you this letter was that I feel I am a precise example of the reader you are mentioning when you say,

"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.' "

Just as you predicted, the first time I made an attempt to read the letter "there was little chance I could see the true beauty in math", a statement so true that I could not only fail to see the beauty in math but I could not even read a letter about someone else promising me that even though I couldn't see the beauty, it was there.

It was quite a shock to me to read the letter last night and realize what a strange coincidental experience it was to randomly come across it a year after diving head first into the world of mathematics. It felt like a testament to myself of the progress I have made in math over the last year, a type of progress that cannot be explained or noticed through grades or high marks but by reading and truly relating to a mathematicians admiration for the beauty in math.

Before college I lived a bit of a bumpy life, it was a long and interesting road getting to where I am now. I will spare you the details as this letter has already turned out to be longer than I expected but I can truly say that finding math has been the best thing that has ever happened to me. In a lot of ways it has set me free.  I am very grateful to have the opportunity to study math at a university, to study something I am passionate about, and to reflect on how my relationship with math has evolved. I also must note that I hope I don't sound naive! I know I have only been doing math for a little over a year, which might sound like child's play to a Doctor of Philosophy in Mathematics. I am ecstatic that I have reached the point where I can appreciate mathematical beauty and I am also confident that math will continue to fascinate me and reveal its beauty for many years to come. Like most things, math is a journey not a destination.

Overall, I just felt the need to write to you because I thought you might enjoy knowing that even 9 years after you wrote it there are still students thinking for the first time:

"Yes, now I can see what he was getting at. Now I too can see the beauty".

Thank you,

Murray Pendergrass


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