A Tour of the Calculus

by David Berlinski

Were it not for the calculus, mathematicians would have no way to describe the acceleration of a motorcycle or the effect of gravity on thrown balls and distant planets, or to prove that a man could cross a room and eventually touch the opposite wall. Just how calculus makes these things possible and in doing so finds a correspondence between real numbers and the real world is the subject of this dazzling book by a writer of extraordinary clarity and stylistic brio. Even as he initiates us into the mysteries of real numbers, functions, and limits, Berlinski explores the furthest implications of his subject, revealing how the calculus reconciles the precision of numbers with the fluidity of the changing universe.



"An odd and tantalizing book by a writer who takes immense pleasure in this great mathematical tool, and tries to create it in others."--New York Times Book Review

Genres: Mathematics, Science, Nonfiction, History, Philosophy, Calculus, Reference

331 pages, Paperback

First published January 1, 1995

About the author

David Berlinski is a senior fellow in the Discovery Institute’s Center for Science and Culture.

Recent articles by Berlinski have been prominently featured in Commentary, Forbes ASAP, and the Boston Review. Two of his articles, “On the Origins of the Mind” (November 2004) and “What Brings a World into Being” (March 2001), have been anthologized in The Best American Science Writing 2005, edited by Alan Lightman (Harper Perennial), and The Best American Science Writing 2002, edited by Jesse Cohen, respectively.

Berlinski received his Ph.D. in philosophy from Princeton University and was later a postdoctoral fellow in mathematics and molecular biology at Columbia University. He has authored works on systems analysis, differential topology, theoretical biology, analytic philosophy, and the philosophy of mathematics, as well as three novels. He has also taught philosophy, mathematics and English at Stanford, Rutgers, the City University of New York and the Université de Paris. In addition, he has held research fellowships at the International Institute for Applied Systems Analysis in Austria and the Institut des Hautes Études Scientifiques. He lives in Paris.

Reviews

[Tom · Rating 1 out of 5]

unreadably frothy. author may have already died from severe case of terminal cuteness.

[James Watson · Rating 1 out of 5]

In writing "A Tour of the Calculus", there are three things that David Berlinski would like you to know, in order:

1. David Berlinski has read more books than you.
2. David Berlinski is well regarded among mathematicians.
3. The motivations and concepts that support calculus as a foundational achievement of modern thought.

Let me offer this praise: Berlinski faithfully and artfully expresses what almost every math teacher misses; The motivation for creating calculus was to understand a world of varying forces, flows, and influences. Questions like "Why do the planets move like they do?", "How fast is a cannonball moving in mid-flight?", and "How much concrete will I need to dam this river?" were not really answerable before calculus. Calculus gives us a way to quantify the nuances of natural phenomenon that was never available before in history. This is exciting, and the author communicates this excitement in the mirror of his own excitement.

This enthusiasm is quickly buried, however, in a mountain of cheeky literary references and irrelevant, boring asides. The author tries to bind all these together with metaphors, which are stretched to ludicrous lengths and ultimately break. These overlong side trips are neither explanatory nor entertaining.

Don't get me wrong. I do not wish mathematics to remain dry and unattainable. Ostensibly, neither does David Berlinski. I so wanted to be able to point to this book and tell others, "See, this is why calculus exists. This is why math is central to every aspect of modern life!" Alas, the book fails in this regard. It fails also in its stated mission to illustrate the formal, stuffy proofs that underpin calculus in a down-to-earth way. Every time such an illustration is attempted, it turns into a bizarre shaggy dog story that is a blend of historical fiction and motorcycle rides.

I cannot recommend this book; either as a calculus tutorial, or as entertainment.

[Rich · Rating 1 out of 5]

Florid, ostentatious, and inexcusably pretentious.
Berlinski's writing does more to obfuscate than clarify, and wearies rather than enlightens the reader. Understanding mathematics requires selectivity and focus. Berlinski demonstrates that writing about it requires neither. Perhaps the most worthless, overwrought book I've ever suffered through.

[Jamie Smith · Rating 3 out of 5]

Okay, I will give David Berlinski credit for taking a different approach to calculus, an attempt to make it less intimidating to the uninitiated or those returning for a refresher. This book is definitely different, and at times it does do a good job explaining various functions and their applications, but its style is ultimately more distracting than useful.

Partway through I started keeping track of some of the book’s oddities. First, for a book that attempts to be an easy introduction, it is sidetracked by Berlinski’s repeated use of odd and stilted vocabulary, words like pullulating, congeneric, regnant, and irrefragable (for this last word, in the context in which it was used almost any of its synonyms would have been a better choice, such as indisputable, indubitable, undeniable, or irrefutable.

And it’s not just individual words, it’s the way he describes things, such as:
- The function P(t) is a simple polynomial, the result of a conjugal visit between the power function t2 and an amiable constant.
- the dominion of the law [of numbers] extends over a realm in which relationships are purged of their impurities in an annealing fire
- Differentiation is mathematically a puritanical operation. Not so antidifferentiation, which retains something of the promiscuous
- how fast is that red and rufous ruby falling? [N.B.: rufous simply means ‘reddish’]
- The position function inscribed on a Cartesian coordinate system, by way of contrast, records direction as well as position and so enters the scene defiantly bisexual.

I have had many experiences with math, but I have never thought of it in sexual terms.

I know, I know, he is trying to be playful, but he can get so caught up in his descriptions that the words get in the way of the examples, as in

For all that, the definite integral yet expresses a number. As a result, area undergoes an inevitable contraction, the quantity (or quality) vanishing in favor of the number that describes it, the whole of something, all that space, the Pedernales, those pecan trees, canceled in favor of a symbol, the lush particularity of this place, these trees, that swollen creek, demoted and then dismissed.

Also, ‘Pedernales’? The only use I could find for that word is as a name, so I don’t know what it is doing here.

I did enjoy his discussion of the origins and history of calculus, which has become so much a part of the modern world that we tend to forget how shocking it was when first introduced. When you think about it, the whole idea of using infinitesimally small units of time or space to determine real-world solutions seems like something out of Alice in Wonderland. No wonder Lewis Carroll was a mathematician. There is a good comment by Bishop George Berkeley (philosopher and namesake of the city and university in California):

Writing in 1734, Bishop Berkeley wasted no time in attacking the very idea of infinitesimals. If they were greater than zero, the definition of instantaneous velocity would not define anything instantaneous, and if they were zero, the definition of instantaneous velocity would not define anything like speed.

I picked up this book because I wanted a refresher for the long-ago calculus classes I took in college. A Tour of the Calculus made me interested enough to want to continue my studies, but I will get a more conventional textbook next time.

[Alex Stockdale · Rating 2 out of 5]

Ugh. Almost finished, but what a slog. Picked this book up in an airport bookstore a few years ago (obviously pre-Kindle), and finally decided I should read it or get rid of it. The subject matter is, of course, fascinating. Berlinski's writing, however, seens almost guaranteed to discourage anyone from reading the book (well, maybe graduate-level humanities students would appreciate it – hard to say, since I am not). I have persevered because the historical context that Berlinski provides *is* interesting, and if you plod along carefully you can tease the essential information out of Berlinski's overblown prose and digressions. I'm giving this 2 stars because it *has* helped me understand the importance of limits in the foundation of the calculus (something I didn't understand when I took it in university - infinitesimals made more sense, but now I understand why they are problematic). However, if you are looking for a book that will help you understand the basics of calculus, I would look elsewhere.

[Ryan · Rating 5 out of 5]

Disclaimer: there is an impossible-to-miss current of casual sexism in this slim volume. This fact is all the more regrettable owing to the book's genius.

I bought A Tour of the Calculus because its back cover compared it to Godël, Escher, Bach. That is a worthy comparison.

[JaeHee · Rating 1 out of 5]

The author thinks he's funny. He isn't.

[GS Nathan · Rating 4 out of 5]

David Berlinski starts the book saying that he would like to feel that that the reader says, "Yes, that's it, now I understand", when he or she finishes reading the book. And, sir, atleast this particular reader can report that you have succeeded.


Calculus and the concepts behind it have always been the stuff that even people formally educated in its methods find difficult to fully comprehend or explain. Yes, it works. Yes, it is very useful to solve real life issues. But some parts make sense while others don't, not in the "normal" way. How does it all hang together? How did it all get built up? What are the foundational concepts and why are they "foundational"? Beyond the techniques of differentiation and integration, what should one really know?


I read the book twice, and I think you have to do that to fully understand. Understanding comes in a sort of a circular manner, as you contemplate how one thing can lead to the other. Each concept seems like a matryoshka doll that hides more than it displays. This circular waywardness of understanding, to me, epitomises the essence of this wonderful, elusive mathematics. From numbers to functions, to infinitesimals, to limits to continuity, to differentiation, to anti differentiation, to the Reimann sum, and thence to the giant steps of the mean value theorem and integration and the generalised understanding of the derivative and the indefinite integral...it is a fascinating journey, one that expands and fills your mind. Re-read it and surprisingly it makes even more sense.


I loved the writing too. Yes, it is a bit purplish, a bit frothy and over the top, sometimes. (And Berlinski himself refers to his English professor, at one point, dryly accusing him of using a great deal of words to state precisely nothing!) But mostly though, the prose is crystal clear, taking the reader gently along from one redoubt to another; bringing a sense of the real worldliness of the abstractions in the calculus, explaining it all in a mannner that the most "unmathematical" among us can easily tag along. It seems to me that Berlinski's writing, be it in turn philosophical, contrarian, lyrical, or contemplative, is suffused with his gratitude at being able to convey to the reader the beauty of what he sees in the calculus.


And especially, the connection that Berlinski experiences with the great mathematicians who built the calculus - Descartes, Newton, Liebnitz, Dedekind, Euler, Bolzano, Cauchy, Lagrange, Reimann - comes through beautifully in the writing as he vividly imagines how they looked like and what they felt when the insight flooded their consciousness. For instance, he writes:


I encountered Bolzano on the Karluv Most on my last night in Prague...a round figure in a brown monk's cowl, walking ahead of me, muttering. As we reached the middle of the bridge, he turned and drew back his monk's cowl. I looked at his open, honest face, with its thick stubble along his chin and cheeks. For a moment I stood like that, embarassed as one is always embarrassed in meeting the dead. Then he said what I knew - what I had always known - he would say. It is what the dead always say, and it is the only thing they say.
"Remember me".


"Long live the sun. May the darkness be hidden", the book begins with these words. And, most certainly, this book does its part to enlighten us! Bravo, Mr. Berlinski!

[Valerie · Rating 5 out of 5]

This is a great addition to any study of the calculus. I used it as part of the precalc class one year, and all the students asked to keep their copies. (Granted, it was a small class.)

[Katie · Rating 2 out of 5]

This book was rather disappointing to me on the subject of calculus. As a current student of calculus, I was looking for an alternative and more in depth approach to the basic principles of calculus and its history. Instead, I felt as though I had been drug through superfluous antecedents and dismal attempts at staying 'hip' or 'readable'. While some brief paragraphs were indeed very helpful at looking at calculus from a different perspective, I rather think my time overall would have been better spent by going over in more depth an actual text on the subject that was perhaps enhanced by outside readings on the history of the concepts.

[Marina · Rating 2 out of 5]

The writing is awful. He tries some snaky stuff and just blows it. Would be a lot better if he just wrote about calculus and stopped trying so hard to be interesting about it as no one is just going to pick up such a book looking for goos writing. No GEB here despite the effort.

[Joshua · Rating 5 out of 5]

This book is one of the single most incredible books I have ever had the pleasure of reading. Berlinski really takes his time in exploring the Calculus, providing insight into it's history and applications as he does so. A phenomenal read: always interesting, rarely difficult, never boring.

[Jesse Broussard · Rating 1 out of 5]

Some enjoyable prose, but mostly a cheap script writer attempting to wax mathematical without waxing intelligent.

[Mohammad Tanviruzzaman · Rating 4 out of 5]

Numbers can express real world; that's the assumption.
Let's take the simplest reality, a straight line. Put a zero somewhere and negatives are on the left and positives are on the right. There are large gaps, though, say between zero and one. Now, define a pair of integers (with division in between) as a rational number. We have filled quite a bit, not the whole line, though, because the square root of 2 is not a rational number. Thence come the irrational numbers -- Dedekind's cut is one way to create them. Rationals and irrationals form reals. Now with the irrationals at hand, we have finally been able to assign a number to every point. (don't forget Zeno's paradox)

Two such lines can start forming the two axes of the 2-D plane, etc. But there is no life yet, nothing moves, so far we have just assigned numbers to the points of the world. To bring life and motion, let us connect numbers with one another. Among infinitely many possible relations, we chose a special type, called function, because it appeared to us that this relation is predominant in the world.

One interesting relation is between time and space -- position function. (Let us assume that both can be represented by real numbers.) How much position changes in a period is speed, actually average speed. But what about the speed at this very moment? (Like I speeded over the limit but my average speed was below limit, should I be ticketed?) Ok, let us make the period smaller and smaller and smaller, goes on to infinity -- but never becomes zero, because we cannot divide by zero. We hope that if we start making the period smaller and smaller, the change in position will go closer and closer to a particular numebr, for example like 1.1, 1.01, 1.001, 1.0001, ..., etc. So, we can guess if we could seize the moment, the change in position would have been nothing other than 1 and that is the limit (can you prove it?).

Limit is like trying to know about someone to whom we cannot connect directly, but we have access to close ones of him; and sometimes (for example he is not a weirdo) we could find all there is to find about him just from his near and dear ones -- without ever really meeting him.
The area under the curve is the same idea, accessing something to which we do not have direct access. We can chop the area under the curve into more and more rectangles and hope that if we keep dividing into more and more rectangles the total sum will go closer and closer to a particular number, for example say with 10 rectangles the sum is 1.1, with 100 rectangles it is 1.o1, with 1000 rectangles it is 1.001, etc., again -- not all -- but in some situations (like if the function is continuous) we can say that the exact sum is actually 1 (Can you prove that the sum is exactly 1?).

Speed was differentiation, and area is integration, they have an opposite relationship almost like addition vs. subtraction or multiplication vs. division, one undoes the other. Almost, because if we differentiate a function, we get its derivative -- a single function, but if we integrate (or anti-differentiate) the derivative, we do not get the original function. Instead, we get a family of similar functions which differ by constant. This relation, one undoes the other, is the fundamental theorem of calculus.

Philosophically calculus is the meditation on continuity. Why bother about continuity? Because, motion, that important concept, we feel at some level is continuous. (What about quantum physics?)

[mpacer · Rating 1 out of 5]

I could not finish reading this. I tried, I really tried. But the author not only included incomplete and seemingly inaccurate maths, he in no way explained anything clearly, and he didn't do so in what I found to be an entertaining manner. I even got most of his references which would be difficult for others with less of a background in the history of mathematics, but even those I didn't enjoy but internally groaned.

I know a good deal of the underpinnings of calculus, and while yes, I may now know more generally what role dedekind played in establishing modern analysis, berlinski's style made it only that much more confusing.

I know I can be difficult to read at times, but I sure as hell hope that I get to the point more rapidly than he. I cannot more heartily anti-recommend this book.

It saddens me to say it because I really wanted it to be good. I like books that aim to both convey maths and their history in an intuitive and accessible manner. In this genre I would include books such as "Unknown variable: a real and imaginary history of algebra"(algebra) or "the lady tasting tea"(statistics) or "fermat's last theorem"(number theory), each of which does an excellent job of conveying its content in an interesting, accessible, and profound way(unlike a 'tour of calculus').

But I'm sorry, this book must go in my bag of books that fail to deserve a place on my bookshelves. In fact because of this book, I'm going to need to get a new bag.

*Le sigh*

[Wanda · Rating 5 out of 5]

This is a beautifully written book, giving both fanciful historical perspectives as well as excellent analogies to support light proofs. Although the prose is sometimes challenging to parse, I was completely drawn into the scenes painted with (grantedly) florid prose, and even burst out laughing at some of Berlinski’s quips. I had such fun doodling my notes in the sidebars, filling out some details of the proofs that were glossed over... What a delightful overview of the development and denouement of the calculus!

[Douglas Weathers · Rating 4 out of 5]

This book wants to be the calculus in layman's terms, but Berlinski's prose is more complicated than a textbook would be. Even though this isn't bad, I feel like it makes the book more difficult to get through than it needs to be. As complicated as his flowery, ornate writing is, Berlinski still writes well, and offers an interesting look at beginner's calculus.

[Joseph · Rating 2 out of 5]

A good survey of the field but AWFULLY written. The baroque language and unnecessary tangents the author uses sometimes obscure his point rather than illustrate it. The author could have easily made his points in a third of the length and in a much better way

[Jesse Keyton · Rating 5 out of 5]

Had me laughing harder than I’ve laughed in a while

[John G · Rating 3 out of 5]

It has been awhile since I read this. I remember clearly the fine explanations of the origin of the limit in differential calculus. This is very important because it represents a lot of modern mathematical weakness wherein formula are not possible and instead we use approximation and logical induction.

As you probably discovered in school, even simple equations with two unknowns can be difficult. And at a higher level of difficulty, 5th degree polynomials can't be solved by anyone. Today the limitations of mathematics have become painfully real as we try to model complex system behaviour in fields like fluid dynamics.

I think this has been good. For no longer do we have the arrogance of Russell's Principia Mathematica or vain attempts to establish mathematics as the pre-eminent logical system. So we can now teach math as the useful set of tools that it is, rather than dressing it up as a comprehensive progression of logical thought.

So even though mathematics cannot solve continuous functions the limit makes a calculus that works just fine in approximating a useful answer.

As mathematics has fallen from its exulted state as the queen of science, so too has science in general lost its authority as it has turned to empirical methods. And scepticism of scientific results is widely held by a population too often led astray by commercially funded science.

Thus reading this book is a nice trip back in time when thinkers were sincere and wrestled with problems that had nice logical solutions.

[William · Rating 5 out of 5]

I took calculus twice in college, got a "B" the first time and an "Incomplete" the second time. I've always wanted to master it, knew I hadn't, "B" or not. Reading this book didn't get me over the hump into mastery, but its fun to read along and follow what he says. I know why the infinitesimal was the path that Newton and Leibneitz followed as they invented calculus, and I recognize why mathimaticians are uneasy with it and were happy when calculus was resated in terms of the limit, not the same as an infinitesimal. g

The imagined monologue by Berlinski's mathematically inclined, New York, cab driver, is a treasure. Am I right?

Now if I can get my retirement life better organizeed, I can get back to my studies. I was never able to use calculus in my work, but I managed to do ok without it. Even with extrapolated voltage thresholds for FET transistors, which are based on the derivative of the gate voltage vs drain current curve changing from increasing to neutral or decreasing. Well, derivatives are the easier half of calculus, where I came up short was in integration, and substituting equations for other equations, to get something that could be differentiated or integrated. Mastery of trigonimetric identities played a significant role it his, and I'd stopped taking pre-calculus in 11th grade for dumb reasons, never mastered trigonimetric identities.

So take my advice, if you haven't taken calculus yet, make sure you're algebra II and Trig. are ship shape and Bristol fashion. You'll be glad you did.

[Upom · Rating 3 out of 5]

In the rushed use of calculus in physics, chemistry, biology, economics, and other subjects, it's easy to forget what an intellectual achievement it really is. "Tour of the Calculus" tries to do something which no other popular math text I've ever read has attempted: it brings poetry to mathematics. Looking both at the obscure characters who made the subject possible, as well as the various definitions, postulates, and theorems that make up the calculus, the book gives a foundational and rather lovely view of calculus, describing the beauty of the subject through imagery from all levels of human experience (art, science, sports, religion, etc.).
That being said, the author did rub me the wrong way a little bit. His florid descriptions and esoteric pop references sometimes verged on the pretentious. The descriptions often turned into molasses, sweet, but thick and slow to read. Still, I commend Berlinski for writing a math book that integrates mathematics and art into one beautiful whole.

[Jay · Rating 4 out of 5]

I've always loved math, but when I took calculus in high school it kind of went over my head, which is unusual. My teachers taught the technical skills to work out calculus problems, but I never learned what was really happening behind the derivatives and integrals; I never learned about instantaneous rates of change or the true relationships between a function, its derivatives, and its integrals.

This book changed that completely. It showed me, in good prose, the underlying theories and concepts of the calculus. It took me through its dual discovery, the necessity of the calculus, and even into the nature of mathematics itself.

If you're curious about math, especially on a theoretical level, then I highly encourage reading this book. Berlinski is very clever, very well informed, and a very good writer.

[Scott Murray · Rating 4 out of 5]

I thought I was getting some in-depth insight into calculus, technical insight I missed in my college course. I did get that, but got a lot of back-story re: the history of the development of calculus, which was usually pretty interesting, and also lyrical, allegorical musings on the nature of functions, limits, and calculus itself, which I had to be in the mood for and frequently was not. This was my fault, the author writes very well, and has clearly thought very deeply about his subject, but I was looking for more math and less philosophy.

I thought some of the proofs could have been more clear (for example, on page 215, why is f(a) a constant?), but in general I could follow along well enough not to lose the thread.

[Patrik · Rating 5 out of 5]

I loved this book. Berlinski managed to make the story of “the” calculus both interesting and engaging. Not only that, but I also feel that I now understand calculus better (at a deeper level) than I did before. Berlinski broke up the field of calculus in its two parts, speed and area, then took a deep dive into each, only to bring them together beautifully through the fundamental theorem of calculus. Yes, the language used and the stories told are occasionally too cute, but I enjoyed them and viewed them as appropriate resting places between the mathematical content. Again, the best book I have read in quite a while.

[Aiman Adlawan · Rating 4 out of 5]

This book is not really showing the reader a 'tour of the calculus' in a sense. It is only picking some underlying concepts and elaborates it in some shallow that is sensible for some beginners who wants to avoid complex explanation.
The author has a lot of unnecessary prose which are very annoying. It made me jump to the next page where the main content is explained. This is not a math book but very essential for understanding some concepts.

[Monty Circus · Rating 1 out of 5]

I was given this as a gift by my older sister (the shit one), another gift straight out of the bargain bin (or perhaps she found it lying in a gutter somewhere). I thought it would help me with my high school calculus class. I was dead wrong. It addled my brain with garbage and I failed out quickly. Thanks sis!

[Kurt Schwind · Rating 2 out of 5]

The book does a better than good job of putting the history and story of the calculus in an easy and entertaining format. However, at times David Berlinski gets very purple in his prose. Enough that I had to take periodic breaks away from reading it to find one of my eyes that rolled so much that it fell out of my head.

[Mark Pedigo · Rating 4 out of 5]

While a Seattle software developer in the late 90s, I loaned this to a coworker (who eventually worked for Microsoft and Google). His response? "Good lord, does that man love to talk." Well, yes, but it *is* a fascinating subject he chose.

[J.F. Ramirez · Rating 4 out of 5]

I stopped at page 185. Good, but I'm sure there is something else I could be reading right now. QED anyone?