Mathematics Made Difficult

by Carl E. Linderholm

Genres: Mathematics, Humor, Nonfiction, Philosophy

207 pages, Hardcover

First published January 1, 1971

Reviews

[Saharvetes · Rating 5 out of 5]

This is one of the most hilarious books about mathematics ever written, and likely to remain a classic for a long time. Unfortunately it is out of print, but it is not entirely impossible to find a copy. The jokes in the book are at varying levels of mathematical sophistication, involving everything from simple puns to advanced graduate-level mathematics, and a fair bit of in-jokes about the foibles of mathematical culture.

The first chapter, of about 70 pages, is on Arithmetic (and includes a lot of Category Theory, naturally); the book eventually covers Algebra, Topology and Geometry. Among other highlights, its category-theoretic construction of the integers and its description of all the mess involved in the terms "improper fraction" and "mixed fraction" stand out for their utter mathematical rigour and chaos. Also notable is its lament on the sad love-life of brackets, where '(' and ')' though paired and meant for each other can never meet for there is always something between them. And so on and on.

Some sample quotes from the book:

* [p.10] "Mathematicians always strive to confuse their audiences; where there is no confusion there is no prestige. Mathematics is prestidigitation."

* [p.28] "With a few brackets it is easy enough to see that 5+4 is 9. What is not easy to see is that 5+4 is not 6."

* [p.25] (On mathematical "beliefs".) "Like the world of a science-fiction story, a system of beliefs need not be highly credible—it may be as wild as you like, so long as it is not self-contradictory—and it should lead to some interesting difficulties, some of which should, in the end, be resolved."

* [p.37] "unfortunately, there is a flaw in the reasoning. [..] to say that each of two numbers cannot be bigger than the other is to repeat the statement that is to be proved. It is not correct in logic to prove something by saying it over again; that only works in politics, and even there it is usually considered desirable to repeat the proposition hundreds of times before considering it as definitely established."

* [Starred exercise] "Show that 17 × 17 = 289. Generalise this result."

[Simon Mcleish · Rating 5 out of 5]

Originally published on my blog here in April 2002.

When I was a mathematics graduate student, this book was passed around the department, delighting those of us working in pure mathematics. Basically, it takes apart the sort of mathematical ideas generally taken for granted, and shows that they are much more complicated than it first seems if you want to make them rigorous. (There is some cheating when ideas from category theory are introduced and make the explanations even more abstract than they need to be.)

There is, of course, a subject for a serious book in this; I can think of two without any effort (Rudy Rucker's Infinity and the Mind and the far older Bertrand Russell's Introduction to Mathematical Philosophy). Mathematics Made Difficult is not in any sense a book which aims to educate and inform its readers. Much of the mathematics is presented in a way which would probably not make a great deal of sense to anyone not already familiar with it (a course in the foundations of number theory is really the minimum needed to understand most of it, and one in category theory for the detail). What is enjoyable about Mathematics Made Difficult is that it is very funny, full of parodies of school textbook problems and bad puns. Mathematics is not easy to turn into humour, and this book is one of the very small number of consistently successful examples.

[Isaac · Rating 2 out of 5]

It is seriously funny at times, especially at the beginning, but the humour drags on by the end and I felt myself skipping sections.

[Benjamin · Rating 5 out of 5]

I read this towards the beginning of my college math degree and most of the complexities beyond basic abstract algebra concepts were lost on me. However, it's plain to see that Linderholm's main point when writing this book was to poke fun at the way that mathematics is written about and make jokes about how informal we are being when we talk about mathematics (i.e the area of a rectangle doesn't make any sense based on it's definition).

Even though I'm sure I didn't pick up on some of the book, I enjoyed reading the whole thing. The sections of the book are prefaced with a ridiculous story pertaining to the current topic, followed by a hypothetical dialogue between the author and a mystery person doubting his claims. These sections are generally understandable to everyone; it's only when the author begins to derive the point he is trying to make with advanced (and I do mean advanced) mathematics that the reader may get lost -- and that's not necessarily a bad thing, because the point is to be overly complex about things we take for granted.

I'll be picking this one up again when I finish my undergrad to see how things have changed. However, I actually learned a good bit from the book and got a lot of laughs out of it as well, and I'd recommend it to anyone who's read any mathematical literature. It's very funny.

[J. · Rating 4 out of 5]

Well, this is certainly one of a kind. It's a mathematical comedy! A spoof of mathematical-ness, but deeply rooted in mathematics itself (an 'insider' spoof, as it were.) Tons of category theory, and tons of puns! One of the weirdest and best things I've ever read. I'm only knocking it down to 4/5 because some of the longer gags run a little thin, but this is a brilliant, brilliant book. I wish there were more like it.

[Gavin · Rating 4 out of 5]

Mathematicians always strive to confuse their audiences; where there is no confusion there is no prestige. Mathematics is prestidigitation. Confusion itself may be taken as the guiding principle in what is done here-if there is a principle. Just as the fractured leg confused the Zen disciple, it is hoped that this book may help to confuse some uninitiated reader and so put him on the road to enlightenment, limping along to mathematical satori.

If confusion is the first principle here, beside it and ancillary to it is a second: pain. For too long, educators have followed blindly the pleasure principle. This oversimplified approach is rejected here.

Great delicacy and tact are needed in presenting this idea in conversation - if the aim is, as it should be, to bewilder and frighten the opponent. His level of sophistication is very important. He may know all about it; then he will utter some crushing reply, like 'So what else is new?' He may even add, 'Just finding out about cyclic groups ?'-or mention some other concept you yourself have never heard of; if he does so, you have lost the advantage and may not get out ~of it without a few scratches on your own escutcheon. On the other hand he may be so ignorant as to be impervious to doubt; you will be laughed at. The idea is much more useful if the intention is merely to annoy...

I am no mathematician. But, so, this is the book I've been looking for, for a decade: troll maths, a comedic work with jokes in higher maths. (Carroll for grownups.) The main gag is to use the awesome machinery of category theory to define and manipulate extremely basic objects like bagels or boolean algebra, and then to intentionally weave in spurious historical explanations, misapplications, misabstractions, and category errors. Here's the first instance, in which elementary logic is defined as a category equipped with ... I understood something like half of the gags.


The whole book is a stream of consciousness flitting between fake worked-problems prose ("Muscular M. Boulangiaire... makes three kinds of loaves in his shop. The first kind is a flat square of side x; the second is an ordinary French loaf, just x units long and one unit wide..."), fake Bronowski megahistory, and needlessly pure abstraction, with little barbs scattered throughout. The barbs are like that bit in Principia where Russell says "1+1=2 is occasionally useful.". As here.

Pyramids, another Egyptian invention, were principally a tourist attraction, like the Eiffel Tower. They were copied in America, where they were used for heart transplant operations, flower shows, and football games... [the Greeks] based their unit of measurement on the curious idea that the more beautiful was a woman, the larger was the number of ships her face could launch... The practical result of this idea was that the Greek women spent much time on the seaside toughening their facial features by pushing boats into the water, and in the end the Hellenes lost interest in women and invented Platonic love.

Popular imagination conjures up the myth of the wicked professor who teaches that 2 + 2 is not 4 precisely because it cannot conceive the truth. The truth is much stranger, more monstrous, more impressive. It is not scepticism about 5 + 4 = 9 that exists, but scepticism about 5, about 4, and about 9.

Is Linderholm an elitist dick? Maybe. He might be mostly pretending to be one. At least it's intentionally hard, unlike the others!

A few off-colour jokes about violent Italians and square-headed Aryans. Even these are half subversions: at various points he spells it 'Arian', as in the ancient heresy, and flip back and fore between the Christian and Indian referents.

Linderholm has not written the perfect work of maths comedy, because there are falsehoods in it. (e.g. The first chapter is in medieval disputation style and comes with all the fallacies of the schoolmen - "mathematicians pretend to count by means of a system supposed to satisfy the so-called Peano axioms. In fact, there cannot be any Peano axioms, since they were really invented by Dedekind. Hence even mathematicians cannot count. Furthermore, the piano has only 88 keys; hence, anyone counting with these axioms is soon played out.".) The perfect work would be hilarious and completely true.

PS: an accidental joke about Causal Language Models, which have climbed surprisingly high doing only stupid sequence completion:

8, 75, 3, 9, _ .

Now all you have to do is look at the numbers, and then in the blank provided write in the number that seems to you logically ought to go there. Now read the numbers again: eight, seventy-five, three, nine, ... What was that you were about to say? Was it 17? Right! The only number any sensible person would put there is 17...

The scientific fact is universally acknowledged that only intelligent people can do these puzzles; moreover, nobody denies that there is a crying need for intelligence in all areas of the national economy. Hence, and one would think the inference would be obvious to any person who can guess the next number even in the easy example we saw just above, all that needs to be done in order to cure a vast proportion of the world's ills is to teach everyone to guess the next number. Because then, naturally, everyone would be intelligent.

[Peter McLaughlin · Rating 4 out of 5]

The most important work of philosophy of mathematics of the twentieth century? No, I’m serious. Yes, it’s a comedy book, but Linderholm took great pains to ensure that the jokes were true. All of the ‘serious’ problems of philosophy of mathematics and most of the ‘serious’ problems of the early twentieth-century foundational crisis appear here, and not under any disguise: e.g., a fully transparent and rigourous presentation of Frege’s ‘Julius Caesar problem’ is given at pp. 25–26. But Linderholm’s philosophical genius was to see just how obviously crazy this way of thinking about maths would be in a vacuum, even if it is all technically correct; and his stylistic genius was to conduct an extended theoretical argument by means of comedy. The key task in foundations ought to be trying to extend existing foundational ideas in ways that are not susceptible to Linderholmisation; but comedy is not taken seriously enough by mathematicians and philosophers, so they aren’t working on it. Alas. It is very funny.

[James · Rating 3 out of 5]

Reading books, like mathematics, is an art to which many pretend, to which some aspire, and to which few attain

A subtractable number is less tractable than a tractable number.

When it shines, it's a great satire of the rigour and writing style of mathematicians. A must-read for anyone with a passing familiarity with category theory and a working knowledge of abstract algebra.

When it doesn't, it's horribly dated with that ever-so-slightly racist humour one'd expect from a 1970s academic.

Beneath the puns and cries for help, there is a terminal object kernel nugget of insight: it's not just business majors that can waffle for 200 pages to say almost nothing. And I, for one, think that's a valuable lesson.

Show that 17 × 17 = 289. Generalize this result.

[Jesse · Rating 5 out of 5]

This (along with the section of Sorrentino's Mulligan Stew) is the best satire of mathematics. Even better, everything in the book is also a true mathematical statement (based only on my knowledge of advanced modern algebra and rudimentary category theory, that is).

[Tinwerume · Rating 2 out of 5]

I loved the premise, but thought almost all of the humor fell flat. It would have been much better if it had just focused on overcomplicating simple proofs.

[Dennis · Rating 3 out of 5]

Understood half of it, enjoyed most of it.

[Aidan W. Murphy · Rating 4 out of 5]

More reader-friendly than most graduate mathematics “textbooks”.

[Roi Orzach · Rating 5 out of 5]

After reading plenty of books where every example was riddled with phrases such as, "the proof is trivial and is left as an example to the reader," "the following is an obvious consequence of," and other proofs from the category of proof by intimidation, this book was an extreme juxtaposition of the above. Everything from 1+1 = 2 to higher mathematics is proved with techniques from abstract algebra to analysis. Being what some might call a math-nut I found this book extremely comedic. This book is a classic and should be read as it makes the reader rethink even the most seemingly simple of proofs. Although, what I enjoyed most about the book is how it showed there is no set proof method in mathematics. I would recommend it to anyone who enjoys math, but wants a more relaxed version of a textbook. Near the end of the book, the mathematics got complicated as it approached the graduate level curriculum. My one negative comment on the book is the high price amazon sells the book for. However, the way the author satires common non-rigorous proof methods, more than makes up for the price. For anyone with a math major, or interest in math I would not only recommend this book, but I would insist upon it.

[Thatsgreateric · Rating 5 out of 5]

Nobody accepts a challenge anymore. If you watch Teen Mom religiously, a glance over this textbook might just save your nth marriage.

[George · Rating 5 out of 5]

Hilarious for those acquainted with the subject matter. Can be enjoyed even by undergraduate students who have taken a course in number theory.

[Ril · Rating 5 out of 5]

The best math book I've ever read, showing us just how many ways can a mathematician fail and how common bad explanations are around us.

[Andreas Furth · Rating 5 out of 5]

This finally achieved the centuries old goal of making mathematics even a tiny bit difficult.