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I recently wanted to use the title of the famous short story "Everything that Rises must Converge" in a poem of mine. However, the mathematician in me insisted on changing it to "Everything that Rises, if the rise is bounded, must Converge".

Are there other literary quotations that are false mathematically, and how can they be changed to make them true?

Note: Attempts to use "To be or not to be" will be dealt with most severely.

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closed as too broad by Daniel Fischer Jun 25 at 13:54

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs. If this question can be reworded to fit the rules in the help center, please edit the question.

Bounded or not, it converges in the extended real line ${\bf R} \cup \{\pm\infty\}$. –  Noam D. Elkies Oct 3 '13 at 3:25
While a nice question, shouldn't it be Community Wiki? –  vsz Oct 3 '13 at 6:08
Just about every use of the word "exponentially" qualifies. –  congusbongus Oct 3 '13 at 6:53
This doesn't quite work because it takes some fiddling, but many run across this in gradeschool: Two wrongs don't make a right. vs -1 * -1 = 1 –  Izkata Oct 3 '13 at 18:25
@Izkata Two wrongs is usually interpreted to mean -1 + -1, which does indeed fail to make positive 1. –  Amory Oct 3 '13 at 20:17

19 Answers 19

up vote 25 down vote accepted

Hosea 1:10

Yet the number of the children of Israel shall be as the sand of the sea, which cannot be measured nor numbered

Both the children of Israel and the sand of the sea are, of course, finite sets. As a fine grain of sand has a mass of approximately $3.5 \times 10^{-10}$ kg according to http://en.wikipedia.org/wiki/Orders_of_magnitude_%28mass%29 , it only takes about 2.5 kg of sand grains to outnumber all the people currently alive on Earth. Even for coarse sand, a truckload should suffice.

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Since that is future tense, it could be true eventually. –  Trejkaz Oct 3 '13 at 6:11
Assuming there are about 10^18 grains of sand in the sea, and the surface area of earth is 5 * 10^14 m^2, the density of Isrealites would be 1,960 Isrealites / m^2. In other words, the surface of the earth would be a writhing mass of Isrealites. –  trav1s Oct 3 '13 at 6:32
Obviously we need to expand into interstellar space. –  Robert Israel Oct 3 '13 at 6:55
Did Hosea even read Archimedes? en.wikipedia.org/wiki/The_Sand_Reckoner –  Sammy Black Oct 3 '13 at 9:23
I'm just going to leave this here for you, @trav1s. what-if.xkcd.com/4 –  Sammy Black Oct 3 '13 at 9:26

A great example of this:

"Sir, in your otherwise beautiful poem (The Vision of Sin) there is a verse which reads 'Every moment dies a man, every moment one is born.' Obviously this cannot be true and I suggest that in the next edition you have it read 'Every moment dies a man, every moment one-and-one-sixteenth is born.' Even this value is slightly in error but should be sufficiently accurate for the purposes of poetry."

     - Charles Babbage, in a letter to Lord Tennyson

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The version of the story I remember reading is that the line originally read "Every minute dies a man, every minute a man is born", and in response to Babbage's critique each "minute" was changed to "moment", which is not a fixed time interval and thus does not contradict population growth. –  Noam D. Elkies Oct 3 '13 at 4:12
I would take issue with Babbage's critique, as it does not say "exactly every minute a man is born" - it only establishes that at least one is born every minute. Indeed, I would assert that "moment" is a poor substitute, as one can choose a moment at which there is no man born. Tennyson should have stuck to his original choice. –  Glen O Oct 3 '13 at 13:46
Suppose there was one death per minute. In the 19th century, life expectancy at birth was maybe about $16$ million minutes, so that would imply a population of about $16$ million people. The census of 1841 (one year before "The Vision of Sin" was published) recorded the population of England and Wales at $15.9$ million. Not bad! –  Robert Israel Oct 3 '13 at 15:29
@RobertIsrael What was the birth rate? Also, how does that imply that? I feel like there are unstated assumptions you're making. –  Random832 Oct 3 '13 at 17:46
Imagine a single individual who upon his/her death is immediately replaced by a newborn, and this process continues indefinitely. Let $N(t)$ be the number of deaths from time $0$ to time $t$. The Renewal Theorem says that with probability $1$, in the long run the average number of deaths per unit time is $\lim_{t \to 0} N(t)/t = 1/\mu$ where $\mu$ is the expected time from birth to death, i.e. the life expectancy at birth. A constant population of $M$ individuals can be regarded as $M$ copies of this process, so the average number of deaths per unit time would be $M/\mu$. –  Robert Israel Oct 4 '13 at 3:08

This one is rather famous:

1 Kings 7:23

Then he made the sea of cast metal. It was round, ten cubits from brim to brim, and five cubits high, and a line of thirty cubits measured its circumference.

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By "line", it actually refers to a cable of some type (typically something about the thickness of paracord, but may vary). Thus, it is saying that a piece of rope 30 cubits long wraps exactly once around it. –  AJMansfield Oct 3 '13 at 14:16
@Random832 Remember, elasticity. Most cables are longer when stretched, as it would when wrapped around the object, that it would be laid out on the ground for measurement. –  AJMansfield Oct 3 '13 at 18:56
@AJMansfield People back then weren't stupid - if it stretched to any significant degree, they would stretch it for measurement as well. –  Random832 Oct 3 '13 at 19:07
I always took this just to be an example of the use of significant digits in action rather than false mathematics. –  Kevin Costello Oct 3 '13 at 21:07

As I noted some years ago, an example is the claim by Thomas Carlyle (1795$-$1881, in Sartor Resartus (c.1833)) that "It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity of the universe." In fact, this echo of John Donne's "if a clod be washed away by the sea, Europe is the less" [Meditation XVII, 1624] is not a fact of mathematics but an error in physics according to Newton's Laws (1687): the recoil must exactly cancel out the pebble as the "centre of gravity of the universe" continues on its unalterable course.

[Later] Proposed correction: "It is a mathematical fact that the casting of this pebble from my hand moves the centre of gravity of the Earth."

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(but only while the pebble is in flight: soon it rejoins the Earth, and then we're back where we began.) –  Noam D. Elkies Oct 3 '13 at 4:32

From John Green's The Fault in Our Stars:

There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million.

The problem is that $[0,1]$, $[0,2]$ and $[0,10^6]$ all have the same cardinality of $\mathfrak{c}$, i.e. have the same number of elements :)

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"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "and then the different branches of Arithmetic--Ambition, Distraction, Uglification, and Derision."

Alice's Adventures in Wonderland, by Lewis Carroll.

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Some may say that this one is accurate... –  Rodrigo A. Pérez Oct 3 '13 at 14:25

Would you count sayings as literary? If so, what about

  • What goes up must come down.

Some would argue this was proven wrong in a way by Sputnik (or take any modern GEO satellite). A strict reading of your question would dismiss this as a physical constraint, not a false statement in a mathematical sense).

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Taking the other side, some might say that Sputnik is constantly coming down. After all, what is orbit but continuous falling? –  Andrzej Doyle Oct 3 '13 at 10:35
So pick Voyager instead... –  Rodrigo A. Pérez Oct 3 '13 at 14:23

Perhaps my favorite is the example of The Hitchhiker's Guide to the Galaxy, a trilogy of five books.

"If a man will begin with certainties, he shall end in doubts." - Francis Bacon

This would be a very poor website were that to be true.

"There's no limit to how complicated things can get on account of one thing leading to another." - E. B. White

Okay maybe that one is correct.

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The test instructions, “Draw a circle around the correct answer.” really means, “Draw a Jordan curve around the correct answer.” - or, even more accurately, “Draw an approximation of a Jordan curve around the correct answer.”

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Two wrongs don't make a right.

Clearly this is a false statement.

If you multiply a negative (a wrong) by another negative, you get a positive (a right).

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But three rights make a left. –  marty cohen Oct 4 '13 at 5:41
And who told you that the group operation was multiplication? Adding two wrongs, indeed, does not make a right! –  Soham Chowdhury Oct 4 '13 at 6:06
All that says, is that if we let $+$ denote a 'right', $-$ a 'wrong', $S\supseteq\{-,+\}$, and $m:\{+,-\}^2\to S$ the compositing function we are referring to, then the statement simply means that $m(-,-) \ne +$. –  AJMansfield Oct 11 '13 at 14:20

from “Animal Farm”: “...some are more equal than others”

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the common expression, “If I’ve told you once, I’ve told you a million times.”

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This isn't wrong! It just means that either I haven't told you once, or else I have told you a million times. What's false is: "In all possible worlds, it holds that if I've told you once, I've told you a million times." –  goblin Jun 25 at 13:58
@goblin: You’re imposing mathematical usage on ordinary language usage. In ordinary language, the premise of such an implication is always taken to be true. It has probably never happened in human history that anyone made this statement, without having told the hearer the item at least once. –  EsperantoSpeaker1 Jun 25 at 14:15
True. I guess it depends on how the original question is interpreted. –  goblin Jun 25 at 14:31
@goblin: Now it looks like you’re resorting to what is described at the following Stack Exchange location: english.stackexchange.com/questions/254612/rhetorical-evasion –  EsperantoSpeaker1 Jun 25 at 14:59
no, I think you're wrong, but I didn't want to start an argument. –  goblin Jun 25 at 15:00

from the poem “Kubla Khan”: “...through caverns measureless to man”

There are unmeasurable sets, but no cavern is unmeasurable.

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In the ordinary language phraseology used in Chemistry, they speak of the “temperature dependence of the rate constant”. This amounts to a “variable constant”, a contradictory notion. The mathematical term for a “variable constant” is “parameter”.

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In ordinary language, a “set” usually means an indexed set (aka “multiset”).

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The adage, “If you lose an hour in the morning, you’ll be looking for it the rest of the day.”

Admittedly, the inconsistency is due simply to metaphorical usage, but still...

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From the Matrix revolutions:

"Everything that has a beginning has an end, Neo." -Agent Smith

Ummmm... what about the natural numbers? And the ordinal numbers? And lets not forget the cardinal numbers. Hell, even $\mathbb{R}_{\geq 0}$ is a counterexample. In fact, given any totally ordered set $T$ that lacks a greatest element, we can always adjoin a first element.

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In ordinary language, a “biased” result is considered useless, but in Mathematics (Statistics), a result can be biased, but still useful.

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In ordinary language “in general” means “most of the time, but possibly with exceptions”, whereas in Mathematics it means “always, without exception”. However, because (introductory) Mathematics textbooks rely so heavily on ordinary language, the ordinary language usage can show up there, as in, “The projection of A x B onto A is, in general, not one-to-one.”

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