Entering the town of Gold Hill, Colorado, you encounter one of the most extraordinary posted signs in the entire USA:
The founding year: 1859.
The elevation: 8463 feet.
The population: 118 people.
And at the bottom—oh, the glorious bottom—these three numbers have been added together, yielding a total of 10,440. You can check it yourself: 10,440 is exactly correct. The arithmetic is flawless.
It’s perfectly right… and profoundly wrong. It’s a memorable token of a common mathematical mistake: carrying out an operation without investigating its meaning.
I could easily spin out 1000 words bagging on this poor sign-maker. But I’m not going to. (For one thing, there’s a chance the error was a deliberate joke, and even if it wasn’t, there’s enough ridicule out there for bad math.) Instead, I want to argue the opposite.
This error isn’t brainless, stupid, or contemptible. Rather, in several ways, the Gold Hill error is a uniquely sophisticated and modern one.
Hop in a time machine. Tour the range of human societies. You’ll find that most consider the idea of “mindless” computation paradoxical, nonsensical—a contradiction in terms.
That’s because, for most of human history, computation was really hard.
As recently as 1000 years ago, simple multiplication—the stuff we teach to 9-year-olds now—required expert equipment. Determining a square root was the special provenance of academics and well-trained merchants.
Back then, “simple” math didn’t feel so simple.
Now, of course, we’ve got calculators, iPhones, and—most importantly—the Hindu-Arabic numeral system. This way of writing numbers, familiar and obvious to us today, arrived as a revelation. It’s far easier to compute with our friendly numerals than it was with, say, Roman numerals.
For most of our history as mathematical thinkers, the Gold Hill error would’ve been tremendously unlikely. Computations took too much effort for you to waste time performing a silly one!
Math and science teachers chide students for “forgetting their units.” Omitting units is like leaving a number naked and unclothed. Do you mean 7 feet? 7 meters? 7 miles? Or 7 jars of jelly?
That’s the basic problem with the Gold Hill sign. The three numbers have very different units (years, people, and feet). These can’t be added.
But for many mathematical cultures, “forgetting” the units would have been unimaginable, because numbers (and operations) were inextricably tied to physical representations.
That is: numbers needed units.
Take square numbers. Why is multiplying a number by itself known as “squaring”? Simple: it’s equivalent to finding the area of a square (with the original number as a side length).
What about “cubing”? Likewise, that’s just finding the volume of a cube (with the given side length).
For Greek mathematicians, a single number represented a length. The product of two numbers stood for an area. And the product of three numbers represented a volume.
What about the product of four numbers? It’d be nonsensical, impossible. In Euclid’s famous geometry book, he carefully avoids ever multiplying four numbers together. For him, it would have violated common sense, gone against his geometric conceptualization of number.
What’s the point of all this? It’s that our very ability to conceive of numbers without units is a pretty remarkable abstraction. The Greeks would find our notion of numbers, even as practiced by little kids, a bit eerie and ethereal. Their vision of math was beautiful but immensely concrete, closely tied to geometric and visual meanings.
And so to a Greek, the Gold Hill sign would be… well, like Greek.
Most of the math that we teach to our students comes from long, long ago. Integers and fractions have been studied since prehistory. All of our geometry was known to the Greeks. Simple algebra dates back to medieval Arabia.
But one particular branch of mathematics is incredibly new—almost shockingly modern in comparison to the lumbering dinosaurs with which it cohabits our curriculum. That young, upstart branch?
Statistics.
Although it’s often used as a synonym for “number,” the word “statistic” in fact has a specific and little-discussed meaning. It refers to a single number that summarizes a whole sample. For example, if I grab five people off the street, ask them how annoyed they are (on a 1-to-10 scale) at being grabbed by me, and then average those numbers, that average is a statistic.
The thing about statistics is that a lot of numbers go into them. Just as honeybees need acre upon acre of flowers to produce a single jar of honey, statisticians need lots of raw data to produce a simple summary.
But where past societies lacked for data, we’ve got a surplus. Heck, it’s 2015; we’re practically swimming in it. If you clicked on this post through Facebook, Google, or Twitter, then your click is another data point in some massive collection in Silicon Valley. We’re so immersed in numbers—drowning in them, really—that we need statistics. We need summaries.
We need totals.
Now, I don’t mean to suggest that finding a total is a new idea. The number at the bottom of the Gold Hill sign is a simple sum. It’s the result of a basic addition operation—a computation as old as time.
But the attitude that produced the Gold Hill error—the practice of summarizing data immediately after presenting it—is a new development in our mathematical culture, a strange necessity created by our glut of information.
So yes, it’s easy to see the Gold Hill sign as an emblem of everything that’s wrong with our mathematical culture. We push students into mindless computations and unnecessary abstraction, and the result, too often, is gibberish.
But if you look deeper, you can see the Gold Hill sign as a bizarre but encouraging artifact of the advances we’ve made in mathematics. We’ve developed (and propagated) technologies for computing with ease. We’ve built (and popularized) an impressively abstract concept of number. And we’ve grown so prolific at gathering and processing data that we practically do it in our sleep.
The Gold Hill sign is still wrong. But it’s wrong for fascinating reasons.
You seriously believe that the sign wasn’t created with the painter’s tongue firmly planted in his/her cheek? On that score, I think you’re in error. And you could have written the column without making that assumption. Of course, I could be totally wrong, but so could you, and I’m surprised you weighed in on this without knowing for sure as if it was clear that there’d been an unconscious error instead of a math joke.
Touché. Added a parenthetical acknowledging the possibility.
I honestly don’t know either way! Could’ve been sign maker having some fun. Could’ve been a real error. (I’m not much of a journalist, apparently.)
I agree completely with your great insights. Math is about abstraction. You can use multiplication with any units you like, you could even compute the content of four cans with radius 10 and height 20 each (which requires multiplying 5 numbers, n, h, r, r and pi).
A more interesting example is a vector space. It makes perfectly sense if it consists of functions instead of arrows in space or plane. Even orthogonal projection is useful. It determines the function in a subspace of least distance! That makes Fourier developments possible, using abstract tricks like complex numbers. Polynomial interpolation, another original geometric concept, is then applied (together with a trick) to make the this fast (FFT) and run your MP3 player.
Yep, math is anything, but not useless.
Excellent job! So refreshing to see a positive take on it.
This is not the only such error in Colorado: here is one for Snowmass Village.
I do not know what rules sign-makers must adhere to in CO…
There’s one in New Cuyama, California as well (see https://en.wikipedia.org/wiki/New_Cuyama,_California).
Now it’s time to have a go at Value Added Modelling. It won’t be long before the UK goes down this mysterious and secret method of scoring teachers. The original method has its good points but the implementation in the US is bizarre, to say the least.
You could read this:
Value-Added Modeling:
The Challenge of Measuring Educational Outcomes
by Barbara Elizabeth Stewart
ABOUT THE AUTHOR
Barbara Elizabeth Stewart, a graduate of St. John’s College in Santa Fe, New Mexico, and of Columbia University, has been a journalist for some twenty years. She has written for The New York Times, The Washington Post and the Observer and the Telegraph in Britain, among other publications. She lives in New York City and has a particular interest in education.
©2006
I have a link to this paper but it appears to be broken.
I have a copy on my machine and can email it if you feel like joining the fray.
The method she describes could work, but the implementation has proved too difficult, and a bastadardized version has been developed, which is along the lines of the Colorado sign.
My math teacher just went over the definition of addition which is: the counting of similar things. (Or something similar to that, I don’t always get the wording right) If that sign was not meant as a joke, then the painter forgot what addition really is! Haha, what a timely post for me!
Euclid never multiplied more than three numbers together? What about his proof of the infinitude of the primes? I think you are short changing him and the Greeks a bit.
The sign is great though, and I appreciate the general point.
I’m getting the Euclid thing second hand – my understanding is that he takes pains to avoid it (perhaps “never in a single step” would be more accurate) but I should really verify this now that I’m citing it.
Take a look at how Euclid actually formulates his proof:
http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html
Yep, doesn’t require more than three unspecified primes for the classic proof. The power of generalizing rears its head again.
I love it! A sign maker with a sense of humor! Good for him or her.
I vote for “joke” over “error”, but as a data guy (and homeschooling dad) I think there are more issues with the presented table than the summation.
None of the labels are complete. “Est.” ? Estimated? Esteemed? Estragons? Same sort of potential confusion with “Elev.” s and “Pop.”s. Now, happening to know that estragon and poppies are kinds of medicinal herbs, I might infer that the unknown label “Elev” is ALSO a kind of herb. I might then suppose I am reading a recipe…
But then, the table lets me down even more. No units of measure. How many poppies? Or how many grams of crushed poppies? Or how many volume-ounces, or drams of solution?
Then, how do we know we have been tasked to calculate a sum rather than, for instance, a total volume? (Product of the three values?) And even if we ARE adding — isn’t that a negative value sign in front of each numeral? Shouldn’t the sum of three negative numbers be, itself, a negative? (For that matter, the product of three negatives is negative) But there is no such negative sign on the last entry…
‘Est.’ is ‘Established’
@Tarn Somervell Fletcher: I did not take @pouncer to be seriously confused about any of those abbreviations, but rather playing a game related to the one the signmaker(s) are playing.
Thank you. But I was then correct that ‘Pop.’ is for poppies? And did you have any guess (I still have only one but it is seriously inconsistent with herbal recipes ) regarding ‘elev’?
It’s a mixture of Cardamom (Elettaria cardamomum) and Lovage (Levisticum officinale).
The percentage of each is a secret known only to those who carry the herbal gene, visit the herbal temple in Scotland twice every four months, and hop on their left foot at least 7 times every Monday.
Speaking of carrying out an operation without investigating its meaning, it has always irked me that math textbooks will teach how to average a set of numbers (add up all the numbers, then divide by how many numbers there are) without explaining that the set of numbers must display a normal distribution for an average to have meaning.
And to further obscure the issue the exercises use numbers that are selected at random (Dave has $13, Mike has $124, and Steve has $4,864. What is the average amount of money they are carrying?). Thus, students master the operation but don’t understand what it means.
Probably textbooks don’t say that because that’s not true. Which is a pretty good reason. The average (arithmetic mean) always has meaning.
For that matter, the standard deviation always has meaning, although its interpretation in terms of probability (related to the cumulative distribution function) is not straightforward unless the distribution is normal.
“For Greek mathematicians, a single number represented a length. The product of two numbers stood for an area. And the product of three numbers represented a volume.”
That’s not true for Euclid, and definitely not for Diophantus. I haven’t read any others, so I can’t speak for them.
Euclid’s number theory proofs are geometric, but I get the feeling that’s simply because geometry was the language that he had. Geometry can work with (positive) rationals and a subset of the reals (anything you can make from the rationals through addition, subtraction, multiplication, division, and square and cube roots), but Euclid restricts numbers to positive integers.
Multiplication is repeated addition, not the area of a rectangle: “A number is said to multiply a(nother) number when the (number being) multiplied is added (to itself) as many times as there are units in the former (number), and (thereby) some (other number) is produced”. But the connection to areas and volumes was definitely recognised, as squares and cubes must have a geometric inspiration.
The proof for the infinity of primes doesn’t use multiplication of more than three, but those three stand for the whole finite set that’s being hypothesised. It’s clear that if there are four, five, or a hundred thousand primes, then all of them are to be multiplied.
Five hundred or so years later, Diophantus accepts positive rationals as numbers, and works with expressions involving powers up to the sixth (his notation, while terrible by our standards, could handle higher powers; the problems he solved just didn’t need them).
There’s a curious mixture of the geometric and number-as-pure-number here. For example, problem 17 of book 6 is “To find a right-angled triangle such that the area added to the hypotenuse gives a square, while the perimeter is a cube”.
I don’t know what you’d call Diophantus’ work, if not algebra. He could multiply polynomials, factorise some of them, and solve quadratics.
Heath’s translations of both are out of copyright, and worth reading. Maybe not cover-to-cover unless you’re exceptionally enthusiastic, but the introductions are very informative.
Gregg: isn’t it enough that the distribution be symmetric?
There is about a half a millennium between Euclid and Diophantus. It would be surprising if the latter’s take on mathematics was unchanged from that of Euclid and his contemporaries. The fact that both are Greek (and the latter an Alexandrian Greek at that) doesn’t mean all that much in terms of the claim you’re trying to refute or undermine.
This is probably not a great analogy, but I can’t help but think about the evidence we have of cultures that counted “One, two, three, many,” a notion that crops up in various places in our language and mathematics today (e.g., “monomials, binomials, trinomials, polynomials), and which may be grounded in the three-dimensionality of human experience (if not abstract thought). I can’t say with confidence that Euclid never thought about multiplying more than three numbers together or that if he didn’t that it was due to geometric/spatial thinking that was restricted to the 3 dimensions of ordinary human experience.
I do believe, however, that the ancient Greek mathematicians were not algebraists and according to various histories of mathematics did not fully trust “number” but rather relied on spatial/geometric reasoning almost exclusively. We know that there were religious aspects of the mathematics of some of the ancient Greeks (the Pythagoreans, most famously). Notions like irrational numbers were very disturbing to them, it seems (the incommensurability of an integer-length square to its diagonals really upset the Pythagoreans, as I’m sure you’ve heard).
That’s hardly the first or only example of people, even highly knowledgeable people, finding certain mathematical ideas unacceptable. See this short 3 part video series by William Dunham on the solution of the cubic equation for evidence that even Gerolamo Cardano eschewed using negative numbers (and hence made his work more complicated than it needed to be). That was in the 16th century! And Gauss apparently invented the term “nonEuclidean geometry” but never published any of his work on the subject. I’m not sure if that reflects his own doubts or just his propensity to be conservative about what he submitted for public scrutiny.
Finally, I’d be very cautious about the statement “Multiplication is repeated addition,” which you offer up as if it were a universally accepted fact. It’s not. There are aspects of multiplicative reasoning that don’t work very well when restricted by the notion of multiplication as being tantamount to repeated addition. Keith Devlin argued in series of articles about 10 years ago that restricted multiplication that way is problematic in various ways and I agree with him. I’m not going to repeat the extensive arguments but I’ll throw out one notion for people to chew over: if multiplication “is” repeated addition, why do we need to pay attention to things like common denominators when we add/subtract fractions, but not when we multiply/divide them? Similarly, why must we be talking about the same units of measure for the former but not for the latter? Compound units like “manhours” are commonplace and seem to cause few people any conceptual or philosophical concerns. Adding men and hours, however, would raise just about everyone’s eyebrows.
“Finally, I’d be very cautious about the statement “Multiplication is repeated addition,” which you offer up as if it were a universally accepted fact. It’s not.”
I was talking about Euclid there. To him, in the arithmetical part of the Elements, it was. His definition of multiplication is repeated addition (book 7, definition 15). That’s a perfectly fine definition if your concept of numbers only includes positive integers. It’s only when you start using fractions that you need to change it, and that didn’t happen until centuries later.
There is some geometric language in the next couple of definitions (the product of two numbers is a ‘plane’ number, and the product of three a ‘solid’). But in the actual proofs, multiplication is done by addition of lengths in a line. It was not an area.
The remaining question is: 118 inhabitants, when? Is someone updating the sign whenever needed?
How strange. I would understand if the sign maker included the number of people/the area (otherwise known as the population density), but this sign just does not make sense to me. It would be interesting to find out exactly what the sign maker was thinking when he made this.
Perhaps they were thinking the sign would be a good conversation starter
Do I need more coffee, or is there a mistake in https://mathwithbaddrawings.files.wordpress.com/2015/08/20150807105425_00023.jpg, the illustration of multiplication in Hindu-Arabic vs. Roman numerals? By my math, 6 X 34 = 204, not 184, and if we assume for some reason that 184 is correct, 184 + 680 doesn’t equal 884.
(Let me guess: you did that on purpose to see if anyone would notice. 🙂
I think the sign is a joke and would be funnier if they had performed other computations than addition on the value. Say, they’d divided the established date by the elevation and then multiplied by the number of people. That calculation could even have units (people-years/mile, or something like that).
I also love that your commenters found two more signs.
This sign is in the hills just a few miles from Boulder, home to the University of Colorado, many science and tech headquarters (Colorado’s Silicon Valley), well known for geeks and pranksters.
Not that correlation equals causation, but in this particular case, I would hypothesize that it increases the probability.
864! I can’t be the only person who noticed, right?!?!
I mean 884, but 204+680. 🙂
“It’s far easier to compute with our friendly numerals than it was with, say, Roman numerals.”
is kind of not true. it’s true that it’s harder to use roman numerals with our modern paper algorithms, but romans didn’t use paper algorithms. They used counting boards (basically an abacus). And while roman numerals are not good for symbolic manipulation, they’re great as records of abacus bead locations. That’s why roman numerals used both iiii and iv for the number 4. iiii refers to 4 stones in the one’s place, while iv refers to the act of adding 4 but subtracting a 1 stone and adding a 5 stone.
Back when arabic numerals and paper algorithms were first appearing in Europe, arithmeticists tried to settle the question of which method was faster by having competitions. You can see a print of one of these competitions in Swetz’ book _Capitalism and Arithmetic_, which is really a great read. Although the question of which method was faster was not settled until the modern era.
The answer is the opposite of what people trained in arabic numerals tend to think. Abacus algorithms (like those recorded by roman numerals), are much, much faster for mental computation than arabic numerals. Part of this is because abacus algorithms tend to be big-endian, rather than small-endian, which gives faster estimates. But the other reason is because keeping an abacus in your head (or more accurately, the finger motions of manipulating an abacus) is much more efficient than keeping arabic numbers in your head. Lightning calculators train with abacus algorithms for exactly this reason. You can see some examples in youtube videos of Flash Anzan, which is a sport for competitive addition. Here are some examples:
9 year olds adding 30 numbers in 20 seconds while playing a rhyming game: https://www.youtube.com/watch?v=_vGMsVirYKs
flash anzan national competition: https://www.youtube.com/watch?v=7ktpme4xcoQ
7 year old abacus training: https://www.youtube.com/watch?v=GQtqlB-jXO0
So is roman numerals are faster and more accurate, why do we use arabic numerals? The answer is that paper algorithms rose in popularity around the same time as the rise of the merchant class during the Renaissance. Merchants wanted ways to keep better records at the same time that the price of paper was falling due to the aftermath of the black plague. Paper provides a much better way of keeping records and checking work. So paper algorithms became much more important.
This is not to say that paper algorithms don’t have advantages. But those advantages aren’t in speed or accuracy. They’re conceptual. Once paper algorithms became popular in Europe, there was an explosion in mathematical algebra. Cardano and Ferrarri solved the cubic and in the 1500s. Calculus followed shortly after in the 1600s, etc. This is because (and I’m hypothesizing here, but it makes a lot of sense) once mathematics transitions from pushing around objects (abacus beads and stones) to pushing around symbols (arabic numerals), certain ideas became more accessible: negative numbers, variables and unknowns, complex numbers — things we can’t see and touch easily but result more naturally from manipulating symbols according to rules.
In short the advantage of arabic numerals is not speed, it’s abstraction.
I’m going to pose a very simplistic question here and ask you ‘Do you think the reason that Roman numerals are easier to compute than arabic numerals is because they are more visually based?’ Human beings tend to respond to visual stimuli and counting by moving objects may have been a better way than using arabic numerals. I remember being taught to use an abacus in school as a young child.
It’s clearly a checksum; the state register must have the entry “Gold Hill — 10440” which allows visitors to know whether or not they’ve arrived at the correct Gold Hill.