Innumeracy Watch

Published

on July 10, 2015

. 47 Comments

While a team of four has just six interconnections, a team of 16 has 120 interconnections. It is near-exponential growth: n(n-1)/2.

— Rich Karlgaard and Michael S. Malone

Wall Street Journal

July 10, 2015

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47 Responses to “Innumeracy Watch”

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1 1 Windypundit
July 10, 2015 at 9:59 pm

…for sufficiently small values of exponential.
2 2 Jeff S
July 10, 2015 at 10:31 pm

In line with a recent U.S. Supreme Court ruling, the term “near-exponential” growth must be interpreted according to the intent of the authors. In the case at hand, the authors clearly intend to have this term apply to the binomial coefficient and triangular numbers.

Math has become too complicated not to be given a liberal reading.
3 3 Roger
July 11, 2015 at 1:20 am

If a mass-market newspaper just said “near-exponential growth”, then what sortof function would you expect? It seems okay to me.

More wrong is the usage of the term “fraternal twin” in this article.
http://www.nytimes.com/2015/07/12/magazine/the-mixed-up-brothers-of-bogota.html
4 4 Jonathan Kariv
July 11, 2015 at 1:56 am

That is pretty awful.

Can’t find the link just now but at some point there was a sporcle quiz on what answers people gave to “1/3+1/4”. 100 people gave 12 different answers…
5 5 William
July 11, 2015 at 10:30 am

@Roger:

Interesting article. Why is the usage of “fraternal twin” wrong?
6 6 Roger
July 11, 2015 at 10:50 am

The term “fraternal twins” is for those who have shared parents and a shared womb, but not genetically identical. The article uses it for guys who grew up in the same family, but are not biologically related at all.
7 7 Jens Fiederer
July 11, 2015 at 11:08 am

Well, they did say NEAR-exponential growth.
8 8 Jens Fiederer
July 11, 2015 at 11:09 am

Roger, I think the ASSUMPTION was that the boys were fraternal twins, since they were the same age and thought to be from the same mother.
9 9 Roger
July 11, 2015 at 3:04 pm

Yes, they were thought to be fraternal twins, and not identical twins. Then someone discovered last year that they must have been switched at birth, and that they are really identical twins, and not fraternal twins. I expect the article to describe them based on what we now know is correct. Instead it alternates between calling them identical twins and fraternal twins.
10 10 David
July 11, 2015 at 11:02 pm

Seems like an okay statement to me. The statement would fall flat on the average reader if it said, “it is quadratic growth: n(n-1)/2.” I agree with your innumeracy statement, but it is the innumeracy of the average reader, not the authors.
11 11 Harold
July 13, 2015 at 5:07 am

From google “exponential/ˌɛkspəˈnɛnʃ(ə)l/
adjective
1.(of an increase) becoming more and more rapid.

Seems to fit to me. Is the problem that it either IS exponential if we take the colloquial meaning of more and more rapid (i.e. not “near” exponential), and just wrong if we take the mathematical definition? Not quite sure, I seem to have missed somenthing.
12 12 Ken B
July 13, 2015 at 9:01 am

Eureka! P = NP
13 13 nobody.really
July 13, 2015 at 10:23 am

1. Eh. This isn’t a statement about mathematics; it’s a statement about vocabulary.

Would you say x*x exhibits exponential growth? Sure.

Would you say x*x/2 exhibits exponential growth? I guess. I agree with Google that “exponential growth” refers to a phenomenon that does not merely grow, but grows at a continually increasing rate. By that characterization, x*x/2 qualifies. But if you’re focused on magnitude, I’d guess you could say the function exhibited half exponential growth. But the term “exponential growth” would seem to be a bad way to focus on magnitude, given that my savings account also exhibits exponential growth — not that anyone could tell.

So would you say that x*(x-1)/2 exhibits near exponential growth? I guess. Given the context and the audience, how else would you describe it?

2. On the topic of people confusing vocabulary disputes for math disputes: Some people express alarm that people disagree about whether to characterize a square as a member of the class of rectangles (“Is a square a rectangle?”), and whether to characterize a circle as a member of the class of ellipses (“Is a circle and ellipse?”). Yes, these may be relevant matters for purposes of taking a multiple-choice quiz, but wholly irrelevant for understanding geometry. Thus the question has become a litmus test for determining what a person thinks is the purpose of studying mathematics.
14 14 Harold
July 13, 2015 at 11:57 am

“Thus the question has become a litmus test for determining what a person thinks is the purpose of studying mathematics.”

The whole point of maths is to have more jokes:
https://www.tes.co.uk/news/school-news/breaking-news/10-best-maths-jokes-ever-nominated-teachers-and-twitter

An extra one: Physicist and mathematician were looking at a building. They see two person walk in and a little later two people walk out. “The building is now empty” they hypothesise. This goes on all day, for every person going in the same number come out. At the end of the day they see two people go in and three people come out. Oh, our hypothesis was wrong says the physicist. “If one more person goes in that building will be empty again” says the mathematician.
15 15 Ben Kennedy
July 13, 2015 at 1:42 pm

Is the issue that this is a polynomial growth rate rather than a true exponential growth rate of the form a^n?
16 16 Nick
July 13, 2015 at 2:13 pm

If this is innumerate, than what is ‘near-exponential’ growth, exactly? How close do two growth processes need to be before they are ‘near’ each other, and what metric are we using to compare growth processes?
17 17 nobody.really
July 14, 2015 at 12:45 pm

Ack! You know, I think I completely misunderstood Landsburg’s point.

If you go back and read the actual article from which the Rich Karlgaard and Michael S. Malone quote derives, you find that they’re discussing how the Apple Watch has failed to find acceptance in the market. They speculate that this gizmo is simply too expensive compared to the alternative smart watches.

These alternatives include the Pebble, the anticipated Windows Watch, and most especially, the Innumeracy Watch. Because it derives its power as a function of the number of innumerate people in the population, the Innumeracy Watch has an almost limitless power supply. The authors would give a more precise estimate of the power supply but the calculation is kinda hard….
18 18 David E. Wallin
July 14, 2015 at 4:27 pm

In their example, 16 has 20 times more interconnections than 4. I suggest that is well less than 162,755 times as much.
19 19 Ken
July 14, 2015 at 10:02 pm

4 out of 3 struggle with math.
20 20 Henri Hein
July 14, 2015 at 11:51 pm

The term “near-exponential” seems nonsensical to me, in any context. I think I would personally have accepted a description of the function as “exponential” — as David points out, readers would probably not have understood more technically correct alternatives. However, by implying something can be almost exponential, the writers are mucking up the average readers’ already dicey understanding of the term.
21 21 Harold
July 15, 2015 at 6:57 am

#17 I believe they are bringing out an illiteracy clock soon, to be followed by an aircraft carrier powered by ill-informed press reports.
22 22 Ken Arromdee
July 16, 2015 at 3:41 pm

“Near exponential” means “is not actually exponential, but has many of the traits of being exponential that people find significant.” By this definition, it is indeed near exponential.
23 23 Henri Hein
July 17, 2015 at 9:55 pm

Exponential has “many” traits? You make it sound like a Dostoyevsky character. It has a pretty specific meaning. If by “people find significant” you mean that people think of anything growing faster than linear as exponential, you are probably correct. However, I still don’t see where “near” comes in. Either something grows faster than linear, or it does not. Either it is exponential, or it is not.
24 24 nivedita
July 18, 2015 at 9:58 am

@Henri Hein, would you find the terms “quasi-polynomial” equally nonsensical? It’s a standard term in computational complexity.

Exponential does not have a specific meaning divorced from context, even in mathematics. It can mean exp(x), the specific function, it can refer to any c^x, it can refer to any function that grows faster than c1^x but slower than c2^x for some c1 and c2, it can refer to any function that grows faster than c1^x but slower than c2^(x^k) for some c1, c2 and k.
25 25 Ricardo Cruz
July 18, 2015 at 5:47 pm

nobody.really wrote:
“Would you say x*x exhibits exponential growth? Sure.”

If you need math tutoring, and you do, call me.
26 26 Harold
July 20, 2015 at 4:41 am

Ricardo, as a taster for your tutoring services can you explain why that does not exhibit exponential growth? I would have thoght it did.
27 27 Michael Stack
July 20, 2015 at 12:28 pm

I’ll be perfectly frank:

1. I assumed X*X (or X^2) exhibits exponential growth.
2. I’d also assume that X*(X-1) is exponential growth.
3. I’d further assume dividing the result by 2 doesn’t change matters.

Assuming #1 describes exponential growth, it seems reasonable to describe the author’s function as “near” exponential.

Based on what I’ve read here (http://www.purplemath.com/modules/expofcns.htm), a true ‘exponential’ function is one where the power is the variable, but at least in conversational terms, I’ve understood exponential growth to mean growth that occurs at *some* power, typically 2 or greater.

It would be very nice for us innumerate readers to know in clear detail why the function listed is or is not exponential. Can somebody provide a clear explanation?
28 28 Harold
July 20, 2015 at 1:11 pm

#27 From your link: “This is the definition of exponential growth: that there is a consistent fixed period over which the function will double (or triple, or quadruple, etc; the point is that the change is always a fixed proportion). So if you hear somebody claiming that the world population is doubling every thirty years, you know he is claiming exponential growth.”

For that to happen the power must be the variable.
29 29 nobody.really
July 20, 2015 at 1:29 pm

Ok, maybe “near-exponential growth” suggests an inappropriate pseudo-scientific precision.

So let’s substitute the familiar phrase “meteoric growth.” ‘Cuz nothing implies things going up, Up, UP better than something that, whenever it comes to our attention, is inevitably falling.

After all, why speak in a way that annoys the math-heads, rather than annoying everybody? No need to be selfish….
30 30 Ricardo Cruz
July 20, 2015 at 3:09 pm

Harold, this article on the Big O notation has all types of functions:

https://en.wikipedia.org/wiki/Big_O_notation#Orders_of_common_functions

x*x=x² is called a quadratic function. It grows much slower than an exponential. Exponentials can be found in compound interest or atomic decays (negative power). They can be found in much physical phenomena because as you may remember stuff like dx=ax gets translated into x=a^t+c.
31 31 Ricardo Cruz
July 20, 2015 at 3:11 pm

Mistake: x=e^at+c. Please someone edit the comment for me, to avoid the embarrassment. :P
32 32 Henri Hein
July 20, 2015 at 3:31 pm

@nivedita:

Honestly, no, I would not have made sense of “quasi-polynomial.”

I understand there are gradations of exponential growth. Even Infinity has cardinality. What I object to is anything that blurs people’s understanding of how fast “exponential growth” really is. Growth rates are not like a color spectrum, or points on a line, but if you speak of them that way, mathematicians especially, people will come to think of them that way.
33 33 nobody.really
July 20, 2015 at 4:00 pm

nobody.really: Would you say x*x exhibits exponential growth? Sure.

Ricardo Cruz: If you need math tutoring, and you do, call me.

Harold: Ricardo, as a taster for your tutoring services can you explain why that does not exhibit exponential growth? I would have thought it did.

Ricardo Cruz: x*x=x² is called a quadratic function. It grows much slower than an exponential. Exponentials can be found in compound interest….

You know of an interest rate that is compounding faster than x*x? That’s a tutoring lesson I can sign up for — please share!
34 34 Ricardo Cruz
July 20, 2015 at 4:19 pm

The exponential was discovered by people studying compound interest.

You can see it is exponential by looking at its function:

x(0) = x0
x(t) = x(t-1) * (1+r)^t

where t are for instance months since the deposit, x is the money, x0 is the money lent, and r is the interest rate.

This is the traditional compound function. It does not mean in the real world it always works like this. I believe some financial applications do not pay interest on interest for instance.

For very small values of time, a quadratic is almost identical. But for big amount, they are completely different. I believe there is a computer science millennium challenge (P=NP) in proving whether you can work out any non-polynomial problem into a polynomial problem. Many computer scientists work on this, and things like cryptography rely on exponential growth to defend themselves. If you could solve them in polynomial time, or, worse, quadratic, internet commerce would not be possible.
35 35 Ricardo Cruz
July 20, 2015 at 4:50 pm

A website to plot functions: http://tinyurl.com/occdmug

If you consider the derivatives:
f = x^2
df = 2x
ddf = 2

Acceleration is constant.

f = 2^x
df ~= 2^x
ddf ~= 2^x

It keeps accelerating, and accelerating. (It is an approximation. You would had to multiply each derivative by log(2), so it would be even faster.)

I confess it is very hard for me to think of exponential growths. Intuitively, it all feels the same to me, which is why I myself try very hard to categorize growth rates correctly.
36 36 Ricardo Cruz
July 20, 2015 at 8:59 pm

I wrote that post a little too fast, and realized I messed up in the interest compound model.

The basic model is:

x(0) = x0
x(t) = x(t-1) * (1+r)

We can all agree with that, right?

This month, you get what you had plus interest.

Now, we want to solve it to get rid of that x(t-1), so we expand it:

x(t) = x(t-2) * (1+r) * (1+r)
x(t) = x(t-2) * (1+r)^2

We need to keep doing it:

x(t) = x(t-3) * (1+r)^3

In the end we can solve that dynamical system and obtain:

x(t) = x0 * (1+r)^t

Thus showing that interest compounding has exponential growth. QED

(Sorry, I did my masters in applied mathematics, but I studied something else during my undergrad. Sometimes I mess up. :P)
37 37 nobody.really
July 20, 2015 at 11:36 pm

According to https://en.wikipedia.org/wiki/Big_O_notation#Orders_of_common_functions

Quadratic: O(n2)

Exponential: O(cn) , c > 1

Perhaps I’m not following the notation, but it would appear to me that a quadratic is a special case of exponential.
38 38 nobody.really
July 20, 2015 at 11:39 pm

Let’s try that again.

According to https://en.wikipedia.org/wiki/Big_O_notation#Orders_of_common_functions
Quadratic: O(n^2)
Exponential: O(c^n), c > 1

Perhaps I’m not following the notation, but it would appear to me that a quadratic is a special case of exponential. Or perhaps, the set of quadratics overlaps the set of exponentials.
39 39 Harold
July 21, 2015 at 5:16 am

#33 From Purple maths site “Exponential growth is “bigger” and “faster” than polynomial growth. This means that, no matter what the degree is on a given polynomial, a given exponential function will eventually be bigger than the polynomial. Even though the exponential function may start out really, really small, it will eventually overtake the growth of the polynomial, since it doubles all the time.”

So x^10 is bigger than 10^x until x = 10, then the exponential is bigger.
40 40 RPLong
July 21, 2015 at 9:24 am

“Perhaps I’m not following the notation, but it would appear to me that a quadratic is a special case of exponential.”

Yes, and 1 is another special of e^x where x = 0, but who among us would call y = 1 an exponential function?
41 41 nobody.really
July 21, 2015 at 9:35 am

#33 From Purple maths site “Exponential growth is “bigger” and “faster” than polynomial growth. This means that, no matter what the degree is on a given polynomial, a given exponential function will eventually be bigger than the polynomial. Even though the exponential function may start out really, really small, it will eventually overtake the growth of the polynomial, since it doubles all the time.”

So x^10 is bigger than 10^x until x = 10, then the exponential is bigger.

HEY — I get that.

So yeah, I suggested that the formula in the article would approximate the growth curve of X^2. But since X^2 is NOT an exponential rate of growth, the fact that the formula might approximate this model doesn’t really help.

I learned something today. But I gotta say, Harold gets the tutor award.
42 42 nivedita
July 24, 2015 at 8:37 pm

Henrei Hein @32, I was responding to your rather expansive comment that near-exponential makes no sense, in any context. I fully agree that expanding this term to cover quadratic functions is not sensible, even for the lay audience.

At the same time, now you seem to be making an even more expansive assertion.. as a person who doesn’t know what certain mathematical terms mean, surely you shouldn’t be judging how mathematicians may use them.
43 43 Harold
July 27, 2015 at 4:57 am

Speaking of rapid growth, I was just reading about Graham’s number, which uses arrow notation. Too complex for me to try and describe here, but details here
https://en.wikipedia.org/wiki/Graham%27s_number

One snippet:
The magnitude of this first term, g1, is so large that it is practically incomprehensible, even though the above display is relatively easy to comprehend. Even n, the mere number of towers in this formula for g1, is far greater than the number of Planck volumes (roughly 10^185 of them) into which one can imagine subdividing the observable universe. And after this first term, still another 63 terms remain in the rapidly growing g sequence before Graham’s number G = g64 is reached.
44 44 Henri Hein
July 28, 2015 at 7:39 pm

@nivedita,

I cannot ask that mathematicians don’t confuse lay people what terminology used in the vernacular means formally? Then we have a fundamental disagreement about the role of experts.
45 45 nivedita
July 29, 2015 at 7:24 pm

@Henri Hein, I’m not sure what you’re saying now. Experts have many roles — the primary role of most mathematicians is to do mathematics, and teach mathematics to other budding mathematicians. This is nothing special about math, the primary role of for example most civil engineers is doing stuff like building bridges that don’t fall down, it is certainly not explaining to lay people why bridges do or don’t fall down, or even explaining that “resonance” or “harmonic” has a technical meaning that may be different from how it is used by lay people, or by experts in other fields.

I am saying, don’t ask mathematicians not to use words in any way they like among themselves — if they’re happy talking to each other about growth rates lying on a continuum, they don’t have to stop just because you don’t understand them. Obviously when a mathematician is speaking to a lay person, they have to talk a common language, so using words that the lay person does not know the meaning of would be bad.

But Steve’s post is not about mathematicians talking to lay people. The original WSJ article doesn’t appear to have any math people in it? It’s just lay people confusing other lay people.

Mr. Karlgaard, the publisher of Forbes, and Mr. Malone, a technology journalist
46 46 Henri Hein
July 30, 2015 at 2:01 am

@nivedita,

I was going to let it go, but here is one last attempt at laying it out for you. I really don’t think it’s complicated.

I don’t understand what “near-exponential” means. In any context. That’s just how it is. If that offends you, I won’t even apologize, except if you convinced me it was due to some neglect on my part.

Mathematics is a discipline with important input to other fields. As such, it’s important that the experts in it communicate its results outside the discipline. If they fail at this, the discipline becomes useless (at least to non-practitioners). If they communicate the results in a way that confuses others, it becomes worse than useless.

I really don’t understand what’s difficult or controversial about any of that.

Regarding the subject change, you seem to be laying that on me. I’m not sure why. You are the one that steered us into intra-disciplinary communication.
47 47 nivedita
July 30, 2015 at 7:53 pm

@Henri Hein, your lack of understanding doesn’t offend me. What does offend me is your view that your lack of understanding means other people shouldn’t use words that they understand perfectly well.

Innumeracy Watch

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