"Exponential Growth", "Exponentially"

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I guess this is how new words supplant old ones.
One of those nasty shocks like learning in childhood that everyone eventually dies
(ok, intentional hyperbole, expressive of a certain degree of exasperation; do not take the above completely literally)
is when you find out that to some people, the phrase "exponential growth" means, simply, extremely or surprisingly fast growth. Now imagine that you're talking about a population that grows at a rate jointly proportional to the present population size and the amount by which the population falls short of the carrying capacity. The population is NOT growing exponentially when it's growing fastest not anywhere near exponentially. But when it's near the beginning of its growth, and growing much more slowly, then it's growing approximately exponentially. Pointing this out seems sure to cause illiterates to find this contradictory.
In childhood, one goes through the reasoning that makes it clear why exponential functions eventually grow faster than polynomials even if in the short run they grow more slowly. Thereafter one understands what "exponential growth" is.

Somehow , it seems, persons who never go throught that line of reasoning come to absorb the phrase "exponential growth" into their vocabulary, and of course, do not even suspect that they are clueless about what it means.
Then someone says something like "We would need an exponentially larger budget in order to ...", so that the word "exponential" now means "very much larger". A new word that is spelled and pronounced the same way as the old one but means something completely different has appeared. What the new word has to do with exponents (those things written in superscript in math) becomes obscure. The whole concept of exponential growth is about the dependence of one quantity on another, but in the new usage, it's about just one quantity. Mike Hardy
I guess this is how new words supplant old ones. One of those nasty shocks like learning in childhood that ... dependence of one quantity on another, but in the new usage, it's about just one quantity. Mike Hardy

I have never expected mathematical terms to be applied appropriately in common speech. Even percentages are often abused. Does 500% bigger mean 5 times as big or 6 times as big?
In the recent thread on the subject, I only objected when people started quoting formulae. If you are trying to talk serious mathematics then I would prefer that you got it right. In ordinary speech, I don't much care. Even in serious maths, the meaning can vary. Evan's definition and mine were not the same yet I would not say that either was wrong. They are just from different areas of a very large subject. There is no ISO standard for mathematical terms. Different subject areas and even different authors vary. Evan's definition was appropriate in the area of program complexity. Mine was appropriate in the abstract study of real (*) functions. (*) The meaning of real here is quite different from its day to day meaning.

There is a phrase which is becoming popular in marketing recently which amuses me: "The fastest growing X". X could be a bank, club, plague, or many other things. I have never checked but it is quite possible that it is being used correctly. Nonetheless I suspect it is intended to mislead. As you point out, it is easier for something small to grow fast. If your four week old bank had 1, 2, 4, 8 customers in its first four weeks then it probably is the fastest growing bank in the country. The established banks might need to get the whole country as customers to match this growth.

I have also noticed "exponential" being used for a single step rather than a trend. It seems to be replacing "Bigger by an order of magnitude". That was better but still rather meaningless. I am also disappointed but I don't fight against it for the same reason that I don't try to knock down walls with my head.
Trying to apply mathematical terms to common speech could have amusing consequences. Imagine insisting that the terms: rational, irrational, transcendental, real, imaginary and complex were appropriately applied to numbers.
Seán O'Leathlóbhair
There is a phrase which is becoming popular in marketing recently which amuses me: "The fastest growing X". X could ... bank in the country. The established banks might need to get the whole country as customers to match this growth.

When I started at HP in 1989, there was a joke going round that Sun was bragging about how their MTBF(1) had doubled, while HP's had only gone up by ten percent. They had gone from two weeks to four, while we had added a little over a month to our year.
Then again, anybody who's read The Innovator's Dilemma knows that when it comes to technologies, you ignore fast-growing, but small, competitors at your peril.
(1) Mean time between failures.

Evan Kirshenbaum + HP Laboratories >You may hate gravity, but gravity
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When I started at HP in 1989, there was a joke going round that Sun was bragging about how their ... The Innovator's Dilemma knows that when it comes to technologies, you ignore fast-growing, but small, competitors at your peril.

My favorite percentage joke is a Dilbert cartoon wherein the boss was demanding an investigation into why employees were calling in sick 40% of the time on either Monday or Friday*.
Don
Kansas City
*Monday and Friday account for 40% of the work week.
Sean O'Leathlobhair wrote in part:
Trying to apply mathematical terms to common speech could have amusing consequences. Imagine insisting that the terms: rational, irrational, transcendental, real, imaginary and complex were appropriately applied to numbers.

From
on re us/trucker rules:

They pointed to studies showing that the risk of crashes rises geometrically after the 10th and 11th hour of driving.

Huh?

Skitt (in Hayward, California)
www.geocities.com/opus731/
Sean O'Leathlobhair wrote in part:

Trying to apply mathematical terms to common speech could have ... transcendental, real, imaginary and complex were appropriately applied to numbers.

From http://news.yahoo.com/news?tmpl=story&u=/ap/20040716/ap on re us/trucker rules: They pointed to studies showing that the risk of crashes rises geometrically after the 10th and 11th hour of driving. Huh?

I don't know what studies they refer to, but it's not an unreasonable thing to claim. Read literally, it says that the risk of crashes starting with the tenth hour(1) is best modelled by an exponential curve. It doesn't say anything about what happens before that. It might also be exponential or it might be some slower function.(2) Of course, the multiplier might be so small as to not be worth worrying about, but then it wouldn't have been worth bringing up. (And, presumably, when they did the "pointing", they did more than simply say "it rises geometrically".)
(1) The "after the 10th and 11th hour" is presumably because those are the old and new (but thrown out) thresholds.
(2) Or faster, of course.

Evan Kirshenbaum + HP Laboratories >Usenet is like Tetris for people
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From on re us/trucker rules They pointed to studies showing that the risk of crashes rises geometrically after the 10th and 11th hour of driving. Huh?

Could you please expand on that "huh"? I figure you probably know the difference between an arithmetic progression and a geometric progression, so that's not it. I don't see a problem with the basic idea that, after a certain point, there are increasingly more accidents per hour, not just more of the same, steady, constant increase.

Best Donna Richoux
From on re us/trucker rules They pointed to studies showing that the risk of crashes rises geometrically after the 10th and 11th hour of driving. Huh?

Could you please expand on that "huh"? I figure you probably know the difference between an arithmetic progression and a geometric progression, so that's not it. I don't see a problem with the basic idea that, after a certain point, there are increasingly more accidents per hour, not just more of the same, steady, constant increase.

Best Donna Richoux
From on re us/trucker rules They pointed to studies showing that the risk of crashes rises geometrically after the 10th and 11th hour of driving. Huh?

Could you please expand on that "huh"? I figure you probably know the difference between an arithmetic progression and a geometric progression, so that's not it. I don't see a problem with the basic idea that, after a certain point, there are increasingly more accidents per hour, not just more of the same, steady, constant increase.

Best Donna Richoux
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