Yesterday's term, "begging the question", meant "making a circular argument". 'Circular' brings us to this week's discussion of terms from mathematics.

circular – using a premise to prove a conclusion that in turn is used to prove the premise: a circular argument.

So says the dictionary. But I find, in sample quotes, that the term is used principally in other ways.

circular argument – an argument between two people that 'goes around in circles', making no progress (1st and 2nd quotes)
circular argument – a 'self-reinforcing' process, in which progress in one area both requires and stimulates progress in another area (3rd quote)

He had been right not to speak to her; there were no more words to be said, only a circular argument going nowhere.
– Maeve Binchy, Scarlet Feather

The moderator was gettitng panicky now. He could see the clock telling him he had six minutes to bump this squabble out of the endless circular argument about gun control and turn it into a news story.
– Thomas Perry Pursuit

We must crack the supply side by making broadband affordable and available to create network access for application services, but to a degree it is a circular argument because application services will drive bandwidth and stimulate demand for broadband.
– Peter McCarthy-Ward, East of England regional director at BT

I wonder, is a circular argument as bad as a circular definition?

I recall something from Jonson, "A net is an object composed of interstitial vacuities (sic)."

Four terms, which in geometry name four related curves, also have related meanings in rhetoric. We saw the first of these yesterday (circle; circular argument), and today we'll look at the other three. On a future day we'll see how the geometrical and rhetorical meanings are related in concept and in etymology. I leave it to readers to post a bit, on the board, telling where the geometric curves appear in everyday life.

ellipse – an oval-shaped curve; a circle that has been 'stretched' (note: not "egg-shaped" as some say; an egg is wider at one end; an ellipse is not). elliptical (rhetoric) – 1. of extreme economy in speech or writing; hence, 2. having a part omitted (see here at 'ellipsis'); 3. deliberately obscure

His description of the Lewinsky affair is just as elliptical (and he doesn't get to it until page 773). – Thane Peterson, reviewing Bill Clinton's new book My Life, in Business Week, June 28, 2004

In his statement Fed[eral Reserve] chairman Alan Greenspan, just reappointed to his fifth term, was less elliptical than usual. – Alex Brummer, of the London Daily Mail, in This is London, 2 July 2004

parabola – a certain geometrical curve (a thrown ball travels in a parabola as it rises and then falls to the ground). parable (rhetoric) – a story illustrating a moral or religious lesson; an allegory)

hyperbola – a certain curve, opening more widely than a parabola. hyperbole (rhetoric) – extravagant exaggeration (This book weighs a ton.)

This research leads us to one hyperbolic conclusion: Neckties are bad for you. OK, the study didn’t go that far. But you can’t blame us for trying. – Beware attack of the killer ties, Salem (Oregon) Statesman Journal, June 9, 2004

Kansas is a state prone to extreme temperatures and hyperbolic meteorologists. – The Wichita Eagle, June 1, 2004

The dictionary definitions for our next two words do not match the usages I find in the quotations.

In proper mathematical use an exponential change need not be rapid.¹ But in popular usage it usually means a large and explosively-rapid increase, whether by very fast growth or by a one-time "jump". MW's definition says "characterized by or being an extremely rapid increase," but the word as actually used refers to a very large change, usually (but not always; see first two quotes) a rapid one and an increasing one.

exponential – characterized by very large increase or other change, particularly a very rapid one; sometimes implies due to interaction with other factors.

Microsoft continues to appeal a huge fine … by the time anything is decided the result will matter exponentially less than it does today. – David Coursey, eWeek, June 30, 2004

As hard as the cowboy life was for Wilcox, life was exponentially more severe for the Fremont [Indians, a thousand years ago]. – Tom Kenworthy, USA Today, June 30, 2004

Melani said, "... It's an exponential change in the need for health care once you get into the over-50 age." – Pittsburg Post-Gazette Sunday, July 4, 2004

Beleaguered by exponential increases in insurance and fuel costs ... – Valerie Hubbard, Richmond Times-Dispatch, June 7, 2004

public libraries … are turning to stronger tactics to track down overdue material. Ignore the traditional overdue notice, and you may hear from a debt-collection agency. … "The value of this service is exponential. It's not the $20 book we get back. It's the $20 book times all the people who will read it after it is returned," Catrambone says. – Ellen Hale, USA Today, June 27, 2004

In scientific usage, 'exponential' growth is basically "the bigger it is, the faster it grows", a sort of snowball effect where prior growth makes for more rapid further growth. More exactly, "as the quantity gets bigger, it grows faster by a proportionate amount." That is, if a 100,000 population is growing exponentially and adds 5,000 in the first year, then when it has become (say) 20% larger its growth the next year will not just be more than 5,000, it will be precisely 20% more (that is, 6,000).
. . . .Exponential change can be negative (shrinkage), as in the decay of a radioactive substance, in which the shrinkage-per-year diminishes each year.

---

¹ Compounding of interest creates exponential growth, which is rapid if the bank pays 15% interest - but is slow if the bank pays 1% interest. Exponential change can be negative (shrinkage), as in the decay of a radioactive substance, in which the shrinkage-per-year diminishes each year.

Wordcrafter is right about 'exponential'. It is almost never used correctly, and has become a fancy-pants word for 'big'.

Real exponential growth almost never occurs in reality, because exponential growth, slow or fast, eventually gets infinitely large. Bacteria colonies approximate exponential growth when they are small and have plenty of nutrients, but they will eventually outgrow their energy and nutrient sources, and growth ceases to be exponential.

One area where things can grow exponentially is mathematical complexity. This is exploited by encryption algorithms in which the number of possible solutions grows exponentially with the size of the key, and adding an extra bit to the key makes the solution exponentially more difficult.

Exponential decay, on the other hand, happens all the time.

Good to see you back, neveu. I'd gotten interested in a question you posed a few weeks ago, researched it, but then forgot about it. Seeing you reminded me, and I'll follow up now.

An example of exponential growth I often give in training sessions is a simple one - and I have never yet had a delegate believe the answer.

Simply take a sheet of paper, one hundredth of an inch thick (and as large in area as you wish) and cut or fold it into two (which makes it a fiftieth of an inch thick). Take that and repeat the process which makes it a twenty-fifth of an inch. Repeat the process fifty times. The question is, how thick will the pile of paper be?

The answer, which few believe until they try it and find that their calculators overflow, is many thousands of MILES.

Try it for yourself - you'll need to divide by 63,360 (the number of inches in a mile) after about the thirtieth multiplication or your calculator, too, will overflow.

quote:

---

... and I have never yet had a delegate believe the answer.

---

If no-one believes you, it might be an idea to give a different example.

Mathematician Benoit Mandelbrot coined the term 'fractal' in 1975, defining it as "a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Right. Many less-technical definitions are nonetheless near-incomprehensible (such as the 19 you'll find by putting define:fractal into a Google search box). Let's see if we can make this clearer.

fractal adj – characterized by having small parts that are miniature replicas of larger parts. In other words, similar, to itself, at different scales.

Branching in nature is often fractal. That means that branching at the twig ends looks very similar to branching near the trunk. The branching is the same at all scales of size.
– Walter Witschey, Computer studies branch out, Richmond Times-Dispatch, June 24, 2004

The earth is round, but the small portion you can see appears flat. In other words, a sphere (or circle) is not fractal: a very small part does not look like a miniature of the larger figure.

fractal noun – a fractal figure or picture

A computer program can make factals recursively. The procedure is to apply a rule to make a simple starting figure more complicated; then reapply the same rule to the resulting figure; then re-reapply; etc.¹ One can add to the mix other elements, such as randomness.

Fractals are important in computer graphics, for they can generate wonderfully detailed images of such natural features or textures such as mountains, clouds, trees and forests. (For these images, credit Vistapro Pictures Vistapro Pictures and Kevin Meinert respectively.) Fractals can also be beautiful as abstract art, and one can buy programs to generate them.

I have simplified quite a bit, but this gives the basic idea.

---

¹ In this example starts with an equal-sided triangle. At each step, a triangle of 1/3 the prior size is added atop the middle third of each side.

iff - if and only if

A useful word - if it is a word. M-W considers it a word, but AHD lists it only as an abbreviation. Apparently the jury is still out on this one.

It's next to impossible to search for usage, since a google search brings up cites to IFF meaning 'International Flavors and Fragrances' or 'Illinois Facilities Fund' or the like.

Wordcrafter's new entry "iff" sent me back to "Symbolic Logic" by I. M. Copi (MacMillan, 1967).

Memory played near but not quite. I did not find "iff," but did see "wff," defined as "well formed formula."

I learned iff many years ago and find plenty of use for it...but so few people who would understand it without lengthy explanation that I regretfully abandoned it except when writing to mathematicians!

but so few people who would understand it without lengthy explanation that I regretfully abandoned it except when writing to mathematicians!
Hmmm, can't say that I have had much occasion to write to mathematicians. Hab must keep better company than I do

This message has been edited. Last edited by: Kalleh,

I'd guess that using "iff" in conversation -- with mathematicians or not -- must be confusing, since it is presumably pronounced in the same way as "if".

I don't think 'iff' is a word, as it's pronounced 'if and only if', i.e. as an abbreviation. In contrast, 'wff' is actually pronounced 'woof' as well as 'well-formed formula'.

The French for 'iff' is 'ssi', which I presume is pronounced 'si et seulement si'.

To search for mathematical uses of 'iff' combine it with another term: {iff exists} brings up a lot.

hyperbole (rhetoric) – extravagant exaggeration (This book weighs a ton.)

QT from the Sun Times had a good column today, talking about hyperboles needing a holiday. I completely agree with him. Here are his examples:

"...the legendary King Arthur..."
"...the legendary Robin Hood..."
"...the legendary Dennis Rodman..."
"...the legendary Hard Rock Cafe..."
"...the legendary Tigers [sic] administrator Graham Richmond..."
"...legendary customer service..."

I don't know about in the USA, but both King Arthur and Robin Hood would correctly be described as legendary in the UK since neither person ever existed.

And customer service in many parts of the world is also legendary - in other words, it's a myth that it ever happened - and it doesn't happen now!

What seems to be happening is that the word legendary is shifting in meaning and is often taken to mean "famous" or "excellent"