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Terms from Mathematics

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July 01, 2004, 06:56
Terms from Mathematics
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)
July 01, 2004, 07:16
Robert Arvanitis
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)."

July 02, 2004, 08:50
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 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.)
July 04, 2004, 20:43
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. 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.
July 04, 2004, 22:42
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.
July 05, 2004, 09:56
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.
July 06, 2004, 00:33
Richard English
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.

Richard English
July 06, 2004, 01:31
... 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. Wink

Build a man a fire and he's warm for a day. Set a man on fire and he's warm for the rest of his life.
July 06, 2004, 09:11
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.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.
July 07, 2004, 08:46
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.
July 08, 2004, 09:08
Robert Arvanitis
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."

July 08, 2004, 14:59
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!
July 08, 2004, 19:26
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 Wink

This message has been edited. Last edited by: Kalleh,
July 09, 2004, 01:40
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".

Build a man a fire and he's warm for a day. Set a man on fire and he's warm for the rest of his life.
July 09, 2004, 11:21
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.
July 12, 2004, 20:53
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..."
July 13, 2004, 01:01
Richard English
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"

Richard English