Joe Niederberger posted Jan 2, 2013 6:48 AM: > > PT III says: > >To provide the reader with some context so that he > or she can see Keith Devlin meant, to see that he is > not confused at all, here is part of what Keith > Devlin actually wrote: > >"What Exactly is Multiplication?" > >http://www.maa.org/devlin/devlin_01_11.html > > This is pretty funny and I can't pass it up. > Following the above ref. to one of Keithe's columns, > we find: > Keith D. says: > - > ------------------------------------------------------ > -- > The MIRA fallacy becomes very apparent when you > u consider my second example, where I take an elastic > band of length 7.5 inches and stretch it by a factor > of 3.8. The final length of the band is 28.5 inches. > But what are the units? What goes after the number > 3.8 in the calculation > > [3.8 - - -] x [7.5 INCHES] = 28.5 INCHES ? > > The answer is nothing. It has no units. In this case, > the 3.8 is a dimensionless scaling factor. > - > ------------------------------------------------------ > --- > > He is writing far too fast here to consider his > words. If no units are involved ("it has no units") > then stretched inches of rubber band and un-stretched > must be exactly the same to any commercial buyer. > Same with puffed rice and uncooked rice. An inch is > an inch. A cup is a cup. If I sell you 1000 yards > stretched inches of rubber band you have no recourse > if you expected 1000 yards UN-stretched. The very > "nature of multiplication" declares you have no case! > > Very funny indeed. > > Cheers, > Joe N > I really don't at all see anything that's wrong (/funny) with Keith D's argument (though he seems, to me, to be stating something quite obvious and not something terribly profound). Consider the following, taken from the examples you've provided:
Rubbber band: - -- unstretched: 1" - -- stretched: 3". These are NOT the same thing at all!
If I, as a commercial buyer, buy a rubber band - and the shopkeeper, as a commercial seller, sells me a rubber band - I guess we would both know whether we are transacting a stretched or an unstretched rubber band?
The 'stretching of the rubber band' occurs because some degree of force - of some appropriate units - is applied to the rubber band: the units of force are, in fact, quite different from the units of length, and the force units do not enter into the commercial transaction at all! The 'stretched rubber band' is entirely a different product, as a matter of fact.
Similarly rice: - -- uncooked: volume 1 unit - -- puffed: volume 3 units Presumably I'd know what kind of rice I was seeking to buy in the first place? The puffed rice is, in fact, an entirely different product.
A cup: - -- whole and unbroken - -- broken I guess I would generally be in the market specifically for a whole and unbroken cup? Then I probably would not accept the broken pieces of the cup - that's a different product entirely! - and the shopkeeper would generally NOT present me with the broken pieces as representing a 'cup'.
On the other hand, if I actually needed some amount of broken glass or ceramic for some strange purpose, I'd be going out to buy precisely that - and the units I'd be dealing with would be quite different.
(I have not looked at the original Devlin column and I don't know what the 'MIRA fallacy' might be).
In India, some ration shops have the reprehensible practice of diluting milk with water - then adding some kind of petroleum product to bring the mix to an appropriate density so that they can sell it as 'milk': I guess the water and the petroleum product together would cost less than an equal volume of real milk.
Doing this is in fact a punishable offense under the Indian Penal Code (IPC): those shopkeepers are actually misrepresenting the product as 'milk': that's probably the least serious of their offenses. I believe there are a whole lot of sections of the IPC under which they'd be charged.
Some shopkeepers pay the food inspectors off and manage to get away with it for a while. That's a whole different slew of offenses. Some do not get caught at all.
One of the leading pharmaceutical companies I understand was charged with doing substantially the same kind of thing with one of its expensive drugs (in a much more sophisticated way, of course). I guess that was the one that got caught: as noted, some never get caught at all.