On his Web site at
http://www.j-bradford-delong.net/movable_type/2004_archives/000734.html
Brad DeLong, who is sometimes on this list, quoted the mathematician
Benjamin Peirce as saying
Gentlemen, that is surely true, it is absolutely paradoxical; we
cannot understand it, and we don't know what it means. But we have
proved it, and therefore we know it must be the truth.
about Euler's equation. Peirce had first proved the equation
i pi
e + 1 = 0
but that is as far as he got. The equation certainly is wonderful:
it relates the five most important numbers in mathematics:
* e, which tells you how quickly a quantity grows when its growth depends
on how much has grown before;
* i, the square root of minus one;
* pi, the ratio of the circumference of a circle to its diameter;
* one, the amount of a single instance;
* zero, which is not there.
Peirce's comment got me thinking: although he was a famous
mathematician, he was unable to express to others the meanings which
underlie mathematics, although there is no doubt that he worked with
them. Since he was able to prove Euler's equation to his own
satisfaction, but not understand its meaning, his remark demonstrates
what I call `Proof Without Understanding'.
This got me to considering the opposite, `Understanding Without
Proof'.
For me, understanding is a step up. When I first heard of Euler's
equation I could neither prove it nor understand it.
But I now think there is a way to gain understanding.
George Lakoff and his colleague, Rafael E. N��ez, say that mathematics
consists of metaphor piled on metaphor. Moreover, the metaphors are
so blended and transformed that people seldom see them.
This idea comes from their book
Where Mathematics Comes From:
How the Embodied Mind Brings Mathematics into Being,
Basic Books, 2000
ISBN 0-465-03770-4
The authors say that mathematics is based on `inference-preserving
cross-domain mappings' that are
... a cognitive mechanism for allowing us to reason about one kind
of thing as if it were another.
Lakoff and N��ez write that infants can see the sizes of groups of up
to four objects. Moreover, infants understand addition and
subtraction prior to the development of language. The authors contend
that arithmetic comes from an inference-preserving extension of this
ability into larger numbers.
Moreover, the Lakoff and N��ez go on to argue that there are actually
four `grounding' metaphors that based on experiences many of us had as
children:
* adding and taking away objects from a collection (playing with
pebbles);
* construction of a larger whole from smaller objects (playing
with blocks);
* measuring the width or height of something (by stretching our
hands to the ends of the object or standing up to see how high
it is);
* moving from one place to another (by crawling or walking).
They say that these experiences provide us with four metaphors that
work with arithmetic: four inference-preserving cross-domain mapping
mechanisms that work consistently with each other and the world.
What do you think?
(A couple of days ago, I wrote a bit on this topic for my Web site
http://www.rattlesnake.com/notions/math-metaphor.html
(Parts of that page duplicate these comments; but on that Web page, I
also give an explanation of why Euler's equation is true -- not a
mathematical proof, but an explanation of most of the metaphors that
go into making it meaningful. For example, I say
... imagine multiplying two with itself some fractional amount,
such as two and a half times. This is hard, since ordinary
multiplication can only operate as an integral whole. However, we
do know that two times two is four, and that two times two times
two is eight. So if we were able to multiple two with itself 2.5
times, the result would be somewhere between four and eight. ...
(I don't go into that here.)
--
Robert J. Chassell Rattlesnake Enterprises
As I slowly update it, [EMAIL PROTECTED]
I rewrite a "What's New" segment for http://www.rattlesnake.com
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