torsdag 15 april 2010

What is a Metaphor, in Mathematics?

According to a methaphor 
  • is a figure of speech in which an implied comparison is made between two unlike things that actually have something important in common. The word metaphor itself is a metaphor, coming from a Greek word meaning to "transfer" or "carry across." Metaphors "carry" meaning from one word, image, or idea to another, 
  • expresses the unfamiliar (the tenor) in terms of the familiar (the vehicle). When Neil Young sings, "Love is a rose," "rose" is the vehicle for "love," the tenor. 
Fine: metaphors are very useful and effective in communication, and poetry of course.
Are there metaphors in mathematics? Figures of speech? Yes, mathematics is full of metaphors:

An equation in mathematics has the form A = B, where B is not identical to A, because the equation A = A is not interesting. Thus an equation A = B rather expresses something like CA = CB where C is a shared aspect while A and B represent something which is different. The basic example is 

                                            2 = 1 + 1

                                         whole = sum of parts (integral of parts per unit step)

like in  

                                      position = sum of increments of position = integral of velocity

                                       velocity = sum of increments of velocity = integral of accelleration.

It is clear that "the whole" as a non-subdivided unity (like 2) is something different than 
"the sum of the parts" (like 1 + 1) because the parts and the summation are visible/present 
in "the sum of the parts" but not in "the whole". One can decompose 2 also as 2 = 0.5 + 1.5.

So "the whole" and "the sum of the parts" share something without being identical. So what
do they share? Yes, they share the number associated with "the whole" (that is 2) and the 
number associated with "the sum of the parts" (that is also 2). Thus 1 + 1 is exactly "as big as"
2, but 1 + 1 carries an additional structure (parts and summation), which is not visible when looking merely on the size of 1 + 1.

So mathematical equations are metaphors, and is it then so,  like in ordinary language, that an interesting equation (metaphor) tells us something of interest? Probably.  About the tenor 
or the vehicle? It can probably go both ways, so that something unfamiliar in something familiar gets exposed, or that something unfamilar is made more familiar.

A Fourier expansion of a function is interesting because it reveals "a construction" or maybe
rather "deconstruction" of the given function in terms of elementary functions. Familiar, simple, trivial functions can be deconstructed by Fourier analysis showing that they are not so simple trivial (make the familiar unfamiliar):   

In the other direction,  seemingly complex functions can be constructed from simple Fourier
expansions, like Weierstrass functions (from unfamiliar to familiar):

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