> In what way is e^i*pi = -1 a metaphor for anything?

Its a metaphor for taking the unit length vector [1,0] represented by the complex number 1+0i and rotating it 180 degrees to -1+0i...

  > Mathematics is funny because it is presented in
  > "reverse", i.e. not the way it was derived.

Its usually presented in both ways in most curricula, sometimes depending on where you read about it or who teaches/tells you about it. Most mathematical books include historical contexts and non-formal accounts of the way results were derived, specially for classic and old results such as Euler's Formula. In most modern topics sometimes the historical context for a theorem is not easy to understand (i.e. discrete signal processing or optimal control) and is only briefly mentioned.

I appreciate what you're saying, but see my response below about semantics.

If you are calling it a metaphor, then aren't you calling ALL equations metaphors? That is doing violence to the meaning of the word "metaphor".

There is for sure a "relation" (or association) between the symbols e^i*pi = -1 and the picture of a unit vector on a complex plane. But that relation is not a metaphor.

You can prove that equality using pure analysis, or pure geometry, using appropriate definitions. The metaphor is the intuition that the two proofs are equivalent in an abstract sense.

You’ll need to convince me that “e^ipi = -1” is actually a “thought" before I’ll buy it as a counter example. This will be an uphill battle.

Can you think about Euler’s identity? Almost certainly. Can you think Euler’s identity? Very unclear.

> You’ll need to convince me that “e^ipi = -1” is actually a “thought"

Then what is it?

A statement of grammar? A picture? A string of characters?

What would a thought in print look like? And what print with any meaning at all would not "be a thought" in ordinary parlance?

> And what print with any meaning at all would not "be a thought" in ordinary parlance?

Exactly my point.

a pattern if repeated enough times.

Haha, sounds like you've met a Lacanian (Lacan was famous for saying, among other things, that "the unconscious is structured like a language"). Such reductionist intellectualizations (aka "man, everything is a pineapple!") are neat to meditate on, but they alone won't give you tangible results or something you can use to forecast events and test hypotheses.

> In what way is e^i*pi = -1 a metaphor for anything?

Well, we have a generalized Euler's formula[1] for this identity:

    e^(ix) = cos x + i sin x

Metaphorically, you can visualize the function e^(ix) tracing the unit circle out in the complex plane.

[1]: http://en.wikipedia.org/wiki/Eulers_formula

Writing the formula down on a piece of paper doesn't by itself count as thought though. I haven't read the book so can't really comment what the author would say, but to load the formula into your brain and think about it in a useful way is very different than plugging it into a calculator or reading the formula.

> e^i*pi = -1

Gee, I think that extending the power series for e^x to the complex domain is a pretty fair interpretation of `metaphor' in the context.

Like a lot of philosophical debates, this comes down to semantics. I don't think there's any useful way in which Euler's identity is a 'metaphor' for anything.

Metaphor has a fairly specific definition; it is a type of analogy. An analogy is a type of association.

As mentioned, I think it's fair to say that all thought is based on associations. But it isn't true that all thought is analogies or metaphors.

There are simply other types of associations. I would call this case a "generalization", an extremely common thought process in mathematics, and an example of a kind of analogy (not a metaphor) where the original domain is a proper subset the new one.

In other words, applying an idea from one domain to another is an analogy, not necessarily a metaphor. To claim otherwise is just being loose with words in a way that has no meaning.

> I don't think there's any useful way in which Euler's identity is a 'metaphor' for anything.

Euler's Identity isn't itself the metaphor, it's the equation we use to teach and understand it that is metaphorical. The letters themselves only mean "Euler's Identity" when imbued with the extra meanings that come from the symbolic framework of mathematics.

The problem is that as we understand them today, mathematical formulas like yours are void of meaningful semantics unless you have any objective abstract notion of "infinite set", which seems not to be the case because "everything is a set", and "everything" is not an abstraction.

So we have to deal with them as pure sequences of signs which are part of the set of deducible formulas.

Your "understanding" (or mine or Euler's) of the formula is most likely a metaphor (well I'd say an analogy in this case) and is what led to its proof.

For interest's sake, the book "Metaphors We Live By" concerns that idea. It's a pretty good book though I personally think they take it a bit far

http://www.amazon.com/gp/aw/d/0226468011/

This goes way back before 2000. See George Lakoff and Mark Johnson’s 1980 book, Metaphors We Live By.

> See George Lakoff and Mark Johnson’s 1980 book, Metaphors We Live By

I highly recommend this book! You'll find yourself nodding in agreement at one line and then realizing that you're agreeing with a deeper truth than you knew. At the very least, you'll start reading newspapers at multiple levels.

Art:Science::Analogy:Isomorphism

I thought quality was at the core of all thought? ;-)

Haha, yes I also enjoyed ZAMM, which again has the tendency to explain "everything" using one concept. It is just a tempting thing, I guess. But the exercise does lead to interesting thoughts, even if the grand thesis isn't true.