Re: Trouble with 5.10 and what I am supposed to do.

Ah, yes, indeed, thanks again!

So after re-reading all the posts innumerable times and dredging the internet for hours and covering reams of A4 paper with

-sized pencil marks of unwavering dubiousness (all the while aggrievating a major case of

crampo mentale), I think I've finally come to have some sort of a handle on this issue.

Basically of course yous had it sussed. (Your posts and susume's in particular deserve special mention.)

What continued to bug me though was the mechanical side of the issue. Surely it should be possible to specify a few unambiguous rules for complex exponentiation that if followed would allow one to mindlessly manipulate the symbols while pursuing one's higher goal - indeed, isn't this the main reason that algebra's so useful?

You pointed out Ex.5.15, and while coming up with a proposed solution for it (posted separately) I realized the following, which puts the whole business in a very clear light I think.

We have by definition

where

is a specific choice of logw.

But look what happens when we put w = e and naively follow the same manipulations:

So

for any z! (Hence my revolutionary discovery that

)

Posts here have shown that this is due to inconsistent choices of k. And having parsed the book many times I realized eventually that Prof. Penrose deals with it by stating the tacit convention of always taking loge = 1 for the special case of

, so

always. (Though he states it after posing the paradox so it wasn't available to solve it.)

However, allow me to introduce the explicit or dickdockian convention, a variant of the tacit convention that I hope is clearer:

Define

for all (nonzero) w except where w = e.

That's it. We don't need the definition (I think) for w = e so let's not include it in the definition.

How does this help? Looking at the original paradox:

is fine, and

is fine also (where

the specific choice of loge in accordance with Ex.5.15). But we're not allowed to write

anymore than we're allowed to write

for arbitrary w.

Similarly, looking above, one can't write

anymore than one can write

, though

is fine, as of course is

.

Anyway as far as I can see (which isn't very) this, coupled with the convention outlined in Ex.5.15, allows us manipulate

and

willy-nilly without having to use tacit assumptions and/or unerring choices and/or principal values et/aut al.

I don't know if any of this means anything to anyone or means anything full stop and I don't anticipate a big rush to adopt the dickdockian convention by the worldwide mathematics community but it's got rid of my

crampo mentale (for the moment anyway).