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Re: More /.{I->-1} craziness, con brio

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  • Subject: [mg106649] Re: More /.{I->-1} craziness, con brio
  • From: Richard Fateman <fateman at>
  • Date: Wed, 20 Jan 2010 06:47:44 -0500 (EST)
  • References: <> <hhhmn8$o9t$> <his71l$lad$> <> <hj131g$bdr$> <> <> <hj40qq$sgp$>

Andrzej Kozlowski wrote:
> And, by the way, any claims to the effect that something like what you 
> "explained" must be going on in the C source code in which Mathematica 
> is written do not count for that would be at best pure speculation which 
> something anyone else can do as well as you .

No, I also read the documentation:

"Module allows you to set up local variables, to which you can assign 
values and then change them. Often, however, all you really need are 
local constants, to which you assign a value only once. The Mathematica 
With construct allows you to set up such local constants. "

also in the documentation for With, we learn that With is faster than 
Timing[Do[Module[{x = 5}, x;], {10^5}]]  is 3X slower than
Timing[Do[With[{x = 5}, x;], {10^5}]].

This is presumably because the local variable x in With  has lower 
overhead. No extra symbol name etc.

I find Mathematica's "Trace" to be quite useless, myself, so I had to 
brace myself to look at what you posted there...

I think that what the Trace reveals is that in the use of "With", the 
symbols, which I have said were "something like I$1234" are, indeed, 
something like I$1234,  but in order to make things less burdensome for 
Mathematica, they are not fully instantiated symbols.
This saves on memory allocation/deallocation. It also means that they
print just like "I" and not like "I$1234".  But that does not change the 
argument that they act like new symbols, local to the body of the With.

Perhaps this example further illustrates the situation:

With[{I=a}, f[I]+g[Complex[0,1]]]   which returns f[a]+g[I].

That is, the local variable "I", which according to me becomes a secret 
something like I$1234, is unrelated to Complex[0,1] or I, even though 
they print exactly the same way.

This totally misses the point of the discussion, which is the illogical 
consequences of the current treatment of the symbol I vs. the constant, 
which also appears to be compound object as well as having the 
appearance of a function -- of Complex[0,1].


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