John VanSickle wrote:
>>     And that theory is simply mathematically invalid (at least with 
>> the current state of mathematics).
> 
> I'm not sure that "mathematically invalid" means anything here.  It 
> would be more precise to say that the model described by the mathematics 
> in question does not apply to the phenomena described.

	I think I was misunderstood. Here are two examples I've seen:

1) Summing a mathematically divergent series and assigning a 
non-infinite value to it. I think I saw it only once and there was some 
physical rationale, but I've been told this isn't all that rare.

2) Consider the integral of f(x)/(x*x) where the lower limit of 
integration is 0. Now the standard way to handle this is to let the 
lower limit be p > 0 (assuming the upper limit is positive). Evaluate 
the integral, and let the limit of p go to 0.

	Except for certain f(x), that limit does not exist, or diverges.

	That won't stop the physicists, though. They "solve" the problem by 
considering the following integral:

f(x)/(x*x + i*t*t)

	t is some positive parameter, and i is the square root of negative 1. 
Now the integrand is no longer singular at x=0, because of the t. Great. 
Integrate, and get your result in terms of t. Now take the limit as t 
goes to 0. It's possible you'll get a finite limit.

	The physicists then proceed to take *that* as the answer to the 
original integral.

	I've seen this "trick" used quite often. To be honest, I don't know how 
mathematically invalid it is. I asked a math PhD friend, and he doubted 
it was valid - but he's not in analysis, so he wouldn't know for sure. 
It's quite possible that it's mathematically valid for a certain domain 
of problems, and the physicists never really state them - just 
shortcutting to the final result. I should go and study some *real* 
analysis (real as in serious, not real vs complex...).

	There's the whole issue of "renormalization" to deal with infinities in 
physics. I won't go into any detail as I've never formally studied it. 
But it's another often cited example of "fishy mathematics".

	So by mathematically invalid, I meant that had they tried publishing in 
a math journal, it would never get accepted. Within the fields of 
mathematics, their results are invalid. Your example of relativity is 
not an example of invalid mathematics. The algebra works out whether 
you're assuming relativity or not. One of them may not correspond to the 
universe, but the algebra is still sane mathwise.

	Physicists, however, get away with this often. Mathematical correctness 
is a secondary concern. If they can abuse mathematics and get a theory 
that better explains the world, then no one cares about the mathematical 
irregularities. Physicists are scientists - they're beholden to 
different criteria than mathematicians. Consistency is not that important.

	So one is left with three possibilities (well, more - but no one 
considers those):

1) The theory that works even though it violates rules of mathematics 
may actually be false/flawed. It's just fortuitous - an intermediate 
theory. It will be replaced by a mathematically consistent theory that 
explains the results better.

2) The theory, *and* the mathematics is valid. It's just that current 
mathematics is not sufficiently advanced to "explain" the weird 
mathematical manipulations in the theory. I wish I could think of a 
better example, but it's as if mathematics had no concept of negative 
numbers, and a physicist introduced them and better explained the 
universe. The mathematicians then finally come up with a system that is 
consistent with the old math, but also encompasses negative numbers.

3) The physical world and mathematics, at some fundamental level, are 
just inconsistent. It's anathema to some, but at the end of the day, we 
have no good reason to insist that the two should be consistent.


-- 
Aibohphobia: Fear of palindromes.


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