xeno said:
Hi Again.
How can infinities cancel out? Except when scientists do not want them to. Isn' Calculus magic it allgets smaller and so we can eventually forget it, ( Try that with homeopathy). I find it amazing how science in general chooses to turn a blind eye to stuff that does not fit in, and also to stuff that it can not reconcile with its present "Model". However it does let me play some pretty good games on my computer so I guess I will have to accept it.
Good name, given the questions that you're asking. Hopefully it won't take an infinite number of steps to get to an answer.
It all comes down to the idea of a limit. For example in the case of calculus we can work out that the gradient of the function f(x)=x^2, by the following technique.
The gradient at x=x_0 is given by
gradient(x=x_0)=((x_0+d)^2-x_0^2)/d
in the limit d->0.
The improtant thing is to work out the answer for finite d. i.e.
gradient(x=x_0)=(x_0^2 + 2*d*x_0 + d^2 - x_0^2)/d
=(2*d*x_0 + d^2)/d
=2*x_0 + d
This is the answer for finite d.
Now we take the limit that d->0 et voila
gradient(x=x_0)=2*x_0
Now, if we go back to the discussion of the infinities cancelling, I'll attempt to give an example of the sort of way in which it can happen.
Imagine that we perform a calculation by discretising space onto a lattice of N points. Say that our answer comes out to
Ans = 12 + (N^2)/5
Clearly for the continuum limit N->Infinity, and our answer becomes somewhat large
Now, kets say that we realise that the answer that we have just worked out is the idealised case, and that the number is not one that we can measure directly. In that case we try to work out what the measurable quantity is. Again we can do this for the discrete case, and we will find that there is generally a correction term, say
Ans_Measure = 12 + (N^2)/5 - Correction
where the correction is equal to say 3 + (N^2)/5.
Then we find that the real measurable quantity comes out to be
Ans_Measure = 12 + (N^2)/5 - 3 - (N^2)/5
= 9
This is quite well behaved in the N->Infinity limit. At this point the panic is over.
I hope that this example illustrates the kind of way in which you can legitimately cancel out one infinity with another.
