Sorry, yes. Absolutely convergent sums do behave exactly as you’d expect normal sums to. You can multiply them together, rearrange terms as you please and add brackets wherever you want (actually you can add brackets wherever you want in any infinite sum I think, but I’ll need to check that). Can’t think of any other properties you might want from a normal sum, but those are the ones that immediately come to mind.
A nitpick - your sum converges to pi^2, not pi. A better example would be Pi = 2 (1 - 1/(23) + 1/(45) - … + (-1/2)^(n-1) / (2n+1) + … . This is absolutely convergent by something called the comparison test (if you’re feeling mathematically inclined, try proving that it’s absolutely convergent - my previous post contains all the information you need). It doesn’t really matter for your argument 'though.
There’s an entirely different problem with your argument. If we agree god cannot change 1 + 1 = 2, it doesn’t neccesarily imply that god cannot change pi as you’re also assuming that god cannot change ‘absolutely convergent sums behave like finite sums’. I can easily define systems in which absolute convergence does not imply convergence. e.g. If we consider only the rational numbers (numbers which can be written as m/n where m and n are integers). Consider the sum of:
-1 + 1/2 + 1/4 -1/8 + …
Where the sign is negative if the place of a term is a perfect square, positive otherwise. This is absolutely convergent in the rationals (the sum of the absolute values converges to 2) but is not convergent (it can’t have a terminating binary expansion). Surely something that is so sensitive to what system we’re working in is open to change?
Ultrafilter, I wonder if one of us is misunderstanding the other.
I’m not suggesting a reality in which Pi-the-circle-ratio lacks SOME infinite series which sums to it; rather, I’m asking you to consider the case in which Pi-the-circle-ratio can no longer be derived by any of the specific infinite series summations that now apply to “our” Pi. Basically: which is more important in identifying some something as Pi–that it be the ratio of diameter-to-circumference of a circle, OR that it be the resultant of one or all of the infinite summing procedures that (in this reality) correspond to the first clause?
Historically, the circle is the important one. Whatever the ratio of the circumference to the diameter is, that’s [symbol]p[/symbol]. If things were different, so that the current series didn’t sum to that…well, there would be other series that summed to it, and we’d be wondering exactly the same thing.
It’s a little bit like calling the same person Bob and Robert. Just different names, y’know?
Did you notice my .5 exponent at the end of the formula?
I don’t understand what a terminating binary expansion has to do with convergence, but I’ll take your word for it. I don’t remember enough mathematics to argue the point.
There seems to be some “object-referent” confusion here. When I answer “no” to “can god change the value of pi” I mean, the number we now call “pi” cannot be changed: it is a number that is defined etc. Can he change the universe in such a way that the number we now call pi no longer represents the ratio of etc then I think the case could be made. As such, I don’t think it could be said, as ultrafilter does, that we’d be wondering the same thing. We could ask the same question, though (assuming only this sort of thing changed, and not our entire language).
and, to answer Lib’s question: synthetic. All knowledge is synthetic. Yay Quine! (Though I must admit I only have a rough sketch of his argument handy.)
I’m coming in way late here, and it doesn’t really matter, BUT:
Without having read Kant or having the official terms to explain this, a circle is an abstract concept, not a physical reality, right? I mean, humans looked at things that appeared a certain way and decided to call them ‘round’ or ‘circle’. WE created circles by selecting certain shapes and declaring that they and only they qualified as circles, while other shapes (such as squares or ovals) didn’t.
The ratio pi is inherent in our concept of circularity, and our concept of circularity is the only existence circularity has. Shapes exist regardless of whether we call them circles or not; our calling them circles or not calling them circles doesn’t affect their reality in any way whatsoever.
What I’m driving at here is that we’ve (humans) created the abstract quality we call circularity. The SHAPE exists regardless, but the fact that it is a circle is our own definition (in nature, all shapes are equal and a circle wouldn’t be any more remarkable than something highly irregular), and by definition, the ratio between circumference and diameter is pi. If it weren’t pi, it wouldn’t be a circle.
So the only way an omnipotent god could change pi would be to change the definition of circle in the minds of all beings who have that concept, wouldn’t it?
Oops, disregard previous post, as I missed the entire second page of postings. This has really become (I think) a discussion of whether pi has real (or synthetic, I think Kant’s term is) or only abstract (analytic) existence. Obviously, from my first comment, I believe the latter. In any case, the math is now over my head (despite having been a math major lo these MANY years ago!) and I really have to start working (I mean job work, where they pay you) anyway. Sorry to intrude so stupidly.