W.D. Clinger,
The meaning of a mathematical expression is whatever we agree on. It's not intrinsic.
By the way, one might want to consider that using infinity in ordinary calculation leads to undefined results, too. For example, 1 * infinity = infinity. 2 * infinity = infinity. Since infinity == infinity, therefore 1 == 2.
The only way you can prevent this is to disallow the operations or declare the result meaningless (which any sane person would). Note, however, that you allow the same type of undefined behavior is your algorithm for operating on an infinite series because you introduce infinity as one of the terms. Even though it's common sense (and basic math) to see that infinity on one side cancels the infinity on the other, just as any normal number would, it's a huge leap of faith -- completely unsupported by any real-world application -- to use infinity this way. It's solely a definitional truth, used because it makes calculations more useful. It is not based on any experience with countable items.
I suspect the reason most mathematicians get so upset with people on this question is that they (the mathematicians) don't understand where the non-mathematicians are coming from.
It's not so much incredulity or stubbornness, as it is insistence that math exists in order to abstract from, manipulate, and then re-relate the answer to reality.
For an ordinary person, taking three apples and splitting them among friends means that each friend gets one apple, not that each friend gets 0.333... of three apples. In the real world, if you split one apple three ways and are relatively exact about the split, one friend gets 0.333, the second gets 0.333, and the third gets 0.334. Apples are not infinitely splittable, nor are they amenable to esoteric notions of mathematical piety.
The fact is that, except in computation, infinity doesn't exist (oh, perhaps the universe is infinite, I suppose, but since we can't measure it, we can't know, and, even if we knew, we can't split it or multiply it, so operations on the universe as a whole are undefined).
An ordinary bloke who gets 0.333... as the answer to his long devision problem may use the word "infinite," but what he means is that it's conceptually required that he be able to keep going that way for ever, but not practically possible, for one cannot continue infinitely (one's pencil lead would give out, if nothing else). It's the point, not at which apperception knocks him cold, but at which he says, "Good enough for the job" and moves on to the next problem.
Those of us with math geekhood in our veins want neat solutions, with rules to handle every possible situation. Secretly, down deep, we want an operation that lets dividing by zero to be meaningful somehow. We also want to be able to use infinity -- a by definition undefined quantity as if it were a regular number we can plug into our existing formulae and go home after a good day's work.
The foundation for using infinity this way simply isn't present, except as a convenience. The numbers work out in the end, and we can usually translate things back to the real world with new information added. That may indeed be sufficient justification for the process, but if so, the philosophy behind it is only pragmatism.
It all yearns towards Heisenberg in the end. What, we can know the circumference exactly but only as long as the poor radius is left irrational? Well, then, by all means declare by fiat the the radius is exatctly one ues. What? We can no longer calculate the circumference? I don't believe you.
But ah, says the mystic robed mathematician, I can indeed tell you the answer. First I must translate your requirements into my arcane and learned scribblings, and perhaps sacrifice a sparrow or two. Then I shall make scribbling leap upon one other and tear each other to bloody bits, using rules only I and the other magi fully understand. No, no, you may not watch; these beasts become irrational, and would attack anything in site. You just sit there, good man, and at the end of the fight, your figure will come stumbling through that door, ready for you to use in your humble, real-world kind of calculations.
But, says the peasant, I have this 14-inch long strap I must use for a radius, so how long must I roll the rim metal to make my perfectly circular wheel.
After argle-bargling for hours, the mathematicians tell him the rim strip must be exactly 2 x [pi] x Radius. The poor peasant can't find [pi] on his measuring string, so he begs the learned lords to explain.
You take your radius, see, says the Lord High Mathematician, and multiply it by two.
I've only got the one.
Well then make another, for when we place the radii lengthwise on the same line, their combined length will equal the diameter. From there we can see that the circumference must equal [pi] times the diameter. Bob's your uncle, laddie. Off you go.
The commoner simple gave up and rolled a stip of metal he thought would probably be long enough. With great care and no help from irrational numbers. "Bit near 3 1/4," he said, and workd from that. He managed to get the wheel complete. His rim, true, was a bit too long, but he trimmed the excess until it was a perfect match.
He then reattached his wheel to his carriage and rode awy from the filth and the muck beside the road, leaving the highly-educated mathematicians still arguing about the impossibily of his having found a workable solution without using [pi]. The peasant, having spent a splendid evening at a pub down the way, came upon them, still arguing, the following day.
Masters, cried the peasant, have you still not solved the problem of how I should meaure my use of tools to form a perfect circle?
We have solved it, fool, said the Highest of the High Muckety Mucks. You are riding upon an approximation of the truth, not the truth itself.
Well, good lord, said the peasant, that's good enough for anyone, in'it?
