[T]he more that physicists stopped worrying about what their complicated equations meant and simply ran the numbers, the more progress they made. Some of their predictions have now been confirmed by experiments to 10 decimal places or more— the most accurate predictions in history.
Real objects cannot have infinite charge or mass or whatever. But when scientists in the 1950s started calculating those quantities with their latest and fanciest theories, infinities kept sprouting up and ruining things. Rather than abandon the theories, though, a few persistent scientists realized that they could do away with the infinities through mathematical prestidigitation. (Basically, they started calculating with and canceling out infinity like a regular old number, normally a big no-no.)
No one liked this fudging, but because it led to such stunningly accurate answers, scientists couldn’t dismiss it. In fact, the reigning paradigm in physics today—which describes the workings of invisible “fields” (similar to magnetic fields)— would not exist without this hand waving. And now physics is stuck with fields: they’ve become more fundamental to understanding the universe than mass or charge. Fields have become the very fabric of reality—even if our understanding of them relies on some unrealistic assumptions.
So what's the problem with infinity? Let me offer a couple of starting points at getting at an answer to that question -- both of them discovered not in modern philosophy, but in Ancient Greek.
The first one is the problem the article cites -- theoretically, actual infinities shouldn't exist. Aristotle explained why potential infinities could exist, and you probably know at least one of the arguments: if you have a number, you can always divide it in half. Thus one can be divided into 1/2, then 1/4, and so on forever. The other kind he recognized was like this: if you have an actual universe with the size of two, you can divide it into half, and then take the second half and divide it in half, so that now you have divide out one and a half from the total; and then again, so that you have divided out one and three quarter from the total; but you will never reach two. Two is thus a kind of infinity, since you can never get to it; but it is not an actual infinity. (As the Stanford article points out, however, this is not consistent with Aristotle's idea that the universe was eternal -- and that there must have been, therefore, an infinite number of days.)
These are not actual infinities, because you have really only the thing you are dividing and it is of determinate magnitude. If the universe is not infinite, you should not be able to get to infinitely large magnitudes. It is only through infinite division that there should be even a potential infinite.
(The concept we learned in grade school -- that given any number, you can always add one more -- is another kind of potential infinity. It deals only with imaginary objects, not "real objects," which we would like to believe are finite).
The other starting point for the problem are Zeno's paradoxes. Zeno's paradoxes of motion show that, if things are infinitely divisible, motion should be impossible. Since a distance is a length, and a length is the kind of thing that should be infinitely divisible (one mile into half a mile, etc.), it should be the case that motion is not possible. For say that you divide the line into an infinite number of points. For motion to be possible, you'd have to pass from one point to the next. But there is no "next" point if there is a true infinity, because you can always divide the distance between them in half. Thus, between any two points are an infinite number of points. (For further discussion, see the second paradox of motion.)
Likewise time should be divisible (a minute into half a minute, etc). This provides its own problem, since we think motion takes place in time (indeed, Aristotle thought that time was the counting of motion). So if we divide time down to the smallest possible increment, either a flying arrow is frozen in time and space, or it moves. If it moves, though, our instant of time must have a start and a finish -- which means it should be further divisible, contrary to our assumption that this was the smallest possible division. Therefore, we should get to a division in which motion isn't possible; but if time is composed of moments in which no motion is possible, how could motion be possible at all?
The rest of Zeno's paradoxes, and the thinking of Parmenides and several others, posed real challenges for any system that includes infinities, but also for any system that includes multiple things. The upshot of both is that reality only makes sense if it isn't really divisible, but finally unified. Aristotle argues against this in Physics I, in part from the obvious rejoinder: well, but we see motion all the time. Thus, motion and time must be real; we all agree on it.
For many years we've followed Aristotle's basic solution by assuming that real (i.e., actual) infinities didn't exist. Now, however, we find that we are able to make scientific predictions that are far more accurate than anything in human history... but only by assuming the infinite with real objects.
That's a problem. It's a problem because it means that the most accurate science in the world is founded on assumptions that we have some good reason to think are impossible.
What does it mean if they're not?