Did you see the World Series last fall? It was fun to watch. It was especially fun for me since Dennis Danielson, my coauthor for A Universe of Earths, is from Canada, and I am from the U.S., and the teams in the World Series were the Los Angeles Dodgers and the Toronto Blue Jays. We both took an interest in the series, and offered commentary back and forth to one another, making the whole thing more interesting for both of us (neither of us being big sports fans normally). The Dodgers triumphed, in the seventh game, in extra innings!
It certainly could have gone the other way. One call at home plate late in game seven went up for review. It involved Will Smith of the Dodgers, and Smith is from my hometown of Louisville, Kentucky! — thus giving me opportunity to gloat over Dennis! The call was not overruled, but the action at the plate was so close that I’m sure that if the umpire had originally made the call in the other direction, that call also would have stood, and the game and the series would have come out in Toronto’s favor.
How often does a close game hinge on a call from a referee or an umpire? — a call that fans of the losing team will bemoan as having robbed their team of its rightful victory? Referees and umpires make measurements in sports: whose foot touched home plate first? And measurements are inherently uncertain.
There is an interesting discussion of this in the 1614 Mathematical Disquisitions Concerning Astronomical Controversies and Novelties of Johann Georg Locher and his mentor, the Jesuit astronomer Fr. Christoph Scheiner. Galileo really hated Mathematical Disquisitions and spend a lot of time in his 1632 Dialogue Concerning the Two Chief World Systems talking about what a lousy book it was. But, it wasn’t a lousy book, and the uncertainty discussion within it is pretty cool. Locher speaks of how the Earth is not a perfect sphere, but the imperfection is too small to have any effect on astronomical observations. He writes:
This globe, the Earth, is an approximate globe, not a geometrically perfect globe. For according to Clavius, the greatest height of a mountain, and the deepest depth of the seas and valleys, is about one German mile. Therefore each of these is, in comparison to the semidiameter of Earth, as 1 part to 860. If we consider a perpendicular distance of two German miles from the deepest depth to the highest peak, then this will be as 1 part to 400, approximately.
That one part is hardly sensible among the 400. Indeed, who could sense a single gold coin missing from 400? How much less sensible is that same distance in comparison to the distance from Earth to the planets, or to the fixed stars? Thus those who persuade themselves that the stars appear larger when viewed from a high mountain, and smaller when viewed from a deep valley, are mad. Even the semidiameter of the Earth itself causes no sensible variation in the apparent sizes of stars (as is obvious since the size of a star does not alter from rising, to transiting, to setting). How much less variation will result from two German miles?
Locher is talking about something that students in science lab classes everywhere learn about — uncertainty in measurement (in this case, of 1 part in 400, or 0.25%). Experimental uncertainty, or error, is not necessarily a result of experimenters doing sloppy work during an experiment. Error is not “bad”. It is an unavoidable aspect of experimental science. All measurements involve uncertainty.
Consider this example:
If I were to measure the distance from my home to the town of Lexington, Kentucky by driving my car there and reading how many miles elapsed on the odometer, I would encounter a problem in that the Lexington metropolitan area does not have a definite boundary. So I would simply have to make an educated guess about when I can say I am “in” Lexington. If two other people make the same measurement they may get significantly different answers owing to the fact that they make different guesses as to when they are “in” Lexington. This is owing to the inherent uncertainty involved in the type of measurement they are making.
Now consider this second example:
Instead of just measuring from my home to Lexington, I decide to measure the distance from the front door of my home to the front step of Lexington’s famed Rupp Arena. In this case there is a very definite boundary to the step of Rupp Arena. There is no guessing as to when you have reached it — you know for sure. However, if you are measuring the distance with a car’s odometer, you have another problem. Odometers on cars only read to the nearest 1/10 of a mile. Thus, if I use my car as my measuring instrument, I cannot know the distance from my front door to Rupp Arena to better than a tenth of a mile. That tenth of a mile accuracy is the experimental uncertainty involved in my measurement. It is inherent in the limitations of my measuring device. I could fit a more accurate odometer to my car, but I would always have some uncertainty in my measurement. The more accurate odometer would reduce the experimental uncertainty in my measurement, but it would not eliminate uncertainty.
So there are two sources of error in any measurement, no matter how carefully it is conducted. These are 1) errors due to the nature of the measurement itself, as seen in the first example, and 2) errors due to limitations in the measuring device and, by extension, to limitations in the experimenter’s ability to read the measuring device, as seen in the second example. You see this in the example of baseball calls. First, determining whose foot was on home plate at what moment is inherently difficult if things are close enough. Second, the tools we use to measure whose foot was on home plate in a close situation (the TV cameras and whatnot) have their own limitations.
If you want to win a sports contest, you have to be good enough to beat your opponent soundly enough that the uncertainties of refereeing don’t matter. Had Toronto beaten L.A. in four straight games, by three runs or more each game, it wouldn’t matter if there was some missed call at home plate. Teams who can thoroughly defeat their opponents don’t get “robbed” of a win on account of a referee’s call. It’s when the teams are closely matched that the uncertain nature of refereeing can make the difference in who wins and who loses.
The same thing occurs in elections, where the counting of votes — a measurement — is always going to include some uncertainty. If the margin of victory is smaller than that uncertainty, then the outcome of the election might be able to be determined by who can monkey around with that uncertainty, and push for these ballots to be disqualified or those ballots to be included. But if Candidate A thrashes Candidate B, winning by a wide margin, then shenanigans involving 0.25% of the ballots are irrelevant. In the U.S., we have such famously contested presidential elections because so often our candidates do not appeal to a wide enough portion of the population to win decisively. And so we fall to fighting over whether or not there is one coin missing in a pile of 400! — like sports fans going berserk because some referee’s call “robbed” their team of their rightful victory.
But uncertainty is an unavoidable aspect of experimental science. All measurements involve it… much to the chagrin of the Toronto Blue Jays!
