Infinity is not a number, it is a place. It lies just to the left of the Undiscovered Country. Nobody knows what it is like there, because nobody has seen it. All we can ever do is approach it and say what it isn’t.
Funny thing about Infinity is that they way you get there matters. If you were to head out toward it on a straight line taking one step at a time, and I were to follow taking two steps forward and one step back, we would not necessarily arrive at the same neighborhood. That would depend on how fast we were walking.
None of us would ever get there anyway, not while we’re stuck in time. Infinity isn’t in time. It sure isn’t in our intuitions. Which explains the latest flap.
There’s a video going round (below) which shows that we can assign the value of -1/12 to the infinite sum 1 + 2 + 3 + 4 + … Those dots don’t end until Infinity, always a key something weird is going on. I’ll assume you’ve watched the video.
The path the gentlemen in the video took was not a straight line, which is how they arrived at -1/12 (See also these videos, particularly the third, for more on the path.). But if you were to go straight—the simple sum, i.e. “the limit”—you’d end up right where intuition suggests, at a whopping big number, unimaginably big. Why the difference? Mathematical truths, like all truths, are conditional on the premises assumed and those premises include the paths.
Well, bizarre, right? Yet why shouldn’t Infinity be bizarre? Why a “flap”? Turns out P.Z. Myers, self-proclaimed “rationalist”, saw the video but could not understand it, and since he could not understand it he concluded it therefore could not be true (a line of argument which he frequently employs). So he put up the post “The sum of all natural numbers is not -1/12.” “I saw [the video] and said to myself that it’s obviously wrong”. All the proof he needed.
Switch on the Wayback Machine and slide back to 1990 when Marilyn Vos Savant explained it’s better to switch doors in the Monty Hall Probability Problem (see this for an explanation), a highly non-intuitive result. Thousands of genuine PhD mathematicians reacted like Myers and said “No way this result can be true because I don’t understand it!” Which proves probability is notoriously difficult—and that academic certification is far from a guarantee of infallibility.
Myers also enlisted the support of his own PhD mathematician, Mark Chu-Carroll, who explained carefully but failed to appreciate the difference between limits (a technique which the proofs in the videos do not use) and Cesaro sums (which they do). Chu-Carroll also forgot the -1/12 result was first given to us by Leonhard Euler, perhaps the fattest mathematical brain ever.
The gentleman in video number three (above) was also careful about explaining how Grandi’s series—1 -1 + 1 – 1 + 1 – 1 + …—could, out at Infinity, be +1 or 0, on or off, spin up or down, and that if we consider the series in a sort of probabilistic sense, it can be given the value 1/2. Sounds a bit like quantum mechanics, no? It is this assignment that makes the magic happen and is why the infinite sum can be -1/12. Anyway, Myers didn’t bother to investigate any of this before going off. Which is what makes him a rationalist.
Enter our friend Lubos Motl, arch defender and knight-errant of string theory (he will not see her virtue impugned), who has the habit of writing even simple numbers in Latex, who took Myers to task in the post “Sum of integers and oversold common sense.” Motl also takes pains at showing there is more than one way to sum a series.
Phil “Bad Astronomer” Plait joined in the fray at Slate, which might not have been the wisest move. If there is any place on the Internet where the people already know all they need to know, this is it. And they already knew the infinite sum could not be -1/12. Poor Phil had to issue multiple corrections for being too glib with his language.
Most civilians and rationalists don’t know there is a (let us call it) tension between the kind of math physicists do and the types mathematicians themselves use. Physicists are a little bolder, even playful. Mathematicians are more staid. Full disclosure: I learned my math from physicists. And you may be surprised to learn that there are even warring camps inside each field about the very fundamentals of mathematics. Too much for us here today, except to note that this latest incident is part of the never-ending war of ideas.
Oh, read the history of the Heaviside function for a fun example (I don’t have a link).
Update: now with even more Infinity!
Mathematics isn’t the only place where we meet Infinity. Take the idea of Omniscience, which is knowing everything, and everything includes Infinity. Suppose one knew 10100 facts, a googol of facts. That’s a lot of knowledge, but still far from infinite knowledge. How about knowing a , a googolplex, of facts? Some estimate there are only about 1082 particles in the universe. If you knew a googolplex of facts you’d be able to name each particle. You’d know where everything is located, including my Kindle which went missing a week ago, so I’d appreciate an email.
A googolplex of facts is already unimaginable, but is just as far from Infinity as 10100 is. We still have a long way to go before reaching Omniscience. And what’s it like when we get there? Boggles the mind to think about, just like, in a much smaller way, the sum above does. The lesson is intuition, particularly knee-jerk reaction, only takes you so far. And usually to the wrong place.
Update The HTML superscripts weren’t rendering properly, so I switched to Latex for the googolplex.
Thanks to our friend Luis Dias for alerting us to this topic.