Unscrambling infinity and chronicling math's forbidden numbers
Unscrambling infinity and chronicling math's forbidden numbers
A reply to The Bob Murphy Show, Episode 287
Sometimes the Twadpockle Report is able to serve up a “scrambled story” straight out of real life, without invoking fanciful metaphors or cryptic references.
This is one of those times.
We are going to talk about infinity, though, so it’s still turtles all the way down. (Aack, sorry! I can’t resist.)
On September 5, 2023, The Bob Murphy Show tackled a tough question about infinity in modern mathematics and briefly speculated about the history of irrational numbers, a deeply strange concept routinely passed off to grade school children as a bread-and-butter banality. Bob Murphy and his guest Ian Deters also touched on how mathematical concepts are formulated by human beings for a purpose, not dictated in the clouds by some sterile process.
It just so happens, I have given a talk to high school math students several times about strikingly similar things: math’s alarmingly vicious history, its deeply unappreciated strange ideas (and why they are utterly unavoidable), and its foundation in human intuition and purpose.
Since I just made a recording of my talk a few months ago, I am delighted to share it with Dr. Murphy and his audience, and to have this timely segue to feature it here in The Twadpockle Report.
Also, while listening to Murphy and Deters discuss the “rearrangement” paradoxes of infinite sums, I was struck by an even simpler gut-level perspective that removes every ounce of mystery from the matter, which I am eager to put down on paper (though my laptop screen will suffice). What story could be more scrambled than rearranging infinity?
This is not the first time Dr. Murphy’s show has sparked an epiphany for me. His Episode 24 had me seeing flaws in Modern Monetary Theory (MMT) in a new light, a few years ago. Give him a listen! Relevant links are at the end of this post.
My math talk
Forbidden Numbers Ate My Brain
The Dark Side of Math and Why We Need It
https://fnamb.primetime.games/
Abstract
Numbers are logical, orderly, absolute. We “turn the crank,” and they follow the rules.
Yet, their history is a tale of heresies, coverups, and revolutions—how can this be?
We will examine some of math’s most dangerous ideas, explore what made them so dangerous, and discover that math is not about “turning the crank” at all—no more than reading is about “sounding it out,” or that art is about BLUE + YELLOW = GREEN.
Rather, math is about shifting your perspective, on purpose, to solve a problem.
It is an inherently creative process, fraught with danger, where the only rule is that the rules will change!
I talk about “intuition” a lot in this recording. If I recorded it today, I would also mention “purpose,” and I would find a way to call it “Austrian math.”
Unscrambling infinity
Speaking of shifting your perspective—on purpose—to solve a problem…
Suppose you heard some babble that this infinite series
1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 …
will add up to anything you want—perhaps twelve—if you just rearrange the numbers. That is, merely SCRAMBLING up the numbers, without changing any of their values, gives you a different answer. For real. Because infinity.
This is called Riemann’s Rearrangement Theorem, and it is a Real Thing™ in serious mathematics. To the uninitiated—including the very smart and highly educated who just don’t happen to be math majors—the nonsense already appears thick enough to doubt the foundations of “higher” math, supposedly the most objective and infallible discipline of human thought, of all time.
I mean, they told you by third grade that adding up the same numbers in a different order doesn’t change the answer, and you can see it just by piling a bunch of LEGOs together, removing some, adding some back, etc. The order in which you move the pieces doesn’t change the final number—it can’t—as long as you ultimately make all the same individual moves.
But now we’re breaking this rule? Because infinity? Then infinity must be broken, pun intended. And if that is broken, what else is broken? For starters, all of calculus and then physics are history (pun not intended). Was any of this funded by Fauci? He was appointed by Trump, you know…
This was the quandary in which we listeners of The Bob Murphy Show found ourselves at the beginning of Episode 287 titled “Ian Deters Defends the Use of Infinity in Higher Mathematics.” What ensued was a calm, lucid discussion of how infinity does indeed change things, all because you can group the numbers so that pockets of them increase at a faster rate than other pockets and so because there is an endless supply, you can manipulate the groupings to converge to a limit…
Well, it was quite a good explanation for such an abstract topic, and I daresay any thoughtful person could grasp it by listening very intently without so much as chewing gum at the same time. But while it was playing, I stumbled on an easier way to see it!
And I don’t need infinity. I don’t even need convergence.
Let’s SHIFT our PERSPECTIVE (yes, I have a wooden club with those words on it). Forget about sums of numbers. Instead, let’s go to soccer practice.
The Soccer Mom Rearrangement Theorem
Coach is making us run “suicides” again. Starting on the goal line, we sprint to the five yard line and back, then the ten and back, then the fifteen, the twenty, and so on. It builds endurance. And character. And puking.
But not for the new kid. He seems to like it and is still going, nearing the other side of the field, while the rest of us lie exhausted in the grass, waiting for our rides home. Is he really running faster than when we started? He’s an odd one. His name is Clark Kent (I guess I lied earlier about no fanciful metaphors).
Clark’s mom arrives. “Where is Clark?” she asks.
“He’s still running,” we explain and gesture toward the field.
She rolls her eyes, then shades them with her hand. “I don’t see him.” Indeed, he has disappeared into the trees past the other end of the field.
“He’ll be back soon,” says a teammate. “He’s just running back and forth from this goal line.” Sure enough, he emerges from the trees and arrives next to us at an unnerving pace. But his mom has glanced down at her buzzing phone. It is only for an instant, but when she lifts her gaze again, he is already out of earshot (his super hearing has not yet developed).
“Aww, you just missed him,” says the teammate.
“That’s okay,” says another. “He’ll be back again right here in a moment, because even though he keeps running farther and farther away every time, he also keeps running back by the exact same amount to this exact same spot.”
“So,” says Clark’s mom, “He is diverging farther and farther away all the time, but also he keeps converging right back to this spot? INTERESTING.”
“PARADOXICAL!” says Coach. “I’m glad you’re here to see it for yourself, or I would have a hard time describing how he is getting farther away all the time but also not really going anywhere, in a way.”
Clark breaks out of the trees again and dashes toward us. His mom takes a breath to speak to him and begins extending her hand, but he pivots ten yards too soon and runs away from us again.
“That’s strange,” says the first teammate. “He broke the pattern.”
The next time, he pivots twenty yards away from us. Then thirty. He is no longer “converging” to the goal line. Instead, his pivot points are receding from us by ten yards each time. We wonder what is happening at the other end. Is he pivoting farther and farther away there too? Or, is he converging to a new spot on the other side of town, as if he has chosen a new goal line far away, with his pivots approaching closer and closer to it?
“This is whacky,” says Coach. “Imagine trying to explain this on the phone. I’d have to invent some new words! Converging and diverging just wouldn’t cover it.”
“Well,” says Clark’s mom, “I’m going to find my son.” She takes out a pocket calculator and drives off. She ultimately discovers him at a malt shop on the other side of town, where a girl named Lois works the counter after school. Even Clark Kent doesn’t run FOREVER.
WHAT JUST HAPPENED?
The first pattern of Clark’s run illustrates the series
1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 …,
not to infinity, but only until he changes his pattern. Let’s call N the last segment he runs in the first pattern. Then his entire (finite) first pattern is
1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 ... + (N-1) - (N-1) + N,
which leaves him out in the trees or beyond, pivoting to run back toward our goal line and begin the second pattern. I’ve dropped a factor of five for convenience.
Since this is a finite sum, mathematicians would not normally discuss its “convergence” but would simply add it all up. It adds up to N, as you can see by imagining all the matching plus/minus pairs canceling out.
Similarly, when we check out at the grocery store, we don’t care about the order in which the clerk scans our items. We don’t care how the total is unfolding with each individual scan. We only care about the final sum, so we can pay and go home. Finite sums we encounter in real life are normally like this.
But Clark’s mom does care how Clark’s finite sums unfold as they expand from left to right, because she needs to find him in real time at his current location. She is very interested in the “partial sums” of this finite series.
Mathematicians normally define a “partial sum” as part of an infinite series, starting at the left and summing to the right, stopping after some finite amount. They need to do this in order to analyze infinite sums with rigor, by reasoning from finite ones as they grow. Infinite sums can then be expressed as limits, averages, etc. of these partial sums as they grow to the right to encompass more and more of the infinite terms.
Clark’s mom, then, is very interested in this unusual “partial sum” of a finite series, because it follows Clark’s motion by growing from left to right as he progresses. Mathematically, it’s the very same object as her grocery store receipt growing line by line at the scanner. It does not contain any new mysteries or require any new analysis, but she cares about it in a completely different way, from a very different perspective, because she is looking at it with the purpose of finding Clark in real time.
Suppose she adds it up in the wrong order, like this:
1 + 2 + 3 ... + N - 1 - 2 - 3 ... - (N-1).
It’s all the same exact numbers and still adds up to N, but the partial sums are wildly different. This series makes it look like he just runs really far out in one long progression and then runs almost all the way back. She will have a hard time following him from this! His “convergence” toward the goal line is completely lost, as is the “divergence” of his pivot points. Totally not what happened, and very boring.
But this has nothing to do with infinity
Of course not! That’s my point.
This “partial sum” business and “rearrangement” trouble is as easy to comprehend as a down-home Kansas soccer mom running late for dinner. No infinity needed.
But if we now turn to the poor mathematicians trying to make rigorous sense of infinite sums, we can see why they are forced to say such crazy-sounding things. It’s not that they want the slings and arrows of tracking partial sums unfolding endlessly from left to right, but that’s the predicament in which they find themselves. And once you track partial sums like this, order matters—infinity or not. Even Clark’s mom knows.
Now let’s cap off with some infinite rearrangement tricks.
Clark’s first pattern, if he had never stopped:
? = 1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 ... = (1-1) + (2-2) + (3-3) + (4-4) ... = 0 + 0 + 0 + 0 ... = 0
Ah, so it adds up to zero (chuckle). Not surprising, as he always ends up back at the goal line (heh, heh). Let’s check our work, like Teacher always says, by doing it another way:
? = 1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 ... = 1 + (-1+2) + (-2+3) + (-3+4) ... = 1 + 1 + 1 + 1 ... = infinity!
But wipe away those tears! This is a perfectly sensible description of reality, not kidding, no joke.
If we think of Clark’s five yard line as our reference point—that first “1” which we leave unpaired—we see him first run to it, and then from there he runs pairs of slightly unbalanced segments that each take him one unit farther out, forever. The description is perfectly accurate and is the same as the previous one, but we’re choosing to track his pivot points moving away from the goal line instead of watching him at the goal line, where we would mark him returning endlessly to zero. We’re just framing the same events differently.
We have merely (prepare to duck!) SHIFTED OUR PERSPECTIVE!!!!!!
Both zero and infinity are correct—physically, pragmatically, realistically correct—even though this series is divergent, the kind that we’re told is pure nonsense. As Coach said earlier, and I quote, “he is getting farther away all the time but also not really going anywhere, in a way.”
Our rearrangement trick requires an endless supply of opposing pairs of numbers, so that if we split them up, there is always a larger number available to the right, to pair up with a smaller opposing number to its left. We can form an endless stream of ones that way, or we can form an endless stream of zeros by keeping the original pairs together to cancel out. Because infinity. But if Clark ever stops running, the numbers to the right dry up, and our trick fails. He stops either at the goal line or at the pivot, for a sum of either zero or N, just one answer. But if he never stops for that malt with poor Lois, we get our infinite supply, and we can see it as both zero and infinity! Because frankly, it is. Both.
B-b-but numbers don’t lie!
Sure, twadpockle. They don’t tell the truth either, because they don’t mean anything. Only people can lie, tell the truth, or mean something.
This is the hushed side of math, overlooked, misunderstood, and underestimated: you can’t just “do the math” without very careful decisions about what you mean, and then a very careful effort to keep that meaning absolutely consistent as you interpret the results. Lose sight of this, and you’re just another overpaid dumbass with a Nobel Prize crashing the economy.
But if you do take care, you have an incredibly powerful tool that has been honed for thousands of years by humanity’s sharpest minds of all time.
References
“Ep. 287 Ian Deters Defends the Use of Infinity in Higher Mathematics,” The Bob Murphy Show.
https://www.bobmurphyshow.com/episodes/ep-287-ian-deters-defends-the-use-of-infinity-in-higher-mathematics/
The podcast episode I am responding to in this post.“Riemann’s paradox: pi = infinity minus infinity,” Mathologer.
Demonstration of summing the first series in this post, to get pi—not twelve, but it’s the same idea.“Ramanujan: Making sense of 1+2+3+...=-1/12 and Co.,” Mathologer.
An identity mentioned in Murphy’s episode. This is a great example of being very careful what we mean before slinging numbers. This weird identity is only true in a very narrow sense from a certain perspective (analytic continuations on functions in the complex plane—I barely know what that even means). If you write this -1/12 claim in a vanilla high school class, you should get zero points.“Numberphile v. Math: the truth about 1+2+3+...=-1/12,” Mathologer.
A more complete debunking of sloppy claims about this identity, which went viral all over the web in 2014.“Conditional Convergence,” Wikipedia.
https://en.wikipedia.org/wiki/Conditional_convergence
This did not play a role in my analysis, because I ended up only needing a finite series, but it is the reason that you can sum the first series to twelve or pi or whatever you want.“Riemann series theorem,” Wikipedia.
https://en.wikipedia.org/wiki/Riemann_series_theorem
Also called Riemann’s Rearrangement Theorem. True of conditionally convergent infinite series like the first one in this post. It did not play a role in my analysis either, for the same reason, but it is the counterintuitive result that Murphy referred to as motivating his Episode 287.“Ep. 24 Murphy Dissects His Discussion of MMT With Warren Mosler,” The Bob Murphy Show.
https://www.bobmurphyshow.com/episodes/ep-24-murphy-dissects-his-discussion-of-mmt-with-warren-mosler/
The other time I had an exciting epiphany listening to The Bob Murphy Show.“Bernie Jackson on a Flaw with MMT Analogies,” Consulting By RPM.
https://consultingbyrpm.com/blog/2020/03/bernie-jackson-on-a-flaw-with-mmt-analogies.html
My reply to Episode 24, which I emailed to Dr. Murphy in 2020 because I didn’t have my own soapbox yet.About “yard lines” on a soccer field,
Some of us played soccer at a small school overlapping with football season.