Have you ever heard of amoeba stalks?
Okay, so there’s this kind of slime mold, only, it isn’t a mold. Not really. Molds are usually in the fungal family, but this mold, or “mold”, is really an animal. Well, wait, no I think they’re some other family actually? Protists, or…wait, is Monera still a kingdom? I feel like it was when I was little, but I think last time I looked at the Wikipedia page for taxonomic kingdoms it wasn’t there, and I remember I was kind of confused about that. But they aren’t fungi is what I’m getting at. They’re these giant stalks made of amoebae.
Normally, amoebae just float around, eating bits of algae or whatever: reaching out with their weird tentacle pseudopod things and absorbing little particles. But when one amoeba senses that it’s run out of food, it sends out this chemical signal. This signal alerts other amoebas that food is low. Then their amoeba buddies that heard the first guy’s signal start scooching closer to their starving friend and firing off their own starving chemical signal – starting this whole chain reaction until they’re all huddled up together into a gigantic clear balloon of ectoplasm and vacuoles and nuclei. The amoebas that are the hungriest all start bubbling up to the top, which makes this growth start appearing on the top of the amoeba-ball. It looks a bit like a nipple.
So, all the hungry guys are in the nipple, and all the well-fed guys are on the bottom. And the hungry guys start doing some chemical reaction thing, which basically is a no-holds-barred everything-must-go laxative kind of evacuation. This turns each of the starving amoebae into hard, empty shells, and has a nice side effect of bathing all the lower amoebae in nutrient-rich slime. Biologists call this portion of their life cycle the slug stage. The bubble structure from before has turned into a writhing, little goo-worm, shooting out a sticky snail trail.
As this is happening, the well-fed amoebae are also slimming down. They basically eject everything except genetic information and stick to the inner walls of the slug. The slug propels its passengers higher and higher until finally, the slug’s propellers are all dried out: dead. The result of this process is a tall stalk with a big ball of freeze-dried amoeba on the tip. Hopefully, later, something bigger will come by, break the stalk, and send the sleeping amoebae somewhere with more food. At this point, the residents of the spore will wake up, and resume business as usual: maybe with a renewed sense of motivation not to be hungry for too long.
Anyway, the reason I bring this up is because I always wondered why the hungry amoebae play this game in the first place. The starving ones have nothing to lose. They’re going to die anyway, so why not just go look for more food instead of using their last dying breath to help the guy who happened to be standing in a crowd with him? I think it’s because they’re selfish.
I know, that doesn’t make a lot of sense but hear me out. So, they know they’ll die no matter what, but they also know that if they’ve self-replicated lately, one of their clones will probably be out there in the crowd too. I mean, they can’t know this, obviously. They don’t have eyes, so if their clone wandered off away from the pack, they have no way of knowing. And more than that, they don’t have brains, so they have no memories. Whether they’ve had a million children, or they’re virgins – whatever that means for an asexually reproducing thing – they won’t remember. They can’t remember. So, for all they know, they’re standing next to their entire family. Maybe their kids or siblings or whatever are well fed. The amoeba doesn’t know. The best thing to do is self-sacrifice. Maybe that amoeba’s genes will live on.
I’m hoping this will explain why I did what I did.
When Jake asked me to be his second author on what would no doubt be his last paper before defending his Ph.D., I was overjoyed. I would dutifully labor away, contributing to his work for little or no reward just to add another publication to my CV. He was an algorithms guy, and I was more into pure math. Category theory, as you probably already know.
At the time, we were both research assistants in the Discrete Computing Lab. It was a real “publish or perish” kind of environment. There was this one student, Sean, I think his name was, who was working there with us for a while. He was a student for a year or so, and he was Jack’s last co-author. There was some issue with his writing, or his lack thereof, or something I was never exactly made privy to. Eventually Sean was sent a firm email suggesting that maybe he and his advisor would both be made a whole lot more comfortable if he applied his accumulated credits to a Master’s degree, and subsequently left the department: “mastering out”, we called it. It was something spoken of in hushed tones between classes. It was always an option, but then again, so is suicide. He left with little fanfare. There was no lab meeting about it. One day, he simply stopped coming into school. Jack was left without a second author, and asked me to fill the slot.
Jack was one of those rare people that just understood things. He could read a paper, really fly through it, and when he was done, he could explain it in such simple terms it was almost infuriating. So, when he asked me to help him with his paper, I knew it wasn’t to contribute anything. I knew he didn’t need any help solving whatever problem he was working on. My contribution would be writing the boring “Related work”, and “Background” sections so we could push this thing out as fast as possible. He would do the heavy lifting.
That’s how it started. I set up a big, annotated bibliography, I checked his grammar and so forth. But after a while, he started asking me to write proofs. I was overjoyed by this. Proofs are important; proofs are like the ingredients that make up the whole pie, you know? He kept asking me to prove these seemingly unrelated concepts, building out a series of lemmas: closures on specific directed graphs, optimal node colorings…for some proofs I had to go back and reference Alfred North Whitehead’s work on the logic of arithmetic. It was weird stuff.
I kept asking him to let me see the whole paper, and maybe then I could be of more help, but he refused. He was careful not to reveal exactly what this was building to. Something about not wanting to bias me in my work. Keeping my math straight so that no one could say I was on his side when this thing came out. He explained that it added to the verisimilitude. I wasn’t pleased about this, as you can probably understand, and it made the math a lot harder than it needed to be. But, despite his best efforts, after a few months, I had a pretty good idea of what he was building to.
There’s this theory: P=NP. Basically, there’s a class of algorithms, P, that you can do in polynomial time: in other words, really fast. There’s another class of algorithms called NP, where that isn’t the case. What this means in real terms is that for P problems, you can just tell a computer to do it, and after a few seconds it’ll shit out a solution; for NP problems, you’re shit out of luck. Either wait a year for the program to find the answer, or settle for a close approximation. P=NP just means that those problems that seem to take a long time can really be solved much faster, we just haven’t figured out how yet. Jack had set out to prove the opposite.
This revelation was disappointing at first. Sure, most computer scientists were pretty sure this result was inevitable, but still. If P=NP, then protein folding, kidney donor matching, and even optimal wedding seating charts could be calculated in the blink of an eye. Tragically, weddings would always take NP time to plan. But being the bearer of bad news had its own kind of charm. No, I’d think. You and your partner can’t get residency at one of your top 3 hospitals efficiently, I would imagine telling prospective medical students (yet another NP-complete problem). Sorry, Garmin, your GPSs will always route round trips slowly. It was liberating to be so evil.
But after weeks of writing, I began to notice something. I didn’t think it was intentional. Maybe Jack had just made a mistake. More likely, I had made some mistake, and all of this just stemmed from my lack of understanding. I arranged for a meeting to try and clarify.
***
“I’m not worried about the schedule or anything,” Jack started, “I think we’re making excellent progress. So, I hope that isn’t what this is about.”
“No, nothing like that.” I looked away. The walls were made of alternating glass and whiteboards. The glass showed the still hallways of the science and engineering building. The whiteboards were covered in equations and diagrams from the previous meeting. More likely the previous ten or so meetings. I think they told the janitors not to clean off the whiteboards because it made the meeting rooms look more “academic” or something. Some dean or manager saw A Beautiful Mind and decided to make it school policy.
“It’s about the paper itself,” I said. “I’m having some serious doubts about it. Like, I’m sure it’s just that I’m not understanding this stuff, so I don’t want it to come off as me saying you’re wrong about this, but I’ve just been noticing a lot of…inconsistencies, maybe? Or not inconsistencies, that’s not the right word. I guess you’d say that there are still some gaps we need to patch up before this thing is really steady enough to be called a proof. It’s those gaps I’m worried about.”
Jack took a deep breath in through his nose. He put his hand on his chin. “Gaps?”
“Well, okay, so like for one example, I think it was…” I quickly flipped through a copy of the paper I had printed out for this exact occasion. On the fourth page, I found the highlighted section. “Okay, here. Equation 6 is true…but we omit that it’s only true for certain cases. And then later on in Lemma 2 we extend it, but then right here,” I point to a letter in subscript below a Greek Zeta, “sometimes this variable falls outside those cases where it’s true. Not all the time! But it happens in a few cases, so we can’t really say that this is…true. Like, in all cases, I mean.”
Jack was following along in the copy I had printed out for him. He raised an eyebrow for just a second, lowered it, and then looked up at me. He just stared, waiting for me to continue.
“…and, we build a bunch of other proofs off of that one. So, I guess we just need to…find a way to fix that?”
Again, silence. He looked down at the paper again, pinched the bridge of his nose, closed his eyes, and sighed. Finally, he said, “Do you think we can bridge those gaps?”
“It’s possible-”
“No. I mean really. Do you actually think it’s possible? You must have tried to do that before you called me in here.”
“Well…no, not really. Actually, I think I was close to a proof showing that…it’s impossible. But that’s the good news! That’s the whole reason I called for this meeting! I think that means we’ve proven the opposite!” I got up, smiling, and wrote on the whiteboard: “P=NP”. Jack seemed unimpressed, so I circled it a few more times. “P equals NP, Jack! P equals N. P.!”
He stood up, snatched a marker from the whiteboard tray, and drew a slash through the equal sign. He looked at me with a quiet intensity. “Don’t say that,” he said. “Never say that”.
I didn’t understand. If I was right, this was the most important discovery of the century. The consequences of this proof were unimaginable. Humanity could only benefit. We had discovered this, and he was angry? “I don’t understand.”
“Let me show you something”.
<center>***</center>
We left the building. He walked with a brisk pace down the city street.
Someone a block away was shouting through a megaphone. “THEY ARE LYING TO YOU!” he cried. “THE DEPARTMENT OF EDUCATION IS FUNDED BY DEVILS!”
As we walked, his sermon faded away. Jack did not seem to notice. We pushed through a crosswalk at a pace slightly faster than a walk. For several blocks, we pretended to ignore the relentless requests for spare change that echoed off alleys and alcoves until we arrived at an imposing brick building just off the river. A metal door: in the spots where mint green paint had long since peeled off, rust scabbed over the wounds. Jack scanned his student ID to open it, and I followed him in.
I had never been to the school’s data center before. I knew it existed, I had used some of the computers for projects; I was vaguely aware that it powered the school’s VPN and did some sort of routing for the larger internet, but I had never actually seen it. I always imagined it would look more crisp and clean and chrome: white tile, bright fluorescent lights, imposing black obelisks holding hundreds of rack-servers. Instead, the floor was mottled concrete, the ceiling, a dusty skeleton of metal trusses, webbed with wires. The lights were bright fluorescents, though.
Jack led me to a far-off corner in front of one of the nondescript metal boxes that housed computers. LEDs blinked green and red in codes too abstruse to decipher. “The server room,” I said. “Very cool.” I looked at the computers through the cage that blocked them from intrusive human hands. “Oh nice. Is that an A100?”
“I’m not here to show you the hardware,” he said. He withdrew a tube key and opened the door to the rack. He pulled a laptop out of his bag and plugged it into the server. White text scrolled down the black screen faster than I could read it.
“Have you ever heard of Philip Maymin?”
I shook my head.
“He wrote this paper. Almost a decade ago. It was pretty controversial. He said that markets are efficient if and only if P=NP.” He paused and looked at me. I had no idea what he was talking about, and it must have shown. “Okay, so his theory was that if there was a pattern in financial data that could be extrapolated in polynomial time, then people would use it to predict the market. But, if such a pattern does exist, and people aren’t using it, then it must mean that the pattern takes such a long time to compute that it isn’t usable.”
I just cocked my head. He seemed angry.
“The efficient market hypothesis – or EMH, if you prefer – says that if enough smart people are trying to benefit themselves at the cost of others, and the others are all also smart, that the market should be perfectly efficient. So, stocks are always traded at fair prices, and no one can get a leg up on anyone else, assuming everyone has as close to perfect information as possible, which, believe you me, those MIT quants have. But, if everyone has perfect information, it means that everyone could find some pattern in old trading data that perfectly predicts future data. The only reason they don’t is because it would take longer than real time to use all the past data to make those predictions. I.e., any algorithm that could perfectly value trades must be in the class NP. He, dubiously, goes on to say that rational agents who are highly motivated to find a P-class solution to such a problem would have found it by now, and because they can’t or haven’t, that means that either the market is not fair, or such a solution does not exist. Are you following?”
I wasn’t.
“He also makes mention – and this is important – he also makes mention of the ‘No Free Lunch’ theorem, which states that if some perfect algorithm was discovered by one agent, all of the others would also discover it, and it would come out in the wash. So, after enough people found whatever perfect trading pattern existed, everyone would just use it, and again, no one would have a leg up. But, and this is a big but, it doesn’t work if only a few insiders know the pattern.” He gestured again to the laptop. “Turns out, markets are not fair,” he said, “but not because P doesn’t equal NP.”
I took a closer look at the laptop. The text whirring by began to make sense. It was numbers and stock tickers. There was a number at the end of every line. It was hard to make out, but it seemed to be getting larger with every second that passed. I suddenly understood.
“I don’t think your fake proof will last very long,” I said.
“Maybe not,” he said. He looked at the computer again. A contentedness washed over his face. “But then, it only needs to last about a decade. This account started with just one dollar. Do you want to take a guess at its value now?”
I squinted but couldn’t read the terminal.
“Hundreds of millions,” he said. “And if it runs long enough, I’ll crack ten billion and change. I can give you access,” he said, smiling, “if you want. If you’re willing to play ball.”
“I don’t think it can be done,” I said. “Maybe, and it’s a slim chance, but maybe it will get past the reviewers, but you know as well as I do that this is going to be unpopular. And if I could figure it out from your notes, what makes you think other people won’t figure it out from your paper?”
“It doesn’t matter,” he said. “I found it first. The longer it stays unproven, the longer the market remains unfair.”
“Then why publish anything at all? Isn’t it better to just let it be? Isn’t this just inviting the world to try and prove you wrong?”
“Afred North Whitehead,” he said. “He set out to prove that arithmetic was both sound and complete. He and Bertrand Russel published the first volume of Principia Mathematica in 1910; it was completed in 1913, but it took until 1930 for Kurt Gödel to figure out their logical inconsistencies. But you know, when people talk about Gödel’s Incompleteness theorem, they inevitably cite Alf and Bert. They get to ride Kurt’s coattails into the annals of mathematical history. So, no. Not really. I’ll have a few more years of profits. Probably not 18, thanks to the internet, but time enough. More importantly, my dissertation proving P doesn’t equal NP will be ground-breaking. I’ll probably have my pick of tenure-track Ivy positions. As for you, if you play your cards right, maybe you can be my Gödel. In a few years or so, you can come along and prove me wrong.”
“And what if I publish on my own?”
“Well, Sean asked me the same question. It’s interesting. The number on that machine actually used to be quite a bit higher. It’s surprising how expensive some jobs are…. No, hitmen aren’t real. Believe me. But there are other services that can be just as convincing.”
The text continued to scroll by agnostic to its meaning or purpose – a white blur of numbers and letters. Characters on a screen that symbolized value. Theoretical value or true value, I wasn’t sure, but value. I stood there in silence for a long time, watching. At one point, the solid rectangle of text bumped up a place-value and became a stair-shape for just a moment. Nines rolled into zeros. Wealth accumulated. For my R.A. services, I was paid $14/hr.
“How much does my silence cost?”
***
The paper was finally in pretty good shape. We had worked out most of the inconsistencies, and buried the fallacies so deep in minutia and appendices that it didn’t matter who the reviewers were. This would get published. Moreover, it was structured in such a way that the reasons we were wrong were so convoluted and difficult to discern, that when the inevitable counter-paper emerged, and discredited us (I was already working on such a paper –however, my portion of the proceeds was held in a trust such that I was forced to wait at least two years before submitting it anywhere) the authors could not say we were intentionally deceptive. Instead, we had made one of those “mistakes” that was so important to advancing science that we would inevitably be credited with the real proof’s discovery. Or so I hoped. There was always the gut-wrenching possibility that someone at Tsinghua University had made the parallel discovery, and their paper was even now in the process of being fast-tracked into The Annals of Mathematics, pending an English translation.
After our advisor’s stamp of approval, the paper was officially camera-ready. There were still some minor formatting issues to sort out: it was a tiny bit over the page limit. There were probably some grammatical errors still hiding in plain sight. Our advisor had this fixation on paragraphs whose bottom line contained a single word. These were all things that were fixable within an hour. It was 4:56 PM, and I was done for the day.
Around this time, it was always horrible to try to ride the trains. Obviously, you couldn’t find a seat; that should go without saying. But even standing room at rush hour in the city was a nightmare. Right at closing time, you were lucky if you could find a hand-sized opening on any of those roof-attached grabby-bars throughout the train. I was about halfway through some book, and really wanted to get at least a good spot to lean back on a wall so both of my hands could be free to hold it and turn pages on my commute home, but this was an unthinkable dream. So, to kill some time, and to celebrate the conclusion of this horrible paper, I decided to have a beer or two at the campus bar. I walked over, grabbed a stool, and sheepishly told the bartender that no, I wouldn’t be needing a food menu. I just wanted something with an ABV higher than 6.5% and silence.
Just over two hours had passed when I closed my tab and shambled to the train station. I had that warm feeling in my belly that accompanies local ales sloshing around in there as I walked down the dark city street. With the sun down, it was cold. I seemed to hit every “do not cross” sign as I walked but took the time to appreciate the graffiti and posters that adorned every light fixture on the corners. I was in no rush.
Blood has this bizarre, indescribable smell to it. When I was younger, we had this unit in chemistry or physics or something. Back in high school – it was a guest lecture, or whatever you would call that for pimply sixteen-year-olds – we had to take human blood in these droppers. I still don’t know how they sourced this blood. I assume that’s what happens when you check the little “organ donor” box on your driver’s license. Anyway, we took the blood in these syringes and dropped it from various heights onto little metal plates at differing angles and we were supposed to measure the splatter the droplets made. We’d take readings, and report back the size and shape of the little crimson splatters that plopped down onto the slides.
When we were doing this whole experiment, I told the cop that brought in the blood that the big puddles of sticky liquid settling into our Petri dishes had a smell that reminded me of the beach. He asked me what the hell kind of beaches I was going to, and I didn’t have a great answer for him. Either way, blood, in my opinion, has a salty, umami kind of scent that smells like the beach. It’s intrusive. It penetrates your nose and doesn’t do that thing that air fresheners do where it just becomes ignorable after a while. It lingers. It has this thick, heavy quality to it that smells like brine and sand and…the beach. I stand by it.
But, so, I had managed to wander back to campus, only a few blocks away from the Science and Engineering Hall, just about to the train station, when I thought that I was smelling something that was out of season. It was one of those smelling bug spray in the middle of winter kinds of moments. As soon as it hits you, it starts a whole chain reaction of thoughts and feelings about warm weather, and shorts, and flip flops even though it’s November. I’m not even sure how much this permeated into my conscious mind. That smell was filling the air, totally impossible to ignore, and I looked down and watched as this slick carmine pool pulsed out of some guy’s neck, as he was spread out, supine in the middle of the road. I was totally unable to grasp what was going on as his blood pooled onto the pavement, permeating the air with that salty, beachy smell. I was so disconnected from reality that I didn’t even realize that I was looking at Jack.
So, I’ll admit it, it was a pretty big ask. When I saw him lying there in the middle of the road, emanating that beachy aroma, I made a calculation. I said that I would only help if he relinquished first-authorship of our paper to me and allowed me to fix the necessary passages to prove that P does indeed equal NP. To hell with the money in the trust. I wanted glory. Given the circumstances, it didn’t seem like a very difficult decision. His options were (1) I could call the ambulance, they could make him stop bleeding out, and he would live: all for the small price of my discovery of a polynomial-time algorithm for the Traveling Salesman Problem. Or (2), I wouldn’t, and he would bleed out; I would go ahead and publish by myself, and still be the sole discoverer of the TSP proof.
So, imagine my surprise when he told me that he didn’t exactly appreciate my terms. I tried to explain it in a way I knew he could understand. I told him that this was a classic Nash equilibrium-type situation, and that no matter what he did, it was in my rational best interest to just wait it out until he told me I was the rightful first author. Either that would happen, or he could just lay there and quietly exsanguinate onto the pavement, at which point, I would just pop open the LaTeX project, quietly rearrange the author order on our paper, roll it back to my secret counter-proof draft, and go ahead and post it to an open access preprint server while I waited for the reviews from Nature to flood in, begging to accept it.
This made a lot of sense to me at the time, and I think it made sense to him too, because between violent, productive coughs, forcing out brownish clots of sputum, he gestured in a way that made it seem like he was all too happy to live another day, and write papers about my proof. At which point, I was all too happy to call the paramedics, alert them to the situation I had discovered on my walk home from campus and addressed – rather heroically, I might add.
So, the EMTs arrived, and scooped him off the pavement. The cops showed up and asked me for a statement. I told them that while I hadn’t seen the collision in person, I was quite sure that he wasn’t the kind of guy that would jay-walk. I was certain that he waited for the little walking man to show up on the sign before entering the crosswalk, and that surely the person that hit him must have driven off into the night, possibly – no, probably drunk, and that they would be much better informed by checking the closed circuit televisions that surely must monitor this intersection at every hour of the day and night. While I wish I could be of more assistance, unfortunately I was just a good Samaritan that called this incident in to help a good friend and colleague in a time of desperate need.
Of course, you know as well as I that Jack Wilton et al. (me being among those et alia. The second, in fact. But this ellipsis will at best be a trivia fact for particularly studious algorithm and discrete math students.) was indeed the discoverer of the proof that the TSP is solvable in polynomial time, and therefore P=NP, and now all of our GPSs and map colorings and knapsacks can be routed and colored and filled with items of non-discrete weights in a much faster manner than before. Not that most people really noticed.
The problem with the logicians is that they rarely consider what happens after their little thought experiments end. Prof. John Forbes Nash Jr., for all his brilliance, failed to consider that after one criminal rats on the other in the prisoners’ dilemma, there is hell to pay. Jack got to the hospital, they stitched him up, and in 5 days he was back at the Discrete Computing Lab. By which point, he had already switched my edit permissions on our paper to read-only. I asked him about the little deal we struck that night, and he just motioned to the stitches on his neck using only his middle finger. I didn’t have a great retort. So, he submitted his proof to Nature, and, well, you know the rest.
But the proof he submitted was the correct one. All of my old edits and questions and counter arguments were taken into account, and his whole efficient market hypothesis was now out the window. In a few months, all the rational agents at the New York Stock Exchange will be reaping the benefits of it. You probably noticed a small bump in your 401ks around that time; that’s why.
So, to circle back to your original question, I guess the way I handle difficult situations in the workplace is to wait until the right time, and then address them head-on. Sure, Jack sabotaged my entire academic career after I forced his hand that night. But my self-sacrifice was really for the good of all mathematics, right?
I mean, I would have loved to be the first author myself, but sometimes we have to destroy ourselves to propel the colony up the slime stalk.
