Some philosophers of science have a habit of fondly, dreamily saying something like this: “You know, science understands much — but we, the understanders, are just jumped-up apes. Who knows, maybe there are things in this world we will never understand. Things that are just too complex, too weird, for our brains to process! Things that’ll leave us scratching impotently like a rat against a glass wall. I don’t mean Lovecraftian ‘flee howling into the safety of a new dark age’ stuff or religion, but just cases where reality is too big and complex for short-lived, ill-communicating hominids to grasp. Maybe we’ll be faced with that, some day in the far distant future.”
As I said: some philosophers of science say that.
I am not a philosopher of science.
I am a research mathematician; hello; that is as far as I usually get before people start to politely disengage. (Do biologists ever get told conversant X was no good in biology and really, what’s biology really useful for? (“Beetroots are more useful than square roots, ha ha ha!”) Then again, do biologists get gazed up at like they’re Grecian gods of dreadfully inhumanly obscure abstraction?)
(Of Grecian gods I’m not sure if I mean Hephaistos the Gimp, or randy Zeus, lover of women, animals, inanimate objects and natural phenomena, all in the same carnal sense. People that look in awe at mathematicians don’t seem to know either; the judging gaze vacillates between a celibate and some unnatural perversion, and then turns away.)
I am a research mathematician, and I do not like those science-philosophers because they are full of caca. “Some day in the far distant future”? Let me tell you about Project Caca.
That’s not the actual official name, of course; funding is difficult to get, so the name was sesquipedalian and polysyllabic in the tradition of “Monitoring systems for time-related reduction of thermal damaging in caffeine intake manifolds”, i.e. waiting for the coffee to cool — the name had a precise and meaningful meaning, and there were maybe twenty people in the world that could understand it without looking for the definitions.
Luckily one was reviewing it, and bang! Me and two graduate students were funded for two years, more depending on results.
In three months we had a draft of a paper.
In six months that draft looked like a Gordian Möbius knot.
In nine months that paper had bloody tentacles spouting from the fractally weirdening insides of it, metaphorically speaking.
The problem was this: in mathematics, it’s difficult to know if something is true. Suppose you want to know if all Putz functions have the Nebbish property. Either you find a big technical circumloquacious proof they are, or you find a non-Nebbish Putz function.
Then again, you can assume a non-Nebbish Putz function and show that with that assumption the structure of mathematics crumbles to the ground, beginning with one equalling zero; that’s a contradiction, and hence there is no such thing and all Putz functions have the Nebbish property. Or, finally, you can find a scrotum-shrivelingly horribly long proof that, yes, we have a non-Nebbish Putz though we don’t know what it is.
Finding, building and checking the proofs can take months, years, lifetimes; I get the dry heaves thinking of all the things that are true but might need Fermatian and Sisyphean labors to actually prove. And often that proof is a single winding tortuous (or torturous) path that, if a single cobblestone is missing, falls apart.
A counterexample or a contradiction, on the other hand, might take just as long to find — or might take a lucky five seconds, coming like lightning out of clear sky.
And usually you can’t know which it will be: True or false? Years of work or a single lucky insight? Within your grasp, or beyond it? Three choices of two; eight possible combinations.
You expect me to say we worked for nine months on a proof and then someone published a one-liner counterexample, right?
Expecting, then, something related to the science philosophy I began with, you probably then expect some Gödel shit, don’t you? That we found a statement that is undecidable? Or that we found out mathematics as currently done is complete and has no undecidables, and we found something worse, an inconsistency, showed that a statement is both true and false?
Get out of here.
It’s on Wiki fucking pedia that mathematics as currently done, ZFC plus the Axiom of Choice, has undecidable statements (there’s a list of examples!), and also cannot prove itself consistent or inconsistent. Finding an undecidable would be nothing new; finding an inconsistency is impossible.
Why is it always either Gödel or Fermat? There’s so much interesting mathematics, and people always go for swordfighting magic mysteries or Gödelian pseudo-quantum shit. I feel like a doctor that’s never asked anything except the name of that bridge between your nostrils, and the identity of the funniest objects found in the human cloaca.
No, what I am about to tell is worse: in twelve months we had a paper.
Said much like one would say, “in twelve months we had a peepee”.
Gödel and Fermat being exhausted, you probably expect a shaggy dog tale now, that we had found nothing and this is all a sad tale of the difficulty of getting funding, and of the unpredictability of mathematical research and ends with me living penniless under a bridge chewing on a graduate student’s shinbone like a troll.
Ah, the predictability of mathematical narratives gets my goat sometimes.
No, we had a paper; not nice, but publishable. A proof — two different proofs actually. Both bristled with inelegant assumptions, and led into Quasimodo-like siblings of already known results in a more general case. Let me explain.
Suppose the known case is this: “If you have $900, you can get a 1963 Ford Galaxie with it.” That’s a real-life theorem I suppose; I ride the bus.
Our result was something like this: “If you have C credits in currency X with the doubling property, one pee-harmonic black goat of the woods and a set of rainbow candles with a manifold Q, you can get a F(X,C,Q) Goatxie with them”. Where F is a function that looks like a hedgehog with a Hölder enema.
We sent the paper to be reviewed by a mid-rank journal; mathematics being mathematics “mid-rank” means the same kind of an impact factor as physics journals which publish high school science club analyses of fart gas, with pictures. (Not bitter; just sour.)
I myself wasn’t sure what the fuck our result actually said or did; my graduate students understood even less.
That’s the problem with mathematics. As a graduate student of a particular field you’re in the dungeon of Nethack with a tiny flickering feeble finger-Maglite to guide you: you can peer at details but you have no idea of how things fit together. As a result when you prove lemmas you’re doing fine, but when you try to write an introduction to a paper your advisor boggles and, having unchoked herself, tells you what you were actually doing all those months. (The equivalent comment from a physicist might be, “What do you mean, ‘mapping the streets of Bucharest’? This is predicting water channeling on Mars!”)
Then you get your Ph.D., and expect yourself to be magically transformed into a higher being of pure understanding and energy, like Bruce Willis and Carl Friedrich Gauss rolled into one, so that you can just point at a problem and say: “That’s the one. Let’s roll.”
Not quite so.
And so I and my two graduate students sent to a mid-rank mathematics journal a paper that proved something that was hopefully new, surely horrendously inelegant, and probably, possibly, hopefully worth publishing.
The journal sent the paper out for anonymous peer review to see if we were full of shit; this is standard practice and not something that happens to just my papers, thank you very much.
We went on; and in the remaining year of the grant got four papers done. One of them was nice, and one actually good. The other two went into the Great Hural Journal of Mathematics and Yurt Sciences, and into a conference book bashed together in memoriam August Legend Pseudonym. Also, got one of the graduate students elevated from the Mount Doom-ish slopes of thesiswork to the Lengian plateau of Ph.D., with its associated new woes and insights. (Such as the human-corpse eating cult of interview committees; again, not bitter, just sour.)
With the other graduate student, we got into an agreement that it might be best to aim for a thesis defence the next year: the same procedure as every year.
Then, a year having passed, I wrote to the mid-rank journal and inquired, politely, about our paper. (A year is not that much in this business; the last time a mathematics paper was rushed out with great fanfare was when Grak discovered the number eleven.)
A day later, the response e-mail… do you think we actually use paper? What century are you living — wait, no, I’m not getting drawn into the “when the millennium turned” spat again. I got punched for that once already; the last time I’ll try outreach while waiting for my kebab and fries.
A day later, the response e-mail came: the editor was very much sorry for the delay, but there had been complications: the referee had died.
Knowing that August Legend Pseudonym, a person who (to continue the metaphor above) defined and named (after himself, naturally) the 1963 Ford Galaxie, had died recently, I said “Hot diggity!” and felt uncertain.
On one hand, it would be nice to think old Pseudonym could have read the paper, could have liked it, had he had the time.
On the other hand, suppose the paper was schlock, dreck, muck, something that’d come back with a three-line comment showing a shorter, simpler way. That’s what every mathematician dreads.
No, scratch that. What every mathematician dreads is a single question mark, next to something you blithely assumed would of course be true. Something that, when you write it open and begin checking, turns into a leak in the bathyscape Trieste, with the Mariana Trench pissing in: something that is so badly, inescapably, unavoidably WRONG that the entirely paper crumples into a red singularity of shame and blips out of existence.
The thought of that possibility, and Pseudonym seeing that, written by me, a young, up-and-coming researcher, would have been… well, not disastrous because mathematicians as a rule are too detached to be malicious, but fucking embarrassing.
But no; Pseudonym was dead, and the referee, Anonym though possibly also Pseudonym, was dead too. The editor said the paper had been sent to a second referee after the sad news, and he would send a reminder to make sure there were no more delays.
I forwarded this to the graduate student and the new Ph.D., sans commentary on the pants-wetting nightmares of authorship, and forgot about it.
A week later an e-mail dropped from the editor: I got ready for a rejection, and read with amusement, then with vague unease and shame, that the second referee had died, too. They were going for a third, and very sorry for the delay but surely I understood.
Out of morbid curiosity, and in a conference a week later, I made inquiries. (Not a conference in the foreign parts, so no cheap beer and exotic sights — but then again, no danger of a rubber-gloved math-flunking yokel sodomising you for fun and security. They always seem to pick, one, the guys with the biggest paws, and two, me.)
I made inquiries — which was easy, Pseudonym having died so recently, so I had a conversation starter — and tried to find out who the other dead referee had been. Morbid curiosity, nothing more. Besides, our circles (to use a mathematical allusion) are so small and compact that most referees, needing to be people who know shit from solid research, are people you sort of know. Meaning you’ve read their papers, seen their presentations, and in one unfortunate case, seen them nude and arrested after making drunken suggestions to the mayoress of Poznan at a posh and lush academic reception.
I found out there had been a second death, indeed: a tall, thin woman I may just as well call Noma-de-Guerre; a mathematician that spoke loudly, though with an abominable eastern accent, waved her hands like a scarecrow on speed when she spoke, and had a brain that would read your thesis over once, call it trivial, and be entirely right, from her own Olympean viewpoint.
It’s scary to run into prodigies. Scarier still when they say they’re not special; they just work hard. Makes you feel like a worm; a worm in the blaze of glory of a goddess of your shared ethereal domain; but still a worm.
She had killed herself; that was the story. Not because of any crap about geniuses being unstable, or too good for this world, or that being the price of brilliance; all that is slander made up by worms like me, unwilling to look up in adulation.
No, Noma-de-Guerre had been courteous, stable, polite, and restrained as long as not talking about her love, our love, the goddess Mathematheia; there was no reason anyone knew, had heard or could reasonably guess for why she had applied for a membership in a gun club, taken three lessons in safety, and then put a pistol to her head and said these last words to the safety attendant: “I apologize for the mess, but smaller calibers would not be certain to kill.”
In addition to that (and a badly slept night for all of us), I gleaned this: Pseudonym had died of a stroke. Old man, too much stress, understandable really. Who of us could imagine being so active at such an advanced age? Which led to an angry scene, this happening in a pub as it did: a tense, drunken Norwegian function theorist, ten years Pseudonym’s junior, took this as a hint that his mathematical virility was flagging. That was defused with a lot of quick talking; the night ended with the Norwegian weeping he had not gotten anything big done his whole life, though he had wanted to; we either comforted him with sympathy and admissions and fears of the same, or (in a few instances) were too young and arrogant to admit we would just as likely be the same, eventually.
The third referee did not coincidentally drop dead; I was, in an obscure way, rather relieved. His verdict was: Do not publish!
Not what I’d hoped for; but much better than the editor writing “Hey, the third guy died too! Do you want a fourth?”
I muttered a few curses, hit forward (that always helps), and after a quick consultation, sent the paper in for a second journal, a sort of lower-mid-rank journal, slightly above J. Dept. Xerox: if they published it, the paper wouldn’t be seen by much anyone, but I really wanted to know if there was something so bloody wrong with it, and I really didn’t want to do that through the jinxed mid-rank journal and that damned snooty dismissive no-comment third referee.
Thus into the Journal of Local Heroes it went; and either because they mean business, or because they mean business in a different sense (i.e. are unprincipled money-grubbing hacks with referees from the St. Unread Academy of Approval), the article was approved for publication in a month without a single comment save one about typesetting.
So, so much for my faint wish of substantiative critique; I cursed again, forwarded the news to my two co-authors, and turned at other business.
In time, I added a pre-print PDF to my homepage, and to the open, sub-rosa and probably not really legal PDF depository homepage of the greater research team on the subject. (If academic papers could be monitored for copyright violations, you wouldn’t need to worry about education in the prison population. Hell, we could organize a whole university on the inside!)
This and more other business I was then yanked from, a few months later, by one more e-mail.
Well, less a mail and more an extended caps-lock-heavy rant which I would have deleted, had it not come from a university address in Bulgaria.
It’s a delicate balance between being called “a shit-puking devil” in the first sentence, and being addressed out of the blue by a remotely familiar colleague of the same general area of interest.
Smelling collaboration (well, one needs to be hopeful) I read the rant; then re-read it; then made a few inquiries.
The Bulgarian was my third referee, the snooty dismissive one; and, one, he had just been forcibly held in a mental institution for a few months after a total shrieking raving mental breakdown, and two, he was not happy with me.
Figures, I thought: he has to have his breakdown just after rejecting my paper. Why couldn’t he have had the giggles, accepted it with tittering praise, and then went total ga-ga?
As I re-read the rant, something like meaning slithered out from between the boulders of badly translated and occasionally incoherent abuse.
Returning from his forced vacation he had found my paper, which he had thought he had rejected, squashed and obliterated… published in the Journal of Local Heroes, with not a letter changed. More specifically, he had found the paper because of finding in his inbox links to it, links wrapped in questions and layers of puzzlement from his local research group. Then, with no pause, he had flown into this incandescent rage, and extruded this mass of abuse towards my e-mail address, me being the corresponding author and thus the traditional target for comment.
At that point a post popped into my mailbox, from the graduate student — he had been cc’d the same abuse, and wanted to ask what it meant.
The ultimate meaning that slithered out, and that I did not forward to my student, seemed to be something like this: I was a stupid shit-beast for publishing such a loaded article, and had no idea what destructive potential was hidden within it, in an ignorant aside of the whole.
I imagined the air being rent open, the vengeful ghosts of past surging forth and Nazi zombies rising from lakes in Bavaria — then shook my head and tried to puzzle out the exact nature of this “destructive potential”.
I could not see it.
It certainly was not an application; we mathematicians have a special grumbling, grudging vocabulary for that, and my Bulgarian referee had used none of that.
Hence I took the article, sent it to my own advisor of long ago (young beliefs in omniscience die hard), and asked her if she saw anything weird in it.
The next day the phone rang, and I was treated to a half-hour harangue of livid audial abuse; my olden advisor had not liked what she had seen. After shouting myself hoarse with “Why?” against her torrent of doom, she finally told me the reason.
I had not let loose the demons of hell through some trick with dimensions and measures.
I had not discovered an application of Jacobians as weapons of mass destruction.
I had not broken mathematics with some Gödelian uppercut.
No, nothing like that.
I had just written, and gotten published, and into circulation and already widening notice, a paper with an innocent, oblivious remark in it that — for those with the eyes and the wits to see — proved that all the problems and questions of our shared field of study reduced to three lines of dismissal, or to a single sentence —
Trivial business; has nothing of further interest in it.
You can expect some abuse when you accidentally show the whole subject collapses into another which has already been solved and exposed to death. Sod philosophers of science and their talk of what we may never understand; understanding can cause funding problems too.