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Free Mathing, like Free Writing

Yesterday afternoon I found myself sprawled on the couch, eyes half open, the concepts of dot product, cross product, projection and determinant swimming around in my head. It’s worth noting here both that I studied up through a MS in math and that I’m no longer a mathematician professionally. I came upon these concepts idly walking memory lane, as happens on Rosh Hashanah.

It felt great—and not just because hey, I know a lot—still got it a decade later, baby!

Doing original research in math rarely felt great when that was my task. Full of dead-ends where you think you’re about to crest a peak to a beautiful vista. An emotional tug-of-war between excitement and despair that probably didn’t help my bipolar disorder any. I often explain, post facto, that languages ended up being a better fit for me because they allow reliable, accreting progress. If I work at a language, I‘m more proficient at the end of the day than I was when I woke up, by 8 new words or a syntactic structure or maybe an inspiring target-language conversation. If I do wake up sweating at 3am—as happens—it won’t be because I recognized all the last day’s work was flawed. When I did math research, I woke up to the demoralizing poof of a good day’s thinking more often than I woke up to a fruitful new idea.

But yesterday, I knew the math I was running wasn’t going to go poof. The thinking felt like improv, where every choice is right and you just keep moving forward. It was goal-less, aimless, intensely associative, fleet, light, inconsequential. I started with a concept—a landmark on the surface of my mental model of Math World. If the flight of math had started with a problem statement, I imagine it would’ve been what a Zen koan is for the one about to sit for meditation: a jumping off point, an invitation to someplace that promises to be cool. Not a challenge demanding a solution. A problem you actively don’t bother solving, even if you know it’s an easy one and within your wheelhouse. An infinite game and a cooperative one between me and math.

One of the mini-games is how many paths forward you can generate from a single unit of math. Another is minimizing the size of your displacement while maximizing the distance wandered. If you get back to where you started, it’s a return to Cavafy’s boring Ithaca, enriched by the travels. If you set out again, in the same spirit from the same spot, oh the places you’ll go!—destination unknowable, as to Bilbo walking out his door. Any dead end reveals itself as a portal: you glide through to the next stop, no banging your head or re-pacing your steps.

I’m going to call this thing free-mathing, by analogy with free writing. It seems impossible: the internal editor is central to doing math. But I think the sense of play, openness, spontaneous generation and infinity can be likened to the state free writing wants to generate.

Free-mathing shares benefits with free writing, too: warming up the mind and getting the gunk out; lowering the activation energy for approaching the Serious Real Work that Must Be Excellent. Both mean you get in reps of concepts and skills you’ll need for a “real”, convergent-thinking math or writing task down the line, whether in five minutes or six months. Both show you what you know and actually don’t know so well as you thought you did, while being rather less critical of you when compared with a set of exam questions or the strategy of self-testing.

And why bother with this, now that I don’t “do” math anymore? I enjoy free-mathing once in a while, and now that I’ve named it maybe I’ll try it more often. I already have the permission from myself not to take my hobby math seriously. Maybe if I practice not taking things seriously there, it’ll make my writing work more approachable, too, the work I take too seriously to do it at the moment. If I freewrite and freemath and my pattern-loving brain sees the connections between the two things, maybe I’ll get more playful about serious writing. Or, supposing (ridiculously, helpfully) that Plato is right that all learning is recollection, maybe I’ll remember the lightness of play on the mind.

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Scientific methods, not method: going back to the sources for Francis Bacon

The natural philosopher Francis Bacon is well-known (among historians of science…) for his Novum Organon that rejected much of Aristotle’s theoretical method (the Organon) and sought to establish methods of investigation that intimately bound empiricism and abstract reasoning. In the blitz history of science given in science classes, he’s one of the people cited as a founder of modern science, clearing the intellectual detritus of reliance on ancient authorities and handed-down notions.

In the passage I’ve translated below, from the preface to the work, we see a more complex Bacon (as is usually the case—let’s return to the sources when we can!) He is hardly triumphalist about the powers of his method, or of human reason, and he has more respect for what the ancients and his predecessors in philosophy achieved than I, at least, expected given what I’d been told. (I suppose Bacon would have approved of me going back to the sources and breaking through the confusions and errors brought on me by received wisdom, even given with the best of intentions.)

I give this translation in the spirit of sharing works-in-progress; there are many things about it that I’m unsatisfied with, among them my somewhat archaizing style and use of “men” because “humans” still feels awkward to this genderqueer-yet-methodologically-conservative classicist.

Original Latin text here (the two paragraphs starting from “Qua re, ut quae dicta sunt complectamur…”).

“It seems neither trust in others’ authority nor our own individual efforts have favored us humans in illuminating the sciences so far—especially since neither demonstrations nor experiments done up to now have been much help. Human reason, contemplating it, finds the edifice of this universe, its structure akin to a labyrinth, where so many impasses in the ways, such deceptive analogies between signs and things, such crooked, knotted coils and tangles reveal themselves everywhere. We must constantly journey through the forests of experience and particular things by the wavering light of the senses, now shining forth, now concealing itself. Even worse—those who offer themselves as trail guides also get turned around and increase the number both of errors and those erring.

In such difficult matters there is little hope of progress for us humans by our own powers or happy good fortune. Nor can excellence of mind or endless rolls of the dice overcome the difficulties. We must step forward along the threads reason lays down: every line of inquiry, starting from the first impressions of the senses, must be supported by reason. But nor should we deal with these matters as if they hadn’t been treated for many centuries, by many hard labors. We are not ashamed of what has been found; we do not repudiate it wholly. And the ancients certainly showed themselves to be admirable men, to the extent that they treated these matters by their own powers of mind and abstract meditation.

Conquered by an eternal love of the truth, we give ourselves over to unsteady and demanding paths and vast deserts; and leaning on and held up by divine aid, we keep our minds steadfast, against the buffeting of opinion, practically arrayed in battle against us, and our own internal hesitations and anxieties, and no less against the fogs and clouds and phantoms of the things themselves. Thus we can gift our contemporaries and those who come after us with more trustworthy and sure signposts. And if we have made any progress at all in this, the way was laid open to us by nothing other than true humbling of the human spirit.

For all those who applied themselves to the invention of arts before us, having taken a cursory look at the things themselves and case studies and the evidence of their senses, invoked their own spirits as if shown to them through oracles, as if discovery came from nothing other than a bit of cogitation. We, though, perpetually sitting with the same phenomena and exhibiting restraint, do not let our intellects roam farther from the things themselves than is necessary to form impressions and images of the concrete things in our minds—and so nothing much is left to our mental powers and excellence. And we have shown the same humility when we teach as we use when we make researches. 

We do not attempt to impose or force a magisterial quality on our discoveries, neither by flashy refutations, nor invoking the ancients, nor by force of authority or any cover of obscurity. This would hardly be difficult for anyone who tried to illuminate his discoveries for his own sake and not others’ benefit. I do truly believe that we have not used any force or set any treacheries for men through our judgments, nor are we planning such now. We are, rather, leading them to the phenomena themselves and the connections between them, so that they may see for themselves what our arguments have in them, what they explain, what they add and contribute to common knowledge. And if we held false beliefs in some matter or fell asleep on the job and payed poor attention, or got lost on the path and screwed up our inquiry, we have endeavored to show these, our errors, too, unadorned and openly, so that our mistakes can be noted and separated out before they infect our scientific enterprise more deeply, and so that our efforts may be continued with fewer impediments and the way forward easier. In these ways, we trust we have formed a true and legitimate marriage between empirical investigation and abstract reasoning, whose wayward and unfortunate split and rejection of each other shook up everything in our human family.”

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Maria Popova: A Dear Mentor I’ve Never Met

Maria Popova is a thought-hero of mine. She’s not an idol—rather, at my best thinking self I try to be her apprentice. No, we’ve never met. And yet, as is manifest in her writing on years of written correspondence between women who admire each other intimately, we can seem to ourselves to know a soul through its verbal emanations.

Popova is a rigorous curator of poetic-scientific feeling. Her teenaged (13 years of consistency!) weekly newsletter is timeless—she puts the date only in the URL, in case you need it for citation—not in a way that endorses the shill of universal humanity, but because its story-threads accrete into a solid world. Her implicit theory of knowing prizes personal experience while including our full intellectual, vicarious, imagined worlds therein, avoiding the smallness of the preciousness and pathos that she does not devalue. She values grand narratives and builds them bottom-up, dialectically and dialogically insisting that the cosmos, and ours, are a rich tapestry of glittering details and a sweeping structure that is true and beautiful scribed in ideal forms. If I may, she’s a Platonist imagist and an Aristotelian observer, availing herself of the logical tools of both.

Her worldviews, through my kaleidoscope, give me confidence that our brilliant hearts can perceive the gestalt, a solid whole with these threads as atoms. Our consciousness can expand to simultaneously comprehend the old woman and the young one in that classic demonstration of shifted perception: we are not limited to watching the shifting tensions in pretty partner dances between micro and macro.

Popova’s poetics of bodies and evidence do have arguments, contra Jorie Graham’s idea that “great poems have so few arguments in them.” But she agrees with Graham in not “want[ing] to make the reader ‘agree’.” Her magnum opus (thus far), Figuring, is a smart book and a full one, loving both structure and flow, vivid and associational while taking real things as its raw material—if we understand feelings, mysteries and phenomena all as reality. Above all it holds all these poles as glimpses of a whole, fading and emerging by tricks of the light, in a decidedly expansive and expanding geometry.

The author’s voice is rarely present in first person in either Popova’s newsletters or Figuring, but only she could have written her work. And so she provides us a model of clear observation by the light of the heart and me with the courage to speak words like “heart” without a reflexive academic flinch. As a curator she is no mere compiler, but gets idea-full figures to play with each other through what feels like a light touch of intuition but I know to be thorough investigation. She runs towards intensity and in that process guides me to a practice of doing so, despite my fears, despite the defense of “rigor!” that we academics so often throw up against the intrusion of abiding care.

BrainPickings is an email newsletter you will actually open every week. Figuring is a bigger and differently rich attention commitment. I’m going to be a terrible curator here and point you to the whole damn archive to find what resonates, with votes in for mentions of Oliver Sacks, Carl Sagan, Johannes Kepler, Emily Dickinson, and the wisdom of trees. Each article ends with a “complement with” section that is Popova curating herself: trust her to guide you, as I do.

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Scholar, would you tell me a story?

“Proportions alternate between infinitesimal and astronomical. The signals are infinitesimal. The sources are astronomical. The sensitivities are infinitesimal. The rewards are astronomical. The human ambition to understand the universe is merely epic, and astronomical trumps epic.” – Janna Levin in Black Hole Blues

    My father, the wonder-full layman, and I, the one who broke up with professional science, read the same pop science books. We know that where they make bald declarations they are wrong in significant ways. From the ones who seem to have their noses thrust up skywards as they declaim we keep a correspondingly disdainful distance. Stephen Pinker is too confident. Malcolm Gladwell pushes too many Unified Theories. The collector and commentator E.O. Wilson of The Diversity of Life is far more compelling than the later E.O. Wilson grasping after ultimate meaning in Consilience. I find Wilson on social insects more beautiful than Wilson on beauty—in his care, his wonder, his generosity towards both subject and audience.

 All the authors I’ve cited put forth simple, big ideas. Simplified, too. They leave out most footnotes or make them endnotes, as I believe they should. As best I can figure it, the ones that inspire wonder and midnight conversation differ from the ones we condemn as condescending and simple-minded mainly in whether they are inviting us to sit in front of or next to them. 

The expositors who profess in their books like they do in the lecture hall are hit-or-miss. If we don’t buy or even are simply skeptical of their One Big Idea, the whole reading experience falls on the rocks. Knowing that 10,000 practice hours does not necessarily make one an expert knocks a whole subgenre of performance psych books out of consideration for what to spend my learning time on. I am so wary of the it is so because prehistoric man… just-so stories that I missed out on gems from behavioral ecology like Joan Roughgarten’s Evolution’s Rainbow for years out of fear of proximity to pop evolutionary psychology. The authorial voice in these works is strong, ruining even meaty evidence and anecdote with tiresome insistence on the Big Idea. It doesn’t matter how much good there is to be found in these books (often very much!): I come away resentful, my curiosity dulled. The author didn’t really trust me to follow along, so I don’t follow up. I imagine that he (often he) resents staying for the Q&A after he has given a talk.

The writers whose words ripple through me long after my first reading have strong authorial voices too. But they modulate them for the audience in the room. Janna Levin, Carl Sagan, Stephen Jay Gould, Atul Gawande, Maria Popova—they invite me to wonder with them, rich in offerings while declining the high throne of authority. Reading their work, it is simplified, too, but we both trust the other knows that. We are vulnerable with each other: we don’t know, we say, and we want so badly to know, knowing we can only ever do so in part. We are no nihilists: we treasure little fragments of knowing. Like the proportions Levin sings, we are small and great, pieces and wholes. Wandering the world of astronomy we are astronomical: that is to say, we dance between the infinite and the infinitesimal and see them together dialectically in freeze-frame. As a reader I am not frozen in the spotlight of the author’s brilliance—rather, the author has a pocket flashlight to illuminate the next few steps, or invites me in as observer when she gets her few hours a year on the radio telescope, or maybe just knows where there’s a working lighthouse beaming onto this thought world I am just meeting.

So, to those, like me, who fret over telling simple stories of their nuanced worlds: show us more than telling us. Give us a map of your terrain and let us loose, but be there when called upon—whether with notes, correctives, countermelodies from other voices, or a URL that grants access to more context and sources. As your reader, I promise that I know you are telling one simple story of many not yet told. I know you have woven this one with care, oh, so much care. But I do not grieve as you do at all the untold stories, because I trust that the path you charted through them for me will lead to others—other stories, other tellers from your same world. I thank you for the courage you’ve shown in inviting simply. I will invite others in turn—my dad, first of all—and slowly, in wonder, I will learn to know better. Only open the door: we have been waiting. We who do not want perfect—who would be insulted if you claimed it. We ask, simply, for a thread of connection into your world of meaning. For a voice to end the silence, even if it trembles.