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How to Learn Like Leonardo da Vinci

Thinking it sounded fun, I registered last winter for a college course entitled “Symbolic Logic (MTH 303).” I had expected a Junior-level math course, even one so harmlessly christened, to pose a challenge, but it was more grueling than I could have guessed – cruelly so, even, because after every unbearable homework problem had been resolved, all one could think was, “Oh – of course. Obviously. How could I not have seen that? Ugh.” We couldn’t even sympathize with our yesterselves.

(Math, incidentally, is only tangential to this post, and an understanding thereof even moreso. Non-math folks, please hang in there.)

What Results from Their Intersection

As it turned out, the course primed us in the rules and symbols of standard mathematical logic (Modus Tollens, DeMorgan’s Laws, Ǝ symbolizing “there exists,” etc.) before using those to cover formal proof language and to devise proofs for scrupulously basic math concepts (proving, for instance, that A+B=B+A). Typically, symbols would be defined, axioms laid down, and we would be tasked with proving either something basic about addition or something indecipherable about an invented universe we didn’t recognize; sometimes they turned out to be real math models, like groups, fields, and rings, and sometimes we would combine the concepts’ axioms to come up with even more theorems.

In a passing remark during one such exercise, the prof spoke what became the most far-reaching, universal lesson I took away from the course: paraphrased much, “Now, we’re not combining these two concepts just for the heck of it; we want to see what interesting things result from their intersection. Otherwise, there’d be no point.” (He may have actually spoken “Ǝ no point”, but I digress.)

Since then, I’ve noticed how run through our universe is with connections and similarities. Just today, a friend of mine remarked how all of the breakthroughs in math are coming from people who can make connections between the separated fields, like the solver of Fermat’s Last Theorem whose victory was achieved with parabolic curves.

Connections Just Waiting to Be Discovered

It really is wonderful how so many different things in life have underlying themes, how so many things in life connect, just waiting for people to discover it. This can involve coming at ideas from unconventional angles; approaching one thought from a different perspective, either one difficult to achieve or simply unconsidered, may reveal some alignment between it and another that leads to a better understanding of both. Some angles are relatively obvious but hard to pull off; I’ve heard of one alum from my college who triple-majored in German, economics, and mathematics. These three subjects, very different, nevertheless intersect and link at multiple points that make an understanding of all three impressive. Imagine reading the logician Gödel and the economist Marx in their original; imagine applying rigorous math logic and higher math understanding to economic models. As in the case of another friend’s double major in music and math, most of us can guess at (though probably not fully grasp) the revelations granted to people with the right perspective.

I was sitting, writing at my text-book; but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by the repeated visions of the kind, could now distinguish larger structures of manifold conformation: long rows, sometimes more closely fitted together; all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis.A whole other species of breakthroughs happens when people realize connections so unconventional that no one thought to look there for anything; these more readily happen to people. Consider August Kekulé, who realized in a flash the structure of benzene with its alternating single and double bonds:

The image of a snake eating its own tail is one of ancient myth: Ouroboros, they named him, never suspecting his structure to be comparable of benzene’s.

This sort of thing also happened to Archimedes, realizing where his body mass and the king’s crown intersected – discovering, in fact, an underlying unity to reality in general. Through something distant, these men with insights find themselves back at the answer to their original problem. Their attention, directed at one thing, realized that it remained pointing at its original subject from another angle; they directed their attention elsewhere and it wound up misdirected back to home base.

Everything Relates to Everything Else

The point here is that knowledge is integral; everything relates to everything else. This underlying unity, disguised by limited perspectives and exacerbated by artificial partitions, is discussed by Dorothy Sayers in her essay “The Lost Tools of Learning,”  in which she contrasts modern education with medieval education:

“Do you often come across people for whom, all their lives, a “subject” remains a “subject,” divided by watertight bulkheads from all other “subjects,” so that they experience very great difficulty in making an immediate mental connection between let us say, algebra and detective fiction, sewage disposal and the price of salmon–or, more generally, between such spheres of knowledge as philosophy and economics, or chemistry and art? . . .

[Much later:] It would be well, I think, that each pupil should learn to do one, or two, subjects really well, while taking a few classes in subsidiary subjects so as to keep his mind open to the inter-relations of all knowledge. Indeed, at this stage, our difficulty will be to keep “subjects” apart; for Dialectic [Medieval stage #2] will have shown all branches of learning to be inter-related, so Rhetoric [stage #3] will tend to show that all knowledge is one.”

Sayers hints at the unity of understanding, and the significance of this unity is that, indeed, two apparently disparate things may have meaningful, illuminating, fascinating things to say about each other. This unity is marvelous, the intelligibility of our universe being scrawled on every wall and corner of itself, faster than we could read. Trying to read all of it, I learned recently, is the properly understood motive of the Renaissance man; in one sense, not trying to master everything, but trying to figure out nature’s thread of gold running through everything, to understand the one thing from every angle. (It was the Renaissance that discovered perspective; Renaissance men that they were, it seems they imported the concept into their understanding of truth and study.)

The Thread that Fascinated da Vinci

It was a motif that fascinated da Vinci, the Renaissance man. That man, who studied so hard at almost everything, was looking for how everything came together, precisely how everything already was together. This is manifest in his creation: He employed anatomy and mathematics in art, broad experience in fable writing, and this unifying idea itself in certain depictions of plants that thread all around the picture the while being, on closer inspection, a single thing. He was enamored of the idea of the “golden thread” that weaves in and out of all life, experience, and being, connecting everything with everything else.

Often, these discoveries lead to creations, the other side of the coin. Creation was the subject of an article I happened to be reading (“Mathematical Creation,” by Henri Poincaré) while writing this post, and I discovered my whole idea already expressed, and elegantly:

To create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority. Invention is discernment, choice. . . . those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another. Among chosen combinations the most fertile will often be those formed of elements drawn from domains which are far apart. Not that I mean as sufficing for invention the bringing together of objects as disparate as possible; most combinations so formed would be entirely sterile. But certain among them, very rare, are the most fruitful of all. To invent, I have said, is to choose . . .

Examples abound. The Comte de Buffon made a name for himself pursuing what ideas, problems, and solutions, came of combining geometry and probability (“What are the odds that a coin tossed on a tiled floor touches no edges?” &c.).

Crossovers in Poetry and Video Games

Or consider one story in the work of the Latin poet Ovid, whose supreme epic Metamorphoses, part compilation and part re-working, chronicled over 250 myths across fifteen books. One (or two) interest/s me: in Book III, the character of Narcissus is introduced, withholding himself from all the pretty girls (and boys). The cursed nymph Echo falls for him, unable to speak anything of her own and only repeating the ends of others’ sentences, and is rejected by him; she ends up in a cave where she wastes around, down to her voice. Afterwards, Narcissus falls in love with his own reflection in a river; “wasted by his passion,” he then dies.

What’s fascinating is that the tales of Echo and Narcissus were separate – Ovid combined them, Ovid took two separate myths and fused them as one. This combining was neither haphazard nor fruitless; the new tale gives a new perspective on Narcissus, allowing him, in a sense, to live out the reflection scene before being consigned to it, insofar as with Echo, as with his reflection, he is dealing with someone/thing that only repeats to him what he has first given it, though with this important distinction: Echo remains a personal (not quite “human”) being. With Echo, he has a physical and separate being dependent on his input and desiring him; he is given a chance before the reflection trap to, as it were, “have it all.” The two stories, then, are almost variations on each other that color and expand the other.

Examples even reside in the realm of video games. (More accurately, perhaps, these unique and excellent fusions thrive there.) One of the most popular games of 2007 was Portal, a first-person-shooter puzzle game. Combining these two genres, which one might beforehand have placed at opposite ends of a gaming spectrum, was manifestly a stroke of genius. Nor were the creators content merely to graph one onto another, interrupting traditional, Call of Duty-style scenes with levels of Tetris, for example; the two are integrated, wedded, inseparable. Portal without puzzle is as unthinkable as is Portal without FPS; both are essential because the creators combined the genres thoroughly, perfectly, with spectacular results. Another of the most popular series of all time is Super Smash Bros., a game series that combined fighting with – pinball. (I read once that this was deliberate; though I am unable to relocate the reference, the underlying pinball nature is so manifest that I would argue that it exists regardless of intent and that though this may detract from the creators’ glory, it serves just as well for this argument of combination.) The lifeblood of Smash Bros is fighting, but done so in a uniquely pinball manner – KOs consist of knocking opponents off the 2D stage, while actual pinball bumpers are in-game items – that is not only distinct but fantastic, a winning formula, the expression of two things combined amazingly. All of the latent, interesting intersections of fighting and pinball flower.

The Mash-Up Mentality

I feel I should also address the obvious perversion of this idea. A common misunderstanding is to aim at dabbling in everything, “a little of this, a little of that.” I think a more insidious abuse is current in the creation half, where people combine issues that actually have (at least in such a respect) nothing to do with each other. These sterile combinations, the antithesis of Poincaré’s mathematical creation, form the mash-up mentality. This mentality, mere cross-over humor, rampant in much Internet pseudo-humor, leads to creations like renderings of Angry Birds and The Avengers as one (or vice versa). Why is that interesting? ‘Well, it just is; you combined the two things’. So-called “Random humor” often operates on this principle, introducing something which has no place in the current context on the pretext that it’s consequently funny, by no virtue but that of the randomness itself.

If I were to make one case out of this (far from original) observation that calls mainly for wonder and pondering, it’s a recommendation that people throughout life try to keep learning – and not just “in their fields”, but in disparate ones, far-flung, apparently unconnected. It may well help them not just know more in general, but even have a new perspective on and further understand their own field. It suggests great potential in misdirecting attention – sending your attention elsewhere and discovering, as though accidentally (it probably will be), you have returned to your origin. It’s a bit like flying in the Hitchhiker’s Guide to the Galaxy: One need only throw oneself at the ground, and miss. This is actually possible in our world, because the ground and the sky have much more in common than we suspect; the analogy breaks down insofar as, here, you can contact the ground and still be flying.

Noah Diekemper

Author: Noah Diekemper

Noah Diekemper studied Latin and Mathematics at Hillsdale College (B.S.) and Data Science at Loyola University Maryland (M.S.). Motivated by a desire to preserve and share knowledge, he contributed to and has also been published in The Federalist, Intercollegiate Review, and The Baltimore Sun.

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