On System 3 Thinking
It's the most basic rule of the stymied researcher: stop thinking about the problem. Do something else. It will come to you if you just stop being conscious of it. ⸻ Peter Watts, Blindsight
Amanda Askell recently tweeted:
System 1 = fast, implicit reasoning
System 2 = slow, explicit reasoning
System 3 = slow, implicit reasoning
For me, system 3 is the real genius of the lot.
This tweet rings true to me, and I think that system 3 is a useful addition to Daniel Kahneman's taxonomy.1 This post will discuss several anecdotes that illustrate why.
#Mathematics
In addition to Askell's breakdown, it will also be useful to keep Jacques Hadamard's definition of the same concept in mind throughout this post. In An Essay on the Psychology of Invention in the Mathematical Field, he defines it as follows: system 3 insights appear (1) "without any relation to [one's] attempts of former days, so that it could not have been elaborated by [their] previous conscious work," and (2) "appeared without any time for thought, however brief".2
Hadamard's book is ripe with examples of the phenomenon, making for a good starting point for this post. The first one comes from Henri Poincare. Hadamard recounts Poincare's attempt to form a theory of Fuchsian groups and functions. Poincare started off with a futile attempt over a fortnight trying to prove that there could be no such functions. He eventually built up a class of these functions and wanted to find an expression for them. Here's what happened next, in Poincare's own words:
I wanted to represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate; the analogy with elliptic functions guided me. I [...] succeeded without difficulty in forming the series I have called thetafuchsian.
Just at this time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. I did not verify the idea [...] but I felt a perfect certainty. On my return to Caen, for conscience's sake, I verified the result at my leisure.
Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty, that the arithmetic transformations of indefinite ternary quadratic forms were identical with those of non-Euclidian geometry.
Hadamard's next example comes from Gauss, who wrote about a theorem that he had tried to prove in vain for years:
Finally, two days ago, I succeeded, not on account of my painful efforts, but by the grace of God. Like a sudden flash of lightning, the riddle happened to be solved. I myself cannot say what was the conducting thread which connected what I previously knew with what made my success possible.
A final example comes from Hadamard's own experience, from when he grappled with a difficult determinant calculation:
On being very abruptly awakened by an external noise, a solution long searched for appeared to me at once without the slightest instant of reflection on my part the fact was remarkable enough to have struck me unforgettably and in a quite different direction from any of those which I had previously tried to follow.
#Arts and music
One of my favorite tales in musical history is the origin story of The Beatles's "Yesterday". In 1964, Paul McCartney woke up one morning with a complete melody in his head. It was so fully formed that he immediately went to a piano in the room and played it through, convinced he must have heard it somewhere before. McCartney then played the song to the other Beatles, Mick Jagger, George Martin, and other musical heavyweigths of the day, asking each of them whether they'd heard the song before. After a few weeks of blank stares, McCartney finally became convinced that the song is original.
Though examples as dramatic as this one are rare, there are several other musicians who have described a similar phenomenon or written songs in less than an hour. Hadamard's book covers the creative process of Mozart and draws parallels between him and the aforementioned mathematicians. When Elton John received the lyrics of "Candle in the Wind" from Bernie Taupin, he immediately sat down at the piano and composed the melody in one sitting. Noel Gallagher has mentioned writing multiple of Oasis's best songs in under 30 minutes. Further examples of lightning-quick music writing are discussed in David Hartley's YouTube video Why are the best songs written so quickly?
Painters also seem to benefit from a similar phenomenon. To leverage system 3 thinking, Salvador Dali famously used to take naps with a key in his hand that woke him up at the moment of falling asleep. Here's a quote from Dali's 50 Secrets of Magic Craftsmanship:
In this posture, you must hold a heavy key which you will keep suspended, delicately pressed between the extremities of the thumb and forefinger of your left hand. Under the key you will previously have placed a plate upside down on the floor. Having made these preparations, you will have merely to let yourself be progressively invaded by a serene afternoon sleep [...]. The moment the key drops from your fingers, you may be sure that the noise of its fall on the upside-down plate will awaken you, and you may be equally sure that this fugitive moment when you had barely lost consciousness [...] is totally sufficient [...] to be revivified by just the necessary amount of repose.
#Are insights found in dreamlike states a form of system 3 thinking?
The example of Dali brings us to the question of whether insights found in dreamlike states also constitute a form of system 3 thinking. Dali is not the only genius who has used the nap technique described above—here's a second-hand account of Thomas Edison's nap routine:
The inventor is said to have napped while holding a ball in each hand, presuming that, as he fell asleep, the orbs would fall to the floor and wake him. This way he could remember the sorts of thoughts that come to us as we are nodding off, which we often do not recall.
Similarly, Ramanujan was famous for claiming that his discoveries were revealed to him in dreams. Finally, we have August Kekulé's discovery of the benzene ring, who recounted the experience as follows:
There I sat [...] and fell half asleep. Again the atoms gamboled before my eyes. Smaller groups this time kept modestly to the background. My mind's eyes, trained by visions of a similar kind, now distinguished larger formations of various shapes. Long rows, in many ways more densely joined; everything in movement, winding and turning like snakes. And look, what was that? One snake grabbed its own tail, and mockingly the shape whirled before my eyes. As if struck by lightning I awoke. This time again I spent the rest of the night working out the consequences.
I'm less sure that these examples constitute instances of system 3 thinking than I am for the examples given in the two sections above: Kekule's discovery could well have been an instantaneous intuitive analogy, rather than the result of a long background reasoning process. Similarly, Edison could have been hunting for intuitive analogies during his naps, rather than fully formed hypotheses or device designs. On the other hand, consider Kekule's quote again: the part about having been trained by visions of a similar kind seem to play a crucial role. Since at least some subconscious background processing seems to have been involved in the discoveries described here, they seem worth including as plausible examples of the phenomenon.
#Is system 3 thinking a disguised form of system 1 thinking?
Is the concept of system 3 thinking actually necessary? Couldn't we find a way to fit the examples I've given into the system 1/2 dichotomy? It sure isn't system 2—it seems unlikely that even Poincare would be able to do cutting-edge mathematical thinking while having to navigate omnibus steps, and he himself testified that he was thinking of other things. Could these have been instances of quick, on-the-spot intuition? This seems more plausible. However, consider the examples of system 1 thinking Kahneman provides in Thinking, Fast and Slow:
Detect that one object is more distant than another.
Orient to the source of a sudden sound.
Complete the phrase "bread and ..."
Make a disgust face when shown a horrible picture.
Detect hostility in a voice.
Answer to 2 + 2 = ?
Read words on large billboards.
Drive a car on an empty road.
Find a strong move in chess (if you are a chess grandmaster).
Understand simple sentences.
Recognize that a "meek and tidy soul with a passion for detail" resembles an occupational stereotype.
The only example here that's remotely similar to Poincare's insights or the making of "Yesterday" is finding a strong move in chess. However, finding a strong move in chess can still be a matter of quick pattern-matching, while putting together an entire proof or song is more like a long and hard sequential reasoning problem. Maybe Poincare and McCartney were so good at their craft that every single sequential reasoning step was akin to completing the phrase "bread and ...", though, making it possible for the brain to construct the entire proof/song in an instant?
This seems unlikely to me for a few reasons. First, there's the incubation time. Gauss mentioned working on his proof for years before it instantaneously struck him—if it was just a matter of pattern-matching, why didn't the pattern fire right away? And second, all the examples here involve temporarily stepping away and leaving the problem aside, which shouldn't help with an instantaneous pattern-matching problem. This makes me inclined to consider system 3 thinking as a distinct category from systems 1 and 2.
#Leveraging system 3 thinking
How can we leverage system 3 thinking for better ideas and higher productivity? I'll let Paul Graham speak for myself:
I think most people have one top idea in their mind at any given time. That's the idea their thoughts will drift toward when they're allowed to drift freely. And this idea will thus tend to get all the benefit of that type of thinking, while others are starved of it. Which means it's a disaster to let the wrong idea become the top one in your mind.
You can't directly control where your thoughts drift. If you're controlling them, they're not drifting. But you can control them indirectly, by controlling what situations you let yourself get into. That has been the lesson for me: be careful what you let become critical to you. Try to get yourself into situations where the most urgent problems are ones you want to think about.
Graham's post takes the importance of system 3 thinking as a premise and provides excellent advice for productive thinking. I suggest that as your next stop after this post.