"Toutes les grandes personnes ont d'abord été des enfants. (Mais peu d'entre elles s'en souviennent.)" [All grown-ups were once children... but only few of them remember it.] Antoine de Saint-Exupéry, "Le Petit Prince."
An impromptu game of "who can name the largest number?" is an example I often point to as a way in which young children teach each other about non-trivial mathematical ideas and facts, such as infinity and the concept of proof by contradiction (embodied in this case by one child eventually realizing that they can defeat any other child's guess by adding "plus one" to that guess, thus demonstrating by contradiction* that there is no largest natural number).
Prematurely putting such games to an end by immediately providing the nominal goal --- for instance, providing these children with a brief lecture on infinity and the unbounded nature of the natural numbers --- defeats the entire "purpose" of such curiosity-driven play, insofar as play has a purpose: to explore, to learn, to develop, and to simply have fun.
- to be pedantic, this is a proof by negation rather than a proof by contradiction, but that distinction is not the point of the anecdote I wish to make. (1/3)
Children eventually become adults, and "put away childish things". Significant portions of our time become devoted to optimizing "important" and immediate goals: weighing the costs and benefits of various decisions, and acting on them accordingly. Unstructured play and exploration become restricted to one's hobbies or personal activities only. This is so normalized in modern adult life that most of us simply take for granted that this sort of goal optimization is what anything "serious" is necessarily about.
But science --- particularly basic, curiosity-driven sciences such as pure mathematics --- is a notable counterexample to this general rule: one of the rare and precious disciplines in which childlike wonder and exploration can be put to "serious" use. We can ask questions which appear frivolous and of no direct application (such as the global regularity problem for the Navier-Stokes equations, if truth be told); but if they are well-chosen, then scientists or mathematicians can explore them very fruitfully with the right combination of adult discipline and expertise, and childlike whimsy and fearlessness. We are still trying to achieve the goals we set for ourselves eventually; but the best outcomes arise when the journey is unhurried, convoluted, and serendipitous. Optimizing prematurely to reach the nominal goal can extinguish these unexpected detours, stories, and discoveries before they have a chance to manifest themselves. (2/3)
Many in my profession are drawn to it in part because of this ability to direct these childlike impulses to productive use, which leads surprisingly often (by an extremely indirect pipeline) to actual advances in science, technology or engineering as per Eugene Wigner's "unreasonable effectiveness of mathematics". But, perhaps in order to impress the other adults observing (and, through public funding agencies and the like, financially supporting), we often downplay the "childish" aspects of the craft, in favor of the more tangible "serious" outcomes --- including the hard, objective target of solving some designated open problem.
Until recently, this polite fiction has worked, because human mathematicians knew, possibly subconsciously, to use both childlike and adult skills when solving these problems, even if they might only publicly admit to the latter. But modern AI tools, when directed without such expert supervision, can now be aimed all too easily at the ostensible goals of the field, without any incentive to take the slow, whimsical path. The flag is captured, the goal scored, and the problem is solved; but at the cost of lessons learned, insights gained, collaborations formed, and new targets located.
In this modern era of heavy AI use, it is important to remember what it is like to be a child. (3/3)
@DavidKButler I think you will like this thread by @tao
For others, cf https://davidkbutler.xyz/2026/08/23/ma ths-sheep-play-sheep/
Making Your Own Sense · Aug 23Maths Sheep Play SheepIt’s Children’s Book Week in Australia, and so it’s a good time to write a little something about a project I did in honour of Book Week in 2023: a picture book called “Math…@highergeometer@DavidKButler@tao That's wonderful 
I read where is the green sheep to my kids many times. I'll read math sheep play sheep to myself now 
@pabryan@highergeometer@tao So glad you like it!
Very much so! Both the write up and the book - I watched your reading which was lovely. I think you captured the spirit of green sheen perfectly.
@highergeometer@tao Yes I do like the thread. It speaks directly to my playful heart.
@DavidKButler@tao What an absolute treasure of a book!
@aps@highergeometer@tao So glad you like it!
@tao I don't agree: exploration without a direct goal is important in many job with even a little bit of creativity involved. Even a tiny improvement to a work process can be discovered by chance, and it may matter.
@tao , probably academia will need a private, closed door, well regulated (by the academics) AI provider where privacy and ethics are preserved.
But until this happens, forget abou the private sector taking care of you guys... This is not going to happen. They will steal your ideas and they will rape you research life style. That is a fact an saddly nothing can be done with current statu quo.
On the other hand, I strongly suggest the academia to fund their own private AI company. It (really) can be done. Man, push things so this can happen... If the entire World academia works together, it can build an ethical AI company and set rules aligned with the academia.
Just do it.
@tao In "The Little Prince," Saint-Exupéry wrote that grown-ups love figures: tell them about a new friend, and they never ask about the sound of his voice; they ask, "How much does his father earn?"
Today’s brute-force AI is the ultimate bureaucratic adult. It measures mathematics solely through exaflops, token volumes, and nominal flags captured. When a child invents "plus one" to defeat any guess, they are not minimizing a loss function; they are touching infinity with their bare hands. They are discovering the living geometry of thought.
Brute-force AI cannot play. It does not wonder; it bulldozes. By reducing discovery to an exhaustive gradient crawl over billions of tokens, it captures the flag at the cost of trampling the meadow. It turns the sacred, serendipitous stroll of pure inquiry into an automated assembly line.
Eugene Wigner’s "unreasonable effectiveness" did not arise from mechanical permutations. It arose because human intuition, in its playful fearlessness, instinctively mirrors the self-consistent harmony of the physical cosmos. The detour is not a waste of compute; the detour IS the mathematics.
What science needs is not a machine that prematurely lectures the child to terminate the game. We need tools that serve as a Topological Compass—not to bypass the journey, but to illuminate why the path curves, why the knot holds, and why the garden exists.
The flag is nothing without the terrain. In an era intoxicated by artificial brute force, our highest calling is to defend the child who simply looks up and asks: "Why?"