Remember what it is like to be a child

"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)