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Terence Tao@tao@mathstodon.xyz

Professor of #Mathematics at the University of California, Los Angeles  #UCLA (he/him).

Home page: https://www.math.ucla.edu/~tao
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Joined: 2026-09-03 12:20:40
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Terence Tao@tao@mathstodon.xyz (2026-09-10 10:17:15)
"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)

Terence Tao@tao@mathstodon.xyz (2026-09-03 12:14:49)
It seems intuitive that a solved problem is unquestionably better than an unsolved problem: a magic button which, when pressed, turns the latter into the former at no cost would thus seem to be a "free lunch" that there would be no hesitation in pressing.

But even such a benign-seeming operation as answering a question can have irreversible costs.  One familiar example is that of spoilers to a movie one is looking forward to watching: knowing key plot points in advance may permanently decrease your ability to enjoy the movie for the first time.  Similarly, leaking information about a competition before it starts can irretrievably damage the fairness or value of that competition.

In the field of AI, problems whose solutions are already publicly available are deemed "contaminated" and of permanently reduced utility for evaluating such models, as the black-box nature of modern AI makes it extremely difficult to distinguish an AI-generated solution that came from genuinely novel reasoning from one that was somehow already present in training data.  For the first time, human-generated open problems in mathematics in particular have become something resembling a non-renewable resource. (1/3)