Games Without Frontiers

Hermann Weyl’s The Open World, published in 1932 and based on the Terry Lectures he gave at Yale a year earlier, has very interesting things to say about the interplay of limitation and freedom in mathematics. Weyl’s book appeared right around the time when the work of Gödel and Turing was starting to expose certain fundamental limitations in the foundations of mathematics. Here, though, I want to comment on the relevance of the dualism between limitation and freedom to ongoing debates about the present and future of mathematics and theoretical science in light of LLMs. The well-known AI researcher François Fleuret tweeted this two years ago:

François Fleuret@francoisfleuret

Mathematics will fall first. Then, a torrent of results will impact everything in theoretical science.

11:42 PM · Dec 2, 2023 · 3.16M Views

180 Replies · 178 Reposts · 2.83K Likes

Fleuret’s take certainly applies to David Hilbert’s formalist view of mathematics as a game of symbolic manipulation according to fixed rules, without any attention paid to meaning. As we watch LLMs perform impressive feats of searching across vast corpora of mathematical facts, it is obvious that, indeed, these systems are poised to be much better players of the glass bead game compared to the best human mathematicians. As we already saw with AlphaZero and with AlphaFold, if a problem can be phrased as a game with fixed (if arbitrarily complicated) rules and if we have a way of evaluating various moves and of iteratively adjusting our strategies for picking a better move, then, indeed, it is only a matter of time before any such problem will, in Fleuret’s words, “fall.”

However, I think it is premature to say the same about metamathematics where, as Weyl says, “the game itself becomes the object of cognition.” This is the realm where we subject the games we play to a process of reflection, question the rules of these games, and invent new games. Intelligence, natural or artificial, is a joint property of the cognizing subject and the environment in which the subject is embedded, and this is where the tension between limitation and freedom comes into play. The key is to dissolve the limitations of closed formal systems by embedding them in larger open systems, where one has the freedom to introduce new axioms, new rules of inference, and new value systems. This act of reflective creation is what James P. Carse called an infinite game, a game without frontiers or fixed rules, a game which is not played with a fixed goal in mind, but with the motivation to keep playing. As such, it will increasingly involve both humans and AI systems in perpetual interaction.

Coming back to limitations versus freedom, when Weyl talks about it, he is referring to the tension between potentiality and actuality, becoming and being, data and process. That is,

the transition from the a posteriori description of the actually given to the a priori construction of the possible. The given is embedded in the ordered manifold of the possible, not on the basis of descriptive characteristics, but on the basis of certain mental or physical operations and reactions to be performed on it—as, for example, the process of counting.

The act of theoretical cognition is to transcend the limitations of the actually given by passing to “the field of possibilities that is open to infinity.” Here there are several options. One can follow L.E.J. Brouwer and other intuitionists and to view the field of possibilities as a potential infinity, something we can never fully access in its entirety, but which we can keep discovering iteratively using constructive procedures. Or one can go the route of Cantor and use the tools of set theory to visualize actual infinities. Weyl rejects Cantor’s platonic sensibilities in favor of moderate constructivism. Towards the end of The Open World he writes the following:

In the spiritual life of man two domains are clearly to be distinguished from one another: on one side the domain of creation (Gestaltung), of construction, to which the active artist, the scientist, the technician, the statesman devote themselves; on the other side the domain of reflection (Besinnung) which consummates itself in cognitions and which one may consider as the specific realm of the philosopher. The danger of constructive activity unguided by reflection is that it departs from meaning, goes astray, stagnates in mere routine; the danger of passive reflection is that it may lead to incomprehensible “talking about things” which paralyzes the creative power of man. … Hilbert’s mathematics as well as physics belongs in the domain of constructive action; metamathematics, however, with its cognition of consistency, belongs to reflection.

Moreover, he highlights the fundamental role of action in the progress of theoretical science:

[t]he task of science can surely not be performed through intuitive cognition alone, since the objective sphere with which it deals is by its very nature impervious to reason. But even in pure mathematics, or in pure logic, we cannot decide the validity of a formula by means of descriptive characteristics. We must resort to action: we start out from the axioms and apply the practical rules of conclusion in arbitrarily frequent repetition and combination. In this sense one can speak of an original darkness of reason: we do not have truth, we do not perceive it if we merely open our eyes wide, but truth must be attained by action.

When Weyl says that the objective sphere is “impervious to reason,” he means that you cannot understand the world by simply thinking about it or by computational simulation detached from experience. In other words, world models in Yann LeCun’s sense can only be arrived at through constructive action, predicting the next token is simply not enough. However, since world models are necessarily formal models, they are subject to the fundamental metamathematical limits of the kind described by Gödel and Turing — see, e.g., the recent work by Cubitt, Perez-Garcia, and Wolf on the undecidability of the spectral gap of certain types of quantum Hamiltonian models. Weyl can be forgiven for not having yet grasped the significance of the work of Gödel when he was giving the Terry Lectures in 1931. Our freedom to take constructive action runs up against the limitations on reflective reason; the limitations of reflective reason can in turn be overcome by further constructive action.