Games Without Frontiers
Hermann Weyl's dialectic of the infinite in the age of AI.
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:
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.
Good to read this, Maxim. I wanted to see where the current thinking in philosophy of mathematics is and what might be relevant to your essay. I quote from SEP:
"In the second half of the twentieth century, research in the philosophy of science to a significant extent moved away from foundational concerns. Instead, philosophical questions relating to the growth of scientific knowledge and of scientific understanding became more central. As early as the 1970s, there were voices that argued that a similar shift of attention should take place in the philosophy of mathematics. Lakatos initiated the philosophical investigation of the evolution of mathematical concepts (Lakatos 1976). He argued that the content of a mathematical concept evolves in roughly the following way. A mathematician formulates a deep conjecture, but is unable to prove it. Then counterexamples against the conjecture are found. In response, the definition of one or more central concepts in the conjecture is changed in such a way that the counterexamples are at least eliminated. Still the thus revised conjecture cannot be proved, and gradually new counterexamples appear. The procedure of revising the definition of one or more central concepts is applied again and again, until a proof of the conjecture is found. Lakatos calls this procedure concept stretching. In recent decades, Lakatos’ model of concept change in mathematics has been revised and refined (Mormann 2002).
For some decades, the view that the philosophy of mathematics should take a historical and sociological turn remained restricted to a somewhat marginal school of thought in the philosophy of mathematics. However, in recent years the opposition between this new movement of mathematical practice on the one hand, and ‘mainstream’ philosophy of mathematics on the other hand, is softening. Philosophical questions relating to mathematical practice, the evolution of mathematical theories, and mathematical explanation and understanding have become more prominent, and have been related to more traditional themes from the philosophy of mathematics (Mancosu 2008). This trend will doubtlessly continue in the years to come.
For an example, let us briefy return to the subject of computer proofs (see section 5.3). The source of the discomfort that mathematicians experience when confronted with computer proofs appears to be the following. A “good” mathematical proof should do more than to convince us that a certain statement is true. It should also explain why the statement in question holds. And this is done by referring to deep relations between deep mathematical concepts that often link different mathematical domains (Manders 1989). Until now, computer proofs typically only employ fairly low level mathematical concepts. They are notoriously weak at developing deep concepts on their own, and have difficulties with linking concepts in from different mathematical fields. All this leads us to a philosophical question which is just now beginning to receive the attention that it deserves: what is mathematical understanding?
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Developments in AI for mathematics would connect with these trends?!
I love this analysis, feels potent and well-articulated. I think this places "the art of the sciences" in question, within the realm of "major vs. minor science" or "nomad vs. State science" as described by Deleuze and Guattari in "Nomadology" from "A Thousand Plateaus":
>> There is a kind of science, or treatment of science, that seems very difficult to classify, whose history is even difficult to follow. What we are referring to are not “technologies” in the usual sense of the term. But neither are they “sciences” in the royal or legal sense established by history...
>> The model in question is one of becoming and heterogeneity, as opposed to the stable, the eternal, the identical, the constant. It is a “paradox” to make becoming itself a model, and no longer a secondary characteristic, a copy.
>> [Discussing a specific approach to architecture and construction] One does not represent, one engenders and traverses. This science is characterized less by the absence of equations than by the very different role they play: instead of being good forms absolutely that organize matter, they are “generated” as “forces of thrust” (*poussees*) by the material, in a qualitative calculus of the optimum.
>> It is instructive to contrast two models of science, after the manner of Plato in the Timaeus. One could be called Compars and the other Dispars. The compars is the legal or legalist model employed by royal science. The search for laws consists in extracting constants, even if those constants are only relations between variables (equations). An invariable form for variables, a variable matter of the invariant: such is the foundation of the hylomorphic schema. But for the dispars as an ele- ment of nomad science the relevant distinction is material-forces rather than matter-form. Here, it is not exactly a question of extracting constants from variables but of placing the variables themselves in a state of continuous variation. If there are still equations, they are adequations, inequations, differential equations irreducible to the algebraic form and inseparable from a sensible intuition of variation. They seize or determine singularities in the matter, instead of constituting a general form. They effect individuations through events, not through the “object” as a compound of matter and form; vague essences are nothing other than haecceities.