Wittgensteining the continuum (& 2): Is the continuum petrified?

I finished my last entry expressing my perplexity by the ‘solution’ mathematicians offered six decades ago to the problem of the continuum: Cohen’s theorem according to which both the assumption that there is some set (say, C) larger than the set of natural numbers but smaller than the set of real numbers, and the assumption that there can be no such ‘intermediate in size’ set, are consistent with the other axioms of standard set theory (Zermelo-Fraenkel axioms). This perplexity was mainly motivated by my traces (some would say piles) of mathematical Platonism: how can C (which is a subset of the set of real numbers) exist in some ‘versions’ of set theory but fail to exist in other ‘versions’? After all, in the ‘versions’ where C exists, it will contain certain specific real numbers, and these numbers, is it not reasonable to think that they constitute a ‘set’ no matter the ‘version’ we are in? As I announced then, a possible remedy (partial at least) for the perplexity can be traced to the ideas of the philosopher Ludwig Wittgentstein (1889-1951).

Wittgenstein’s philosophy of mathematics represents a distinctive approach that diverges sharply from traditional Platonist or formalist views. Rather than seeing mathematics as a discovery of timeless truths or as merely a system of symbolic manipulation, Wittgenstein regarded mathematics as a human activity grounded in linguistic and social practices. One of the key concepts in his later work, particularly in Remarks on the Foundations of Mathematics, is the notion of ‘petrification’ —a metaphorical term he uses to describe the rigidification of mathematical practices into seemingly immutable forms. In his early period, especially in the Tractatus Logico-Philosophicus, Wittgenstein shared some affinities with the formalist view, where mathematical propositions were seen as tautologies or logical structures. However, by the time of his later writings, including the Philosophical Investigations and the already mentioned Remarks, his views evolved dramatically. He began to see mathematics not as a static body of absolute truths but as a collection of rule-governed activities embedded in forms of life.

The concept of petrification emerges from this shift in thinking (see Pérez-Escobar et al., 2024). For Wittgenstein, mathematical rules, once established and followed, tend to solidify into dogmatic practices. A method of calculation or proof that is initially introduced as a practical tool becomes treated over time as an unchangeable norm. In this way, what begins as a fluid, evolving process of human agreement becomes “petrified” into rigid structures that resist questioning. This petrification is not inherently negative; it is necessary for the stability and reliability of mathematics. However, Wittgenstein warns that this rigidity can obscure the origins and nature of mathematical practices. Through the metaphor of petrification, Wittgenstein highlights how mathematical certainty is not a result of the correspondence of propositions to mathematical reality, but a function of the community’s agreement in applying rules. In this view, mathematical necessity is not metaphysical but grammatical—it stems from the way we use language within particular forms of life. For example, the certainty that “2 + 2 = 4” would not be a revelation about numbers existing in a Platonic realm, but a reflection of how we have trained ourselves to use these symbols in everyday practices.

The main problem for applying this concept to our case is, following Pérez-Escobar et al., that Wittgenstein deployed the concept of ‘petrification’ by using very simple examples, like the sums of small integers, and it is not clear if the same strategy would function in the case of more complicated mathematical practices, like those of advanced set theory. One of the virtues of the paper I am citing is that it extends the scope of the petrification concepts not only to simple rules, but to mathematical proofs (or better, techniques of proof), that are hence understood as a kind of empirical regularity (about a cultural practice), more than as an abstract algorithm existing in the Platonic heaven. By the way, the idea that the foundation of mathematical knowledge is our empirical inductive familiarity with some operations, rather than a pure, a priori intellectual intuition, was defended in the nineteenth century by James Stuart Mill (in his System of Logic). And Popper’s disciple Imre Lakatos (in his book Proofs and Refutations) also developed a philosophical approach according to which our understanding of mathematical concepts consists in (quasi-empirically) fallible hypotheses.

Pérez-Escobar et al. argue in particular that petrification must not be seen as an all-or-nothing matter, but more as a process susceptible of developing in different directions and even to different degrees. It is, after all, a historical process. In this way, in the case of the continuum hypotheses problem, the techniques have not still (and may never petrified) in only one exclusive way, but the community of set-theories is still divided into many questions about what procedures are legitimate for which problems, and even more about what are the (philosophical) implications of each strategy. An analogous, and more familiar case, is that of alternative geometries. We might think of the combination of Zermelo-Fraenkel theory plus the axiom of choice plus the continuum hypothesis (say, the ‘standard’ set theory) as the analog to Euclidean geometry, in which there are parallel lines which never cross, whereas other set theories in which the continuum hypotheses is not valid would be equivalent to a Riemannian geometry, in which there are no parallel lines, for example. This is not totally incompatible with Platonism, for after all in each one of those theories (or their ‘universes’) there is what there is. Hence, it would be a kind of pluralistic Platonism as the one the authors mention, though I am not sure whether they end accepting the existence of such a mathematical multiverse.

References

Lakatos, I., 1976, Proofs and Refutations, Cambridge University Press.

Mill, J. S., 1843, A System of Logic, John Parker.

Pérez-Escobar, J.A., C. J. Rittberg and D. Sarikaya (2024) Petrification in Contemporary Set Theory: The Multiverse and the Later Wittgenstein Kriterion, doi: 10.1515/krt-2023-0016

Wittgenstein, L., 1956. Remarks on the Foundations of Mathematics, Blackwell.