[The following remarks were delivered today by Andy Arana at the beginning of a joint Paris-Davis workshop on the philosophy of mathematics, and are posted here with his permission and that of Curtis Franks. The photo above shows Aldo at a cook-off with Marco Panza at the last instalment of that workshop series in Davis, and is courtesy of Robert May.]
These days people are quick to take positions, “hot takes”. ZFC is the right foundation of mathematics, first-order logic is the true logic, the nature of mathematical knowledge is intuitionist, the axiom of projective determinism is true, nominalism is true. And so on. These are bold statements, and they draw attention. The result is a literature that moves quickly and as a result yields little change, little persuasion, little clarification, little wisdom. It comes to seem dogmatic: one’s reputation is connected with one’s intellectual position, and thus changing your mind comes with the loss of professional status.
Aldo was not like this. Aldo’s work was about exploring new possibilities, alternatives to views that could otherwise calcify into mere dogmas. One can think of them as playing a role in an intellectual infrastructure to which spaces are regularly wrongly closed off. “that approach won’t work, for such and such a reason”: Aldo would provide technical results that opened those approaches up again. Whether you take those approaches, that’s up to you. But it’s not their impossibility that should stop you.
I want to start briefly by discussing Aldo’s early work. He worked on the foundations of defeasible consequence and non-monotonic logics, of Quine’s New Foundations set theory, and of non-well founded set theory. Rather than try to summarize the results, I want to tell a story about why he worked on these things, a story that I think will illuminate an order to Aldo’s approach to problems. Indeed, here I want to quote his student Curtis Franks, who kindly shared some personal thoughts about Aldo with me. On this early work of Aldo’s, Curtis writes,
“One might say that there were some technical gimmicks to work out, and he had the relevant tools to discover and present these things. But this is wrong. Aldo was interested in the results that were the most beautiful, the ones that unpacked the most hidden connections, the ones that made us rethink the greatest number of our inhibiting preconceptions. Because he didn’t care much about, and possibly didn’t even understand, ideas about which set theories were correct, he was able to feel his way to the mathematical relationships that disclosed the greatest number of such insights.”
Aldo applied this method with particular focus to logicism and in particular to the thesis that arithmetic is logic. Frege’s logicist program had the following two stages: 1) to define numbers as extensions of concepts, and 2) to derive logically the theorems of arithmetic from that definition. That these two stages cannot both be carried out is more or less consensus today. Where there is consensus, though, there is need for light: for what possibilities are being closed out by the consensus? I want to talk about three projects of Aldo that opened such possibilities for logicism.
I want to begin with “Frege’s New Science”, written with Robert May. Aldo broke the question of whether Frege’s logic could carry out metatheory into two parts: 1) can metalogical questions about Frege’s logic be posed in Frege’s logic itself? and 2) is metatheory necessarily model theory, in which one varies the meanings of propositions in order to prove for instance independence results? Aldo and Robert answered “no” to 2), in light of Frege’s insistence that one cannot reinterpret the meanings (references) of non-logical terms of axioms, since axioms express thoughts. They argue that Frege saw, if not particularly clearly, a way to develop metatheory in which one replaces the reinterpretability of meanings with a kind of permutability of nonlogical vocabulary on which no meanings are changed. While the question of what vocabulary is nonlogical arises, Aldo and Robert sketch an argument (at least nearly) available to the Fregean that this question too can be handled by a permutability argument. Thus this article opens space for a new approach to understanding Frege’s metatheory by pursuing a technical development.
Next, I want to turn to “Frege’s Other Program”, also written with Robert May. Here Aldo and Robert explore a different possibility for reckoning the two logicist stages of defining numbers as extensions of concepts, and of deriving logically the theorems of arithmetic from that definition. As Aldo and Robert put it, one can attempt “to show in a nonlogical theory of extensions, where numbers are concepts, not objects, that Peano Arithmetic can be derived”. In doing so they identify a non-logicist but still broadly Fregean program for deriving arithmetic. This program clarifies the causes of the contradiction entailed by Basic Law V, providing a new counterexample to Hume’s Principle. Here again new spaces are opened, by investigating a extensional theory of arithmetic without supposing Hume’s Principle. One can then consider to what extent such a theory could be judged a vindication of logicism.
Next, I want to turn to Aldo’s 2010 article “The Nature and Purpose of Numbers”, notable not least because it appeared in one of American philosophy’s top journals, the Journal of Philosophy. This article investigates a possible version of Fregean logicism, one that differs from other developments in that it does not reduce arithmetic to set theory. Aldo takes cardinal properties of the natural numbers as the starting point, and derives structural properties of the natural numbers from them, rather than the other way around as is done in typical set-theoretic reductions of arithmetic. Aldo’s idea is that “in keeping with the broadest and most general construal of logicism, cardinality notions… deal directly with properties and relations of concepts—rather than matters of existence of objects such as numbers—cardinality notions properly can be regarded as having a logical character.” They are logical notions, Aldo argues, because relations of concepts are quantifiers: more precisely, they are, Aldo argues, generalized quantifiers. The ordinary existential and universal quantifiers can be thought of as relations of concepts: the existential quantifier as the collection of all nonempty subsets of the domain of quantification, and the universal quantifier as the collection of all subsets of the domain that contain the domain as their only member. But these are not the only two relations that yield quantifiers, on the theory of generalized quantifiers, and in particular one can consider the Frege quantifier, which holds between two concepts F and G when there are no more Fs than Gs. In fact, these are all first-order quantifiers, Aldo argues. He then shows how the Frege quantifier can be taken as logically basic, and shows how one can derive the basic features of the natural numbers from such a logic.
I want to note in particular Aldo’s way of characterizing his accomplishment in this article. “Accordingly, we take the logicist claim that cardinality is a logical notion at face value, and rather than arguing for it (perhaps by providing a reduction to some other principle), we set out to explore its consequences by introducing cardinality, in the form of the Frege quantifier, as the main building block in the language of arithmetic.”
This passage illustrates beautifully the idea that Aldo’s approach to problems was to prescind from particular theoretical stances, and to explore the consequences of these stances. With a clearer understanding of these consequences, of their fruits, one can better evaluate the costs and benefits of particular positions. The job of the logician is to explore these consequences. And Aldo was a logician.
In this 2010 article, Aldo also noted how the Frege quantifier, a first-order quantifier, can be given a generalized Henkin interpretation. In Henkin’s work, one restricts the range of quantification to just a subset of the power set of the full domain. Models for second-order logic can then be specified by giving both the domain and a universe of relations over the domain. Aldo’s novel insight was that such an interpretation can also be given for first-order quantifiers. Within the context of Fregean logicism, this permits one to prescind from concerns about whether second-order logic is logic (another one of those dogmatic debates).
His 2013 article in the Review of Symbolic Logic, “On the general interpretation of first-order quantifiers”, expanded on this novel insight. In the words of Curtis Franks: “60 years or so after Henkin’s groundbreaking work on generalized models, Aldo observed what no one else ever noticed, namely, that the notion of a generalized model can be formulated already for first order languages. The irony is sharp: The “first-order case” of the Henkin construction becomes an extension (not a restriction) of the familiar second order case, where the notion of models given by filters over the full power set construction is more intuitive.” Curtis judges this to be Aldo’s deepest work.
In an article entitled “Life on the Range: Quine’s Thesis and Semantic Indeterminacy”, published this summer, Aldo pursued the consequences of this technical development for the evaluation of Quine’s dictum that to be is to be the value of a bound variable. As Aldo notes, this dictum flows from Quine’s view that second-order logic is “set theory in sheep’s clothing”. Since second-order logic on Quine’s view has ontological commitments, it is not really logic. Aldo observes that his work on generalized models puts pressure on Quine’s views. Since the first-order quantifiers can be interpreted to be extensions of second-order quantifiers, the ontological commitments of second-order logic are also ontological commitments of first-order quantifiers. In the closing sentence of this article, Aldo writes that “this last realization can contribute to the establishment of second-order logic on the same safe footing as first-order logic.”
I want to finish by drawing from Curtis Franks’ remarks one last time. Curtis writes, in closing,
“I cannot enumerate the ways that Aldo influenced me, but the one that I am the most aware of is pretty simple. It seems to me that most professional philosophers spend more time advancing their own research programs than they spend learning. To me this is completely unreasonable, and I am pretty sure that my attitude about this derives from Aldo’s influence. Why would I, or anyone really, care more about what I have to say than about what some 20 or so brilliant historical figures have already said. Do I love logic, math, and philosophy, or do I love professional credits? Everyone in our world initially loved the former, and it is a disgrace, Aldo taught, to abandon this idea. And, he taught me, if you persevere in your love for the most beautiful ideas in mathematics, philosophy, and logic, your own contributions will trickle in at the right time. Those ideas will not come close to being the most interesting things you have to talk about. But they will not only be true, they will be beautiful.”