Description

Mathematics is a pivotal area of inquiry. It is also particularly intriguing. We cannot bump into nor observe numbers, geometrical points or other mathematical entities. So, what are they? And how can it be successfully applied across the sciences, if mathematical entities are not a part of the physical world? And yet, mathematics is among our securest fields of knowledge. But how can this be if mathematical entities cannot be observed? These are deep and perennial questions, whose answers remain elusive.

A more recent question is how mathematics and logic relate. Logicism offers a straightforward answer: mathematics just is logic. It offers natural and seemingly compelling answers also to the previous questions. Mathematical entities are entities whose existence is a consequence of the laws of logic. We come to know truths about them through our logical knowledge, and mathematical knowledge is secure because logical knowledge is. Finally, it is no wonder that mathematics is widely applicable across the sciences. For logic is concerned with the laws governing the most generally applicable notions.

The programme of deriving the basic laws of arithmetic in pure logic was originally developed by Gottlob Frege (1884, 1893), who thought numbers were logical objects, and Bertrand Russell (Russell & Whitehead 1910), who took them to be properties of properties of objects. Both projects failed – Frege’s owing to its inconsistent foundations, and Russell’s owing to its reliance on the nonlogical principle that there are infinitely many individuals. Furthermore, by relying on the Fregean conception of numbers contemporary neologicists (Wright & Hale 2001) ultimately exposed themselves to devastating objections. The lore of the land is that these accumulated failures led to Logicism’s demise, despite its promise.

Notwithstanding, the troubling objections faced by the Fregean and neoFregean logicist projects do not apply to Russell’s. This has led Klement (2012, p. 157) to state that ‘By comparison to the litany of problems that continue to plague [Fregean] neo-logicisms, [a Russellian neo-logicism] actually seems like a far less daunting route for a twenty-first century logicist to explore’. Yet, despite the auspicious route afforded by neoRussellian logicist programmes, hardly any have been proposed, most probably owing to the absence of a technical result capable of sustaining them.

The present project fills this gap by developing and defending a Russellian-inspired philosophy of mathematics. Based on the Russellian conception of numbers, I have recently produced (Jacinto 2024) a novel result showing that the arithmetical axioms are all derivable from Finitary Plenitude – a seemingly logical truth of higher-order modal logic, consistent with it being necessary that there be only finitely many individuals, and according to which every finite cardinality property could have been instantiated.

My result strongly suggests that arithmetic is nothing but higher-order modal logic. It thus changes the philosophical landscape by revealing that logicism about arithmetic remains a force to be reckoned with and unveiling arithmetic’s previously overlooked inherently modal nature.

While this result re-establishes the prospects of logicism about arithmetic, the present project constitutes a much larger and complex enterprise: to found arithmetic, rational and real analysis, and set theory in logic through Upper Logicism, characterised by the following theses:

UL1: Predicates have a semantic role distinct from those of individual and plural terms, and quantification into predicate position is legitimate and irreducible to first-order singular or plural quantification.

UL2: Natural, rational and real numbers, and sets, are all higher-type entities (and so properties or relations) whose canonical applications are contained within their nature, and the primitives of their standard theories are all truly characterizable as higher type-entities definable in pure modal type theory.

UL3: The metaphysically necessary truths formulated in pure modal and type-theoretic languages are all logically true.

UL4: Finitary Plenitude is necessarily true.

UL5: For every standard, true deductive theory T of arithmetic, rational or real analysis or set theory, there is a deductive system of pure modal type-theory which includes Finitary Plenitude as an axiom, has only metaphysically necessary truths as theorems, and includes as theorems translations of T’s theorems by interpreting T’s primitives according to UL2.

UL6: The truths of arithmetic, rational and real analysis, and set theory, are all metaphysically necessary truths expressible in a pure modal type theory by interpreting them according to UL2.

UL1 excludes interpreting higher-order quantification as restricted first-order singular or plural quantification over individuals, sets, classes, pluralities, etc. It thus leads to distinctive views on the interaction between modality and higher-order quantification: while sets and pluralities are commonly thought to have their instances essentially and be extensionally individuated, in general the values of higher-type variables have their instances inessentially and are not extensionally individuated.

UL2 is directly connected to Frege’s constraint (Panza & Sereni 2019), according to which the canonical applications of mathematical entities to the world must be contained within their nature, by requiring that a mathematical entity be equated with a property or relation only if the former’s canonical applications be the application conditions which are contained within the latter’s nature. The appeal to modal resources fits naturally with UL2 and Frege’s constraint. For instance, the canonical application of natural numbers to counting must be sensitive to the fact that properties may have different numbers depending on the circumstances (e.g., though Mars has only two moons, it could have had more).

UL3 is a joint consequence of the views that the logical truths include the necessary truths formulated solely in terms of logical expressions (Shapiro 2005), and that among the latter are all of pure type theory’s primitives; and UL3 and UL4 jointly imply that Finitary Plenitude is a logical truth.

Together, UL2-5 imply that the theorems of every standard true deductive system for arithmetic, rational analysis, real analysis and set theory are all logically true once these theories’ primitives are interpreted as logically definable higher-type entities. Further, together, UL2, UL3 and UL6 jointly imply that the truths of arithmetic, rational and real analysis, and set theory are all expressible by logical truths obtained by interpreting them as higher-type entities.

Overall, Upper Logicism is a novel synthesis between mathematics, logic and modal metaphysics. Among the project’s core aims is to establish this paradigm-shifting, unification of the three disciplines, traditionally thought to be distinct.

Research Team

Bruno Jacinto (Principal Researcher), Joan Bertran San Millán (Team member), José Mestre (Team member), Tabea Rohr (Team member), António José Teiga Zilhão (Team member), Sébastien Gandon (Team member), Nicholas Jones (Team member), Agustin Rayo Fierro (Team member), Christopher Scambler (Team member), Hannes Leitgeb (Team member), Sonia Roca-Royes (Team member), Francesca Boccuni (Team member), New Hire (Postdoctoral Fellow), New Hire (Master Student), Fernando Ferreira (Consultant), Øystein Linnebo (Consultant), Timothy Williamson (Consultant), Crispin Wright (Consultant)

Publications

Events

The following three major events will take place during the project. These will mark the project's progress, disseminate the project's research, enable novel collaborations, motivate further research and serve as inputs to relevant cutting-edge research being done by people outside the project. 1. Opening Workshop: The opening workshop will be a two-day event taking place six months after the start of the project. All the team and consultants will be present. It will serve to get all the project's participants acquainted with each other's work, to strengthen previous collaborations and foster new ones, to incubate new ideas and to motivate the project's participants to produce work of excellence. 2. Masterclass+Workshop. 18 months into the project there will be a two-day masterclass on Upper Logicism and topics in its vicinity given by chosen project members and consultants, followed by a two-day project workshop. The master class target audience are master and PhD students from across the globe. Participants will have to pre-register to the event and will have access to all its contents. The main aim of the masterclass is to get the new generations acquainted with and working on Upper logicism as well as topics in its vicinity. The workshop will allow team members and consultants to consolidate the research presented at the first workshop. 3. Final conference. Thirty months into the project there will be a three-day conference celebrating the project with the presence of all the project's members and consultants. There will be a call for papers for the conference and we will select the best 12 papers on topics related to Upper logicism. Six papers resulting from the project's research will furthermore be delivered at the conference. The expectation is that interest and research into Upper logicism will be at a high point by the time of the conference, thus yielding an extremely vibrant event with the top experts on Upper logicism or topics related to it.