Logical Relativism Through Logical Contexts

We advance an approach to logical contexts that grounds the claim that logic is a local matter: distinct contexts require distinct logics. The approach results from a concern about context individuation, and holds that a logic may be constitutive of a context or domain of application. We add a naturalistic component: distinct domains are more than mere technical curiosities; as intuitionistic mathematics testifies, some of the distinct forms of inference in different domains are actively pursued as legitimate fields of research in current mathematics, so, unless one is willing to revise the current scientific practice, generalism must go. The approach is advanced by discussing some tenets of a similar argument advanced by Shapiro, in the context of logic as models approach. In order to make our view more appealing, we reformulate a version of logic as models approach following naturalistic lines, and bring logic closer to the use of models in science.

Indeterminism in physics and intuitionistic mathematics

Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to “speak” of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientific determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We suggest that intuitionistic mathematics provides such a language and we illustrate it in simple terms.

A Two-Part Defense of Intuitionistic Mathematics

The classical interpretation of mathematical statements can be seen as comprising two separate but related aspects: a domain and a truth-schema. L. E. J. Brouwer’s intuitionistic project lays the groundwork for an alternative conception of the objects in this domain, as well as an accompanying intuitionistic truth-schema. Drawing on the work of Arend Heyting and Michael Dummett, I present two objections to classical mathematical semantics, with the aim of creating an opening for an alternative interpretation. With this accomplished, I then make the case for intuitionism as a suitable candidate to fill this void.

A Two-Part Defense of Intuitionistic Mathematics

The classical interpretation of mathematical statements can be seen as comprising two separate but related aspects: a domain and a truth-schema. L. E. J. Brouwer’s intuitionistic project lays the groundwork for an alternative conception of the objects in this domain, as well as an accompanying intuitionistic truth-schema. Drawing on the work of Arend Heyting and Michael Dummett, I present two objections to classical mathematical semantics, with the aim of creating an opening for an alternative interpretation. With this accomplished, I then make the case for intuitionism as a suitable candidate to fill this void.

Intuitionism

Ultimately, mathematical intuitionism gets its name and its epistemological parentage from a conviction of Kant: that intuition reveals basic mathematical principles as true a priori. Intuitionism’s mathematical lineage is that of radical constructivism: constructive in requiring proofs of existential claims to yield provable instances of those claims; radical in seeking a wholesale reconstruction of mathematics. Although partly inspired by Kronecker and Poincaré, twentieth-century intuitionism is dominated by the ‘neo-intuitionism’ of the Dutch mathematician L.E.J. Brouwer. Brouwer’s reworking of analysis, paradigmatic for intuitionism, broke the bounds on traditional constructivism by embracing real numbers given by free choice sequences. Brouwer’s theorem – that every real-valued function on a closed, bounded interval is uniformly continuous – brings intuitionism into seeming conflict with results of conventional mathematics. Despite Brouwer’s distaste for logic, formal systems for intuitionism were devised and developments in intuitionistic mathematics began to parallel those in metamathematics. A. Heyting was the first to formalize both intuitionistic logic and arithmetic and to interpret the logic over types of abstract proofs. Tarski, Beth and Kripke each constructed a distinctive class of models for intuitionistic logic. Gödel, in his Dialectica interpretation, showed how to view formal intuitionistic arithmetic as a calculus of higher-order functions. S.C. Kleene gave a ‘realizability’ interpretation to the same theory using codes of recursive functions. In the last decades of the twentieth century, applications of intuitionistic higher-order logic and type theory to category theory and computer science have made these systems objects of intense study. At the same time, philosophers and logicians, under the influence of M. Dummett, have sought to enlist intuitionism under the banner of general antirealist semantics for natural languages.