Challenges to predicative foundations of arithmetic

Melisa Vivanco, 2023

Some of the most influential programs in the philosophy of mathematics started from the philosophical study of natural numbers. On the one hand, our arithmetic intuitions appear earlier and more direct than other mathematical (and non-mathematical) intuitions. On the other hand, while arithmetic admits one of the first axiomatizations with wide acceptance within the mathematical practice, the study of natural numbers sets a methodological precedent that will later seek to be replicated in other areas, in particular, in the study of the most complex numerical structures. This course will address the main issues in the philosophical discussion on arithmetic. Among these topics are the ideas of the various classical doctrines on the foundations of arithmetic, from the milestone of Gödel’s incompleteness theorems to recent doctrines on the semantics of numerical expressions and arithmetic sentences. The class will cover debates about metaphysics and the epistemology of numbers and arithmetic truths.

Paul Lorenzen -- Mathematician and Logician, 2021

In this article we give an overview, from a philosophical point of view, of Lorenzen’s construction of the natural and the real numbers. Particular emphasis is placed on Lorenzen’s classification in the tradition of predicative approaches that stretches from Poincaré to Feferman.

2007

An arithmetic of rational numbers is developed organically from the idea of distinction. By explicitly representing distinctions as they unfold naturally, a hierarchy of increasingly complex arithmetic systems are shown to grow out of the same root. Conflating some distinctions produces the binary arithmetic of Laws of Form by Spencer-Brown, while conflating other distinctions produces the boundary arithmetic of Jeffrey James. The present paper thus provides a broader contextual framework within which these systems can be related and seen as different branches in the same tree growing from the seed of a primordial distinction. INTRODUCTION What are numbers, really? If everything is made of number, as Pythagoras declared, then it is necessary to understand the nature of number to understand the essence of all things. In one sense, numbers are to mathematics as the atom is to classical physics. They may be seen as a fundamental irreducible basis for everything else. When we learn to c...

Grazer Philosophische Studien, 1998

Studies in Logic and the Foundations of Mathematics, 1998

One of the richest and most salient applications of a non-classical logic is the matter of how mathematics operates within its province. Historically, this is most evident in the case of intuitionism, insofar as the intuitionistic standpoints with respect to deduction and mathematical practice are tightly bound together. Yet even in the case of Robert Meyer's relevant arithmetic R#, that a robust and compelling theory of arithmetic can be erected on relevant foundations speaks to the maturity of relevant logics. Accordingly, as connexive logic matures as a field, the topography of mathematics against a connexive backdrop becomes more and more compelling. The contraclassicality of connexive logics entails that the development of connexive mathematics will be more complex---and, arguably, more interesting---than intuitionistic or relevant accounts. For example, although formally undecidable sentences in classical Peano arithmetic remain independent of its intuitionistic and relevant counterparts, there exist undecidable sentences of classical arithmetic that will become decidable modulo any reasonable connexive arithmetic. In, e.g., Peano arithmetic, the Gödel sentence G is undecidable. Classically, this entails that the sentence ~(G->~G) is likewise undecidable. Of course, in a connexive logic L and connexive arithmetic L#, L# will prove ~(G->~G), witnessing that some classically undecidable statements in number theory become decidable connexively. Although this example is extremely simple, it demonstrates that there are many subtle questions that uniquely arise in a connexive mathematics. In this paper, I wish to make a few comments on how mathematics---in particular, arithmetic---must behave if formulated connexively. We will first consider some relevant historical and philosophical topics, such as Łukasiewicz' number-theoretic argument against Aristotle's Thesis, before taking a foray into the formalization of modest subsystems of arithmetic in Richard Angell's PA1 and PA2, observing some of the pathologies that will greet arithmetic in these settings.

The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.

Bulletin of Symbolic Logic, 2004

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume's Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege's Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA-one having to do with Hume's Principle, the other, with the underlying second-order logic-and investigates how much of Frege's Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension.

Epistemology versus Ontology, 2012

We discuss both the historical roots of Skolem's primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant's philosophy of mathematics. * Is paper is loosely based on the Skolem Lecture that I gave at the University of Oslo in June, 2010. The present paper has profited, both with respect to what it now contains and with respect to what it no longer contains, from the discussion following that lecture.

Mathematical Theory and Modeling, 2014

As a fundamental requirement, a natural number system must necessarily satisfy Peano axioms. These axioms altogether justify whether or not a number system is a natural number system. In this paper, a set of axioms have been proposed and established to augment and extend the existing Peano axioms for the natural number system.