The Number Concept: Axiomatization, Cognition and Genesis

 

Andrew Arana (Department of Philosophy, Kansas State University)

Purity in arithmetic: formal and informal issues

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This talk will survey several such issues, and discuss ways in which logical considerations shed light on these issues.

Andrey Bovykin (Department of Mathematics, University of Bristol)

The Next Generation of Ideas in Metamathematics of Arithmetic

I am going to talk about several major topics in modern metamathematics of arithmetic: Arithmetical Splitting (the programme to find a rich variety of mutually incompatible first-order arithmetical theories), the theory of templates and some examples from meta-meta-mathematics of arithmetic, the idea of finding barely accessible logical complexity classes, the “Atlas of Truth and Strength”, and the quest to find new sources (new reasons) for unprovability of arithmetical statements. I will keep the talk friendly and accessible to all non-logicians, but some prior knowledge of the basics of unprovability (Paris-Harrington, Boolean relation theory, ordinal analysis, phase transitions) will help the listeners understand the recent developments and discoveries.

Véronique Izard (Laboratoire Psychologie de la Perception, UMR 8158, CNRS, Université Paris Descartes)

Two key premises of the concept exact numbers: Exact equality and successor function

Infants are endowed with two systems for representing numerical quantity. With the first system (analog magnitude representations), they can represent the cardinal of large sets in an approximate way. In addition, infants are able to track individuals in small sets of objects (parallel individuation system), and by doing so they can solve some numerical problems. However, neither of these systems has enough power to represent integers, i.e. large exact numerical quantities. I will present research looking at the acquisition of integer concepts, focusing on two essential premises: the relation of exact equality between numbers, and the successor function. Based on studies involving young occidental children, and people from a non-western culture, the Mundurucu, I will argue that these two mathematically equivalent properties are acquired separately, and inferred from different sources of evidence. Consequently, children or people lacking formal instruction in arithmetic may entertain globally incoherent theories of number.

C.S. Jenkins (Department of Philosophy, University of Nottingham)

Arithmetic and Naturalism

Some philosophers call themselves ‘naturalists’. This term can mean different things, but is often associated with the rejection of a priori knowledge (that is, knowledge which is independent of experience). Naturalists will often argue that, despite appearances, arithmetical knowledge is a straightforward case of empirical knowledge comparable to knowledge of (say) biology or facts about the location and colour of objects like tables and chairs, rather than any special kind of a priori knowledge. In Grounding Concepts, I argued that one can accommodate the a prioricity of arithmetic in a way consistent with saying that arithmetical knowledge ultimately derives from experience. This paper discusses the impact of my suggestion on certain debates between naturalists and their opponents.

Matthew Katz (Department of Philosophy and Religion, Central Michigan University)

Writing and Reading Mental Magnitudes

There is at present a lively debate in cognitive psychology concerning the origin of natural number concepts. The system of mental magnitudes, an innately given cognitive mechanism that represents cardinality, lies at the center of this debate. Contributors to the debate have argued that mental magnitudes cannot be the sole innate source of natural number concepts, because it represents cardinality approximately while natural number concepts are precise. I argue that this view rests on a failure to appreciate the difference between writing and reading representations. While it is true that the system of mental magnitudes reads its representations approximately, it nevertheless writes them precisely. Acknowledgement of this difference allows for an account according to which mental magnitudes are the sole innate source of natural number concepts.

Felix Mühlhölzer (Philosophisches Seminar, Georg-August-Universitt Göttingen)

On the reference, interpretation and application of arithmetical terms

The deep difference that can be seen between the ‘reference’, the ‘interpretation’ and the ‘application’ of arithmetical terms, and of mathematical terms in general, is often blurred in the literature. The paper tries to clarify this difference and to show the consequences of such a clarification with respect to structuralism, Hilbertian finitism and Skolem’s paradox.

Charles Parsons (Department of Philosophy, Harvard University)

The Kantian legacy in twentieth century foundations of mathematics

A picture of the fate of Kantian views in the foundations of mathematics that was the received view in English-speaking circles from 1920 into the 1950s held that Kant’s philosophy of geometry had been fatally undermined by the development of non-Euclidean geometry and its application in physics and that as regards the rest of mathematics, a logicist view leaving no room for Kantian intuition had been established by Principia Mathematica and related developments.

Systematically, this picture ignored the fact that two assumptions of Principia, the axiom of reducibility and the axiom of infinity, were contested from the beginning. Historically, it ignored dominant tendencies in continental Europe, in particular the “intuitionism” of Brouwer and the work of Hilbert and his collaborators toward founding mathematics by axiomatization, formalization, and consistency proofs by Hilbert’s “finitary method”.

To show that the ghost of Kant had not been laid to rest after all, we discuss some philosophical views of Brouwer, Hilbert, and the latter’s collaborator Paul Bernays (with a preliminary on neo-Kantianism). Brouwer’s philosophy is not in substance especially Kantian, but he makes use of a somewhat Kantian intuition of time that seems not to have been maintained by mainstream neo-Kantians. Hilbert, in explaining the finitary method, also appealed to a somewhat Kantian intuition. In the 1920s Bernays was giving a somewhat Kantian philosophical scaffolding to Hilbert’s program and was still close to the Kantian philosopher Leonard Nelson. Later he changed his views considerably, but there remains a Kantian residue, surprisingly not completely giving up on the role of intuition in geometry.

Panu Raatikainen (Theoretical Philosophy, University of Helsinki)

Neo-Logicism and its Logic

The larger part of the philosophical discussion related to the neo-logicist program has focused on the status of Hume’s Principle and other abstraction principles. Much less attention has been paid to the rather undisturbed use of second order logic in the program. I shall demonstrate beyond dispute that there is a definite sense in which Hume’s Principle does very little in the derivation of Frege’s theorem, which is the core of the neo-logicist program, and that it is the background logic assumed that in fact does most of the mathematical work; it turns out that the latter smuggles in certain quite strong set-existence postulates. And that is obviously very problematic, if the project is to vindicate logicism in some sense.

Daniel Sutherland (Department of Philosophy, University of Illinois at Chicago)

Kant on the Conditions of Arithmetical Cognition

Kant’s theory of mathematical cognition includes ordinal elements, and he accounts for those ordinal elements by appealing to succession and temporal representation. Nevertheless, Kant’s theory also includes cardinal elements, and for those, Kant appeals to spatial representation. This paper will explore why Kant might have thought cardinal elements and spatial representation are required for mathematical cognition.

Sabetai Unguru (The Cohn Institute, Tel-Aviv University)

The Greek Concept of Number and Its Demise

The paper shall deal with the main differences between the Greek and the modern, seventeenth century, concept of number. It shall point out, the philosophico-mathematical underpinnings of both concepts and the fundamental, historical changes, marking the passage from the ancient to the modern view. The principal historical figures embodying the story told in the paper are Plato, Aristotle, Euclid, Apollonius, Diophantus, Viète, Fermat, and Descartes.