Introduction

This is the introduction to the special issue on Robert K. Meyer and the philosophy of arithmetic.

Thinking in colors

This article reveals the philosophical grounds of the aesthetics of color, analyzes the correlation between the structures of philosophical and artistic comprehension of coloristics. Interaction of philosophy and art as the forms of cultural identity manifests in the sphere of intellectual understanding of the perception of color and its semantics in painting. In the hidden logic of contemplation of color, can be traced the outlines of the problematic of transcendental and intelligible in art conditions for the aesthetic approach towards chromatic space. Color creates the visual beauty, thus it is apparent why the aesthetic knowledge seeks to clarify to which extent we can assess the experience of color – the result of coloration of light. The art itself creates the so-called color ontology of the world. First the first time, the beauty of color and its perception are analyzed in the context of correlation between art and transcendental traditions of philosophizing  (Descartes, Kant, early Husserl –  his work “The Philosophy of Arithmetic”) that allows matching the key to a new interpretation of the tradition of color. Determination of its meaning requires comparing history and structure of the philosophical and artistic metaphor of color. It is demonstrated that the phenomenon of color is of crucial significance for the aesthetics, as it implies not only comprehension of the problem of correlation between nature and art, but also cognition of the beauty of color, its universal value for all forms of art, profound structures of perception of coloristic phenomena, picturesque unveiling of the color harmony of the painting.

Solving the Caesar Problem—with Metaphysics

According to neo-Fregean Platonism, abstraction principles—such as the principle that the direction of line a is identical to the direction of line b iff a and b are parallel—may in some cases be regarded as introducing new singular terms (e.g., “the direction of line a”) and as fixing the truth-conditions of genuine identity statements featuring them. If neo-Fregeanism is to vindicate Frege’s idea that a plausible philosophy of arithmetic can and should treat the natural numbers as a species of object, it must address the so-called “Caesar Problem”: the problem of explaining in general terms which objects given in other terms they are to be distinguished from. This chapter pilots a novel solution to the Caesar Problem via the notion of a real definition: a definition whose purpose is not to explain a meaning, but to characterize the essential nature of the thing introduced.

Logicism and Logical Consequence

According to Crispin Wright’s neo-logicist reconstruction of Frege’s philosophy of arithmetic, the truths of arithmetic are logical consequences, in the semantic sense, of second-order logic, augmented with an analytic axiom (Hume’s Principle). Neo-logicism thus views arithmetic truths as analytic, being the logical consequences of an analytic axiom. This chapter argues that the semantic relation of second-order logical consequence that is most naturally suited to the practice of arithmetic is proof-theoretically complete, and that given this, Gödel’s incompleteness result shows that there are arithmetical truths which are not derivable in Wright’s proof theory augmented by Hume’s Principle. The chapter thus challenges Wright’s programme of neo-Fregean logicism.

“If Numbers Are to Be Anything At All, They Must Be Intrinsically Something”: Bertrand Russell and Mathematical Structuralism

Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind’s philosophy of arithmetic: a position we now describe as non-eliminative structuralism. But Russell’s response was deeply negative: “If [numbers] are to be anything at all, they must be intrinsically something” (Principles of Mathematics, §242). Nevertheless, Russell also played a significant positive role in making possible the emergence of structuralist philosophy of mathematics. This chapter explains Russell’s double role, identifying three positive contributions to structuralism, while laying out Russell’s objections to Dedekind’s non-eliminative structuralism.

When Logic Gives Out. Frege on Basic Logical Laws

At the end of Grundgesetze Frege tells us that the Urproblem of arithmetic is the question of how we apprehend logical objects. The success of Frege’s logicist enterprise thus essentially depends on the provision of a satisfactory answer to the question of how we can justifiedly hold certain basic truths to be logical, since for Frege it is only through those laws that we can come to grasp logical objects by purely logical means. And yet, despite its crucial importance, the question of what kind of justification we might provide for basic logical laws is one that Frege never fully addressed. In this chapter, I critically examine, and dismiss, three justificatory strategies briefly canvassed, but not wholly endorsed by Frege (respectively, the appeal to constitutivity, self-evidence, and sense-compositionality). I close by discussing a position that I label pragmatic foundationalism, a position that includes a strong externalist component. I claim that pragmatic foundationalism provides the best attempt one could make, on behalf of Frege, towards answering what he took to be the fundamental problem facing a mature philosophy of arithmetic.

Arithmetic, philosophical issues in

The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point where proof by induction enters that arithmetic stops being trivial. The propositions of elementary arithmetic – quantifier-free sentences such as ‘7+5=12’ – can be decided mechanically: once we know the rules for calculating, it is hard to see what mathematical interest can remain. As soon as we allow sentences with one universal quantifier, however – sentences of the form ‘(∀x)f(x)=0’ – we have no decision procedure either in principle or in practice, and can state some of the most profound and difficult problems in mathematics. (Goldbach’s conjecture that every even number greater than 2 is the sum of two primes, formulated in 1742 and still unsolved, is of this type.) It seems natural to regard as part of what we mean by natural numbers that they should obey the principle of induction. But this exhibits a form of circularity known as ‘impredicativity’: the statement of the principle involves quantification over properties of numbers, but to understand this quantification we must assume a prior grasp of the number concept, which it was our intention to define. It is nowadays a commonplace to draw a distinction between impredicative definitions, which are illegitimate, and impredicative specifications, which are not. The conclusion we should draw in this case is that the principle of induction on its own does not provide a non-circular route to an understanding of the natural number concept. We therefore need an independent argument. Four broad strategies have been attempted, which we shall consider in turn.