Number Sentences and Specificational Sentences. Reply to Moltmann (in Philosophical Studies, 173(8), 2173-192)
Abstract: Frege proposed that sentences like ‚The number of planets is eight‘ be analysed as identity statements in which the number words refer to numbers. Recently, Friederike Moltmann argued that, pace Frege, such sentences be analysed as so-called specificational sentences in which the number words have the same non-referring semantic function as the number word ‘eight’ in ‘There are eight planets’. The aim of this paper is two-fold. First, I argue that Moltmann fails to show that such sentences should be analysed as specificational sentences. Second, I show that even if they are to be analysed in this way, Moltmann’s proposed specificational analysis is unsatisfactory. [Draft]
Singular Terms Revisited (in Synthese 193(3), 909-36)
Abstract: Neo-Fregeans take their argument for arithmetical realism to depend on the availability of certain, so-called broadly syntactic tests for whether a given expression functions as a singular term. The broadly syntactic tests proposed in the neo-Fregean tradition are the so-called inferential test and the Aristotelian test. If these tests are to subserve the neo-Fregean argument, they must be at least adequate, in the sense of correctly classifying paradigm cases of singular terms and non-singular terms. In this paper, I pursue two main goals. On the one hand, I show that the tests‘ current state-of-the-art formulations are inadequate and, hence, cannot subserve the neo-Fregean argument. On the other hand, I propose revisions that are adequate and, hence, can subserve this argument. [Draft]
Das Theodizee-Problem (in Puster, R. 2013: Klassische Argumentationen der Philosophie. Paderborn: Mentis, 221-39)
Abstract: An introductory overview over the problem of evil.
Hume’s Principle Revisited: Numbers as Dependent Objects (in Grazer Philosopische Studien 82, 353-73)
Abstract: Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.
Number Sentences and Specificational Sentences Reconsidered (Draft 2016-03-31)
Abstract: Sentence pairs like ‚The number of moons of Mars is two‘ and ‚Mars has two moons‘ give rise to a certain puzzle. On the one hand they seem to be truth-conditionally equivalent. On the other hand, though, it is puzzling how this could be the case. For on Frege’s influential analysis of the former as an mathematical identity statement its truth requires the existence of numbers. And linguistic theory tells us that the truth of the latter does not require the existence of numbers, but only that of Mars and its two moons. How, then, can these two sentences be equivalent given that their truth of only one of them requires the existence number? Recently, it has been argued that, pace Frege, sentences like ‚The number of moons of Mars is two‘ are to be analysed as so-called specificational sentences and that, thus analysed, their truth does not require the existence of numbers but, in fact, imposes the same requirements as ‚Mars has two moons‘. If so, the aforementioned puzzle would be resolved.
In this paper, I show that the specificational analysis proposed by Katharina Felka fails to have the desired ontologically deflating consequence. Thus, even if her analysis was linguistically superior to Frege’s, the equivalence of `The number of moons of Mars is two‘ and `Mars has two moons‘ would still be no less puzzling as it is on Frege’s own.
Ontological Reduction by Analysis: A Case Study (Draft 2016-03-08)
Abstract: According to Crispin Wright, ontological reduction by analysis is a radically misconceived endeavour. In this paper, I investigate whether Wright’s dictum applies to recent attempts of reducing the ontological consequences of ostensibly number-wise committed sentences like ‘The number Martian moons is two’. In particular, these attempts aim to reduce such sentence’s ontological consequence by analysing them as so-called specificational sentences in a way that would reduce their number-commitments to those—i.e. to none—of number-wise innocent sentences like ‘Mars has two moons’.
The aim of this paper is two-fold. First, I demonstrate that the extant specificational analyses of such sentences are either linguistically implausible or fail to be ontologically reductive. Second, I show how these difficulties can be overcome by proposing a novel and superior specificational analysis. This analysis crucially relies on an analysis of expressions like ‘the number of Martian moons’ on which they function as higher-level descriptions.
The Fregean Misconception of Number(word)s as Object(word)s
Abstract: In §57 of Grundlagen, Gottlob Frege famously proposed that a sentence like (1) The number of books Frege wrote is three, be analysed as a first-level identity statement, i.e. as a sentence in which ‘the number of books Frege wrote’ and the numeral ‘three’ function as object-denoting expressions and ‘is’ expresses first-level identity aka identity between objects. Recent years have seen the advent of anti-Fregean analyses of (1), which are united in the claim that the numeral ‘three’ is not an object-denoting expression but rather has the same semantic function as it has in (2) Frege wrote three books. Although not without merit, the extant anti-Fregean analyses of (1) are unsatisfactory. In my paper, I develop a novel and improved anti-Fregean analysis of (1). I argue that (1) is a second-level identity statement in which ‘the number of books Frege wrote’ and ‘three’ are first-level expressions and ‘is’ expresses second-level identity aka identity between first-level (Fregean) concepts. My argument will be based on three main claims. First, that ‘three’ in (2) functions as a predicational adjective much like, say, ‘interesting’ in ‘Frege wrote interesting books’. Second, that a phrase like ‘an odd number of’ as it occurs in a sentence like (3) Frege wrote an odd number of books, is a complex higher-level quantifier in whose context ‘number (of)’ is a second-level predicate which applies to numbers conceived of as first-level concepts. Third, that my anti-Fregean analysis of (1) is supported by the fact that ‘number (of)’ in (3) functions in the indicated way.
Abstract: Numerical realism is the view that there exist genuinely mathematical objects of a certain kind, viz. numbers. According to an influential type of argument in its favour, we should believe numerical realism because (i) we have strong reasons to believe that certain mathematical theories are true, and (ii) at least some of these theories carry ontological commitment to numbers (in the sense that their truth requires that numbers exist). The thesis investigates the merits of the argument’s second central assumption in the restricted case in which the theory in question is an arithmetical theory and the numbers in question are natural numbers.
The investigation will be conducted against the background of three methodological assumptions. First, that arithmetical theories formulated in a formal language such as, most pertinently, the standard regimentation of the theory of so-called first-order Peano Arithmetic should only be taken to genuinely carry commitment to natural numbers if they are genuine first-order theories in the following sense: certain expressions of the language of regimentation such as ‚0‘, ‚|N‘, and ‚∀n‚ are, respectively, individual constants, individual-level predicate letters, and first-order quantifiers. Second, that whether these formal expressions should be taken to be individual constants, individual-level predicate letters, and first-order quantifiers depends on whether ‘zero’, ‘is a natural number’ and ‘every natural number’ are the natural language counterparts of individual constants, individual-level predicate letters, and first-order quantifiers. Third, that this second issue primarily turns on whether in certain constructions such as, for instance, ‘One plus three equals four’ or ‘The number of planets is eight’ the numerals function as arithmetical singular terms. That is, it depends on whether these numerals are, in Fregean terminology, object-denoting expressions that denote numbers if they denote anything at all.
The investigation divides into two parts. In the first part, I examine an argument in favour of the existence of arithmetical singular terms. This argument relies on the reliability of certain tests for singular-termhood proposed in the neo-Fregean tradition. I carefully reconstruct the tests’ existing formulations and propose some needed modifications. In response to a sceptical challenge against the test’s reliability, I show how these tests can be demonstrated to be reliable. However, this demonstration turns on a crucial assumption.
In the second part, I show that, unless this assumption is discharged, they are ultimately powerless to determine whether numerals in the relevant contexts are arithmetical singular terms or whether they are, in Fregean terminology, concept-denoting expressions. The view that numerals in the pertinent contexts are concept-denoting expressions is not new. However, a new type of argument has recently been given in support of this view. For the first time, arguments of this type heavily rely on results from the field of natural language semantics and syntax. I therefore address the cogency of one argument of this novel type. Crucially, this argument involves the assumption, recently defended by Thomas Hofweber and Friederike Moltmann, that numerals in constructions like ‘The number of planets is eight’ are concept-denoting expressions. I carefully reconstruct this argument and, after thorough examination, conclude that it fails to establish its conclusion.