Online research in philosophy

Abstract A number is the number of a class which is an objective, nonactual, mathematical object. The concept of class is analyzed and it is concluded that a number is the number of a pure founded class. A tempting strategy of explaining numbers away is rejected. Some well?known definitions of numbers are analyzed and it is concluded that this analysis purports the thesis that the unique notion of number does not exist. Numbers are conventional. Nevertheless, an argument is offered purporting the thesis that von Neumann's ordinal numbers are the ordinal numbers. Accordingly, the corresponding von Neumann's cardinal numbers are the numbers.

Platonism about mathematics (or mathematical platonism as I will mostly call it) is typically defined as the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. If true, mathematical platonism would be of great philosophical significance: it would be a counterexample to common physicalist views, and it would put great pressure on the epistemology of mathematics. The view would also be of significance for mathematical practice.

Is the number an absolute intelligible reality? The author investigates the number and its nature in Plotinus. works trying to solve the following question: what number is considered intelligible - the number in general or the number in particular? Three answers are given over this study. Thus, if the number is generally defined as intelligible (as Plotinus sometimes does), than the number in general is an intelligible reality (a general intelligible number, therefore, exists). On the other hand, if we make a distinction between numbers (the plural) and number (the singular), it seems that, for Plotinus, only the particular number could be considered clearly intelligible, while the number as a generic reality is not so. Actually, the final solution comes out from the agreement between these two divergent theses. This agreement is based on the idea of the total number: a number that is in the same time particular and general, a number which is the object of the final part of the present study.

Singular-term semantics has been intractable. Frege took the referents of singular terms to be their semantic values. On his account, vacuous terms lacked values. Russell separated the semantics of definite descriptions from the semantics of proper names, which caused truth-values to be composed in two different ways and still left vacuous names without values. Montague gave all noun phrases sets of verb-phrase extensions for values, which created type mismatches when noun phrases were objects and still left vacuous names without values. There is a single type of value for all noun phrases that dissolves the difficulties which have beset singular-term semantics.

The following Principle of Substitutivity holds for the former, but not for the latter sentence: (PS) The truth value of (the proposition expressed by) a sentence that contains an occurrence of t1 remains constant when t2 is substituted for t1, provided that t1 and t2 are codesignative singular terms. It is an undeniable fact that different sentences behave differently when it comes to which substitutions preserve their truth value. What is curious is that this fact has been presented by the philosophical tradition as a puzzle. To be more precise, what is supposed to be puzzling is the breakdown of PS in some sentences. Meanwhile, it is assumed that everything is as it should be, that nothing needs to be explained when we observe that the substitution of 'the number of planets' for 'nine' in 'nine is greater than seven' guarantees the preservation of truth value, in spite of the fact that the subject matter of the former sentence and the subject matter of 'the number of planets is greater than seven' are radically different. The former sentence expresses a claim about numbers and their relationships, whereas the latter sentence makes an assertion about our solar system.

According to the standard view, alethic (or modal) statements are intensional in that the Principle of Substitution (PS) fails for them -- e.g. substituting 'nine' in "Necessarily, nine is composite" with the co-referring 'the number of planets' turns this statement from true to false. It is argued in the paper that we could avoid ascribing intensionality to alethic statements altogether by separating between singular and functional uses of definite descriptions: on the singular use the description given above amounts to 'the actual number of planets', which is salva veritate substitutable to 'nine' in all alethic statements; on the functional use, in turn, that description is really a function from possible worlds to numbers, and thus the Principle of Substitution is not violated in this case either, since such a function cannot be held to be co-referential with 'nine'.

Frege wanted to define the number 1 and the concept of number. What is required of a satisfactory definition? A truly arbitrary definition will not do: to stipulate that the number one is Julius Caesar is to change the subject. One might expect Frege to define the number 1 by giving a description that picks out the object that the numeral '1' already names; to define the concept of number by giving a description that picks out precisely those objects that are numbers. Yet Frege appears to do no such thing. Indeed, when he defends his definitions, he does not argue that they pick out objects that we have been talking about all along-the issue never comes up. The aim of this paper is to explain why. I argue that, on Frege's view, our numerals do not, antecedent to his work, name particular objects. This raises an obvious question: If (like 'Odysseus') the numerals do not name particular objects, how can Frege write (as he does) as if sentences in which numerals appear state truths? One central concern of this paper is exegetical-to answer these questions. But my aim is not solely exegetical. For these questions direct us to something that, I believe, creates only an apparent problem for Frege but an actual problem for many contemporary philosophers: the assumption that singular terms appearing in statements about the world must actually have referents. Another aim of this paper is to suggest that the problem-as well as a solution that can be found in Frege's writings-should be of import to contemporary philosophers.

In his groundbreaking Grundlagen, Frege (1884) pointed out that number words like ‘four’ occur in ordinary language in two quite different ways and that this gives rise to a philosophical puzzle. On the one hand ‘four’ occurs as an adjective, which is to say that it occurs grammatically in sentences in a position that is commonly occupied by adjectives. Frege’s example was (1) Jupiter has four moons, where the occurrence of ‘four’ seems to be just like that of ‘green’ in (2) Jupiter has green moons. On the other hand, ‘four’ occurs as a singular term, which is to say that it occurs in a position that is commonly occupied by paradigmatic cases of singular terms, like proper names: (3) The number of moons of Jupiter is four. Here ‘four’ seems to be just like ‘Wagner’ in (4) The composer of Tannhäuser is Wagner, and both of these statements seem to be identity statements, ones with which we claim that what two singular terms stand for is identical. But that number words can occur both as singular terms and as adjectives is puzzling. Usually adjectives cannot occur in a position occupied by a singular term, and the other way round, without resulting in ungrammaticality and nonsense. To give just one example, it would be ungrammatical to replace ‘four’ with ‘the number of moons of Jupiter’ in (1): (5) *Jupiter has the number of moons of Jupiter moons. This ungrammaticality results even though ‘four’ and ‘the number of moons of Jupiter’ are both apparently singular terms standing for the same object in (3). So, how can it be that number words can occur both as singular terms and as adjectives, while other adjectives cannot occur as singular terms, and other singular terms cannot occur as adjectives?

Abstract   A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets , eight , as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. Content Type Journal Article Pages 1-38 DOI 10.1007/s11098-011-9779-1 Authors Friederike Moltmann, IHPST (Paris1/ENS/CNRS), Paris, France Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116.

Abstract   A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets , eight , as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. Content Type Journal Article Pages 1-38 DOI 10.1007/s11098-011-9779-1 Authors Friederike Moltmann, IHPST (Paris1/ENS/CNRS), Paris, France Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116.