Consul wrote: ↑September 1st, 2022, 1:39 pmI don't think a numerical truth such as "The number of US states is 50" implies that there (really) is such an entity as the number 50.Ontological realists about numbers needn't regarded them as (abstract) objects, because they can regard them alternatively as (universal) properties: the property of being one/two/three/…
Note that there is a distinction between numbers as nonlinguistic mathematical entities and numerals as linguistic entities (purporting to represent numbers); so the existence of numerals doesn't entail the existence of numbers (as abstract mathematical objects).
However, there is a further distinction between tokens and types of numerals: Numeral-tokens are concrete (mental or physical) objects, whereas numeral-types are abstract (nonmental and nonphysical) objects. Numeral-types are as (ontologically) abstract as numbers, but there is still a difference between them insofar as the former are (arguably) language-dependent and the latter are not. (By saying so, I'm not implying that there really are abstract numeral-types and numbers.)
However, if number-properties are regarded as Platonic, transcendent universals, they are ontologically abstract too. They are ontologically concrete only if they are regarded either as Aristotelian, immanent universals or as immanent property-particulars.
David Armstrong, who acknowledges immanent universals in his ontology, objects that pure number-properties such as being twelve are mathematical abstractions from number-involving physical quantities such as being twelve kilograms in mass. According to him, a pure number-property such as being twelve is a "false abstraction" that isn't instantiated by any plurality of things—as opposed to "impure" number-properties such as being twelve kilograms in mass, which are instantiated by pluralities of microphysical things.
Let's assume for the sake of the argument that there (really) are such items as pure or impure number-properties. Are nonexistent things among the things which can have such properties? For example, do Sherlock Holmes and Dr. Watson together have the property of being two despite the fact that they both don't exist (have never existed)?
(I can say truly that Sherlock Holmes and Dr. Watson are two fictional persons without thereby implying that there really are such numerical properties as the property of being two; but, as I say above, I presuppose their existence just for the sake of the argument.)
This raises the general ontological question as to whether nonentities/nonexistents (can) have any properties at all, including number-properties (numerical properties).
The Austrian philosopher Alexius Meinong is famous for his independence principle, according to which Sosein (essence) doesn't entail Dasein (existence)—except for the Dasein (existence) of the Sosein (essence) itself—such that nonexistent objects or persons can have (possess/exemplify/instantiate) properties just like existent ones:
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"…Das alles ändert nichts an der Tatsache, dass das Sosein eines Gegenstandes durch dessen Nichtsein sozusagen nicht mitbetroffen ist. Die Tatsache ist wichtig genug, um sie ausdrücklich als das Prinzip der Unabhängigkeit des Soseins vom Sein zu formulieren[.]"
—
"…All that doesn't alter the fact that the being-so (Sosein) of an object is not affected by its non-being, so to speak. The fact is important enough to formulate it explicitly as the principle of the independence of being-so from being."
[© my translation from German]
(Meinong, Alexius. Über Gegenstandstheorie. In Untersuchungen zur Gegenstandstheorie und Psychologie, edited by Alexius Meinong, 1-50. Leipzig: Barth, 1904. p. 8 )
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Richard Routley (aka Richard Sylvan) accepts Meinong's principle with his neo-Meinongian theory of objects or "items" called "non(e)ism".
Decide for yourself!—but, as far as I'm concerned, I strongly reject Meinong's principle, simply because I think it is ontologically incoherent!
Properties (real ones, not unreal semantic "shadows" of concepts or predicates) are "adherences" or "inherences", which is to say that it is part of their essence that there exists something to or in which they adhere or inhere, with nonexistent objects not being possible substrata of properties qua adherences or inherences.
For in the case of nonentities/nonexistents there simply isn't anything to or in which properties can adhere or inhere; and the idea of an unhad or unborne (unpossessed/unexemplified/uninstantiated) property makes no coherent ontological sense either. Therefore, nonentities lack properties in general and (pure or impure) number-properties in particular.
Of course, according to some ghost story, there may be some mass of ectoplasm with the properties of having a volume of 50cm3 and of weighing 100mg; but, as I already said, what exists only according to some fiction, doesn't exist at all.
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"The position arrived at – hereafter called (basic) noneism, also spelt and pronounced “nonism” – is thus neither realism nor nominalism nor conceptualism. It falls outside the false classifications of both the ancient and modern disputes over universals, since these classifications rest upon an assumption, the vulgar prejudice Reid refers to, which noneism rejects.
By far the fullest working out of these noneist themes – which are firmly grounded in commonsense but tend to lead quickly away from current philosophical “commonsense” – is to be found in the work of Meinong, especially in Meinong’s theory of objects, central theses of which include these:
M1. Everything whatever – whether thinkable or not, possible or not, complete or not, even perhaps paradoxical or not – is an object.
M2. Very many objects do not exist; and in many cases they do not exist in any way at all, or have any form of being whatsoever.
M3. Non-existent objects are constituted in one way or another, and have more or less determinate natures, and thus they have properties. In fact they have properties of a range of sorts, sometimes quite ordinary properties, e.g. the oft-quoted golden mountain is golden. Given a subdivision of properties into (what may be called) characterising properties and non-characterising properties, further central theses of Meinong’s can be formulated, namely:
M4. Existence is not a characterising property of any object. In more old-fashioned language, being is not part of the characterisation or essence of an object; and in more modern and misleading terminology, existence is not a predicate (but of course it is a grammatical predicate). The thesis holds, as we shall see, not merely for “exists”, but for an important class of ontological predicates, e.g. “is possible”, “is created”, “dies”, “is fictional”.
M5. Every object has the characteristics it has irrespective of whether it exists; or, more succinctly, essence precedes existence.
M6. An object has those characterising properties used to characterise it. For example, the round square, being the object characterised as round and square, is both round and square.
Several other theses emerge as a natural outcome of these theses; for example:
M7. Important quantifiers, in fact of common occurrence in natural language, conform neither to the existence nor to the identity and enumeration requirements that classical logicians have tried to impose in their regimentation of discourse. Among these quantifiers are those used in stating the preceding theses, e.g. “everything”, “very many”, and “in many cases”. A similar thesis holds for descriptors, for instance for “the” as used in “the round square”.
The theory of objects – or of items, to use a more neutral term – to be outlined integrates, extends, and fits into a logical framework, all the theses introduced from the Epicureans, from Reid and especially from Meinong."
(pp. 2-4)
"A theory of items – which is what noneism aims at – is a very general theory of all items whatsoever, of those that are intensional and those that are not, of those that exist and those that do not, of those that are possible and those that are not, of those that are paradoxical or defective and those that are not, of those that are significant or absurd and those that are not; it is a theory of the logic and properties and kinds of properties of all these items."
(p. 7)
"There is a very widespread assumption, implicit in most modern philosophical theories, which settles the truth-values of very many of these statements, namely the Ontological Assumption (abbreviated as OA), according to which no (genuine) statements about what does not exist are true. Alternatively, in a more careful formal mode formulation, the OA is the thesis that a non-denoting expression cannot be the proper subject of a true statement (where the proper subject contrasts with the apparent subject which is eliminated under analysis into logical or canonical form).
It is the rejection of the Ontological Assumption that makes a proper theory of items possible and begins to mark such a genuinely nonexistential theory offfrom standard logical theories. According to the OA – to state the Assumption in a revealing way that exponents of the Assumption cannot (readily) avail themselves of – nonentities are featureless, only what exists can truly have properties. All standard logical theories are committed, usually through the theory of descriptions they incorporate, to some version of the Ontological Assumption."
(pp. 28-9)
"Philosophers of almost all persuasions seem to agree that statements whose (proper) subject terms do not have an actual reference somehow fail. But though these philosophers agree that such statements fail they disagree on how to characterise this failure. According to the strongest affirmation of the featurelessness of nonentities, that of the early Wittgenstein and of Parmenides, such statements are not just meaningless, they can’t even be made or uttered; according to Plato such statements are nonsense; according to Strawson they are not truth-valued; and Russell, as well as standard logic, tells us that they are all false. The lowest common denominator of these pervasive positions is given by the following formulation of the Ontological Assumption: it is not true that nonentities ever have properties; it is not true that any nonentity has a genuine property."
(p. 30)
"The Ontological Assumption – and thereby all the positions alluded to – was explicitly repudiated by Meinong’s and Mally’s Independence Thesis, namely
(III) That an item has properties need not, and commonly does not, imply, or (pre)suppose, that it exists or has being. Thus statements ascribing features to nonentities may be used, and are used, without involving any existential or ontological commitment. (The basic independence thesis)"
(p. 31)
"The Independence Thesis, that items can and do have definite properties even though nonentities…"
(p. 37)
(Routley, Richard. Exploring Meinong's Jungle and Beyond. [1980.] Vol. 1 of The Sylvan Jungle. Edited by Maureen Eckert. Cham: Springer, 2018.)
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