Abstract
Two fundamental categories of any ontology are the category of objects and the category of universals. We discuss the question whether either of these categories can be infinite or not. In the category of objects, the subcategory of physical objects is examined within the context of different cosmological theories regarding the different kinds of fundamental objects in the universe. Abstract objects are discussed in terms of sets and the intensional objects of conceptual realism. The category of universals is discussed in terms of the three major theories of universals: nominalism, realism, and conceptualism. The finitude of mind pertains only to conceptualism. We consider the question of whether or not this finitude precludes impredicative concept formation. An explication of potential infinity, especially as applied to concepts and expressions, is given. We also briefly discuss a logic of plural objects, or groups of single objects (individuals), which is based on Bertrand Russell’s (1903, The principles of mathematics, 2nd edn. (1938). Norton & Co, NY) notion of a class as many. The universal class as many does not exist in this logic if there are two or more single objects; but the issue is undecided if there is just one individual. We note that adding plural objects (groups) to an ontology with a countable infinity of individuals (single objects) does not generate an uncountable infinity of classes as many.
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Notes
Aristotle, De Int. 17a39.
For a more detailed account of formal ontology, see Cocchiarella (2007, Chap. 1).
The WMAP satellite was a joint venture in 2001 of NASA and Princeton University. There are three experimental “proofs” of the big bang: the redshift of the galaxies, the cosmic back-ground microwave radiation, and nucleosynthesis of the elements. (See Greene, 2004, pp. 429ff.)
Dark energy is a force that acts in opposition to gravity and presently is causing the expansion of the universe. Dark matter, which makes up most of the mass in the universe, cannot be detected by any emitted radiation. Its existence is inferred by its gravitational effect. Determining the nature of both dark energy and dark matter is one of the most important problems in modern cosmology.
This does not mean that the mathematics of the continuum cannot be applied to space and time—just as the discreteness of the operations of a digital computer does not mean that we cannot solve calculus problems with computer programs.
For different cosmological models of a finite universe see Luminet et al. (1999).
A fundamental property of string theory is that it is supersymmetric, incorporating not only the reference frame symmetries of relativity theory and the quantum-mechanical guage symmetries of the strong, weak, and electromagnetic forces, but also a supersymmetry principle that relates the properties of particles with a whole number of spin (bosons) with those with a half a whole (odd) number of spin (fermions). (See Greene, 1999, Chap. 9.)
See Tegmark (2003). Although the universe is only 13.7 billion years old and the light that is now reaching us from the most distant stars took that many years to reach us, those stars, because of the expansion of the universe during that time, are now more than 13.7 billion lightyears away. They are in fact now about 40 billion lightyears away.
Tegmark (2003).
Ibid.
Having all the same laws of nature the different regions would amount to an equivalence class of “possible worlds,” which is a structure characteristic of S5 modal logic. For a formalization of the logic of actual and possible objects in S5, see Cocchiarella (2007, Chap. 2).
In Everett’s original version of the axioms of MWI no account was given of how the branching into different parallel worlds takes place. Later proposals by Graham and De Witt introduce the complicated notion of a measuring device that results in observations (by humans or automata) upon which the splitting into parallel worlds is based. See De Witt and Graham (1973).
Penrose (2004, p. 784).
Penrose (2004, p. 784).
The modal logic is S4 because the accessibility relation between possible worlds is a partial ordering determined by the wavefunctions that split each universe into its related parallel universes. The result in effect is a branched-tree model of the universe something like what is described in McCall (1994).
For a fuller account of MWI see De Witt and Graham (1973).
The ‘M’ of “string/M-theory” is sometimes said to stand for “Membrane theory,” sometimes for “Matrix theory,” and sometimes even for “Mystery theory” or “Mother of all string theories.”
Greene (2004, p. 349). As Greene notes, “the point-particle description is merely an idealization and … in the real world elementary particles do have some spatial extent” (p. 157). Some scientists have tried work with “blobs” or “nuggets,” but, because of quantum-mechanical probability and the impossibility of faster-than-light speed transmission of information, it was found to be too difficult to construct a QM theory that works with such objects (Ibid., p. 158).
See Greene (2004), Chap. 12 for a fuller discussion of these points.
Ibid., pp. 370f. Originally, there were five competing string theories each with six extra spatial dimensions; but then, in 1995, the mathematical physicist Edward Witten showed that from the point of view of another seventh dimension all five theories are equivalent and can be translated into one another. The result is 11 dimensions in all, counting time as well. (For more on this see Greene 2004, Chap. 13)
Greene (2004, p. 412).
Added note November, 2007: Steinhardt and Turok have recently modified their cyclic theory so as not to require another three-brane in hyperspace a millimeter away from our three-brane. A critical feature of this new version is the decay of dark energy in the future. For details see Steinhardt and Turok (2007).
A finite expansion of an infinite space results only in an infinite space, just as adding a finite cardinal number to an infinite cardinal number results in that infinite cardinal number.
Greene (2004, p. 407).
Here the equivalence class of possible worlds is the class of universes that exist at some cycle or other. We can include the objects in the other three-brane world here as well. A separate issue is how the two three-brane worlds are themselves treated. That is, are they included in the domain of objects as well?
See Greene (2004, p. 410).
See, e.g., Quine (1953, p. 121).
Dedekind (1988). Classes are equipollent if they can be put into a one-to-one correspondence with each other.
A systematic survey of definitions can be found in Tarski (1924).
In set theory, \(x\preceq y\) just in case there is a subset z of y, i.e., \(z\subseteq y,\) such that x is equipollent to z. Then, \(x\prec y\) if \( x\preceq y\) but \(y\ \npreceq \ x.\)
Strictly speaking, formulas with free predicate variables can also be allowed as substituents. What is excluded are formulas with bound predicate variables. See Cocchiarella (2007, Chap. 4), for a fuller account of nominalism as a formal theory of predication.
As an alternative to classes, e.g., sets, as abstract entities, the nominalist could adopt “classes as many” as plural objects, which we describe below under conceptualism.
Henkin (1962).
See Campbell (1981) for an account of tropism as an ontology.
It is perhaps inappropriate to refer to these particulars as abstract objects, but that is the standard terminology today.
See, e.g., Williams (1953) for such a position.
See Cocchiarella (2007, Chaps. 4 and 12) for accounts of logical and natural realism, respectively, as formal ontologies.
Because the logical truths of first- and higher-order logic are not recursive, logical realism cannot have an effective rule that would exclude properties and relations that logically cannot have instances.
See Cocchiarella (2007, Chap. 12), for a description of a formal ontology based on this distinction between natural kinds and natural properties.
Greene (1999, p. 146).
Ibid., p. 151.
Mutations, of course, are changes in the informational content of the hereditary units.
See Shannon and Weaver (1949).
Gatlin (1972, p. 196).
Ibid., p. 202.
See Cocchiarella (2007, Chap. 4), for a fuller discussion of conceptualism as a formal ontology.
The “predicative-impredicative” terminology goes back to Bertrand Russell’s theory of ramified types.
Being the tallest person in the room is an unproblematic impredicative concept, one might argue, because the number of people in the room is finite and can be mentally surveyed.
See, e.g., Quine (1953, p. 127).
See Cocchiarella (2007, Chap. 4, Sect. 4), for a description of this mechanism.
This is in fact Quine’s position in 1953, p. 127.
This result was first noted by Alfred Tarski many years ago.
Finitude is not a first-order notion, and it is for that reason that the model must be second-order.
See Cocchiarella (2007, Chap. 2), for a formalization of Diodorus’s and Aristotle’s notions of necessity and possibility in terms of tense logic.
For a semantics of this see Cocchiarella (2007, Chap. 2).
See Cocchiarella (2007, Chap. 12).
See Cocchiarella (2007, Chap. 12), for a development of conceptual natural realism and Aristotelian essentialism.
For a fuller discussion of this extension of conceptualism to conceptual intensional realism see Cocchiarella (2007, Chap. 5, Sect. 3).
See Cocchiarella (2007, Chap. 11), for a development of the logic of plurals.
For a variety of other examples of irreducible sentences that express irreducible plural references and/or predications see Cocchiarella (2007, Chap. 11). It is noteworthy, incidentally, that Lésniewski’s ontology, which is also called a logic of names, is reducible to the logic of classes as many. See Cocchiarella (2007, Chap. 11), for the details of this reduction.
See Russell (1903, Chap. VI).
For details about theorems in the logic of classes as many see Cocchiarella (2007, Chap. 11).
The universal class, were it to exist, would be the extension of the complex common name “object that is self-identical.” Thus, to say that a class A is co-extensive with the universal class means only that an object belongs to A if, and only if, that object is self-identical. It does not follow in the free logic of classes as many that then the universal class exists if A exists. For formal details, see Cocchiarella (2007, Chap. 11), Sect. 3.
Where the number of individuals is finite and greater than 1, i.e., where there are n many single objects in the universe, for some positive integer n > 1, then, by a simple inductive argument it can be shown that the number of classes as many of objects, single and plural, is 2n − 1, and hence that there are 2n − (n + 1) plural objects. Of course 2n − 1 > n, where n > 1. But where the number of single objects is \(\aleph_{0},\) we cannot show that the number of classes as many is \(2^{\aleph_{0}}-1\) (which is just \(2^{\aleph_{0}}.\)) The attempt to derive a contradiction by Cantor’s argument of assuming a 1-1 mapping f of all of the classes as many of individuals into the class of all individuals fails because the Cantor diagonal class of individuals x such that \(x\notin f(x)\) must be known to exist (or equivalently have a member) in order to derive a contradiction. If it exists, then a contradiction follows, and what this shows is that the Cantor class as many, like the Russell class as many, does not exist in the logic of classes as many (which is “free” of existential presuppositions).
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Cocchiarella, N.B. Infinity in Ontology and Mind. Axiomathes 18, 1–24 (2008). https://doi.org/10.1007/s10516-007-9028-6
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DOI: https://doi.org/10.1007/s10516-007-9028-6