4.Frege on Concepts and Objects
10. Abstract and Concrete Objects
One important issue which I have postponed until now is that of the distinction between ‘abstract’ and ‘concrete’
objects. (As we shall see in Chapter 10, the epithet ‘abstract’ is not always used to register a difference from the
‘concrete’, though that is how I am understanding it at present.) I assume that numbers, sets and directions are uncontroversially abstract, while books and children are indisputably concrete. Of course, it may be asked how I know that numbers are abstract, when nothing I have said about them so far determines what they are. Indeed, it has been argued that numbers could not be ‘objects’ at all.84My own view is that the natural numbers, at least, are sorts or kinds (of sets) and so a fortiori abstract.85 However, even if this is not accepted, perhaps we know enough about numbers to know that they would have to be abstract whatever they are—perhaps because there are too many of them for them to be concrete.
An obvious suggestion is that concrete objects are, while abstract objects are not, denizens of space-time (or, which perhaps amounts to the same thing, are/are not subject to causality).86This has been queried, however—for instance by Bob Hale87—on the grounds that objects such as languages are plausibly abstract and yet come into existence and undergo change and so presumably exist in time. (It won't do to classify them as abstract on the grounds that they only exist in time and not also in space—even if it were altogether plausible to say this of them—for we
84 See Paul Benacerraf, ‘What Numbers Could Not Be’, in P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics: Selected Readings, 2nd edn. (Cambridge: Cambridge University Press, 1983); but see also Wright, Frege's Conception of Numbers as Objects, 117 ff., for criticism.
85 See my ‘Are the Natural Numbers Individuals or Sorts?’, Analysis, 53 (1993), 142–6, and Chapter 10 below.
86 See e.g. Reinhardt Grossmann, The Existence of the World: An Introduction to Ontology (London: Routledge, 1992), 7.
87 Hale, Abstract Objects, 49.
should want to classify immaterial Cartesian egos as ‘concrete’ despite ascribing only temporal, not spatial, existence to them.) Hale proposes instead, developing a suggestion of Harold Noonan's,88that abstract objects can be distinguished by reference to certain features of the criteria of identity which govern them. Specifically, he proposes:89
(A4)F is an abstract sortal iff, for any R that grounds F, either (i) R cannot hold between spatially located items at all or (ii) R can hold between things which are spatially, but not temporally, separated
where R is an equivalence relation and
R grounds F iff, for any statement of identity linking F-denoting terms, there is some statement to the effect that R holds among certain things, the truth of which is (logically) necessary and sufficient for the truth of that statement of F-identity.90
As an example of a grounding relation Hale cites the relation of parallelism between lines, which qualifies as such ‘in virtue of the fact that lines have identical directions iff they are parallel’.91 From this it appears that Hale is thinking primarily in terms of two-level (‘Fregean’) rather than one-level identity criteria (though he acknowledges that at least some sortals must be governed by one-level criteria,92 and it is clear, indeed, that he intends (A4) to prescind from the distinction between one-level and two-level criteria).
Limitations of space prevent me from discussing the interesting reasoning behind Hale's ingenious proposal, but it appears in any case to be fatally flawed. This is most easily seen if one considers what it implies about concrete sortals (assuming that a sortal is ‘concrete’ if and only if it is not ‘abstract’). Negating the right-hand side of (A4), we see that by Hale's account a sortal F qualifies as concrete iff there is some R that grounds F such that (i) R can hold between spatially located items and (ii) R cannot hold between things which are spatially, but not temporally, separated. Now consider the relation ‘x and y coincide in their boundaries’. This is clearly a relation which serves to ‘ground’ the abstract sortal ‘part of a geometrical figure’, for it is evident that if x and y are parts of a geometrical figure (for example, semicircular parts of a circle), then they are, of logical necessity, identical parts if and only if they coincide in their boundaries. However, this is a relation which can also hold between spatially
88 See H. W. Noonan, ‘Dummett on Abstract Objects’, Analysis, 36 (1976), 49–54, and ‘Count Nouns and Mass Nouns’, Analysi s, 38 (1978), 167–72.
89 Hale, Abstract Objects, 61. The numbering is Hale's.
90 Ibid. 59.
91 Ibid.
92 Ibid. 57.
located items (for instance, Switzerland coincides in its boundaries with the mereological sum of its cantons), but cannot hold between things which are spatially separated (and so a fortiori cannot hold between things which are spatially, but not temporally, separated). By Hale's account, therefore, the sortal ‘part of a geometrical figure’ is wrongly classified as concrete. (To this it might perhaps be objected that the sense in which concrete objects, such as pieces of terrain, may ‘coincide in their boundaries’ is different from the sense in which abstract objects, such as geometrical figures, may do so: but that presupposes that the distinction between abstract and concrete entities has already been satisfactorily drawn.)
However, rather than attempt to refurbish Hale's proposal, let us look again at the previous suggestion that abstract objects are those that are not denizens of space-time. The supposed difficulty was that objects such as languages are plausibly abstract and yet also plausibly come into existence and undergo change. But perhaps we need to make a distinction, which can best be brought out by analogy with a related case, that of biological species. These too are said to come into existence and undergo change—indeed, that they do so is crucial to the theory of evolution. How then can species names denote universals, which are abstract entities and so on the present proposal timeless? The solution is to distinguish between biological species, which are concrete particulars consisting at any time of the mereological sum of their currently existing members (individual tigers or individual oaks), and biological sorts or kinds, which are universals instantiated by the members of those species.93 Thus we can say that the horse species at one time did not exist and has evolved over millennia as its individual members have gradually taken on different morphological features, but that the kind horse which all of these past and present individual horses instantiate never ‘came into’ existence and has not itself undergone change. In like manner, we may say that ‘English’, construed as denoting a kind of language, does not refer to an ephemeral and changeable entity, but that what have come and gone and been subject to change are the concrete processes of linguistic communication which, over the centuries of English history, have all qualified as manifestations of English. On this view, in as much as ‘English’ denotes something abstract it denotes a kind (a universal), not a particular. To the extent that we happily identify various sub-kinds of Englishsuch as American English and Old English—this view seems reasonable, since only kinds (not particulars) can have sub-kinds.
93 See my ‘Noun Phrases, Quantifiers, and Generic Names’, and cf. D. L. Hull, ‘Are Species Really Individuals?’, Systematic Zoology, 25 (1976), 174–91. See also Chapter 8 below.