Abstract
In an interview with Fara,1 Quine was asked what the main tenets of his philosophy were. Quine named two, naturalism and extensionalism. Naturalism is the more famous of the two nowadays, and has a big impact on contemporary debates in all philosophical disciplines in the Anglo-Saxon tradition. The relevance of Quine’s naturalism in ontology will be discussed in chapter 6, dealing with the epistemology of ontology, i.e. a naturalised epistemology of ontology.
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Notes
Quine et al. 1994.
Quine published very little on ethics, only the paper “On the nature of moral values”, reprinted in Quine 1981a, 55–66, the items “Altruism” and “Tolerance” in Quine 1987a, and a few scattered remarks, see Quine and Ullian 1970, 136–138; Quine 1974, §13.
Quine 1985a, 32.
The most outspoken proponent of this view is Roger Gibson. Gibson has described Quine’s philosophy in relation to a central theme, namely “the naturalistic-behavioristic thesis of language”, see Gibson 1982, §2.2, and Gibson 1988 §1.2. Gibson offers an adequate presentation of this aspect of Quine’s philosophy, and Quine explicitly applauds Gibson’s work. This is however a one-eyed reading of Quine in which some vital elements in the structure of Quine’s philosophy are neglected. Quine’s philosophy is presented in an impoverished way because his sense of precision is of little importance for naturalists.
See for example Quine 1994a, or Quine 1998, 33, or Quine 1995a, 90: “Extensionality is much of the glory of predicate logic, and it is much of the glory of any science that can be grammatically embedded in predicate logic.”
Quine 1969b, 23; Quine 1981a, 102; Quine 1995a, 75.
Quine 1950a, 268.
Van was a name for Quine that was the least obvious of a whole series of combinations based on his full name, but that was often used colloquially. For example, Carnap opened his letters to Quine with the words “Dear Van” (see Creath 1990). This Van is not an English first name, but the first part of his mother’s name “Van Orman”.
Quine 1960, 116. See also Quine 1987a, 232.
Quine presents Whitehead’s confusion in his typical style, see Quine 1960, 117: “Whitehead once defended the view, writing e.g.. that ‘2+3 and 3+2’ are not identical; the order of the symbols is different in the two combinations, and this difference of order directs different processes of thought.’ It is debatable how much this defense depends on confusion of sign and object, and how much on a special doctrine that numbers are thought processes”, see also Quine 1966b, 4–6.
See Quine 1960 §53, with the title: “The ordered pair as philosophical paradigm”; see also chapter 3.6
See Quine 1970a, 62, or Quine 1963, 12. In this last work Quine offered a more economical version by the single schema Fy t= (3x: x=y n Fx).
This was proven by Gödel, see Van Heijenoort 1967, 589.
See e.g.. Quine 1953a, 143.
The failure of substitutivity of identicals may be circumvented in modal logic, by an ambiguity in the scope of the quantifiers in modal contexts. This was pointed out by Smullyan 1948. Quine later conceded that his argument was not conclusive, see Quine 1966a, 174: “Looking upon quantification as fundamental, and singular terms as contextually defined, one must indeed concede the inconclusiveness of referential opacity that rests on interchanges of constant singular terms.” Though Quine avowed that his argument is inconclusive, he was still opposed to modal logic, mainly because of the essentialism behind it (o.c.. 176), but also because of the complication of the substitution of identicals: see o.c.. 175: “The effect of this [Smullyan’s] interference is that constant singular terms cannot be manipulated with the customary freedom, even when their objects exist.”
For further reading see: Quine 1943; Quine 1947; “Reference and modality”, in Quine 1953a, 139–159; Quine 1960, 141–156; 166–169; 195–226; “Three grades of modal involvement”, in Quine 1966a, 158–176; “Quantifiers and propositional attitudes”, in Quine 1966a, 185–196; “Intensions revisited”, in Quine 1981a, 113–123; “Worlds away”, in Quine 1981a, 124–128; Quine 1990a, 67–74; Quine 1994a“; Quine 1995a, 90–99. I will only discuss the problems about intensional contexts insofar as they have repercussions for what Quine believes there is. I am mainly interested in Quine’s own ontology and the framework in which this ontology is embedded, and less in the deviant frameworks he criticises.
This is Gödel’s completeness theorem, see for example Chang and Keisler 1990, 33.
See Quine 1985b, 163: “Identity is accommodated as one of the two-place predicates”; see also Quine 1987a, 90; 156.
See Quine 1960, 230.
See Quine 1970a, 64: “We turn now from identity theory to set theory. Does it belong to logic? I shall conclude not. The predicate `e’ of membership is distinctive of set theory, as is _’ of the logic of identity”; see also chapter 4.1.
See Quine 1970a, 62, or Quine 1966a, 111; Quine 1995a, 52.
See Quine 1970a, 62: “For, let us picture two such versions of identity as `x =1 y’ and `x =2 y’.. By (2)[axiom schema], then, `-(x =1 y.Fx.-Fy)’ holds under all substitutions for Tx’ and `Fy’.. So `-(x =1 y. x =2 x.-(x =2 y)).. But, by (1) [x=x], x =2 X. So.-(x =1 y.-(x =2 y)).. Similarly -(x =2y.-(x =1 y)).. In short, x =1 y. x =2 y.”
See Quine 1970a, 63. The formula is written in a modern logical notation. For similar examples see Quine 1953c, 117; Quine 1960, 230; Quine 1963, 14; Quine 1981a, 109; 130; Quine 1995a, 52.
Quine 1970a, 64.
Quine 1970a, 64.
See Quine 1960, 231, where, after explaining the above construction, he writes “Also it gives a kind of justification of one’s tendency to view `=’, more than other general terms, as a ”logical“ constant.”
Quine 1963, 14.
Quine 1963, 15. See also Quine 1950b, 71, or Quine 1969a, 55.
This gap is also discussed in Geach 1967. Geach attacks the notion of absolute identity. He thinks that it is not possible to guarantee that objects that are identical in a theory, i.e.. that cannot be discerned by means of the lexicon of the theory, will be identical under all extensions of a theory. Geach proposes a relative identity instead. One should say “… is the same A as…” instead of “… is the same as…” (Geach 1967, 10). Leach’s article is interesting for the fact that he stresses that “shifts of ideology lead to an indecent pullulation of entities in our ontology” (Geach 1967, 10). There are few traces of Quine’s conception of identity in his early writings. Geach may have been right in thinking that Quine had endorsed “’absolute identity”. Quine had a strong belief that ontology was much firmer than ideology (see Quine 1966a, 245, quoted by Geach). It is plausible that Quine thought that identity was a relation between the objects of our theory, and that because we have a good grasp on these objects, we know well what identity is. In later years Quine lost his faith in the absoluteness of ontology. Geach published this article in which he defended the relativity of identity a year before Quine 1969b.
See Quine 1950b, 71: “Our maxim of identification of indiscemibles is relative to a discourse, and hence vague in so far as the cleavage between discourses is vague.” There is hardly any disagreement between Geach’s view and Quine’s later view on the matter. Quine recognises being indebted to Geach in Quine 1974, 116 fn.
Quine clumsily refers to this problem in Quine 1976a, 498–499: “[T]here seems even to be something not quite right about distinguishing between coexisting electrons in space. Thus consider two boxes, which I shall call the east box and the west box, and two electrons, x and y.. Common sense recognizes four ways in which x and y could occupy the boxes: they could both be in the east box, or both be in the west box, or x could be in the east box and yin the west, or vice versa. But statistical findings show,…, that these last two apparent possibilities have to be counted as just one: the fact of x being in the east and y in the west must be identified with the seemingly opposite fact of x being in the west and yin the east.”
Quine had already introduced two of the three grades of discriminability in Quine 1960, 230: “Thus let us call two objects absolutely indiscernible (in a notation) if some open sentence with one free variable is fulfilled by only one of the two objects, and let us call them relatively discernible if some open sentence with two free variables is fulfilled by the two objects in only one order.” Absolute discernability will become strong discriurinability, and relative discernability will be moderate discriminability.
See Quine 1981a, 130.
In the article Quine presents an alternative characterisation of weak discriminability. Ivan Fox had suggested satisfaction by an irreflexive predicate as a criterion of discriminability. This criterion is equivalent to weak discriminability. One might suggest that in this case we do not have to take recourse to an interpreted identity predicate. See further Quine 198la, 132.
Quine 198la, 133. Quine ends: “What I have called moderate discriminability, however, is the only intermediate grade that I see how to define at our present high level of generality.”
Quine had read a large part of Principia Mathematica already in 1928, see Quine 1986, 8. The work was at that time the central textbook for logic, and certainly in Harvard. The new results on the European Continent were not known, and Frege was temporarily forgotten. Quine has discovered Frege only in 1940 when completing Quine 1940. Even at that time could not find a copy of the Begriffsschrift,see Quine 1986, 21.
Whitehead and Russell 1928, 91: “Following Peano, we shall call the undefined ideas and the undemonstrated propositions primitive ideas and primitive propositions respectively. The primitive ideas are explained by means of descriptions intended to point out to the reader what is meant; but the explanations do not constitute definitions, because they really involve the ideas they explain.” Quine sometimes adopts this term in his philosophical writings without being to explicit about its precise meaning. Quine was more explicit in his doctoral thesis Quine 1932, 1: “The expression ”primitive idea“ has been used in the literature in two distinct senses: as referring to an undefined range of variables, and as referring to an undefined but definitely symbolized constant, usually in practice and operation. For convenience let us designate these notions respectively as passive and active primitive ideas.… Unlike the passive primitive ideas, the active primitive ideas are given explicit expression in the postulates of the system.” The passive primitive ideas are propositions, propositional functions and terms or individuals. They do not appear directly in the system. The active primitive ideas are assertion, material incompatibility, universal quantification and particular quantification. In Quine 1963, 13, Quine speaks only of the “primitive predicates” of a theory.
Quine gave a presentation of these basic ideas of PM in Quine 1941 b.
Whitehead and Russell 1928, 127.
See Whitehead and Russell 1928, xvii; 91.
The term “atomic proposition” only appeared in the preface to the second edition of 1928, see Whitehead andRussell 1928, xv.
Whitehead and Russell 1928, 92.
Whitehead and Russell 1928, 15; 138. It is an empirical matter of fact which of the propositional functions really exist. Logic starts from the hypothesis that certain propositional functions exist. There are no explicit ontic decisions or commitments in PM.
Whitehead and Russell 1928, 22. The authors are here somehow negligent about the precise notation which should be “/ is identical with 9”. The formulation “x is identical with y” is strictly spoken not a propositional function, but an ambiguous value of the propositional function where the variable ambiguously designates an object, see o.c.., 14–15.
See Whitehead and Russell 1928, 37; Quine 1963, 241–242.
It is not really relevant what these objects are, see Whitehead and Russell, 161: “It is unnecessary, in practice, to know what objects belong to the lowest type, or even whether the lowest type of variable occurring in a given context is that of individuals or some other. For in practice only the relative types of variables are relevant; thus the lowest type occurring may be called that of individuals, as far as that context is concerned.”
Whitehead and Russell 1928, 51.
See also Linsky 1999, chapter 4 and references therein.
This is not the terminology of Whitehead and Russell 1928, but the later generally accepted Continental idiom. In Principia Mathematica the bound variables are called apparent variables, and the free variables are real variables.
Whitehead and Russell 1928, 50–52; 164–166.
For examples see Whitehead and Russell 1928, 164 (3).
Whitehead and Russell 1928, 56; 166–167.
A technical reason for the indispensability is that one cannot use least bounds without the axioms, and these are needed in the theory of the real numbers, see Quine 1963, 249–250.
See Whitehead and Russell 1928, 59–60, VII Reasons for Accepting the Axiom of Reducibility..
Whitehead and Russell 1928, 60.
Whitehead and Russell 1928, 168.
Whitehead and Russell 1928, 49.
See Whitehead and Russell 1928, 72–73. Strictly speaking, the identity relation is not presented as an intensional relation. The authors give another example, namely the intensional belief contexts: “A believes that `x is a man’ implies that ‘x is mortal” does not necessarily have the same truth-value as “A believes that `x is a featherless biped’ implies that ‘x is mortal”’. From this it is clear that the two propositional functions cannot be the same, because if they were, by means of the identity definition of Principia Mathematica, we would have to conclude that the belief-sentence has the same truth-value. In the system the belief-context is a mere second-order propositional function.
Whitehead and Russell 1928, 40.
See Whitehead and Russell 1928, 74–76; 190 *20.01.
A case in point is Whitehead’s conclusion in his foreword to Quine 1934a, the published reworked version of the doctoral thesis: “Dr. Quine does not touch upon the relationship of Logic to Metaphysics. He keeps strictly within the boundaries of his subject. But — if in conclusion I may venture beyond these limits — the reformation of Logic has an essential reference to Metaphysics. For Logic prescribes the shapes of metaphysical thought.”
Propositions not exhibiting this form were banned from the system.
Whitehead and Russell used both “function” and “propositional function”. Normally “function” was used for “propositional function” but it also could have a different meaning. Apart from propositional function descriptive functions are mentioned, and these are the contemporary mathematical function such as “sin x”, see Whitehead and Russell 1928, 15. In Quine 1932 the primitive idea is “function”, and it has the role of the propositional function in Principia Mathematica.. The descriptive functions are called “relative descriptive functions”, see Quine 1932, 56–57; 193ff.
Quine 1932, 10.
Quine 1932, 15. In the preface written in 1989, Quine writes that this primitive idea should have been eliminated by Wiener’s definition of pairing, see Quine 1932, ii.
It must not be assumed that variables have a dominant role in the system of Quine 1932. There are no genuine variables in the system.
See Quine 1960, 195.
See Quine 1932, 45–51.
Quine describes this as an important point of his doctoral thesis in Quine 199 lb, 266. Quine 1932, 4.
F-.L9 # 3’(p, 3’0j[11/, Sp’(W, 3’0] Pp.
This word was suggested by Whitehead, see Quine 1985a, 11.
Quine had little sympathy for “the metaphysical jungle of Aristotelian essentialism”. He defined it as “the doctrine that some of the attributes of a thing (quite independently of the language in which the thing is referred to, if at all) may be essential to the thing, and others accidental”, see Quine 1966a, 175–176, or also Quine 1953a, 22; 155–156; Quine 1962, 140; Quine 1966a, 51. In Quine 1960, 199, Quine mocks at essentialism: “Mathematicians may conceivably be said to be necessarily rational and not necessarily two-legged; and cyclist necessarily two-legged and not rational. But what of an individual who counts among his eccentricities both mathematics and cycling? Is this concrete individual necessarily rational and contingently two-legged or vice versa?”
Quine 1932, 138–140. Identity is defined as a derived notion. Two sequences are identical if the one belongs to the unit function of the other. The unit function of a sequence is the product function of the essence of this sequence. The product function tp of a function of functions yr is the function that is satisfied by all the sequences X, so that X satisfies all the functions that satisfy yr. This is a complicated way of saying that two sequences are identical if they satisfy the same functions.
Quine 1932, 146. In his comment on this theorem, Quine says that the theorem expresses the fact that intensional functions are excluded.
See Quine 1932, 5.
Quine 1932, iii.
Quine 1932, 97.
Seefn74.
Quine 1932, 95.
See Quine 1932, 97.
See Quine 1932, 141–142.
In standard logic the quantifier has become a primitive idea. The connection of quantifier and quantity has been revitalised by Mostowski in his theory of generalised quantifiers. In addition to the existential and the universal quantifier, also other quantifiers are added, e.g.. “M”, a quantifier expressing that most objects of the universe have a certain property. For a discussion see Sher 1991.
Quine 1932, 40.
That propositions, or classes, were reckoned to exist can be seen in the later preface, see Quine 1932, iii: “Quantification emerges as in Peano: ‘3x Fx’ is ‘3, where 3 is a function taking classes as arguments. Similarly universal quantification emerges as ‘U. This is a line I have found advantages in reviving…, but with ’x: Fx’ontologically deactivated now as an innocent general term or predicate, a `such that’ clause.”
In his “Quine”, Burton Dreben discusses the content and the historical context of Quine 1932 see Dreben 1990. Drehen mentioned the ontology of the system, and said that Quine was committed to an ontology of sequences. In his reply, Quine did not refute this, but diminished the importance of ontology in his earliest work, see Barrett and Gibson 1990, 96: “Struck by my `dread word ”ontology“’ in A System of Logistic and by my protracted belaboring on that subject in later years, Burt perhaps overestimates the philosophical intent of pages 12 and 28 of that early book. I was stipulating my range of variables and my usage of `sequence’.”
Apart from the initial part, the cleavage is also further regularly expressed, see Quine 1932, 37; 44; 69–70; 77–78; 269–272.
Quine 1932, 9.
Quine 1932, 6. See also Quine 1936b; Quine 1963, 253–254, Quine 1966b, 25. Quine thinks that Russell’s and Whitehead’s failure to note this is due to their blurring of the use-mention distinction, see Quine 1963, 254: “One senses from a reading of Russell how he was able to overcome this point: the trouble was his failure to focus upon the distinction between ”propositional functions“ as attributes, or relations-in-intension, and ”propositional functions“ as expressions, viz. predicates or open sentences. As expressions they differed visibly in order, if order is to be judged by indices on bound variables within the expression. Failing to distinguish sharply between formula and object, he did not think of the maneuver of letting a higher-order expression refer outright to a lower-order attribute or relation-inintension.”
Quine 1932, 6: “Whenever a function cp is expressed which would ordinarily be construed as non-predicative, we are to reinterpret the notation as denoting rather the predicative function yr which is extensionally coïncident with (p.”
Quine 1934a, 8: “The propositional function, around which revolved all the difficulties considered above, has been entirely excluded from the present system. I begin directly with classes and relations, which play the rôles both of the classes and relations and of the monadic and polyadic propositional functions of PM.… What has been described… as the ‘predication’ of a monadic propositional function thus gives way to the membership of a term in a class.”
See Quine 1932, iii; Quine 1934a, 7.
Quine 1934a, 36.
See chapter 4.2 and 4.3.
Quine 1934a, 32.
See Quine 1934a, 32. The formal expression of this is theorem 6.22.
See Quine 1934a, 1.3; 3.8.
Since functions gave way to classes, the propositions of the form `4 x’ were replaced by ’.Xe a’. The new term for predication was membership, see Quine 1934a, 6, 26–27.
See Quine 1934a, 27.
Quine 1934a, 28.
This passage draws on Quine 1934a, 33–34.
Quine 1934a, 33.
Quine 1934a, 34.
Reprinted in Quine 1966b, 83–99.
This will be worked out in chapter 4.2.
The axiom states that for an infinite set of disjunct non-empty sets there is a set containing exactly one element from each of the sets.
Quine 1966b, 83.
Quine 1937, 89.
See Quine 1937, 81.
For a further discussion on individuals see chapter 3.6.
Quine 1940, D10..
Quine 1940, *202..
The index refers to pages 11 16, 29, 33, and 73f that deal with truth-functionality; and to pages 120f that deal with extensionality in connection with classes: “If there is any difference between classes and properties, it is merely this: classes are the same when their members are the same, whereas it is not universally conceded that properties are the same when possessed by the same objects.… For mathematics certainly, and perhaps for discourse generally, there is no need of countenancing properties in any other sense.”
Quine 1995a, 40.
See Quine 1960, 1–2; Quine 1995a, 15–16.
Carnap 1928, 92.
Quine 1960, 4–5.
The two conceptions of a physical object, viz. the physical object as a body in daily experience or as a part of space-time already appear in an inchoate form in “Identity, ostension, and hypostasis”, based on lectures Quine has presented in 1949, see Quine 1950b, 65–68.
See also chapter 6.2.
The distinction is clearly made in Quine 1987b, 66–67.
Quine 1966a, 228; see also Quine 1966a, 224–225
Quine 1974, 54. See also Quine 1966a, 251; Quine 1974, 1; 85; Quine 1979a, 159; Quine 1981a, 9; Quine 1990b, 7; Quine 1993, 113; Quine 1995b, 254.
In most works, there is a difference in use of the two terms. The difference was less clear in Quine’s initial work. The first time `body’ was mentioned was in Quine 1939a, 701: “Bucephalus, then is a certain four-dimensional body stretching through part of the fourth century B.C. and having horse-shaped cross-sections.” In Quine 1940, 120 Quine mentioned physical objects, he spoke about a `physical body’ with as an example Continental United States (of arbitrary depth). In later texts there is a clear difference between the two. `Body’ may even be considered to belong to our ideology, it is a predicate that can denote a subset of the set of physical objects, see Quine 1981a, 125: “Momentary objects are declared to be stages of the same body by consideration of continuity of displacement, continuity of deformation, continuity of chemical change. These are not conditions on the notion of identity; they are conditions on the notion of body. Most of our common predicates, like `coin’, denote only bodies, and so derive their individuation from the individuation of the predicate `body’… Despite men’s stubborn body-mindedness, there are good reasons for the more liberal ontology of physical objects.”
See Quine 1974, 54: “When the time comes for the precision of physical science, the notion of body can give way to the more inclusive, more recondite, and more precise notion of physical object.”
Quine 1974, 55; see also Quine 198la, 13; Quine 1995a, 24.
Quine 1960, 126. For similar considerations see “What price bivalence?”, in Quine 1981a, 31–37.
Mereology is the study of the relation part-whole in the spatial sense. It studied the way in which various parts of space or space time could be combined into new parts. These parts provided the ontology for a total conceptual system. Goodman’s The Structure of Appearance is an elaboration of this project. The project itself was initiated by Ldniewski.
Quine 1981a, 124. Quine has always stuck to this characterisation, e.g. Quine 1976a, 497: “let us understand a physical object for a while, simply as the aggregate material content of any portion of space-time, however ragged and discontinuous”; Quine 1995a, 41: “Better simply to admit as a physical object the content of any portion of space-time…”, or also Quine 1960, 171; Quine 1966a, 242–243; 259; Quine 1970a, 30; Quine 1974, 89; Quine 1981a, 34; Quine 1985b, 167; Quine 1987b, 122; Quine 1998, 30.
We may of course wonder which of these objects is really Mount Rainier. Quine has a way to accommodate for vagueness by regarding extension families rather than straightforward extensions of terms, see Quine 1985b, 168: “The extension of the term `desk’ is conventionally thought of as the class of its denotata, thought of as physical objects. Realistically we may recognize rather an extension family, as I shall call it. It is a family of vaguely delimited classes, each class being comprised of nested physical objects any of which would pass indifferently for one and the same desk. When we bring formal logic to bear on discourse of desks, then, we adopt the fiction that the extension is some one arbitrary and unspecified selection class from that family of classes; it selects one physical object from each. Similarly and more obviously perhaps, for mountains.”
A special kind of physical objects are `epochs’. These are taken as a part of space-time “exhaustive spatially and perpendicular to the time axis” during some time, see Quine 1960, 172.
In Quine 1960, 249, Quine presents point masses as a `limit myth’: “When one asserts that mass points behave thus and so, he can be understood as saying roughly this; that particles of given mass behave the more nearly thus and so the smaller their volumes.” Point masses are presented as the objects at the limit of a process of shrinking massive bodies. Quine thinks mass points are on a par with differentials and should be regarded as symbolic ideal objects. Two provisos can be made. First, it is not sure whether the most elementary particles in physics, i.e.. quarks, electrons and the like, are point masses or not. Quine seems to suppose that matter is continuous without further justification. At the moment string theory are in vogue, objects between the discrete and the continuous. Second, mass points may be considered as points in space. The mass point is a point akin to the geometrical centre of an object. It is the `weighed’ centre of the object.
Barrett and Gibson 1990, 334. Quine does not speak about points of space-time but about their proxies, quadruples of real numbers.
See Quine 1976a, 498–499, quoted in footnote 33.
Dalla Chiara and Toraldo di Francia 1995 plead for a revalorization of intensional set theory on the basis of these identification problems. Their analysis however is not convincing, and uses the conclusion as a premisse: “Consider the physical predicate `electron’, for which physicists have a precise definition, which in turn defines a precise intension.” It is not at all obvious that a definition may define an intension. If the identification problems become too pressing, Quine is prepared to expel them from his ontology.
In the text “On multiplying entities”, Quine still wrote, see Quine 1966a, 260: “In describing physical objects as the material contents of space-time regions, I do not mean to concede any ontological standing to the regions themselves. It is just my way of showing how broad a scope I intend for my notion of physical object. We may take the material content and let the regions go.”
Quine 1976a, 499.
Quine 1976a, 500. The same idea is expressed in “Facts of the matter”, 164; Quine 198la, 17; Quine 1990a, 35.
Quine 1976a, 500.
This is in fact the way elementary particles are characterised in quantum mechanics. A particle is characterised by a multiple of quantum numbers such as its spin, strangeness, muon number,…
This procedure was introduced in Carnap 1937.
See Quine 1939a, 701; Quine 1940, 120–121.
Quine 1950b, 67.
Quine 1950b, 66. This river-kinship appears to be a primitive relation. On the next page, Quine introduces `river’ as a primitive predicate denoting certain parts of space-time: “Such ambiguity is commonly resolved by accompanying the pointing with such words as `this river’, thus appealing to a prior concept of a river as one distinctive type of time-consuming process, one distinctive form of summation of momentary objects.” Quine further speculates on conceptualisation. By repeated ostension, aided by our `tendency to favor the most natural grouping’, we may by induction grasp the idea of a river, see Quine 1950b, 67–68. See also chapters 6.2; 7.2.
Quine 1995a, 36–39
In the evolution to modem science, the concepts space, time, and physical object have never lost their interdependence. One could wrongly suppose that Quine thinks that the space-time frame in the course of the scientific evolution got separated from the idea of physical objects. This relates to a central controversy in the philosophical discussion about space, namely whether we need a relational or an absolute conception of space or space-time. In the absolute conception space-time exists independently of the objects, while in the relational conception space-time exists by grace of the ordering of physical objects. In a reply to Smart, who wrote on Quine’s conception of space-time, Quine confesses himself to relationalism, see Hahn and Schilpp 1986, 516: “Smart rightly represents me as having espoused a relational theory of space-time. But he goes on to say that in a later publication I seem to lean toward an absolute theory.… I meant only acceptance of their [the physicists] doctrine that space-time is curved rather than Euclidean, but I continued to regard this as a doctrine about the distances and relative motions of particles and other bodies.”, and also Quine 1969a, 149; Quine 1974, 133; Quine 1987a, 38.
Quine 1995a, 37.
A Minkowski space resembles a four-dimensional Euclidean space, but there is a different metric on the space. Distances between points in this space are differently calculated. We may have positive, negative, and zero distances.
Quine explicitly mentions Whitehead, see Quine 1960, 172 fn3: “Einstein’s discovery and Minkowski’s interpretation of it provided an essential impetus, certainly, to spatio-temporal thinking, which came afterward to dominate the philosophical constructions in Whitehead and others.”
Quine 1985a, 68–71: “I gazed down incredulously on the Principality of Monaco. My dream of small countries was beginning to be fulfilled.… I delighted in the miniature state Vatican City… With my penchant for small countries I was bent on visiting San Marino…I was dropping of at Verona to go north and visit another little country, Liechtenstein… It [Luxemburg] was my fifth miniature country…” In Quine and Ullian 1970, 56–57 Quine even mentioned mistakes over the size of Monaco in several atlases.
Quine 1985a, 129–130.
See e.g. Quine 1981a, 199–202 for a review of The Times Atlas..
Quine 1986, 3; emphasis added.
Quine 1948, 4.
There are few passages where Quine explicitly mentions a gap between the two notions, see Quine 1979a, 160: `Bodies are assumed, yes and they are the things, first and foremost. Beyond them is a succession of dwindling analogies.… It is only in our somewhat regimented and sophisticated language of science that has evolved in such a way as really to raise ontological questions. It is an object-oriented idiom. Any idiom purports to tell the truth, but this idiom purports, more specifically, to tell about object“; see also Quine 1981a, 9–10; Quine 1984a, 25. Especially round 1980 there seems to be a gap between the two notions. In later work the two notions are again coupled, see chapter 7.2.
For comments on Quine’s notions of quality space and kind, see also van Brakel 1999; Shain 1993; Hacking 1990; Broughton 1981; Wilder 1972.
Another way to deal with colour and sound is to regard them as mere physical objects. Colour may be seen as a bulk term. A certain colour, say red, may be considered to be the part of space-time encompassing all the red objects. Red may be considered to be on a par with a river. The only difference here is that a river is a connected part of space-time, while a colour is a scattered part of space-time. It is clear from the previous section that this is no real objection. Analogously we may identify a sound with a physical object, namely the sound wave that causes the auditory perception. The metrical space for identifying the chromatic and auditory qualities is the space-time frame. See Quine 1950b, 69; Quine 1960, 91; Quine 1969a, 14; 31; Quine 1974, 87.
The first elements can in fact be found in “The scope and language of science” from 1954, see Quine 1966a, 231.
Carnap 1928, 130. Quine refers to this passage, see Quine 1995a, 11–12.
Quine once uses purely topological notions. In Quine 1974 he connects the similarity relation with the topological notion of neighbourhood, see Quine 1974, 17.
Quine 1966a, 232.
That the distance concept is quite central in Quine’s thought is seen from the fact that he even thinks of specifying a distance concept for measuring distances between languages or theories. One may measure a “conceptual distance between languages”. This distance is defined in Quine 1981a, 41–42: “Given a pair of sentences from the two languages, sentences that are acceptable translations of each other, select a shortest equivalent of each of the sentences within its language. Compare these two shortest equivalents in respect of length, and compute the ratio. When this has been done for every pair of sentences that are acceptable translations of each other, strike the average of all those ratios. This measures the conceptual distance between the two languages.” Quine sees problems with taking averages over an infinite set of sentences, and with the indeterminacy of translation In Quine 1960, 23, Quine criticises Peirce’s notion of truth by questioning the notion of limit that depends on “nearer than”, which is not defined for theories.
The importance of the notion of norm in Quine’s philosophy should not be underestimated. It was already clear in chapter 2.4 that phonemes are defined by means of norms. Quine sometimes expresses his lingering hope that also for stimulations a set of norms can be found that are the building blocks of all experience, see Quine 1969a, 90: “Now outside the realm of language also there is probably only a rather limited alphabet of perceptual norms altogether, toward which we tend unconsciously to rectify all perceptions. These, if experimentally identified, could be taken as epistemological building blocks, the working elements of experience. They might prove in part to be culturally variable, as phonemes are, and in part universal.”, or see also Quine 1987a, 16.
Quine uses such a norm in Quine 1960, 85: “Such a norm will not be a mere point in quality space; it will sprawl freely, rather, in the dimensions that do not matter to redness.”
In the case of colour there is one exception, see Quine 1975b, 70–71: “Natural selection may be expected to have encouraged similarity standards conductive to rough and ready anticipations of experience in a state of nature. Such standards are not necessarily conductive to deep science. Colour is a case in point. Colour dominates our scene; similarity in colour is similarity at its most conspicuous. Yet, as J.J.C. Smart points out, colour plays little role in natural science. Things can be alike in colour even though one of them is reflecting green light of uniform wave length while the other is reflecting mixed waves of yellow and blue. Properties that are most germane to sophisticated science are camouflaged by colour more than revealed by it. Over-sensitivity to colour may have been all to the good when we were bent on quickly distinguishing predator from prey or good plants from bad. But true science cuts through all this and sorts things out differently, leaving colour largely irrelevant.”
One cannot really be sure that one reconstructs the prelinguistic quality this way because the experiment may affect the prelinguistic quality space, see Quine 1960, 83–84.
Quine 1960, 83.
The two quality spaces are disconnected, see Quine 1960, 84: “Connexity is bound to fail: no chain of subliminal differences will reach from sounds to colors.”
See Quine 1960, 84. Quine uses the forms `ball’ and `kerchief.
Quine 1960, 85: “Like the norm of red, that of `red’ extends freely in some dimensions: thus pitch and volume of utterance are indifferent to whether an utterance is an utterance of `red’.
Quine 1960, 84.
Quine 1969a, 114–138. The text first appeared in a Festschrift for Hempel in 1967. In Quine 1960, 80–90, there is a shorter discussion of quality spaces.
Kinds can be ontologically acceptable objects; namely sets, see Quine 1969a, 118: “There is no call to reckon kinds as intensional. Kinds can be seen as sets, determined by their members. It is just that not all sets are kinds.” In the earlier “Speaking of objects” kinds were either sets or attributes, see Quine 1969a, 21: “Yet not kinds in the sense of classes,… Kinds rather in the sense of attributes.”
Quine 1969a, 117. In Quine 1960, 53, Quine opposes two notions of identity, viz..
ualitative identity or resemblance versus numerical identity.
Camap 1928, 107–108. 175 Carnap 1928, 127–128; 178.
See Carnap 1928, 73: “We therefore speak of an extensional method of construction. It is based upon the thesis of extensionality; in every statement about a concept, this concept may be taken extensionally (i.e., it may be represented by its extension [class or relation extension]). More precisely: in every statement about a propositional function, the latter may be replaced by its extension symbol.” Quine of course sympathised with this extensionalism. In 1934 Quine commented on the typescript of The Logical Syntax of Language, a work equally in the same extensionalist vein. Carnap gradually lost his faith in extensionality and was in Meaning and Necessity committed to intensional concepts. Quine has never followed Carnap in this direction. Their growing divergence can be read in Creath 1990, and eventually resulted in the notorious Carnap-Quine debate at the end of the forties.
Quine has used a related more naturalistic notion “trace”, see Quine 1974, 24–25.
Carnap 1928, 102.
Carnap 1928, 177. Identity is counted as a basic logical concept.
Carnap 1928, 179.
Carnac 1928, 179–180.
For the explanation see Carnap 1928, 128–134; for the technical detail see ibid.., 180–182.
Carnap 1928, 112–119.
borrow this term from Quine 1995a, 16.
Goodman 1954, 74.
The year 2440 is one of the first `years’ that have appeared in the utopian literature. It was the title of the book L’An 2440, which was written in 1770 by the French author Mercier.
For a precise characterisation of the term `projectible’ see Goodman 1954, 82.
Quine 1969a, 116; Quine and Ullian 1970, 87.
Quine 1969a, 117. The quoted passage is followed by evidence of etymological relations between the two notions.
It is not necessary to posit the similarity relation as an object, it may be considered as a predicate only, see Quine 1969d, 91, and Quine 1981a, 78: “retention of the two-place predicate `is similar to’ is no evidence of assuming a corresponding abstract entity, the similarity relation, as long as that relation is not invoked as a value of a bound variable.”
Quine 1969a, 118.
Quine 1969a, 118–119; see also Quine 1966a, 231; Quine 1974, 17–18; Quine 1975b, 6970 Quine 1981a, 56; Quine 1987b, 69.
See Quine 1969a, 119: “The set of all red things and the set of all colored things can now both be counted as resembling one another more than some things do, even though less, on the whole, than red ones do.”
Quine 1969a, 119.
Quine 1969a, 120. Quine also refers Goodman’s more elaborate `difficulty of imperfect community“.
Quine finds reason to introduce a polyadic similarity function to accommodate for these respects, see Quine 1974, 18–19.
Quine 1969a, 121: “I shall suggest that it is a mark of maturity of a branch of science that the notion of similarity or kind finally dissolves…”; 123: “At any rate we have noticed earlier how alien the notion is to mathematics and logic.”
See e.g.. Quine 1966a, 57; Quine 1969a, 122–123; Quine 1974, 19; Quine 1981a, 56; Quine 1987b, 71.
See Quine 1987b, 73; Quine 1995a, 21; Quine 1995b; Quine 1996, 159–162. The term goes back to Leibniz. The harmony is restricted to mankind. Communication with extraterrestrials might fail because there is no correspondence between the standards of similarity, see Karlsson 1997, 221.
Quine 1969a, 125: “each man’s spacing of qualities is enough like his neighbor’s”; Quine 1974, 23: “we can count on considerable social uniformity in perceptual similarity standards”
Quine 1969a, 133.
Quine 1969a, 136.
Quine 1969a, 136: “But there would remain still need also to rationalize the similarity notion more locally and superficially, so as to capture only such similarity as is relevant to some special science.”
See also Over 1976; Carr 1976; Moline 1972.
See e.g.. Quine 1969a, 144.
See Quine 1960, 222–226; Quine 1966a, 71–74; Quine 1969a, 130–138; Quine 1974, 8–15; Quine 1975c, 92–93.
See Quine 1974, 9.
See Quine 1974, 14.
Quine 1969a, 137; see also Quine 1974, 10. Similarly Quine 1960, 223 for “would prompt assent”, or in Quine 1987b, 63–64.
Quine 1960, 224.
See Quine 1969a, 130.
Quine 1974, 14: “Nor am I bent on finding a respectable place for the general dispositional idiom in a regimented theoretical language.”
See Quine 1974, 11–12.
Quine mentions this definition of meaning in Quine 1990a, 52: “we could define the meaning of an expression as the class of all expressions like it in meaning.” See also Quine 1960, 201; Quine 198la, 46, Quine 1979b, 140.
Quine 1948, 11. McX is a strawman adversary in Quine’s argument.
See Quine 1940, 147: “the meaningfulness of an expression — the eligibility of an expression to occur in statement at all, true or false — is a matter over which we can profitably maintain control”
Quine 1953b, 48–49.
The two notions are equated in Quine 1953b, 51.
The chapter on grammar deals with the grammar for languages in general. The logical languages are in detail discussed, see Quine 1970a, 22–26. See also Quine 1970c, 389; Quine 1987a, 202–203; Quine 1987c, 7.
Quine 1970a, 16.
Quine 1953b, 49.
See Quine 1979b, 132–133.
See Quine 195 lb, 21; Quine 1970a, 8–9; Quine 1987a, 22. Another example is “the author of Waverly” and “the author of Ivanhoe”, see Quine 1981 a, 43. An older example is of course Frege’s distinction between the Morning Star and the Evening Star, see Quine 1948, 9.
See Quine 1990a, 59: “Lexicography has no need of synonymy, we saw, and it has no need of a sharp distinction between understanding and misunderstanding either. The lexicographer’s job is to improve his reader’s understanding of expressions, but he can get on with that without drawing a boundary. He does what he can, within a limited compass, to adjust the reader’s verbal behavior to that of the community as a whole, or of some preferred quarter of it. The adjustment is a matter of degree, and a vague one: a matter of fluency and effectiveness of dialogue.”
Quine 1948, 12. Also in Quine 195 lb and in Quine 1953b, 48 meanings are abandoned as “obscure intermediary entities”.
Quine 1979a, 166–167.
Already in 1951 the thesis was foreshadowed in Quine 1953b, 60–64, and in a draft version of a part of Quine 1960 that appeared as Quine 1962. The thesis was strongly endorsed in Quine 1960, 72: “There can be no doubt that rival systems of analytical hypotheses can fit the totality of speech behavior to perfection, and can fit the totality of dispositions to behavior as well, and still specify mutually incompatible translations of countless sentences insusceptible of independent control.” Quine since then never really changed his argumentation, but over the years the stress on the thesis has weakened. In Quine 1987c, 9, Quine says that it is very unlikely that rivalling manuals will ever be constructed, but the thesis is not to be doubted, see also Quine 1990b, 5. Remarkably, in Quine 1995a, 82, indeterminacy of translation is rather presented as a conjecture than as a thesis. In his response to Hintikka (1997, 568), Quine says of the indeterminacy of translation: “I view it less as a dogma than as an argued thesis”. In contrast, the inscrutability of reference is “a simple theorem”.
In Quine 1991b, 265, the two reasons for Quine’s distrust of mentalist semantics are clearly expressed.
See Quine 1969a, 19; 27; 80; Quine 1970a, 8; Quine 1970b, 5; Quine 1987b, 75; Quine 1987a, 130.
See Quine 1969a, 27.
The dissolution of the spatial structure in which entities might be located is somehow reflected in the fact that intensions are called “creatures of darkness”, see Quine 1966a, 188, or Quine 1974, 125, where Quine speaks of “the dark side of attributes”, or Quine 1970a, 10, where Quine calls propositions “shadows of sentences”.
For Quine, propositions and meanings are on a par, see Quine 1969a: “Now a proposition is the meaning of a sentence.… More precisely still, it is the cognitive meaning of an eternal sentence…”
Quine 1966a, 265.
This is in clear disagreement with Quine’s later views on the matter. Later, Quine will consider `p’ or `q’ in logic no longer as variables for sentences, but as schematic letters, see chapter 4.1.
Quine 1966a, 269.
Quine 1995a, 108.
Quine 1960, 244. See also Quine 1969a, 140.
Quine 1969a, 19.
Meaning and essence are related, the meaning of a term is the essence of some kind of objects, see Quine 1966a, 51: “Meaning is essence divorced from the thing and wedded to the word.”
See Quine 1960, 194: “I find no good reason not to regard every proposition as nameable by applying brackets to one or another eternal sentence”, similarly see Quine 1970a, 2; Quine 1990a, 78.
See Quine 1960, 208–209; Quine 1987a, 22–23.
Quine 1960, 31–35.
See Quine 1960, 8 for the following metaphor relating to the intersubjectivity in communication: “Different persons growing up in the same language are like different bushes trimmed and trained to take the shape of identical elephants. The anatomical details of twigs and branches will fulfill the elephantine form differently from bush to bush, but the overall outward results are the same.”
In another context Quine speaks of the external momentary stimulation as “the set of [a person’s] triggered receptors”. The physical objects here are not the light rays but the receptors of a person. See Quine 1981a, 50.
See Quine 1960, 46.
Quine also introduced a less restricted relation `cognitive equivalence’, which is roughly based on the notion of stimulus synonymy. Not only sentences but also terms may be cognitively equivalent, if they can be freely substituted in occasion sentences. Also standing sentences can be cognitively equivalent if they can be transformed into each other by means of cognitive synonyms. See “Use and its place in meaning”, published in Quine 1981a, 43–54; Quine 1979b; Quine 1990a, 53–54.
Quine had the greatest difficulty in finding an intersubjective standard of identity of stimulus meaning, and concluded that it could only be defined for each speaker at a time. See ~. Quine 1969a, 157–160.
Other repudiated objects are `units of measure’, see Quine 1960, 244; `facts’, o.c.., 247; infinitesimals’ and ‘ideals objects’, o.c.., 248–250; Quine 1966a, 22–23.
The reductions discussed in this chapter are ontological reductions. The ontology or universe of objects of a theory is reduced. This means that one countenances a subclass of the first universe as the reduced universe. The reductions have repercussions for the ideology in the sense that the predicates are construed to range over the objects of the restricted universe. The list of predicates is however not restricted. This notion of reduction can be clearly distinguished from more common reductions, such as theory reductions from chemistry to physics, or from psychology to physiology. Ontological reductions are reductions within a theory.
In chapter 5.4 a further aspect of Quine’s notion of reduction will be discussed. See also !Croon 1992; Grandy 1979; Chihara 1973; Tharp 1971; Jubien 1969.
Quine 1934a, 16; Quine 1940, 201–202; Quine 1950a, 297–299; Quine 1960, 257–262; Quine 1963, 58–59; Quine 1966a, 25; Quine 1966b, 47; 110–113; Quine 1995a, 61.
Quine 1960, 262–263; Quine 1966a, 26–27; 212–214; Quine 1963, 74–138; Quine 1987a, 137–139.
See Quine 1963, 81: “Any objects will serve as numbers so long as the arithmetical oerations are defined for them and the laws of arithmetic are preserved.”
The difference between the two methods engendered a deep crisis in the philosophy of mathematics. The two methods are not equivalent. For example, in von Neumann’s progression 2E 4, while in Zermelo’s 20 4. Therefore the question which of the reductions is the right one is a legitimate question. There is no straightforward intuitively plausible answer to the question what numbers really are. This problem was put forward by Paul Benacerraf in Benacerraf 1983a.
Some care must be taken not to violate the vicious circle principle. We might consider for example a class of two elements (1, 2), but this is an element of 2. Frege’s formulation was inconsistent, but the flaw can easily be avoided. After mending the error, Quine still calls it Frege’s version, see Quine 1963, 82 fnl.
For the formal expression see Quine 1940, D39 or Quine 1963, 82.
See Quine 1963, 124–130.
Quine’s version is a little more intricate. He wants the real number x and the complex number x+Oi to be the same set, see Quine 1963, 237–238.
Hahn and Schilpp 1986, 318.
See Quine 1987a, 133.
Davidson 1980, 179; see also Quine 1966a, 260; Quine 1970a, 31; Quine 1981a, 11–12.
Quine 1985b, 167.
The belief itself may be explained in terms of dispositions, see Quine 1987a, 20: “A belief, in the best and clearest case, is a bundle of dispositions. It may include a disposition to lip service, a disposition to accept a wager, and various dispositions to take precautions, or to book passage, or to tidy up the front room, or the like, depending on what particular belief it may be.”
More precisely, Quine has a “double standard”. The intensional idioms have no place in austere science, but on the other hand they are indispensable and welcome in daily use, see Quine 1960, 221.
See Quine 1969a, 145.
Quine 1960, 212–213. See also Quine and Ullian 1970, 11; Quine 1970a, 32.
In Quine 1987b, 101–120, perceptions are treated as propositional attitudes. “x perceives a” is a monadic predicate true of a person or an animal in case the person would assent to the observation sentence a. “x perceives y” is a dyadic predicate, and x takes a person or animal as values, while y takes observations sentences as its values. These predicates are perfectly acceptable in a physicalist universe. See also Quine 1990a, 61–67.
This does not bother Quine too much, semantics is always an unholy domain, see Quine 1960, 213–214.
Quine 1960, 216.
Quine 1979a, 164.
In Quine’s autobiography the qualms remain, see Quine 1986, 31: “Twelve years later still, in Quine 1976a, I recognized that developments in particle physics itself lent some support to a wholly abstract ontology.”, see also Hahn and Schilpp 1986, 402.
Quine 1976a, 503; Quine 1979a, 164; Quine 1987b, 136.
Quine 1995a, 71.
The reduction of physical objects via quadruples of reals to sets has an analogue. In Quine 1969a, 42–43; Quine 1970a, 56, Quine describes a similar reduction of the expressions in protosyntax via Gödel-numbers to sets.
Quine 1976a, 503.
See above. This idea is also hinted at in Quine 1970a, 4.
See e.g. Quine 1997, 567.
Quine 1948, 4; see also Quine 1960, 245.
Quine 1969a, 147–152; see also Quine 1970a, 4.
See Quine 1969a, 152. The problem of transworld-identity remains open as ever.
Quine 1969a, 152.
The notion `criterion of identity’ is not only used by Quine, but was already introduced by Carnap, see Carnap 1947, 125, and also Church was interested in a criterion of identity for proposition, see Anderson 1998, 157.
Quine’s system in “Set-theoretic foundations for logic” had Urelements, see Quine 1936g, 85: “The null class presupposes a distinction between classes and individuals [This should be Urelements given Quine’ s later characterisation of individuals, L.D.] (non-classes) which is inexpressible in terms of membership, since the null class is like an individual in lacking members. Repudiation of the null class enables us to explain the variables of F as representing objects generally, classes and otherwise, and then define classes simply as objects y such that (3x)(xey).” Quine is also astonished by Jensen’s proof of the consistency of NF, with the weakened extensionality axiom, relative to PM, see Davidson and Hintikka 1969, 278–291; 349–352, and Hahn and Schilpp 1986, 593.
See section 4.2.
The empty set is a set containing no elements. Also Urelements have no elements, and therefore the difference between the two can become blurred. In the quoted passage on pages 133–134 Quine mentions that the difference can be assumed as primitive. The intuitions between the two concepts are different. The empty set can be regarded as the set (x: Urelements are usually not reckoned as sets, but as objects extraneous to set theory that form a basic stratum in a set-theoretic hierarchy.
Also Maddy 1990b provides an elaboration of such a `radically impure hierarchy’.
In ZFC, this pure set theory is most common; one needs a basis to build the cumulative hierarchy and the empty set is the least problematic basis. In ML, for technical reasons related to the axiomatic construction, one need not reflect on this foundation axiom, and therefore there is little reason to suppose that all sets are pure. Quine, in accordance with his nominalistic sympathy of that time, wrote in Quine 1940, 122: “It would even be possible, compatibly with the projected formal developments and indeed with the whole of mathematics, to repudiate concrete objects altogether — to recognize just classes, each of which has classes in term as members or else no members whatever.… This exclusively abstract ontology has little naturalness to recommend it…”
See Quine 1963, 141.
Recently modifications of ZFC have been proposed in which the axiom of foundation is replaced by so-called anti-foundational axioms. The violation of the axiom of foundation has been built in. See Aczel 1988 and Barwise and Moss 1996.
See Kanamori 1996, 28.
For defences and a critical assessment of this idea of a cumulative hierarchy, see Boolos 1983 and Boolos 1989, Wang 1983, Parsons, 1983.
The translation of the idea of a cumulative hierarchy is worked in Boolos 1983. Schoenfield 1977 is less explicit but essentially the same.
Quine 1990a, 40.
Quine 1963, x, see also o.c..,5; Quine 1966a, 111; Quine 1966b, 27, quoted in next fn.
See Quine 1963, 286; Quine 1970d, 248. At the beginning of the sixties there was an explosion of mathematical papers on set theory after the results of Cohen, who with his method of forcing proved that the continuum hypothesis is independent of the ZFC axioms. The mathematical community had taken up ZFC as a basis for further work. This community was not really interested in alternatives for ZFC, but wanted to extend ZFC to the higher infinite. Quine’s Set Theory and its Logic, which was a comparison of various axiomatisations, was reviewed unnecessarily sharply by Martin. Quine replied to this review and stressed the divergence in philosophical views about the cumulative hierarchy and the axiom of foundation. With hindsight, there is some irony in a passage written in Quine 1941, 27: “But the striking circumstance is that none of these proposals to block Russell’ s paradox], type theory included, has an intuitive foundation. None has the backing of common sense. Common sense is bankrupt, for it wound up in contradiction. Deprived of its tradition, the logician has to resort to mythmaking. That myth will be best that engenders a form of logic most convenient for mathematics and the sciences; and perhaps it will become the common sense of another generation.” Quine has never really digested the myth of the generation of the sixties, viz.. the cumulative hierarchy of sets.
Hahn and Schilpp 1986, 590–591.
See also Quine 1981a, 103.
In “On the individuation of attributes, Quine explicitly writes that there is ”a good prior standard of identity of physical objects“, but he is not really explicit about this standard, see Quine 1981a, 102.
Barrett and Gibson 1990, 319. One might suppose that the use of the term transcendental is ironically because it occurred in a reply to Strawson. This may be the case, but anyway the space-time frame is always supposed to be available a priori, see Quine 1985b, 168: “Space-time is a matrix that stands ready to cast objects forth as needed in the course of introducing logical order into one or another branch of science of discourse.” (my cursivation).
Quine 1960, 253.
Neurath 1983, 54: “What matters is that all statements contain references to the spatiotemporal order, the order we know from physics. Therefore, this view is to be called `physicalism.’”
Carnap 1922, 63–64, my translation; German text: “Die Grundsätze über den formalen Raum sind offenbar apriori… Die Gründsätze des Anschauungsraumes sind gleichfalls apriori. In diesen Grundsätzen des Anschauungsraumes haben wir die von Kant behaupteten synthetischen Sätze apriori vor uns. Dasselbe gilt aber nicht allgemein für die aus ihnen abgeleiteten Lehrsätze, sondern nur, soweit sie den topologischen Raum betreffen.”
Carnap 1937, 150.
Carnap 1922 mentions multidimensional spaces. Quine does very seldom mention the bare possibility. They are mentioned in Quine 1987a, 196.
A tensor is a mathematical concept, and is the analogue of a scalar field and a vector field in a higher dimension. In a four-dimensional space the value of a scalar field in a certain point is a real number, and the value of a vector field is a quadruple of real numbers. The value of a tensor field in a certain point consists of 16 real numbers.
Quine acknowledges that space-time may be curved, see Quine 1987a, 120.
The existence of black holes has been doubted for a long time, but is at the moment accepted in the field. There is enough empirical evidence to affirm their existence. In the binary star Cygnus Xl a black hole has been discovered.
It is also not very clear how to define real singularities of space-time In their environment there is an infinite gravitational tidal potential that tears all matter apart. A precise definition is not evident.
Misner et al. 1973, 820.
See chapter 3.3.
Quine 1981a, 16; Quine 1984a; 22; Quine 1983b, 420; Quine 1995a, 40.
Quine 1984a, 22–23; Quine 1987b, 130.
In contemporary philosophy of mathematics there is a tendency to widen the gap between mathematics and empirical sciences. Philosophy of mathematics studies mathematics as a separate realm, and not as a part of a total theory of the world. In Maddy 1997 the gap is very wide, and it is stated that philosophy of mathematics ought to explain the research of mathematicians, and that no extraneous considerations should play a role, let alone invoke revisions in the mathematical practice. In Resnik 1997 the gap is smoothed out, but the author is convinced that a naturalised epistemology of mathematics is necessary. The epistemology of mathematics, i.e.. the growth of the mathematical structures, requires a specific investigation apart from more general epistemic reflections.
Quine 1974, 123.
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Decock, L. (2002). Extensionalism. In: Trading Ontology for Ideology. Synthese Library, vol 313. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3575-9_3
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