Nominalism and Realism

See Quine and Ullian 1970, 65–82; Quine 1990a, 95. In Quine and Ullian 1190 Quine mentions the five first mentioned items, and calls them `Virtues’. The last two items together with `simplicity’ are mentioned in Quine 1990a.Google Scholar

See Quine 1974, 138.Google Scholar

A similar definition can be found in Quine 1939b.Google Scholar

Quine 1939a, 708.Google Scholar

Quine 1939a, 709.Google Scholar

Quine and Goodman 1972, 173. In this passage the paradoxes of set theory are presented as a reason for taking a nominalistic stance. The importance of these paradoxes in Quine’s thought on ontology cannot be underestimated, and I will elaborate this theme into greater detail in chapter 4.2.Google Scholar

Nominalistic programmes still exist in the philosophy of mathematics. One of today’s best known nominalists is Hartry Field. Field started his philosophical career by writing some articles on Quine’ s philosophy. Later he worked out a nominalistic position in philosophy of mathematics. His Science Without Numbers (1980) is an attempt to rewrite Newton’s theory of gravity in nominalistic terms. A recent survey of some nominalistic strategies in philosophy of mathematics is Burgess and Rosen 1997.Google Scholar

It is illuminating to read the comment in footnote 4: “According to quantum mechanics, each physical object consists of a finite number of spatio-temporally scattered quanta of action. For there to be infinitely many physical objects, then the world would have to have infinite extent along at least one of its spatio-temporal dimensions. Whether it has is a question upon which the current speculation of physicists seems to be divided.” Of course, this comment is not valid, because many bosons, e.g. light-particles can occupy the same position. The question is still not resolved for other reasons.Google Scholar

Quine and Goodman 1972, 177.Google Scholar

Ibid., 178. I have transformed the notation to present day logical notation.Google Scholar

Quine uses this term in his “Reply to Nelson Goodman” in Hahn and Schilpp 1986, 163.Google Scholar

Ibid., 182.Google Scholar

Ibid., 183, fn 12.Google Scholar

See Quine 1940 and Quine 1963.Google Scholar

Quine and Goodman 1972, 184.Google Scholar

Ibid., 194.Google Scholar

See chapter 3.3.Google Scholar

Goldbach’s Conjecture states that every even number is the sum of two prime numbers. The conjecture has never been proven, and is since Wiles’ proof of Fermat’s Last Theorem one of the favourite examples of an unrefuted and unproved conjecture.Google Scholar

Nominalistic strategies using modality are feasible. For an overview of several such strategies, with references to the original literature, see Burgess and Rosen 1997.Google Scholar

I am not convinced that this is an exclusive alternation, because it is my intuition — since infinity and modality come close to semantic primitives, I have only my intuition to rely on–that the notion of infinity has a tinge of modality. The most natural idea of infinity lies in the idea of a continued discrete process, say counting. This is in a sense a modal notion, because it involves the possibility of going on at any stage in the process. I am at a loss about the way to flesh out the difference between actual infinity and the guaranteed possibility of continuing. From a naturalistic point of view, both notions are equally dubious. This point is also made by Charles Parsons, see Hahn and Schilpp 1986, 374–375.Google Scholar

Quine 1940, §58.Google Scholar

See Quine 1940, 283.Google Scholar

Quine 1940, §54.Google Scholar

A primitive concatenation predicate C is also used to define expressions in Quine 1953c, 117; Quine 1960, 185; 190.Google Scholar

See Quine 1940, 288.Google Scholar

See Quine 1940, 296.Google Scholar

Quine 1986, 26. The problem that classes of expressions needed in proofs may be empty as an empirical matter of fact is also mentioned in Quine 1966a, 41–42; Quine 1987a, 217–218.Google Scholar

Quine and Goodman 1972, 173.Google Scholar

Some occurrences of the word are: Quine 1939a, 704; 708; Quine and Goodman 1972, 173; Quine 1960, 103; 126; 132ß 138; 175; Quine 1966a, 47f, 54; 197f, Quine 1970a, 27; Quine 1981a, 68; Quine 1998, 30.Google Scholar

In Quine’s early texts, predicates were considered to be syncategorematic. In Quine 1970a, 27, this view is entirely abandoned, and this is a reason to put aside the old terms. In later texts Quine defined the lexicon as the collection of predicates, and these were the terms that were meaningful. Therefore they could no longer be syncategorematic. The other terms occurring in the sentence were considered meaningless. They constitute the structure of the sentence, and it should not come as a surprise that Quine mentions the particles as an example of syncategorematic terms in Quine 1970a. The particles are logical connectives and parentheses.Google Scholar

In “Five milestones of empiricism” the second step in the evolution of empiricism is the shift from terms to sentences. The first step was John Home Tooke’s replacement of Locke’s ideas by terms. Contextual definition is a clarification of the old idea of syncategorematic words; see Quine 1981a, 68: “The medievals had the notion of syncategorematic words, but it was a contemporary of John Home Tooke who developed it into an explicit theory of contextual definition; namely, Jeremy Bentham. He applied contextual definition not just to grammatical particles and the like, but even to some genuine terms, categorematic ones. If he found some term convenient but ontologically embarrassing, contextual definition enabled him in some cases to continue to enjoy the services of the term while disclaiming its denotation. He could declare the term syncategorematic, despite grammatical appearances, and then could justify his continued use of it if he could show systematically how to paraphrase as wholes all sentences in which he chose to embed it.”Google Scholar

Ibid., fn 5.Google Scholar

Quine and Goodman 1972, 175.Google Scholar

In the text Quine and Goodman use the term “primitive predicate”, see o.c.,178 fn 8, quoted in the main text. In Quine 1939b, 202 it is already indicated that a nominalistic reduction might involve semantic primitives: “The effective consummation of nominalism… would consist in starting with an immanent (non-transcendent) universe and then extending quantification to classes by some indirect sort of contextual definition. The transcendent side of our universe then reduces to fictions, under the control of the definitions. Such a construction would presumably involve certain semantic primitives as auxiliaries to the logical primitives.” At some other, but rather few, occasions, Quine uses the term “primitive”. He nowhere gives an account of what `primitive’ means.Google Scholar

Ibid., 176.Google Scholar

Ibid., 178 fn 8.Google Scholar

A short discussion about nominalism between Goodman and Quine is published in Hahn and Schilpp 1986, namely Goodman’s text “Nominalisms”, o.c.,159–161, and Quine’s reply, o.c.,162–163. It is clear from these texts that the authors were never exactly on the same line about nominalism, but the status of predicates or ideology is not discussed.Google Scholar

See chapter 4.Google Scholar

Quine 1953a, 173–174.Google Scholar

See Quine 1976a, 500: “The admission of numbers and other abstract mathematical objects is an eventuality that has to be faced, melancholy though it be.”Google Scholar

See Quine 1986, 26: “Nominalism… was the statement of our problem. It would be my actual position if I could make a go of it. But when I quantify irreducibly over classes, as I usually do, I am not playing the nominalist. Quite the contrary”, or Quine 1960, 243 fn 5; Quine 1966a, 75.Google Scholar

See chapter 4.1.Google Scholar

Quine 1986, 15.Google Scholar

In Quine 1936a, 1936c, and 1936e, Quine elaborated on Schönfinkel’s ideas.Google Scholar

Quine 1960, 269.Google Scholar

See for example Quine 1948, 17: “Here we have two competing conceptual schemes, a phenomenalistic one and a physicalistic one. Which should prevail? Each has its advantages; each has its simplicity in its own way. Each, I suggest, deserves to be developed.”Google Scholar

Despite the apparence, Quine’s position was not really inspired by the American pragmatists. His use of the word “pragmatic” stemmed from Carnap, see for example Quine 1991b, 272: “…. I am not clear on what it takes to qualify as a pragmatist. I was merely taking the word from Carnap and handing it back”.Google Scholar

Quine 195 lb, 46.Google Scholar

Quine 1948, 19.Google Scholar

This is still Quine’s position in his most recent book. As for physicalism, he describes two directions in which it may be pursued, the one is the way it is done in theoretical physics, the other is naturalism. Quine chooses the second with the decisive words: “Such is my option”, see Quine 1995a, 16. Sets are members of the physicalist ontology, see Quine 1995a, 40: “A physicalist ontology… consists just of the physical objects, plus all the classes of them, plus all classes of the foregoing, plus all classes of any this whole accumulation, and so on up”, see also Quine 1992, 9.Google Scholar

In Quine 1974 Quine introduced a similar but slightly different maxim, viz,relative empiricism: “Don’t venture farther from sensory experience than you need to”, see Quine 1974, 138. Nominalism is presented as “a manifestation of relative empiricism”.Google Scholar

Quine sometimes uses the term “universal”, and uses it in the sense of “attribute” or “class”, see Quine 1981a, 182–184; Quine 1987a, 225–229.Google Scholar

Before introducing the various positions in more detail Quine refers to the old discussion, see Quine 1948, 13: “Classical mathematics, as the example of primes larger than a million clearly illustrates, is up to its neck in commitments to an ontology of abstract entities. Thus it is that the great mediaeval controversy over universals has flared up anew in the modern philosophy of mathematics. The issue is clearer now than of old,… But this standard of ontological presupposition did not emerge clearly in the philosophical tradition, the modern philosophical mathematicians have not on the whole recognized that they were debating the same old problem of universals in a newly clarified form.”Google Scholar

Quine 1948, 15.Google Scholar

See Hilbert 1925, 196–197.Google Scholar

The term was introduced in the lecture “On platonism in mathematics”, held in Geneva in 1934, see Bernays 1983.Google Scholar

In “Whitehead and the rise of modern logic” from 1940 Quine uses the phrase “a Platonic ontology of universals”, see Quine 1966b, 18. In Quine 1948 uses the term “realism”. In Quine 1953c, the term “platonism” is used, see Quine 1953c, 127–129. The title of §48 of Quine 1960 was “Nominalism and realism”, but the terms were declared synonymous, see Quine 1960, 233: “… realists (in a special sense of the word), or Platonists (as they have been called to avoid the troubles of `realist’)”Google Scholar

Quine 1948, 14.Google Scholar

Gödel 1983.Google Scholar

Quine 1948, 14.Google Scholar

position is the abandoning of the principle of bivalence. Quine has always defended this principle, see e.g. Quine 1981a, 31–37.Google Scholar

Predicativism has been introduced in chapter 1.3. Predicative set theories were initially proposed by Poincaré and Weyl, two of the mathematicians mentioned in Quine 1948. Quine pictures the theory a bit more fully in Quine 1953c, 123–127.Google Scholar

Quine 1953a, 14–15; 127. For a discussion of the higher infinities in set theory, see Maddy 1990 and Maddy 1997. Where Quine is unsympathetic to these ever greater infinite sets, Maddy takes the opposite position by defending the maximal extensions of ZFC.Google Scholar

Quine 1953c, 129.Google Scholar

See Quine 1960, §48.Google Scholar

See Quine 1953c, 129. Also in section 2 of Quine and Goodman 1972, infinity is renunciated.Google Scholar

This problem was also clearly stated by Hilbert, see Hilbert 1925, 185–186.Google Scholar

See Quine 1953c, 129.Google Scholar

See Quine 1991a.Google Scholar

Inquiries into the size of the universe of abstract sets that is needed in mathematics and the empirical sciences are quite popular today. A reconstruction of mathematics in terms of restricted set theories is envisaged. Work in this direction is done by among others Hellman, Penrose, and the already mentioned Feferman.Google Scholar

I have only a vague impression of this period by reading some of the more important texts that appeared at that time. Very little historical work has been dedicated to this period. A lot of articles and books have appeared on the foundational crisis in mathematics, but it is difficult to find a single article that deals with the decline of the great schools, and the rise of new currents such as platonism, nominalism, structuralism, or the proponents of the use of modal notions. Quine’s impact on the philosophy of mathematics in this transient period cannot be underestimated. The determination of this impact goes beyond the purpose and the scope of this work.Google Scholar

For an informal but bold expression of his Platonism, see Quine 1987a, 11: “the notation is not itself the apple of the mathematician’s eye. The notation is the medium and very much the instrument, but the apple is something intangible out beyond.”Google Scholar

See Field 1980.Google Scholar

Quine 1960, 272. See also Quine 1974, 35: “In turning away from the ideas and looking to the words, we are taking the nominalist strategy.”Google Scholar

Quine 198la, 44: “An expression, for me, is a string of phonemes - or, if we prefer to think in terms of writing, a string of letters and spaces. Some are sentences. Some are words. Thus when I speak of a sentence, or of a word, I am again referring to the sheer string of phonemes and nothing more.”Google Scholar

See Quine 1966a, 41; Quine 1987a, 216–219.Google Scholar

See Quine 1987a, 217.Google Scholar

Gödel-numbering is a standard technique for translating languages to the natural numbers. To each expression in the language a natural number is assigned. Languages that can be learned can only have a finite collection of symbols. We number the symbols of the language. For example, we let “a” correspond with 1, “b” with 2, “c” with 3, and so on. Words or text fragments can be treated as concatenations of symbols. We consider them as n-tuples of symbols. The word Quine is treated as the concatenation of “q”, then “u”, “i”, “n”, “e”. This is the n-tuple q,u,i,n,e. We replace the n-tuples of symbols by n-tuples of natural numbers by writing for each symbol the number that is assigned to it. The word “Quine” is represented by 17,21,9,14,5. We can associate a unique natural with this n-tuple. We take the first prime number, i.e. two, and take it to the power of the number on the first position in the ntuple. We multiply this with the second prime number to the power of the number on the second position. We do this until we have multiplied it with the n-th prime to the power of the number on the n-th position. The word “Quine” is thus associated with the number 217321597141 15.Google Scholar

See Quine 1960, 191: “A sentence is not an event or an utterance, but a universal: a repeatable sound pattern, or repeatedly approximable norm.”Google Scholar

Quine 1953b, 49–52; Quine 1960, 85–90; Quine 1966a, 90, Quine 1970a, 16; Quine 1970c, 394; Quine 1981a, 44–45; Quine 1987a, 14; 149–152Google Scholar

Quine 1953b, 49. This is in the lecture “The problem of meaning in linguistics”, that was presented in 1951, i.e. nine years before the radical translation argument appeared in Quine 1960.Google Scholar

Quine 1953b, 51.Google Scholar

Ibid.Google Scholar

Quine 1960, 85.Google Scholar

Ibid., 86.Google Scholar

Ibid., 88–89.Google Scholar

Ibid., 89: “… that law affords no basis for any particular snipping of phonemes to length.”Google Scholar

Quine 1970a, 16 Similar passages can be found in Quine 1981a, 45; Quine 1987a, 150. In Quine 1970c, 394, this point was expressed slightly differently. It was said that the interchange of distinct phomenes changes the (stimulus) meaning of observation sentencesGoogle Scholar