Tarski’s Nominalism

Greg Frost-Arnold

I. Introduction: Textual sources for Tarski’s nominalism
Alfred Tarski was very reticent to express his views on traditional topics in metaphysics and epistemology in print: scouring his entire corpus for any explicit pronouncements directly concerning such issues yields virtually nothing.  Furthermore, his definitions of truth and of related semantic notions—likely the most fruitful Tarskian innovations for philosophers—are, in Tarski’s own estimation, independent of the perennially vexed questions of philosophy:
we may accept the semantic conception of truth without giving up any epistemological attitude we may have had; we may remain naïve realists, critical realists, or idealists, empiricists, or metaphysicians—whatever we were before.  The semantic conception is completely neutral towards all these issues. (1944, 362)

The sentiment in this quotation is not unusual; as Patrick Suppes notes, Tarski “was extraordinarily cautious and careful about giving any direct philosophical interpretation of his work” in semantics (1988, 81).  Furthermore, Frits Staal, a friend and Berkeley colleague of Tarski’s, recalls Tarski expressing “[m]ore than once… that he did not like to talk much about philosophy because he thought it was like giving an ‘after dinner’ speech—in other words, it was not rigorous” (Feferman and Feferman 2004, 318).  If we take Staal’s testimony at face value, it appears that Tarski not only believed his work in semantics to be independent of traditional philosophical concerns, but he also regarded such concerns with some skepticism.
	Nonetheless, Tarski did hold definite views on certain perennial philosophical questions. He shared his stance with others in conversation and correspondence, although virtually never in publications.  Much of the scattered, fragmentary evidence of Tarski’s philosophical stance has been ably collected and analyzed by Jan Wolenski (1993; 2002) and Artur Rojszczak (1999; 2002).  One aspect of Tarski’s philosophical thought that emerges clearly is that Tarski considers himself a nominalist and an anti-Platonist (cf. Mycielski 2004, Feferman 1999).  An extremely valuable resource in this regard is (Mancosu forthcoming).  An expression of his attitude can be found in an audio tape of a session during the 1965 Association for Symbolic Logic Meeting, in which Tarski describes himself as “an extreme anti-Platonist.”  He elaborates, with a joke:
I represent this very [c]rude, naïve kind of anti-Platonism, one thing which I could describe as materialism, or nominalism with some materialistic taint, and it is very difficult for a man to live his whole life with this philosophical attitude, especially if he is a mathematician, especially if for some reason he has a hobby which is called set theory.  
(Quoted in Mancosu 2005, 341)

Tarski here makes explicit what other commentators have noted (Mostowski 1967, 81; Wolenski 1993, 322): Tarski’s ontological views are in tension with his set-theoretical practice, since a central component of modern Platonism about mathematics is the commitment to a literal or realistic construal of set theory, including at least the assertion that sets exist.  But what position, exactly, is Tarski staking out with these labels of ‘anti-Platonism,’ ‘materialism,’ and ‘nominalism’?  Because each of these ‘ism’s has been around for centuries, and had various advocates, their meanings are not univocally determined.
	Fortunately, recently uncovered documents in the Rudolf Carnap Collection shed substantial light upon the specific form of Tarski’s nominalist, anti-Platonist position.  During the 1940-1941 academic year, several eminent analytic philosophers converged upon Harvard: along with Tarski were Rudolf Carnap, Willard Van Orman Quine, Carl Hempel, Nelson Goodman, and, for the fall semester only, Bertrand Russell.  We know from Carnap and Quine’s autobiographies that this group met repeatedly over the course of the school year, but their published reminiscings are regrettably brief, so little was known about the details of what this group of philosophical heavyweights discussed. Fortunately, Carnap had the lifelong habit of taking dictation notes in Stolze-Schrey secretarial shorthand, and he kept a detailed record of the discussions at these meetings.  Paolo Mancosu has recently presented an insightful overview of these notes, and my own view of these matters is indebted to his analysis (Mancosu 2005).  A complete edition of Carnap’s 1940-41 notes, with an English translation, will appear shortly (Frost-Arnold, forthcoming).
	 This essay aims to answer three related questions about Tarski’s ‘nominalism with a materialistic taint’ through an examination of Carnap’s dictation notes.  First, what is Tarski’s view?  Second, what are the rationales or justifications for his view—and are they defensible?  Third, how does Tarski attempt to reconcile his nominalist philosophical scruples with mathematics, since mathematics deals with paradigmatically abstract objects, such as numbers and sets, whose rejection is a standard sine qua non of current nominalism?  This question becomes more pressing if, like Tarski (and his interlocutors Carnap and Quine), one takes modern natural sciences seriously, since they are suffused with mathematics.  I conclude by considering the relationship between Tarski’s nominalism and more recent forms of mathematical nominalism.

II.  Tarski’s conditions for intelligibility [Verständlichkeit]
Carnap’s discussion notes from 1940-41 cover a wide range of topics.  The participants were not dedicated to a single issue; rather, they discussed whatever captured their interest that day.  Nonetheless, a plurality of Carnap’s discussion notes during the spring semester deal with what he and his collaborators call—most briefly—‘finitism.’  However, this topic is not identical with the cluster of claims philosophers usually associate with the label ‘finitism’—namely, the mathematical project associated with Hilbert and his school (specifically, investigating which inferences employed in classical mathematics can be re-cast into a finitistically acceptable form).  Tarski, Carnap, and Quine were fully aware that Hilbertian finitism, at least in its original incarnation, was long dead as a research program by 1940, when their conversations began.  The notion of finitism at issue in these conversations first appears when Tarski proposes rather strict requirements a language must meet in order for it to qualify (in Tarski’s eyes) as verständlich, that is, understandable or intelligible.  It is here that we find the nub of Tarski’s anti-Platonism, his ‘nominalism with a materialistic taint.’ 
	Tarski’s proposed conditions on intelligible languages vary somewhat from meeting to meeting.  Carnap records the first version of it as follows.
January 10, 1941.
Tarski, Finitism.  Remark in the logic group.

Tarski: I understand at bottom only a language that fulfills the following conditions:

1. Finite number of individuals.
2. Reistic (Kotarbinski): the individuals are physical things.
3. Non-Platonic: Only variables for individuals (things) occur, not for universals (classes etc.) 
(RCC 090-16-28)

Three weeks later, Tarski offers a similar, though not identical, characterization of a language he finds completely understandable.  (I have placed my construal of the relation between the following characterization and the previous one in square brackets.)
Finitism.
Tarski: I properly understand only a finite language S1: 
only individual variables,					[identical to condition 3. above] 
whose values are things,					[identical to 2. above] 

whose number is not claimed to be infinite (but perhaps also not the opposite) 
		[modified version of 1. above]  
Finitely many descriptive predicates. 			[new condition]
(RCC 090-16-25)

	Though terse, these conditions are fairly clear, and they flesh out Tarski’s 1965 description of his position as ‘nominalism with a materialistic taint.’  Let us couch Tarski’s proposed conditions for an intelligible language in the modern idiom of model theory.  (Though Tarski is considered the father of modern model theory, many of its central concepts—including the central notion of ‘truth in a structure’—had not taken their canonical form by 1941 (see Hodges 1986).)  We begin with the model-theoretic conception of an interpreted language L, which is an ordered triple <L, M, >.  L carries the syntactic information about the language: what the symbols of the language are, the grammatical category to which each symbol belongs, which strings of symbols qualify as grammatical formulae and which not, etc.  The semantic scheme  determines the truth-values of a compound expression formed using logical connectives, given the truth-values of its constituents.  M is an interpretation or model that fixes signification of the nonlogical constants of L.  Specifically, M = <D, f>, where D is a nonempty set, and f is an interpretation function which assigns members of D to singular terms, assigns sets of ordered n-tuples Dn to n-ary relation symbols, and a member of DnD to each n-ary function symbol.  Now let us use this apparatus to rephrase Tarski’s conditions.  Tarski is describing a certain type of (interpreted) language L that has the following four characteristics, which I will henceforth refer to as Tarski’s ‘finitist-nominalist’ (FN) conditions:
(FN 1)	L is first-order.			[Anti-Platonic requirement]
In a fully understandable language, one cannot quantify over properties or relations, only over individuals.  Note that in the original formulation of this requirement, Tarski says that ‘variables occur only for individuals, not for universals (classes etc.).’  The parenthetical end of that claim is surprising, for it is possible to treat classes as individuals, as is done in first-order set theory.  So Tarski’s first formulation actually outstrips what I have called (FN 1), since it would rule out first-order set theory, whereas (FN 1) as stated does not.  However, this discrepancy is not of great importance for identifying the content of the concept of Verständlichkeit for Tarski, because the next condition (FN 2) will rule out ‘classes etc.’  That is, (FN 2) makes redundant any content of Tarski’s earlier formulation of his first condition that goes beyond (FN 1).  What is noteworthy is that Tarski considered admitting variables for classes, as well as variables for properties and relations, to constitute Platonism.
One might be tempted to interpret Tarski here as claiming that any string that contains (the symbolic correlate of) ‘For all properties, …’ is not a grammatical formula of L; it is, after all, natural to think of being first-order as a grammatical property.  However, Tarski appears not to think of his proposed conditions as grammatical restrictions.  For immediately following the quotation above, Tarski explains that he is willing to construct proofs using sentences containing higher-order variables according to the rules of a proof calculus.  However, he claims that he nonetheless does not truly understand these sentences, at least not in a full or unqualified sense.  Specifically, Tarski says: 
I only ‘understand’ any other language [i.e., a language that does not meet the conditions on intelligibility] in the way I ‘understand’ classical mathematics, namely, as a calculus; I know what I can derive from what… With any higher, ‘Platonic’ statements in a discussion, I interpret them to myself as statements that a fixed sentence is derivable (or derived) from other sentences. (RCC 090-16-25)  

This quotation shows that (FN1) should be understood as a restriction on what is intelligible or understandable, not on what is grammatical.  The notion of genuine ‘understanding,’ which a ‘calculus’ alone cannot guarantee, will be discussed at some length at the end of this section.
(FN 2) D consists of ‘physical objects’ only.		[Reistic requirement]  
In the elaboration and discussion of (FN 2) within the notes, numbers are specifically disallowed from the universe of discourse.  Furthermore, because of (FN 1), not even Frege’s reconstruction of numbers as properties of properties is allowed in a finitist-nominalist language.  Taken together, (FN 1) and (FN 2), along with the standard treatment of existence in usual systems of symbolic logic, yield a modern formulation of nominalism.  The current canonical charaterization of nominalism is that only concrete things exist; abstract objects of any sort (properties, relations, numbers, propositions, sets, etc.) do not exist.  We cannot make any first-order existence claims about abstract objects within a language L whose domain of discourse contains no abstract objects—except to say that such objects do not exist.  And higher-order existence claims, according to (FN 1), are unintelligible at best, even if we allow them as grammatical strings of L.  Tarski restricts the universe of discourse for a fully intelligible language to ‘physical objects’ only—but what, exactly, are the physical objects?  The discussants do not show much interest in settling upon a specific interpretation.  Three options are proposed: (i.) elementary physical particles (electrons etc.), (ii.) mereological sums composed of elementary particles (or quanta of energy), so that the objects referred to by the names ‘London’ and ‘Rudolf Carnap’ will qualify as physical objects, and (iii.) spatial and/or temporal intervals (this suggestion derives from the co-ordinate languages of Carnap’s Logical Syntax) (RCC 090-16-23).  
Tarski’s third condition on intelligible languages appears in two distinct forms:
(FN 3a: restrictive version) D contains a finite number of members;  [finitism]
or
(FN 3b: liberal version)  No assumption is made about the cardinality of D.
As we saw above, the restrictive policy is Tarski’s initial proposal, but in later conversations, he clearly leans towards the liberal policy (see 090-16-04 and -05).  In his autobiography, Carnap attributes the restrictive version to Quine and the liberal version to Tarski and himself (1963, 79).  Presumably, this restriction is why the discussants call their project the construction of a ‘finitist’ language—and we see clearly how different Hilbert’s finitism is from Tarski’s.
	The last restriction Tarski proposes for a finitist-nominalist language can be couched as follows:
(FN 4) L contains only finitely many descriptive predicates.
Tarski offers no justification for (FN 4), and it never plays an explicit role in subsequent discussions, so I will not dwell upon it further.  Presumably, these four finitist-nominalist restrictions do not single out one language uniquely — several different (interpreted) languages could satisfy (FN 1-4).  I have labeled the above four conditions ‘finitist-nominalist (FN) conditions,’ since the first two are nominalist, and the third and fourth finitist; let us call any language satisfying them a ‘finitist-nominalist’ language.

	The FN conditions, as Tarski says, are intended to identify languages that are understandable, verständlich.  But what does Tarski mean by the word verständlich and its cognates?  Unfortunately, there is very little relevant information on this question in the 1941 notes.  (What little there is will be discussed in the next paragraph.)  It is frustrating that the discussion notes lack an account of what intelligibility is, since according to all parties involved, attaining it is the acknowledged central motivation for constructing a finitist-nominalist language.  More specifically, the notes fail to explain why a language violating any of Tarski’s finitist-nominalist criteria is not (fully) understandable.  Why, for example, would a sentence beginning with the apparently understandable phrase ‘There exists a property such that…’ be unintelligible?  The notes from the final day of collaborative work on the finitist-nominalist language highlight how unclear and imprecise the concept of Verständlichkeit remains for the participants.  Carnap writes:
We agree the language should be as understandable as possible.  But perhaps it is not clear what we properly mean by that.  Should we perhaps ask children psychological questions, what the child learns first, or most easily?  (RCC 090-16-05)

So Carnap himself does not know what is meant by verständlich, even six months into the project, and the dictation notes show no response to his query from the other participants.
Our interpretive prospects are not hopeless, for there is some material in the discussion notes that provides insight into what verständlich means for Tarski, Quine, and Carnap.  In particular, Tarski clearly contrasts understandable languages with purely formal calculi, which enjoy only a ‘second-class’ type of intelligibility—if any at all.
Tarski: I fundamentally understand only a language that fulfills the following conditions:
[GF-A: Here are the three finitist-nominalist restrictions; see full quote above]…
I only “understand” any other language in the way I “understand” classical mathematics, namely, as a calculus; I know what I can derive from other [sentences] (rather, I have derived; “derivability” in general is already problematic).  With any higher “platonic” statements in a discussion, I interpret them to myself as statements that a fixed sentence is derivable (or derived) from certain other sentences.  (He actually believes the following: the assertion of a sentence is interpreted as signifying: this sentence holds in the fixed, presupposed system; and this means: it is derivable from certain basic assumptions.) 
(RCC 090-16-28)

That is, there is no genuine understanding of sentences couched in a language not meeting (FN 1-4), such as the language of ‘classical mathematics’.  The contrast between ‘intelligible language’ and ‘uninterpreted calculus’ also appears, albeit more briefly, elsewhere in the discussion notes.  Thus (to put the point in the terminology of Carnap’s Logical Syntax), merely knowing the formation and transformation rules of a calculus does not constitute genuine understanding of the language corresponding to that calculus, i.e., the language that that calculus is intended to model formally.  That is, understanding what a sentence s means is not merely knowing which sentences are deducible from s and from which sentences s is deducible.  This basic view has found modern expression in John Searle’s Chinese Room thought experiment, which purports to show (inter alia) that a computer cannot understand a (natural) language, because a computer’s operations are restricted to the realm of syntax (Searle 1980).  Regardless of what the detailed content of Verständlichkeit might be for Tarski and Carnap, it at least requires that a language be more than an uninterpreted calculus or empty formalism.  And uninterpreted calculi and empty formalisms are usually contrasted with meaningful or interpreted languages.  After all, one cannot understand a sequence of spoken or written characters unless that sequence is meaningful.  
It appears we can take Tarski to hold that meaningfulness is a necessary condition for intelligibility.  A further piece of evidence corroborating this view comes from the German translation of Tarski’s monograph on truth; although Tarski did not translate the entire essay himself from the original Polish, he did assist with the translation (Feferman and Feferman 2004, 99), so the language found in the German version is either approved by Tarski, or his own.  The concept of Verständlichkeit appears in Wahrheitsbegriff in precisely the connection discussed above: the contrast between an uninterpreted or purely formal calculus and an intelligible language.
It remains perhaps to add that we are not interested here in ‘formal’ languages and sciences in one special sense of the word ‘formal,’ namely sciences to the signs of which no meaning [inhaltlicher Sinn] is attached. … We shall always ascribe quite concrete and, for us, intelligible meanings [verständliche Bedeutungen] to the signs which in occur in the languages we shall consider. ([1935] 1983, 167) 

Besides the sharp opposition between merely formal and intelligible, meaningful languages found here, certain other aspects of this quotation merit attention.  First, Tarski requires that the meanings of constants in the language be ‘concrete’ [ganz konkrete]—and the current characterization of nominalism is the claim that only concrete things exist.  Second, there is an apparent discrepancy between Tarski and Carnap insofar as Carnap thinks of languages (and expressions within a language) as intelligible or not, whereas the above passage treats meanings [Bedeutungen] as what is or is not intelligible.  In any case, we should perhaps avoid placing too much weight upon word and phrase choice, for although Tarski was a meticulous writer, he was not the primary translator for Wahrheitsbegriff, so the choices made in the translation may not always reflect his views with complete faithfulness and precision.
With this admittedly partial characterization of Verständlichkeit in place, the next question is: why and how does requiring a language to meet the four finitist-nominalist restrictions guarantee the intelligibility (and thus the meaningfulness) of sentences in that language?  Unfortunately, in Carnap’s notes, that question is neither asked nor answered directly.  However, there is some material in the discussion notes that addresses the broader but related issue of what motivates these restrictions.  By examining the rationales supplied for imposing these restrictions, we can see more clearly what benefits Tarski believed conditions (FN 1-4) would achieve, and we thereby indirectly gain a better understanding of Verständlichkeit, and the project as a whole. 

III.  Rationales for the finitist-nominalist conditions
The official year of birth for modern Anglophone nominalism is generally taken to be 1947, with Goodman and Quine’s Journal of Symbolic Logic article “Steps Toward a Constructive Nominalism.”  In a footnote in that article, the authors acknowledge that the initial impetus and strategy for their nominalist project was proposed in 1940 by Tarski, and discussed with him, Carnap, and the authors (1947, 112).  Thus, the discussions at Harvard in 1940-41 can be seen as an important wellspring of current nominalism.  In the previous section, Tarski’s finitist-nominalist criteria (FN1-4) were outlined and examined in some detail.  A pivotal question that was not addressed then, but will be in this section, is the following: What justifies these finitist-nominalist criteria?  This section discusses two rationales presented by Tarski, Carnap and Quine for undertaking the finitist-nominalist project: their hostility towards Platonic metaphysics, and what I will call ‘the argument from natural science.’
1. Overcoming Platonic Metaphysics
One rationale for pursuing the finitist-nominalist project that appears—both implicitly and explicitly—in the Harvard discussions is the desire to purge metaphysical elements from discourse intended to be cognitively significant, and from language used for scientific purposes in particular.  It is well known that the logical empiricists and their allies (e.g. Russell and Wittgenstein) hold a very negative view of metaphysics.  The group of Polish philosophers from which Tarski came, the Lvov-Warsaw School, also tended to share this anti-metaphysical animus (Simons 1993, Wolenski 1993), though as a group, they were neither as fervently nor as unanimously anti-metaphysical as their Viennese contemporaries.
	The anti-metaphysical drive is closely connected to the notion of Verständlichkeit discussed in the previous section.  One characterization of metaphysics that is widespread among the logical empiricists and their intellectual kin is the following: if a string of symbolic marks x is metaphysical, then x is meaningless.  (The converse does not hold: the string ‘)bPQ))’, which is meaningless in standard formalizations of predicate logic, is not metaphysics.)  And presumably, if a given word or sentence is meaningless, then it is not understandable or intelligible.  The connection to the finitist-nominalist project is clear: by modus tollens, if every word and every sentence in a given language is ‘fully understandable,’ then there are no metaphysical words or sentences in that language.  This argument is never explicitly articulated in the discussion notes; in particular, we never find the conditional claim ‘if a string of symbols is meaningless, then that string is not understandable.’  Nonetheless, given that this conditional seems patently true (how could one understand meaningless nonsense?), it seems reasonable to connect Carnap, Tarski, and Quine’s discussions of intelligibility to their shared aversion to metaphysics qua meaningless utterances and inscriptions.  And if, as just suggested, intelligibility entails meaningfulness, then the construction of an intelligible language will yield a language free of metaphysics.  In short, given the unintelligibility of meaningless discourse, a fully intelligible language would also be a language free from metaphysical impurities—and such a connection was perhaps at least implicit in the minds of the Harvard discussants.
	But the attacks on metaphysics in Carnap’s discussion notes are not merely implicit.  There are explicit references to objectionable metaphysical theses as well.  Tarski and Quine hold that the adoption of FN1 and FN2 would prevent a pernicious slide into a certain kind of metaphysics, which they call ‘Platonism,’ naming the position after the grandfather of all metaphysicians.  Recall that Tarski labels FN1 (the requirement that variables only range over the individual domain, not over properties or relations) the ‘non-Platonic’ requirement in the first articulation of his proposal (RCC 090-16-28).  The participants do not offer a detailed or precise characterization of Platonism, but it involves at least higher-order logic, and probably (full) set theory. (FN1 rules out higher-order logic, and adding FN2 to it rules out (first-order) set theory.)  For example, in a discussion of “general set theory,” Tarski asserts: “With the higher levels, Platonism begins.” (RCC 090-16-09).  (What Tarski means by ‘the higher levels’ is not unequivocally fixed by the context; he may mean only transfinite sets, or he may have in mind anything higher than the ur-elements.)  And even earlier, in a discussion with Russell and Carnap, Tarski asserts: “A Platonism underlies the higher functional calculus (and so the use of predicate variables, especially higher levels)” (RCC 102-63-09).  So, in these records, Tarski associates Platonism most closely with second- and higher-order logic, though set theory (presumably even in its first-order variety) is also caught up in the Platonic viewpoint.
We find a similar conception of Platonism in Quine’s contemporaneous writings.  In a December 1940 lecture at Harvard (and thus before Tarski introduces (FN1-4)), Quine distinguishes mathematics from logic as follows: “‘logic’ = theory of joint denial and quantification,” while “‘mathematics’ = (Logic +) theory of .”  Quine then claims that “mathematics is Platonic, logic is not” (RCC 102-63-04).  Why should the set-membership relation introduce Platonic commitments?  Quine explains that “there are no logical predicates,” but ‘’ is a predicate.  He then claims: 
Predicates first bring ontological claims (not because they designate, for they are syncategorematic here, since variables never occur for them; rather): because a predicate takes certain objects as values for the argument variable; so e.g. ‘’ demands classes, universals; thus mathematics is Platonic, logic is not” (ibid.).    

That is, if there are any true statements of the form ‘PQ,’ then there must be at least one class (provided ‘’ is given the intended interpretation).  For Quine, accepting the existence of at least one class is tantamount to accepting Platonism.  This position is related to, but stronger than, the one he published a year earlier in “Designation and Existence,” for according to the position Quine outlines there, a nominalist could hold ‘PQ’ to be true, provided the nominalist does not ineliminably quantify over the Q-position.  As a historical matter, Quine harbored suspicions of set theory even before Tarski proposes constructing a finitist-nominalist language, so Quine was presumably a receptive audience for Tarski’s proposal.  In short, first-order set theory is Platonic, along this line of thinking, because it forces us to admit the (ineliminable) existence of classes as values of variables.
So why do Tarski and Quine also suspect higher-order logic of being metaphysics—even when the domain of discourse consists solely of (concrete) individuals?  In a letter to Carnap dated May 1943, reflecting on the Harvard discussions, Quine writes:
I argued, supported by Tarski, that there remains a kernel of technical meaning in the old controversy about [the] reality or irreality of universals, and that in this respect we find ourselves on the side of the Platonists insofar as we hold to the full non-finitistic logic.  Such an orientation seems unsatisfactory as an end-point in philosophical analysis, given the hard-headed, anti-mystical temper which all of us share. (in Creath 1990, 295)

Presumably, the ‘kernel of technical meaning in the old controversy’ is composed of two decisions: (i) whether (contra (FN 1)) to adopt a higher-order logic, and (ii) whether (contra (FN 2)) to allow non-physical individuals into the domain of quantification (as discussed in the previous paragraph).  For Quine, by this point in his career, a commitment to higher-order logics brings in its wake a commitment to the ‘reality of universals.’  Why?  In his “Designation and Existence,” published a year before the Cambridge discussions, Quine articulates his famous dictum “[t]o be is to be the value of a variable” (1939, 708).  In the same article, he uses this dictum to characterize nominalism within the framework of modern logic: a language is nominalist if its variables do not range over any abstract objects.  And properties and relations, which are quantified over in second-order logics, are (for Quine and many others) paradigmatically abstract entities.  In short, a language is metaphysical if it quantifies over abstract entities; in first-order set theory, those abstracta are sets, and in higher-order logic, those abstracta are properties and relations.
What is missing from both the discussion notes of 1940-41, as well as from published writings before and after, is an explanation of why admitting abstracta (whether they be sets, relations, or anything else) as values of variables is philosophically objectionable.  That is, why does allowing the existence of abstracta violate ‘the hard-headed, anti-mystical temper’ shared by all the Harvard discussants?  This becomes more troubling when we note that classifying higher-order logic and/or set theory as metaphysics does not mesh well with the characterization of metaphysics offered elsewhere by the logical empiricists and their allies.  Both the explicatum and the explanandum of the term ‘metaphysics’ vary over time and between different thinkers.  Nonetheless, most logical empiricists, most of the time, strongly resist classifying logic and mathematics as metaphysical.  (It is a sign of this that special exceptions are made in their accounts of meaning and knowledge to account for logic and mathematics.  For example, Wittgenstein’s distinction in the Tractatus between pseudo-propositions that are nonsense [unsinning] and those that are senseless [sinnlos] places logic and mathematics in a separate category from metaphysics, even though both ‘say nothing about the world.’) So not only do Tarski and Quine omit an explanation of why classes and relations are metaphysical, but such a view appears to clash with the view of metaphysics presented by many of their philosophical peers. 
Returning to Tarski and Quine’s view of higher-order logic and set theory, it may very well be that there is no further justification for their animus.  In Goodman and Quine’s published paper on nominalism, the authors admit that their “refusal [to countenance abstracta] is based on a philosophical intuition that cannot be justified by appeal to anything more ultimate” (1947, 105).  Of course, we should not assume Quine and Goodman speak for Tarski as well; however, if Tarski did attempt to ‘justify’ his rejection of abstracta ‘by appeal to anything more ultimate,’ that justification is not recorded in the Harvard discussion notes, and it did not impress Goodman and Quine enough to include it in their article.  We today could impute to Tarski (and/or the other participants) some further justification, but such an attribution would be conjectural, given the evidence available in Carnap’s dictation notes.  

2. Second rationale for the FN restrictions: natural science
Another type of justification Tarski and Quine offer for pursuing the finitist-nominalist project can be called the ‘argument from natural science.’  The previous rationale supported (FN 1-2) (viz. the language is first-order and its domain contains only physical objects), which we could consider support for nominalism; the following, however, is only a justification for finitism (FN 3).  Tarski begins his defense of (FN 3) with a reasonable assertion: the number of individuals in our world “is perhaps in fact finite” (090-16-25).  If the universe does contain only finitely many physical things, and if (FN 1-2) hold, then it immediately follows that D has finitely many members—and this is the restrictive version of (FN 3).  If we wish rather to leave open the question of whether the number of physical things is finite or not, and we accept (FN 1-2), then the liberal version of (FN 3) follows.  Note that without (FN 1-2) or something similar, (FN 3) becomes much more contentious.  As explained previously, (FN 1-2) prevent the two most common ways of introducing mathematical objects into a language, and mathematical infinities are usually considered paradigmatic examples of infinite totalities.
Carnap replies to Tarski’s claim by suggesting that there are infinities. These come in two varieties: logico-mathematical and physical.  The usual mathematical infinities will directly violate the nominalist criteria (FN 1-2).  As examples of empirical, physical infinities that will not fall afoul of (FN 1-2), Carnap suggests space and, with more conviction, time.  He claims: 
even if the number of subatomic particles is finite, nonetheless the number of events can be assumed to be infinite (not just the number of time-points…but the number of time-points a unit distance away from each other, in other words: infinite length of time.) (RCC 090-16-24) 
 
Carnap’s suggestion to use events or spatiotemporal intervals instead of physical objects for the domain of a language of science obviously violates the letter of (FN 2), but Carnap likely believes it does not violate its spirit—for spatiotemporal events are still part of the natural, physical world, unlike numbers and their kin.  (However, Kotarbinski, whom Tarski invokes when he proposes (FN 2), explicitly denies that events are acceptable for the reist (1929/1966, 432).)  So, Carnap is suggesting, if we expand (FN2) to allow the domain to contain not just physical objects but rather any entity that is (broadly speaking) part of the physical world, then (FN 3) does not force itself upon us—provided there are an infinite number of events.
	Tarski responds to Carnap’s challenge in two related ways.  The first engages Carnap on his own terms; the second suggests that Carnap’s critique has missed part of Tarski’s motivation for introducing (FN 3), at least in its mature, liberal version.  First, Tarski replies directly to Carnap’s suggestion that space and time will provide us with infinities, even if there are only a finite number of physical objects in the universe.  Tarski asserts that space and time, contrary to appearances, may actually be finite: “perhaps quantum theory will give up continuity and density” for both space and time, by quantizing both quantities (090-16-23).  Furthermore, Tarski says, time and space could both be circular, in which case there would not be an infinite number of finite spatial or temporal intervals.  In short, Tarski claims that developments in quantum and relativistic physics may in fact show that space and time are actually finite.
Second, Tarski suggests that arguing that there is actually an infinite quantity somewhere in nature misunderstands the rationale behind the liberal version of (FN 3).  Presumably (though Tarski does not state this fully explicitly), we should not assume the number of physical things in the world is infinite, because this is an empirical matter.  What Tarski does say is the following: “we want to build the structure of the language so that this possibility [viz. that the number of physical things is finite] is not excluded from the beginning” (090-16-23).  The basic idea is simple: the form of the language we use to describe the empirical world should not prejudge the number of entities in the universe, and Tarski’s scheme happily leaves this question open.  Put otherwise, ‘How many spatial or temporal intervals are there?’ is just as empirical a question as ‘How many subatomic particles are there?’  If one accepts (FN 2), and if one also wishes to incorporate at least full (first-order) arithmetic into one’s language, then one would be committed to an infinite number of physical objects.  To couch Tarski’s worry in Carnapian terms: how many entities there are in the universe—as well as the topological structure of (actual) space and time—are synthetic matters, and by adopting (FN 3), we prevent them from becoming analytic ones.  That is, questions about the number of things in the universe or the topology of space and time should be determined by the structure of the world, not by the structure of the language used to describe that world.

IV. A finitist-nominalist language of science and mathematics
1. The demand for a ‘total language of science’
In further formulations of the group’s project, an additional condition is placed on the language(s) they are attempting to construct.  They wish to formulate a language that simultaneously meets Tarski’s conditions on intelligibility, and is also rich enough to use for investigating the logic of science, including metalinguistic investigations of classical analysis and set theory (they sometimes call such a language a “nucleus language”). 
Jan. 31, 1941
Conversation with Tarski and Quine on Finitism
…
We together: So now a problem: What part S of M [the metalanguage of science and mathematics] can we take as a kind of nucleus, so that 1.) S is understood in a definite sense by us, and 2.) S suffices for the formulation of the syntax of all of M, so far as is necessary for science, in order to handle the syntax and semantics of the complete language of science.  (090-16-25)

Similar sentiments are expressed a few months later, as the discussants’ year together drew to a close.
June 18, 1941
Final Conversation about the nucleus-language, with Tarski, Quine, Goodman, and Hempel; June 6 ’41

Summary of what was said previously.  The nucleus language should serve as the syntax-language for the construction of the complete language of science (including classical mathematics, physics, etc.).  The language of science thereby receives a piecewise interpretation, since the n. l. [nucleus language] is assumed to be understandable.  (090-16-05)

On the one hand, the finitist-nominalist conditions place restrictions on an interpreted language’s richness; this condition, on the other, restricts a language’s poverty.  
Carnap, Tarski and Quine realize it may not be possible to construct a language that simultaneously satisfies this criterion as well as (FN 1-4).  For immediately following the first of the two quotations above, we find:
1. It must be investigated, if and how far the poor nucleus (i.e. the finite language S1) is sufficient here.  If it is, then that would certainly be the happiest solution.  If it is not, then two paths must be investigated:

2a.  How can we justify the rich nucleus (i.e., infinite arithmetic S2)?  I.e., in what sense can we perhaps say that we really understand it?  If we do, then we can certainly set up the rules of the calculus M with it.

2b.  If S1 does not suffice to reach classical mathematics, couldn’t one perhaps nevertheless adopt S1 and perhaps show that classical mathematics is not really necessary for the application of science in life?  Perhaps we can set up, on the basis of S1, a calculus for a fragment of mathematics that suffices for all practical purposes (i.e., not something just for everyday purposes, but also for the most complicated tasks of technology). (090-16-25)

In short, they suspect that a metalinguistic analysis of classical mathematics and physics may require a richer language than what the finitist-nominalist criteria allow.  Further, if that suspicion is borne out, then either this richer language must be shown to be understandable, or the weaker mathematics sanctioned in finitist-nominalist languages must be shown to be sufficient to deal with all sophisticated practical applications.  Unfortunately, we are not informed whether this new condition trumps the finitist-nominalist conditions or not.  That is, the notes do not contain Tarski’s answer to the following question: if the mathematics recoverable in languages meeting (FN 1-4) is insufficient for practical purposes, then what should be discarded—the demand for a single, intelligible metalanguage of science, or the finitist-nominalist strictures on intelligibility?  Thus it is difficult for us to ascertain the relative importance the participants attach to these competing conditions. 
 However, none of the participants assert that we should abandon the investigations of those portions of (e.g.) set theory that fail to meet the finitist-nominalist criteria.  Set theory can still progress, even if parts of it are not fully intelligible—Tarski suggests that set theory then becomes a purely formal (i.e. uninterpreted) calculus, indicating which sentences can be derived from others, as we saw above (RCC 090-16-28).  But that is not a barrier on proving theorems within ZFC.  Thus the Tarskian nominalist would presumably allow mathematical practice to continue unabated and unimpeded, while simultaneously maintaining that some or all of the claims forwarded in the course of that practice are not, strictly speaking, intelligible.  In sum, Carnap, Tarski, and Quine (and occasionally Goodman and Hempel) are attempting to construct a formal language that simultaneously meets the stringent finitist-nominalist constraints (FN 1-4) and is rich enough to serve as a metalanguage for all of scientific discourse, including (at least elementary) mathematics.  Since these two conditions pull in opposite directions, this is a difficult goal to reach, as current nominalists can attest.  Let us now turn to the 1941 group’s attempt to reconstruct arithmetic in a language meeting Tarski’s conditions on intelligibility. 
	Carnap, Tarski, and Quine realize from the outset that one of the most pressing and difficult obstacles facing any attempt to construct a finitist-nominalist language for science will be the treatment of mathematics.  Can a language simultaneously meet Tarski’s criteria for intelligibility and contain (at least a substantial portion of) the claims of classical mathematics?  A sizeable portion of Carnap’s notes from the Harvard meetings deals with attempts to answer this question.  The discussants focus on the simplest case, viz. classical arithmetic.  A number of potential pitfalls present themselves: first, what is the content of assertions about numbers?  Can we assert anything about them at all, given that the only elements in our domain of discourse are physical objects?  Second, how should we handle numerals that purportedly refer to numbers that are larger than the number of concrete things in the universe?  That is, suppose there are exactly one trillion physical things in the universe; what should we then make of the numeral ‘1,000,000,000,001’ and sentences containing it?  Finally, what theorems and proofs of classical arithmetic are lost (or possibly lost) in a language that meets the finitist-nominalist conditions?  I shall deal with each of these questions in turn. 

2. Number 
	As seen previously, one of Tarski’s requirements for a language to be understandable is that abstract entities are not allowed to serve as denotata of names.  So in such a language, the numeral ‘7’ cannot name a natural number, considered as a basic individual object—for the natural numbers (understood as individual objects) are excluded from the domain of discourse.  And as mentioned above, since a FN language must also be first-order, Frege’s reconstruction of numbers as second-order properties is forbidden as well.  But Tarski, Carnap, and Quine want this language to include, at the very least, substantial portions of arithmetic.  So how do they re-interpret numerals under this linguistic regime?
Tarski’s strategy for introducing ordinal numbers is the following:  “Numbers can be used in a finite realm, in that we think of the ordered things, and by the numerals we understand the corresponding things” (090-16-25).  Virtually the same proposal is outlined in the part of Carnap’s autobiography that deals briefly with the year at Harvard:
To reconcile arithmetic with the nominalistic requirement, we considered among others the method of representing natural numbers by the observable objects themselves, which were supposed to be ordered in a sequence; thus no abstract entities would be involved.  
(1963, 79)

Let us illustrate this idea with a concrete example.  Suppose, in our domain of physical things that have been ‘ordered in a sequence,’ Tom is the eighth thing, John is the fourth, and Harry the eleventh.  (Assume the numeral ‘0’ is assigned to the first thing.)  Then the arithmetical assertion ‘7+3=10’ is (put roughly) re-interpreted as ‘Tom + John = Harry.’  Put model-theoretically, the interpretation function f of an interpreted language meeting the finitist-nominalist requirements assigns to numerals of L objects in D: f(7)=Tom, f(3)=John, and f(10)=Harry.  (Arithmetical signs such as ‘+’ are defined via the axiomatization of Peano Arithmetic in Carnap’s Logical Syntax, §14 and §20.)  
	This heterodox view of ordinal numbers raises serious questions.  First, from whence does the sequential order of the physical objects spring?  The ordering is imposed, it appears, by stipulation; Tarski says: “we want the (perhaps finitely many) things of the world ordered in some arbitrary way” (090-16-23).  We may assign any member of the domain of physical things to the numeral ‘0,’ and we may choose any other member of the domain to be its successor, to be assigned the numeral ‘1.’  The sentence ‘0+1=0’ will come out false under any such stipulation, regardless of which physical objects we choose to ‘stand in’ for the numbers 0 and 1 (assuming the cardinality of D is greater than one).  The relation is a successor of need not reflect anything ‘in the order of things,’ spatial, temporal, or otherwise.  There is a second worry about this proposal to reinterpret our usual number-language under a finitist-nominalist regime.  Let us suppose that the sentence ‘Tom has brown hair’ is true.  Then, since the name ‘Tom’ and the numeral ‘7’ both name the same object (model-theoretically, the interpretation function assigns both individual constants the same value), it appears that the sentence ‘7 has brown hair’ will be true.  But numbers cannot be brunettes.  So this finitist-nominalist interpretation of numerals will make true many assertions about numbers that, intuitively, we do not want to come out true. Tarski, Quine and Carnap do not even consider this problem (at least, there is no record of it in Carnap’s discussion notes). 
Technical refinements could perhaps avoid at least some of these unwanted truths. One such refinement is suggested in (Field 2001, 214-215).  His basic idea, couched in our terms, is the following. Note that the ordering of physical objects of D should be viewed as arbitrary, and that alternative orderings of the elements of D would still respect the truths of classical arithmetic captured in the original model.  We can finesse this fact to eliminate certain unwanted truths: while ‘7’ may be assigned to a brunette under one assignment of physical objects to numerals, it will be assigned to a blonde under another, and to various hairless physical objects under other assignments.  However, under all these assignments, ‘7+3=10’ is true.  This suggests the following refinement to Carnap, Tarski, and Quine’s proposal to re-interpret numerals in a finitist-nominalist language: a (mathematical) sentence  is true (in L) if and only if  is true for all assignments of physical-object-values to numerals (satisfying certain conditions: for example, we do not want to include the assignment in which all numerals are assigned to a single object in D).  This line of thought shows that perhaps there is a way to interpret ‘7’ that meets (FN 1-4) and certifies substantial portions of arithmetic as true, without committing us to the truth of sentences such as ‘7 has brown hair.’ Of course, if the number of objects in the physical universe is finite, then this proposal will not capture all of standard arithmetic.

3. Interpreting numerals that are ‘too large’
Now we come to a problem concerning mathematics in finitist-nominalist languages that Tarski, Quine and Carnap did recognize and address themselves. Suppose there are only k items in the universe.  Carnap poses the question: “How should we interpret [deuten]” the number expressions k+1, k+2, …, “for which there is no further thing there?” (-06)  Initially, the group considers three options (employing the usual notation that x is the successor of x):
	(a.) k = k = … = k	
	(b.) k = k = … = 0	
	(c.) k = 0, k = 0, …	

In each of these three cases, at least one of the Peano axioms is violated.  If (a.) is adopted, then there exist two numbers (i.e., there exist two physical objects) that have the same successor; if (b.) or (c.) is adopted, then the number assigned to ‘0’ will be a successor.  None of these options is especially palatable, since none captures the truths of classical arithmetic substantially better than the others.  A surprisingly large portion of the discussion notes is devoted to working through proposed solutions to this problem.  Strategies other than (a.)-(c.) are also considered: Carnap suggests identifying numbers with sequences of objects instead of objects, so that there is no ‘last element’ forced upon us; however, Quine and Tarski consider sequences (at least in the form needed to recover full classical arithmetic) just as problematic as sets, numbers, and other abstracta.
There is another problem reconstructing mathematics under a finitist-nominalist regime: as Tarski notes, under these conceptions of number “many propositions of arithmetic cannot be proved in this language, since we do not know how many numbers there are” (RCC 090-16-25).  That is, suppose that we do not know how many physical objects there are in the material universe; this ignorance will be reflected formally in a refusal to allow any assumptions about the cardinality of the domain of L.  Then, there will be arithmetical assertions we can prove under classical arithmetic, but are unprovable in a finitist-nominalist language.  If we allow ourselves no assumption about the cardinality of the domain (or just the assumption that at least one object exists, as Tarski suggests), we cannot even prove that ‘1+1=0’ is false.  So not only are ‘intuitively true’ arithmetical sentences declared false in this language, but chunks of previously provable assertions are no longer susceptible of proof.  There were other suggestions for how to deal with numbers that are ‘too large’ in a finitist-nominalist regime besides (a.)-(c.); however, none meet with substantially more approval from the discussants.

V. Conclusion: Nominalism then and now
There is still a live philosophical debate surrounding nominalism; the most-discussed recent defense of nominalism is likely Hartry Field’s Science Without Numbers (1980).  As mentioned earlier, the project that Field and other current nominalists see themselves pursuing appeared in public in Goodman and Quine’s “Steps toward a Constructive Nominalism”—and the Harvard discussions of 1940-41 prompted this article.  To conclude this essay, I will briefly examine how these discussions, the ‘grandfather’ of the present debates, relate to current arguments pro and contra nominalism.
Although Goodman and Quine say, in their seminal article, that their “refusal” to admit the existence of abstracta “is based on a philosophical intuition that cannot be justified by appeal to anything more ultimate,” modern nominalists do provide a justification for this refusal.  The leading modern argument for nominalism can be cast as a simple syllogism, whose major premise is a concise statement of the causal theory of knowledge.
   P1.  We can only have knowledge of things causally related (or relatable) to us.
   P2.  Numbers and other abstracta are not causally related (or relatable) to us.
   Therefore, we cannot have knowledge of numbers or other abstracta.
Both premises have been challenged by various philosophers.  More criticism has been leveled at the first, presumably because many philosophers consider ‘standing outside’ the causal order a defining feature of an abstract object.  Another variant of this syllogism replaces P1 with a statement of the causal theory of reference: 
   P1R.  We can only successfully refer to things causally related (or relatable) to us.
The conclusion is modified accordingly: we cannot refer to numbers or other abstracta.  And presumably we cannot say much of significance about items to which we cannot successfully refer.  Broadly speaking, neither the causal nor the referential form of this syllogism seems to have won large numbers of converts to the nominalist cause—at least in part because causal theories of knowledge and reference are not terribly popular nowadays.  Causal theories of knowledge and reference did not appear in an explicit, fully-fledged form until the 1960s and 70s, so it is not surprising that Carnap’s 1940-41 notes do not contain explicit statements of the views expressed in P1 and P1R.  However, Tarski (and his sympathizers) would perhaps not deny that we only know about or refer to entities that are somehow causally connected or connectible with us.  After all, if something is a physical object, then (with some exceptions) it is causally connectible to us; and if something is an abstract object, then it is not causally connectible to us.  Though a causal theory of reference does not appear in Carnap’s discussion notes, a related notion may be at work implicitly in the collaborators’ minds.
The primary objection to nominalism today is the ‘indispensability argument,’ usually attributed to Quine and Putnam.  Shortly after the co-authored 1947 Journal of Symbolic Logic paper appeared, Quine rejected nominalism.  (Goodman did not.)  Hilary Putnam and the post-nominalist Quine argued for the existence of mathematical abstracta on the grounds that relinquishing such abstracta would force us to relinquish much of modern science, and that we should be unwilling to pay that price for maintaining nominalist scruples.   Current nominalist projects, such as Hartry Field’s, usually consist of ‘reconstructive’ projects that attempt to rebut the indispensability argument.  In such nominalist projects, a certain field of natural science is recast in a form that does not appeal to any ‘abstract’ entities.  Field claims that if empirical science can be reconstructed nominalistically, then belief in mathematical objects becomes “unjustifiable dogma.” (1980, 9)  The literature on the indispensability argument is vast, and I will not comment upon its current status.  
	Given that the indispensability argument did not appear as such until around 1970, an explicit desire to rebut it cannot be a motivation for undertaking the 1941 project.  However, there is a precursor of the modern indispensability argument in Carnap’s notes.  We encountered it above in IV., when discussing the lower bound on the poverty of a finitist-nominalist language’s expressive power; the relevant section is reproduced here: 
“If S1 [the finitist-nominalist language] does not suffice to reach classical mathematics, couldn’t one perhaps nevertheless adopt S1 and perhaps show that classical mathematics is not really necessary for the application of science to life?  Perhaps we can set up, on the basis of S1, a calculus for a fragment of mathematics that suffices for all practical purposes (i.e. not something just for everyday purposes, but also for the most complicated tasks of technology).  (RCC 090-16-25)

 This is not precisely the program Hartry Field has pursued, but it is similar: in both cases, the aim is to show that a proper subset of mathematics, which is nominalistically acceptable, is sufficient for all applications of mathematics in science.  This leads us to an interesting question: what is the relationship between the modern indispensability argument and the Tarski-Carnap-Quine demand that their ‘understandable’ language be sufficient to express (at least a substantial portion of) mathematics and natural science?  To answer this question, we need an explicit statement of the indispensability argument.  The following is a typical current formulation, due to Colyvan:
1. We ought to have ontological commitment to all and only those entities that are indispensable to our best scientific theories.

2.  Mathematical entities are indispensable to our best scientific theories.

Therefore:

3.  We ought to have ontological commitment to mathematical entities.
(2001, 11)
 
The 1941 project differs from the modern one primarily in the first premise: there is no normative claim concerning ontological commitments explicitly forwarded in the discussion notes.  Nothing in the texts decisively rules out attributing this position to the participants as an implicit belief, but this (potentially anachronistic) interpretation is certainly not forced upon us, either.  Instead, we can view the discussants as replacing normative-cum-ontological issues with the goal of a unified language of science.  Whether failure of a language to meet that aim, regardless of the language’s other merits, automatically disqualifies it in the discussants’ eyes is not discussed in Carnap’s notes, as mentioned above.  We know that Quine, a decade after the Harvard discussions, opts for disqualification: his grounds for eventually repudiating nominalism are that we cannot recover (in a natural way) a sufficient amount of mathematics to do science if we abide by nominalist strictures.  Tarski’s description of himself as a ‘nominalist with a materialistic taint’ in the 1965 ASL meeting suggests that he did not follow Quine.

	Acknowledgements
Over the last two years, Steve Awodey, Paolo Mancosu, and Laura Ruetsche all helped me grapple with the content of Carnap’s dictation notes that form the basis of this essay.  More recently, Karen Frost-Arnold and Doug Patterson helped hammer my ideas about Tarski’s nominalism into a more presentable form.

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