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What is Frege's Relativity Argument?

Published online by Cambridge University Press:  01 January 2020

Palle Yourgrau
Palle Yourgrau*
Affiliation:
Brandeis University, Waltham, MA02254-9110, USA
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Extract

Sets are multitudes which are also unities. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics.

Kurt Gödel (Hao Wang, A Logical Journey: From Gödel to Philosophy)

In what sense can something be at the same time one and many? The problem is familiar since Plato (for example, Republic 524e). In recent times, Whitehead and Russell, in Principia Mathematica, have been struck by the difficulty of the problem: ‘If there is such an object as a class, it must be in some sense one object, yet it is only of classes that many can be predicated. Hence, if we admit classes as objects, we must suppose that the same object can be both one and many, which seems impossible.' It is, however, in Frege's great work, The Foundations of Arithmetic (henceforth, Grundlagen), that many see the final resolution of the old question: how can something be at the same time one and many?

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 27 , Issue 2 , June 1997 , pp. 137 - 172
Copyright
Copyright © The Authors 1997

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References

1 Cambridge, MA: The MIT Press 1996

2 Whitehead, Alfred and Russell, Bertrand Principia Mathematica, 2nd ed. (Cambridge: Cambridge University Press 1925), 72Google Scholar

3 Frege, Gottlob The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number, Austin, J.L. trans. (Evaston, IL: Northwestern University Press 1980)Google Scholar

4 See Begriffsschrift: A Formula Language Modeled Upon that of Arithmetic, for Pure Thought, in Heijenoort, J. Van ed., Frege and Gödel: Two Fundamental Texts in Mathematical Logic (Cambridge, MA: Harvard University Press 1970) 5–82Google Scholar; and Furth, M. ed. and trans., The Basic Laws of Arithmetic: Exposition of the System (Grundgesetze) Berkeley and Los Angeles: University of California Press 1967).Google Scholar

5 A distinction familiar since Plato. See, for a brief discussion, Wedberg, Anders Plato's Philosophy of Mathematics (Stockholm: Almqvist and Wiksell 1955), 22–3.Google Scholar

6 This is another of Frege's doctrines precisely rejected by Wittgenstein: ‘A proposition of mathematics does not express a thought’ (Tractatus Logico-Philosophicus, Pears, D.F. and McGuinness, B.F. trans. [London: Routledge and Kegan Paul 1974], 6.21).Google Scholar

7 But ‘Equality [identity] gives rise to challenging questions which are not altogether easy to answer’ (Frege, ‘On Sense and Meaning,’ Geach, P. and Black, M. eds. and trans., Translations from the Philosophical Writings of Gottlob Frege, 3rd ed. [Totowa, NJ: Rowman and Littlefield 1980], 56)Google Scholar. To resolve these problems Frege distinguished two elements in the signification of a singular term — the object referred to or ‘meant,’ and the ‘conceptual’ mode of presentation of the object in thought or language, which he calls a ‘sense.’ In ‘The Path Back to Frege,’ 118-20, in Yourgrau, P. ed., Demonstatives, Oxford Readings in Philosophy (Oxford: Oxford University Press 1990)Google Scholar, however, I have argued that Frege really needs to distinguish three elements in the use of a referring term, if his semantic theory is to hold together. The third element is the (nondescriptive) ‘grasping’ of senses that Frege himself recognizes in his late essay, ‘Thoughts,’ in Salmon, N. and Soames, S. eds., Propositions and Attitudes (Oxford: Oxford University Press 1988), 33–55Google Scholar. Frege's failure to incorporate this third, nondescriptive, element into his theory of sense and reference leaves the theory vulnerable to a vicious infinite regress. The force of this problem seems to have escaped Martins, David in his discussion of ‘The Path Back to Frege’ in his recent Review of Demonstratives, ‘Demonstratives, Descriptions, and Knowledge: A Critical Study of Three Recent Books,’ Philosophy and Phenomenological Research 54, 4 (1994) 947–63 at 954CrossRefGoogle Scholar. See P. Yourgrau, ‘The Epistemology of Sense and Reference - A Reply to Martins’ (Unpublished).

8 These extensionally individuated objects resemble in certain respects the sets of current axiomatic set theory (or rather, theories), but they also differ from sets as currently understood. Tyler Burge explores in some detail these differences, and introduces comparisons also with other contemporary logico-mathematical notions such as ‘deviant’ set theories and model-theoretic accounts of the lambda calculus. He concludes that ‘as of now, it seems fair to say that Frege’s notion [of the extension of a concept] has not found an intuitively attractive, mathematically fruitful counterpart’ (‘Frege on Extensions of Concepts, from 1884-1903,’ Philosophical Review 43, 1 [1984) 3-34, at 32, n. 19). See, also, Parsons, Charles ‘Some Remarks on Frege's Conception of Extension,’ in Schim, M. Studies on Frege I: Logic and Philosophy of Mathematics Reihe ‘problemata’ Band 42 (Gunther Holzboog: Friedrich Frommann Verlag 1976) 265–77.Google Scholar

9 One must be careful, however, to distinguish the notion of ‘being n in number (relative to some concept-as-unit),’ and (of a concept-as-unit) being ‘assigned’ (uniquely) a number. Strictly, then, the first order concept, ‘card in the deck of cards,’ is not itself ‘assigned’ the number one; rather, this number is ‘assigned’ to a second order concept under which this first order concept uniquely falls. For Frege, to say that there are n Fs, whether the Fs be objects or concepts, is always, as it is with the assertion of existence, in effect to attribute a property to a higher order concept, under which the objects or concepts in question fall. Thus Frege does not really, after all, ‘save,’ in the literal sense, the doctrine that something can be simultaneously ‘assigned’ both the number one and also other numbers. He does, however, by means of the RA, account for the fact that anything can be at once one in number and also some other number, n, in number, relative to some concept-as-unit. Further, he accounts simultaneously for the ‘unity’ and the ‘complexity’ of a number, like two, which, although it is the sum of one plus one, does not contain the number one ‘inside’ itself, ‘twice over.’ (Seen. 40.)

10 See Benacerraf, Paul ‘What Numbers Could Not Be,’ in Benacerraf, P. and Putnam, H. eds., Philosophy of Mathematics: Selected Readings 2nd ed. (Cambridge: Cambridge University Press 1983) 272–94Google Scholar. One could still ask why the natural numbers could not be (abstract) ‘objects’ whose essential properties were exhausted by their relations to the other members of the progression. Indeed, this is exactly the view of natural number that has been attributed to the original mathematical platonist, Plato, by Cherniss, Harold: ‘What distinguishes each of the ideal numbers [of Plato's) from all the rest is its position in this series …. As soon as the essence of each idea of number is seen to be just its unique position as a term in this ordered series, it is obvious that the essence of number in general can be nothing but this very order’ (The Riddle of the Early Academy [Berkeley and Los Angeles: University of California Press 1945), 36)Google Scholar. Further, one could challenge Benacerraf's assumption that the only essential properties possessed by the natural numbers are those structural features investigated by mathematicians. Thus Jerrold Katz argues: ‘In providing non-mathematical properties of numbers such as necessary absence of spatio-temporallocation, causal inertness, etc., realism [or platonism) makes it easy to refute identities like “17 =Julius Caesar” [which Frege gave as a principal reason for introducing his explicit definition of natural number in terms of extensions of concepts] …. The good name of mathematics does not suffer because our knowledge of the numbers goes beyond number theory. Mathematicians do not intend that theory to be more than a theory of the arithmetic structure of the numbers’ (‘Skepticism About Numbers and Indeterminacy Arguments,’ in Morton, A. and Stich, S.P. eds., Benacerraf and His Critics [Oxford: Blackwell 1996] 119–39, at 133-4Google Scholar; insertions mine). What Katz neglects to say, however, is that the basic point he is making originates with Plato, for example in the discussion of the ‘divided line’ of Book VI of the Republic. According to Plato, in the realm of the ‘intelligible’ even the ‘geometers and calculators’ require the ‘dialectical reason’ of the philosophers in order to give a proper account of the mathematical intelligibles — in particular, in order to discharge their hypotheses. (For more on the ‘divided line,’ see below.)

11 There remains the question of whether logicism should itself rely on a formal account of logic. Peter Hylton writes in this regard: ‘If logic is taken to be a formalism then Godel’s theorem shows at once that mathematics is not identical with logic, i.e., that logicism is false. From a Russellian point of view, however, there is no reason to identify logic with a formalism. Rather Godel's theorem seems, from this point of view, to show that the theory of propositional functions — logic itself — cannot be completely formalized’ (Russell, Idealism and the Emergence of Analytic Philosophy [Oxford: Clarendon 1990], 287n.). See, further, Lachterman's, David fascinating study ‘Hegel and the Formalization of Logic,’ Graduate Faculty Philosophy Journal 12, 1-2 (1987) 153–236.CrossRefGoogle Scholar

12 Wright, Crispin Frege's Conception of Numbers as Objects (Aberdeen: Aberdeen University Press 1983)Google Scholar; Boolos, George ‘The Standard of Equality of Numbers,’ in Boolos, G. ed., Meaning and Method: Essays in Honor of Hilary Putnam (Cambridge: Cambridge University Press 1990) 261–77Google Scholar

13 Heck, Richard ‘Frege's Principle,’ in Hintikka, J. ed., From Dedekind to Gödel (Dordrecht: Kluwer 1995), 120–1Google Scholar

14 George Boolos, ‘Is Hume's Principle Analytic?’ (forthcoming in a Festschrift for Michael Dummett, edited by R. Heck)

15 According to Boolos, one of the principal reasons why Hume's Principle cannot be regarded as a principle of logic is precisely because HP can be used to derive the theorem that every finite natural number has a successor — i.e., that there are infinitely many finite natural numbers. One must ask oneself, however, if the truths of arithmetic are necessary. If they are not necessary, then somewhere in the axioms or proofs of theorems in arithmetic some contingent facts that are true only of our own ‘possible world’ must be recorded. This is indeed true of theories in empirical science, where the contingent information is ultimately supplied by experiments founded on sense experience. But of course proofs in arithmetic are precisely not ‘experiments’ in this sense, and for this very reason they are, if valid, valid for all ‘possible worlds.’ The theorems of arithmetic, then, are if true, necessarily true. Whatever formal purposes, then, are served by the ‘finite domains’ invoked by Boolos, it is difficult to see what philosophical sense can be made of the suggestion that there are genuine ‘possible worlds’ where not all of the infinitely many finite natural numbers happen to exist. One way out of this dilemma would be to reject, contra Frege (and Cantor, and Gödel) the actual infinity of the natural numbers [not to say, the ‘greater infinity’ of the real numbers], in favor of mere potential infinity. This seems to be the suggestion of Harold Hodes: ‘The notion of infinity required by mathematics is merely that of a potential infinity’ (Hodes, Harold ‘Logicism and the Ontological Commitments of Arithmetic,’ Journal of Philosophy 81, 3 [1984] 123–49, at 149)CrossRefGoogle Scholar. But potential infinity, as a substitute for actual infinity, rests upon either a (traditional) temporal foundation, or on a modal basis (as Hodes recommends). I have argued, however, in The Disappearance of Time: Kurt Gödel and the Idealistic Tradition in Philosophy (Cambridge: Cambridge University Press 1991), 128-46, that both the temporal foundations of potential infinity (assuming Einstein's Theory of Relativity) and the modal alternative (as outlined, for example, by Charles Parsons), are shaky.

16 See, for instance, ‘Sets and Numbers,’ Nous 15 (1981) 495-513.

17 The issue of Frege's mathematical (and conceptual/semantic) platonism has been a matter of considerable dispute of late. Some, for example Sluga, Hans in Gottlob Frege (London: Routledge and Kegan Paul 1980)Google Scholar, and Weiner, Joan in Frege in Perspective (Ithaca: Cornell University Press 1990)Google Scholar, have argued that Frege is in fact a kind of Kantian (transcendental) ‘idealist.’ Burge, Tyler by contrast, in ‘Frege on Our Knowledge of the Third Realm,’ Mind 101 (1992) 633–50CrossRefGoogle Scholar, maintains that Frege is indeed a platonist; and Dummett, Michael argues, in ‘Frege's Myth of the Third Realm’ (in Dummett, M. Frege and Other Philosophers [Oxford: Clarendon 1991] 249–62)Google Scholar, that not only was Frege a platonist, but that this constituted, in effect, one of his major blunders. My own view is that Frege is indeed a platonist. This is not the place to defend this claim, but an example from Hans Sluga will serve as an illustration. Sluga cites Frege's comment in Grundlagen that ‘In arithmetic we are not concerned with objects which we come to know as something alien from without through the medium of the senses, but with objects given directly to our reason and, as its nearest kin, utterly transparent to it’ (emphasis added). Sluga comments: ‘These remarks closely resemble the traditional rhetoric of rationalist thought, but their most direct link is with Kant's comment about the metaphysics of Nature which is envisaged in the Critique of Pure Reason. “In this field nothing can escape us,” Kant writes. “What reason produces entirely out of itself cannot be concealed, but is brought to light by reason itself“’ (Gottlob Frege, 115). In my opinion, however, the entire context of the Grundlagen makes it clear that what is really being echoed here is Plato's Phaedo, 79d. In describing how the soul communes with the Forms or Ideas, Socrates says, ‘[W]hen the soul investigates by itself it passes into the realm of what is pure, ever existing, immortal and unchanging, and being akin to this, it always stays with it whenever it is by itself and can do so …. [I]t is in touch [then] with things of the same kind’ (Grube, G.M.A. trans. Plato: Five Dialogues [India polis: Hackett 1981] 93–155, emphases added)Google Scholar. I suggest, then, that if Frege can be described as a Kantian transcendental idealist on the basis of the passage cited from the Grundlagen, then Plato, too, should be so described. But presumably Sluga does not wish to paint Plato, too, with the selfsame Kantian-idealist brush!

18 After coining this term I was interested to discover that Nietzsche had, apparently, anticipated its use.

19 ‘Sets, Aggregates, and Numbers,’ Canadian journal of Philosophy 15,4 (1985) 581-92

20 Oxford: Clarendon 1990

21 Menzel, Christopher ‘Frege Numbers and the Relativity Argument,’ Canadian Journal of Philosophy 18, 1 (1988) 87–98CrossRefGoogle Scholar; Moravcsik, Julius Plato and Platonism (Oxford: Blackwell 1992)Google Scholar

22 Tiles, Mary The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise (Oxford: Basil Blackwell 1989)Google Scholar

23 An opponent of the view that some pluralities, viz. sets, escape the RA, while aggregates do not, would appear to be Linda Wetzel. She writes: ‘[That every plurality consists of exactly one definite number of things — for example, a trio consists of exactly 3 things —] is dubious. [Rather,] a plurality of things consists of exactly one definite number of things of a certain sort. So even though a plurality (of pie pieces, say) may consist of some one definite number, 2, of things of a certain sort (half-pies), it may also consist of some one definite number, 4, of things of a different, sort (quarter-pies) …. So Frege was right … [that] something can be, say, both a pair and a quadruple at the same time …. But he was wrong in concluding that therefore being a pair is not an objective property of external things, not a physical property … In summary … Frege's relativity argument does not show that … pluralities are not aggregates’ (‘Expressions vs. Numbers,’ Philosophical Topics 17, 2 [1989]173-96, at 182-3).

24 Indeed, after noting these lines of Frege's, Blanchette, Patricia in ‘Frege’s Reduction’ (History and Philosophy of Logic 15 [1994] 85–103)CrossRefGoogle Scholar, concludes that ‘the account of what it is to assign a number to a concept forms the basis of the analysis of number in general. In particular, to say what it is to “assign” zero to a concept, or to say that zero is the number which belongs to a concept, will be to give an account of the number zero’ (97).

25 Cambridge, MA: Harvard University Press 1991

26 As Wetzel writes in ‘Expressions vs. Numbers’: ‘[T]here is often or always more than one individuating property that applies to a given aggregate …. Yet … [i]t may well be that there is very often only one individuating property that seems natural and appropriate to apply (like being a rabbit, e.g., as opposed to being a one cubic inch undetached rabbit part’ (181).

27 See Simons, Peter ‘Numbers and Manifolds’ and ‘Plural Reference and Set Theory’ in Smith, B. ed., Parts and Moments: Studies in Logic and Formal Ontology (Munich: Philosophia Verlag 1982) 160–260Google Scholar; and Boolos, George ‘To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables),’ Journal of Philosophy 81 (1984) 430–49.CrossRefGoogle Scholar

28 Kessler, Glenn ‘Frege, Mill and the Foundations of Arithmetic,’ Journal of Philosophy 77 (1980) 65–80CrossRefGoogle Scholar

29 Plato does not actually provide, in the Republic, the ‘unhypothetical first principle.’ For some discussion of what he may have had in mind, see Baltzly, Dirk ‘“To An Unhypothetical First Principle” in Plato's Republic,’ History of Philosophy Quarterly 13, 2 (1996) 149–65.Google Scholar

30 For a clear account of the relationship between Frege’s approach to the foundations of arithmetic and those of Dedekind and Peano, see Gillies, Donald Frege, Dedekind and Peano on the Foundations of Arithmetic (Assen, The Netherlands: Van Gorcum 1982)Google Scholar. For a brief account of Frege's kinship with Gödel on this (and related) issues See Yourgrau, P. ‘Kurt Gödel,’ in Edwards, P. ed., The Encyclapedia of Philosaphy Supplement (New York: Macmillan 1996) 220–2.Google Scholar

31 There are a number of lines of comparison that can be traced. (1) The Platonic Form or Idea is the quintessential ‘one over many’; Frege writes, similarly, in Grundlagen, that ‘The concept has a power of collecting together far superior to the unifying power of [Kantian] synthetic apperception’ (61, insertion added). (2) For Plato, ‘participating in’ or ‘sharing in’ a Form represents a fundamental aspect of the structure of reality. Frege states something analogous: “‘a falls under the concept f” is the general form of a judgement-content which deals with an object a’ ( Grundlagen, 83). (Wittgenstein gives the significance of this: ‘The general propositional form is the essence of a proposition. To give the essence of a proposition means to give the essence of all description, and thus the essence of the world’ (Tractatus, 5.471-5.4711). Similarly, Gödel comments that ‘The basis of everything is meaningful predication, such as Px, x belongs to A, xRy’ (Wang, A Logical Journey, ch. 5). (3) A Platonic Idea would seem to be an intensional entity. See, for example, Richard Sharvy, ‘Euthyphro 9d-llb: Analysis and Definition in Plato and Others,’ Nous 6 (1972) 119-37, at 120. But Fregean concepts, and also extensions-of-concepts, are individuated extensionally. Frege does, however, countenance intensional entities in his philosophy, viz., the ‘senses’ of singular terms and functional words referred to earlier (n. 7). So, Fregean ‘senses,’ too, fulfill part of the role played by Platonic Forms or Ideas. (See P. Yourgrau, ‘The Path Back to Frege,’ 125-6.)

The relationships between concepts and ‘senses’ are complex. An object, a, may ‘fall under’ a concept, which may uniquely characterize it, and a may be the referent of a singular term in virtue of being uniquely ‘determined’ by the ‘sense’ of that term. It is not entirely clear what the relationship is between ‘falling under a concept’ and ‘being determined by a “sense.”’ Both relations, moreover, are in important respects disanalogous to Plato's relation of being an instance of a Form, F, since for Plato, an instance of a Form always ‘falls short of,’ is a mere ‘image’ or ‘shadow’ of, the perfect reality of the Form Itself. A closer analogue, in contemporary thinking, to the relation that Plato has in mind is that between a ‘token’ and its ‘type.’ For a helpful discussion of this last relationship, in an arithmetic context, see Linda Wetzel, ‘Expressions vs. Numbers.’ Further, functional-expressions themselves have, it seems, senses; and it is a matter of dispute whether the senses of these terms are themselves functions. See, for a negative answer, Dummett, The Interpretation of Frege's Philosophy (Cambridge, MA: Harvard University Press 1981), ch. 13.Google Scholar

32 Indeed, according to Frege, ‘It is applicability alone which elevates arithmetic from a game to the rank of a science’ (Grundgesetze, 167 in the translation in P. Geach and M. Black, Translations from the Philosophical Writings of Gottlob Frege). Maddy misinterprets this remark, suggesting that Frege's concern here is with ‘applications in natural science,’ a view that she believes relies, ultimately, on ‘naturalism’ (!) (Maddy, Penelope ‘Indispensability and Practice,’ Journal of Philosophy 89, 6 [1992] 275-89 at 275)CrossRefGoogle Scholar. Once again, however, neglect of Frege's relationship to Plato inhibits a correct understanding of his ideas. For Plato the ‘function’ or ‘end’ (telos) of something is what it is ‘good for,’ or ‘the good of it,’ and it is a thesis of Socrates in the Republic that the highest Form, the Form of the Good, has ultimate ontological force, or ‘determines’ Being, itself. Frege, then, literally ‘builds into’ the very ‘definition’ (or ‘essence’) of natural numbers their use or function, i.e. what they are ‘good for,’ viz., counting (the instances of) concepts. Indeed, Peter Simons, in ‘Frege's Theory of Real Numbers,’ brings out the fact that Frege extended this approach to real numbers: ‘[For Frege] … the applicability of real numbers in measurement must be built into their essence …. One of the features of Frege’s theory of natural numbers in [Grundlagen] was precisely that he attempted to build the application of natural numbers in counting into their definition’ (31).

33 Of course, given this explanation on Frege's part, he must admit that, strictly, number is not universally applicable, since after all it is only concepts that are ‘really’ numbered.

34 On the Greek word, ‘aitia,’ I am in agreement with Richard Sharvy, who writes: ‘Nowadays, we try to make sharp distinctions among reasons, causes and explanations. But Vlastos points out [“Reasons and Causes in the Phaedo”] that the Greek word aitia ‘has a much wider signification than that of the English word “cause.”’ My solution to this problem of translation is to use the word “‘cause’ quite deliberately in an extremely general sense, and thus to include all the others as kinds of “causes”’ (‘Plato's Causal Logic and the Third Man Argument,’ Nous 20 [1980] 507-30, sec. 1, at 526-7).

35 Frege does not, of course, explicitly cite the Phaedo, there. He does, however, in his ‘Notes for Ludwig Darmstaedter’ (in Frege, G. Posthumous Writings, Hermes, H. et al., eds. [Chicago: University of Chicago Press 1979] 253–7)Google Scholar refer to an analogous passage in the Hippias Major. His ‘Notes’ open with the words: ‘I started out from mathematics. The most pressing need, it seemed to me, was to provide this science with a better foundation. I soon realized that number is not a heap, a series of things, nor a property of a heap either, but that in stating a number which we have arrived at as the result of counting we are making a statement about a concept (Plato, The Greater Hippias)’ (253).

36 It is easy to imagine Plato himself writing these words from the Grundlagen: ‘[A] sensible impression of a sort does correspond to the word “triangular,” but then the word must be taken as a whole. The three in it we do not see directly’ (32). Likewise: ‘Can it be that a dog staring at the moon does have an idea, however ill-defined, of what we signify by the word “one”?’ (41-2)

37 ‘It would indeed be remarkable if a property abstracted from external things could be transferred without any change of sense to events, to ideas, to concepts’ (Grundlagen, 31).

38 Thus we know that (a): if you ‘physically add’ (i.e. bring together) one dog with one (other) dog you ‘end up with’ — in general, and cetis paribus — two dogs; and we also know that (b): If you ‘really,’ i.e., arithmetically, add the number one (itself) to the number one, the resulting sum is the number two, itself. In the spirit, then, of the Euthyphro, we ask: Is (a) true because of (b)? or: Is (b) true because of (a)? Frege, clearly, with Plato, believes the former.

39 For an illuminating discussion and defense (!) of ‘self-predication’ in the Theory of Forms, see Sharvy, ‘Plato's Causal Logic and the Third Man Argument.’

40 Tait, W.W. in ‘Against Intuitionism: Constructive Mathematics is Part of Classical Mathematics’ The Journal of Philosophical Logic 12 (1983) 173–95CrossRefGoogle Scholar, rejects Aristotelian accounts of Plato, such as Burnyeat's: ‘Aristotle [in Metaphysics M] in fact said that Plato's (form) numbers are finite sets of pure units. But Plato explicitly says that the forms have no parts, e.g., Phaedo 78b-c …. But, as Plato clearly says in Parmenides, this is inadequate. As a set of pure units, 2 is no longer a form. It has a content, and one is left still with the question: what is common to 2 and any other two element set?’ (193, n. 10)

I am inclined to agree with Tait about the ‘unity’ of the Platonic Form, but Burnyeat is also making an important point, viz., that Plato wished both to account for the mathematicals as mathematicians actually manipulated them and to uncover the ontological basis, i.e. the Forms, of the mathematicals themselves. It is not clear, however, that the Theory of Forms, as it stands in the Republic, can simultaneously fulfill both functions, in arithmetic or in geometry. As Burnyeat puts it: ‘Can a line be drawn through a vertex of the Form of Right-Angled Triangle? Can the Form Square turn up in triplicate and in different sizes? Pythagoras’ theorem is true. What is it true of?’ (229) Similarly, in the case of arithmetic, Plato, in my opinion, never really resolves the fact that although the Form of Two is a ‘unity’ and as such has no ‘parts,’ the mathematician's number two is in some sense something ‘complex,’ viz., the sum of one plus one.

Now, for Frege, too, ‘our concern here is to arrive at a concept of number usable for the purposes of science’ (Grundlagen, 69). But although the number two is also, for Frege, in some sense a complex object — viz., an extension ‘containing’ (as it turns out too large) an infinity of concepts — and although it is the value of the plus function for the inputs of one and one, it does not for all that ‘contain’ the number one, and again one, as two of its ‘parts.’ That is, the number two — the extension of the concept, ‘equinumerous with the concept, “identical to zero or identical to one”’ — contains nowhere inside itself the number one — viz., the extension of the concept, ‘equinumerous with the concept, “identical with zero.”’ All that the set-like number two ‘contains,’ for Frege, as ‘members’ (not ‘parts’) is concepts that have, each of them, two instances; it contains, therefore, no numbers, in particular, not the number one (’twice over’). Skeptics of the alleged ‘unity’ of the Christian ‘Trinity’ should bear in mind Frege's powerful defense of the ‘unity’ of Three-Itself, which accounts simultaneously for the fact that 3=1 + 1 + 1.

41 ‘Platonism and Mathematics: A Prelude to Discussion,’ in Graeser, A. ed., Mathematics and Metaphysics in Aristotle (Bern and Stuttgart: Verlag Paul Haupt 1984) 213–40, at 235Google Scholar

42 ‘Russell's Mathematical Logic,’ reprinted in Feferman, S. et al. eds., Kurt Gödel: Collected Works vol. 2 (New York and Oxford: Oxford University Press 1990) 119–41, at 137Google Scholar; emphases and insertion added.

43 Linda Wetzel has drawn my attention to the fact that Bigelow, John in The Reality of Numbers: A Physicalist's Philosophy of Mathematics (Oxford: Clarendon 1988)Google Scholar, puts forward an account of numbers as ‘universals,’ an account that has strong Platonic echoes (in spite of Bigelow's self-described ‘physicalism’). Bigelow, however, still seems to subscribe to the RA, as do Maddy and Menzel. He writes, for example, that ‘aggregates can be divided into parts in many alternative ways. A property, in contrast, cannot be supplied with instances in many alternative ways. It has the instances it has, and that's that …. Properties are, in this way, like sets rather than aggregates. An aggregate can be broken into parts in many ways, but a set has the members it has, and that's that’ (43; except for ‘sets,’ emphases added). I have argued at length, however, that a set, too, can be ‘broken apart in many ways.’ And an aggregate, say, of sheep, ‘has the sheep-parts it has, and that's that.’

44 Similarly, Wang records Gödel as saying: ‘Numbers appear less concrete than sets. They have different representations and are what is common to all representations’ (A Logical journey, ch. 8, at 254).