Peirce's Realistic Approach to Mathematics: Or, Can One Be a Realist without Being a Platonist?
Claudine Engel‑ Tiercelin, Université de Paris
(Charles S. Peirce and the philosophy of science (papers from the Harvard sesquicentennial Congress), ed. by Edward C. Moore, The University of Alabama Press, Tuscalosa and London, 1993 p.30-48)
Peirce's position on the problem of universals is that of a sophisticated realism inherited from the Avicennian‑Scotistic tradition,which may he briefly characterized by its opposition to metaphysical realism (or Platonism) and various forms of nominalism. Against the idea that there could be either a reality totally independent of our ways of knowing or verifying (an incognizable ding an sich) or realities that could be reduced to singular existents (sense‑data, concepts, determinate individuals), Peirce's realism is a unique and ambitious analysis of all the aspects‑logical, physical, and metaphysical‑of the problem of universals, combining scholastic and categorical elements with scientific, pragmatistic, and straightforward metaphysical concerns.
In this chapter I consider how Peirce's realism fits his approach to mathematics, which is often presented as a somewhat incoherent mixture of Platonistic and conceptualistic elements. Without denying these, I shall attempt to claim that the subtlety of Peirce's position not only helps to clear up some of these so‑called inconsistencies but offers many insights for contemporary ways of dealing with the mathematical aspects of the problem of universals.
Peirce's Definition of Mathematics:
Or, How to Apply Realism to Mathematics
Briefly one could say that Peirce's definition of mathematics has two main characteristics. First, it does not cover a special domain of entities.
It is not defined by the specificity of its objects (space, time, quantity) or by the nature of its propositions (analytical, a priori) or by the kinds of truths it can exhibit. Against Hamilton and De Morgan, Peirce denied any dependence of mathematics on space, time, or any form of "intuition" (CP 3.556). As to the analytic or synthetic nature of mathematical propositions, he said almost nothing, convinced that the real issues were elsewhere and that they had to be thought through and formulated in new terms (and especially through the distinction of corollarial and theorematic forms of deductive reasoning). If mathematics has nothing to say about truth, it is because it is not‑contrary to logic‑a science of facts but a science of hypotheses and abstractions.
But for the son of Benjamin Peirce, a second characteristic of mathematic, was that it was also a science of reasoning, more specifically, "the science which draws necessary conclusions" (C/' 3.558; 4.229). This is a very wide definition of mathematics, since not only is all mathematical reasoning diagrammatic, but all necessary reasoning is mathematical reasoning, no matter how simple it may be (NEM 4:47). In that sense, when Peirce affirmed the fundamentally iconic, observational, and experimental character of deduction, he not only defined mathematical deduction as such, he accounted for all kinds of deduction, thus reviving the whole conception of logical necessity.
Such a definition presents obvious difficulties for any kind of realism. First, how is one to adjust the idea of mathematics as a purely formal and ideal system to realism? What is the status of these entia rationis? Are they pure conventions, arbitrarily chosen, which never refer to reality? Are they simple tautological or analytical statements, incapable of being qualified as true except as concerns the meaning of the expressions they involve? Then why insist on the practical side of mathematics? Why take so seriously the problem of its application? If one accepts the notion of applied mathematics as something which is needed by all sciences, what is to warrant that such idealizations have the objective validity which justifies their being used by these other sciences?
Second, if Peirce's realism is a realism of indeterminacy‑which implies, on the one hand, that it should be possible to think without contradiction not only about reals but also about possibles and, on the other hand, that indeterminacy renounces any idea of absolute or infallible
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necessity and exactitude‑how is such indeterminacy going to be taken care of by mathematics? According to Susan Haack, one can measure the importance of the issue by noticing Peirce's many hesitations about answering the question of whether or not fallibilism should be extended to mathematics. Fo .Haack (1979:37), Peirce never really made up his mind, at times declaring that the necessity of mathematics prevents our being wrong about our mathematical beliefs and that we are only fallible as far as our factual beliefs are concerned (CP 1.149, written c. 1897); at other times he emphasized that fallibilism does extend to mathematics (CP 7.108, 1892; 1.248, 1902) and that mathematical inferences are only probable after all. Clearly enough, for Peirce, mathematical necessity was perfectly compatible with the notion of an ideal system in which one reasons. about possibles (hence indeterminates) and not about real cases. Not only does mathematics allow for individuals that are not perfectly determinate in all respects (NEM 4:xiii), but also one can think about the infinite. In that sense, the basic principles of Peirce's realism that are attached to indeterminacy (possibility and generality, or firstness and thirdness) seem to be obeyed. But can one be satisfied with such a definition of necessity?
Peirce's Conventionalism and Conceptualism
Before trying to make sense of Peirce's realism in mathematics, we have to ask if it makes sense to speak of Peirce's realism at all. Indeed, a number of instances seem to suggest that he adopted a conceptualistic, or nominalistic, position in which pure mathematics is the domain of hypotheses and ideal creations. Moreover, its necessity is apparently not due to any characteristics of its objects or to any particular nature of,its propositions that would afford a specific objectivity.
Conceptualism always has realism in its background. Thus, after defining a pure diagram as that which is "designed to represent and to render intelligible, the Form of Relation, merely" and asserting that "an intelligible relation, that is a relation for thought, is created only by the act of representing it" (NEM 4:316 n. 1), Peirce added that if we should some day find out the metaphysical nature of relation, that would not
mean that it would thereby be created, for the intelligible relation surely existed before, in thought or in the way God represented the universe. But this does not necessarily conflict with conceptualism. As David Wiggins (1980:139) says:
Conceptualism properly conceived must not entail that before we got for ourselves these concepts, their extensions could not exist autonomously, i.e., independently of whether or not the concepts were destined to be fashioned and their compliants to be discovered. What conceptualism entail, is only that, although horses, leaves, sun and stars are not inventions or artifact,, still, in order to single. out these things, we have to deploy upon experience a conceptual scheme which has itself been fashioned or formed in such away as to make it possible to single them out.
And this is, indeed, the spirit that animates such Peircean assertions as the following: "All necessary reasoning is reasoning from pure hypothesis in this sense; that if the premise has any truth for the real world, that is an accident totally irrelevant to. the relation of the conclusion to the premise, while in the kinds of reasoning that are more peculiarly topics of logical [rather than mathematical) discussion, it has all the relevancy in the world” (NEM:4:164; cf. CP 4.233, NEM 4:270).
If rnathematics, is "the study of pure hypothesis regardless of any analogies they may have in our universe" (NEM 4:149; CP 3.560), which is particularly clear in the case of arithmetic (NEM 4:xv‑xvi), and if "it certainly never would do to embrace pragmatism in any sense in which it should conflict with this great fact" (NEM 4:157), it would seem that mathematical necessity was derived, not from some necessity in things, but merely from the link of logical consequence between premises and conclusion (CP 4.232) and from the hypotheses, conventions, and rules which the mathematician has chosen to adopt (cf. defs. 32 and 33 in MS 94 [NEM 2:251]). 1
So mathematical systems are purely formal. The meaning of the terms appearing in the postulates, hypotheses, and theorems is totally irrelevant as such: "A proposition is not a statement of perfectly pure mathematics until it is devoid of all definite meaning, and comes to this‑that a property of a certain icon is pointed out and is declared to belong to anything like it, of which instances are given" (CP 5.567). For example,
Peirce's Realistic Approach to Mathematics l 33
the only definitions that have to be retained as conforming to the "dignified meaninglessness of pure algebra" (CP 4.314) are those implicit definitions that postulates impose on their terms (see CP 3.20). In their turn, the postulates will be considered as implicitly defining the objects) to which they apply, in the exact sense in which Riemann declared that the axioms of geometry provide a definition of space (see Murphey 1961:235).
No doubt all this contributes to qualifying Peirce's position as ant-realistic in the sense given by Durnmett (1987) that mathematical propositions have apparently no predeterminate meaning or truth, or such that it should suffice to discover them. Their meaning is such as is postulated, then demonstrated. In mathematics, there are no propositions which could benefit from truth conditions that were utterly independent of our capacity to recognize them as such, or whose truth conditions could be realized without our being able to recognize that they are so. In other words, one should always be able to define the truth conditions of any mathematical statement in terms of its conditions of assertibility. This is clearly stated in MS 94: "The meaning of any speech, writing, or other sign is its translation into a sign more convenient for the purposes of thought; for all thinking is in signs. The meaning of a mathematical term or sign is its expression in the kind of signs in the imaginary or other manifestation of which the mathematical reasoning consists. For geometry, this (expression) is [in] a geometrical diagram" (NEM 2:251).
This is also why one must ordinarily attach great importance to the mathematical procedures of demonstration, to the modus operandi (CP 4.429; NEM 2:10‑11). That is, meaning is not given from the start but, on the contrary, is determined by the demonstration. To reason is not to use meanings; it is to construct them, to manipulate them in order to determine them. And the analysis of the way these constructions work may help in clarifying, if not in constituting, such a determination of meaning (CP 3.363). Hence the great importance of the iconicity of reasoning (CP 2.279; NEM 4:47‑48) but not only of that (Engel‑Tiercelin 1991), for indeed, there are two other essential procedures in the determination of the meaning of mathematical statements. These are abstraction and generalization, both of which allow a better grasp of the status of the entice rationis the mathematician works upon. Hypostatic
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abstraction has a decisive role (NEM 4:49) because it is that operation by which something "denoted by a noun substantive, something having a name," which belongs to the category of substance as such, is transformed into an assertion, and the reality of which "can mean nothing except the truth of statements in which the real thing is asserted" (NEM 4:161‑62; see CP 4.234). Thus, to say that numbers are real is not to reduce them to some singular existing entities but merely to indicate that there are statements in which numerical expressions are used to describe classes adjectivrly (CP 4. 154‑55). Thanks to such abstractions as numbers, lines, or collections, "it thus becomes possible to study their relations and to apply to these relations discoveries already made respecting analogous relations" (CP 3.642). It thus becomes unnecessary to assume some kind of objects preexisting in some kind of mathematical universe.
All this tends to suggest the image of a conceptualist and conventionalist who is more eager to present mathematics, not as a realm of objects to be discovered or of independent truths describing alreadygiven facts and transcending all possibility of verification or refutation, but as a body of rules, practices, mental constructions, procedures of decision, and methods of demonstration from which mathematics derives its instrumental value and the ground of its necessity.
Arithmetic and the Temptation of Platonism
It would be an exaggeration, however, to say that Peirce's position is utterly clear on all these points. It seems indeed difficult to deny that some of his analyses, mostly in arithmetic but not only there, do not sound conceptualistic at all, but realistic, in the most traditional or Platonistic sense of the word.
Thus Peirce did not hesitate to speak of the "innate" propositions of mathematics, preferring that rather than such a term as "a priori," which "involves the mistaken notion that the operations of demonstrative reasoning are nothing but applications of plain rules to plain cases" (CP 4.92). It is, of course, in arithmetic that the temptation of Platonism is the strongest, numbers being at times qualified as "ideas" belonging as such to a different "universe of experience" from that of facts and laws;
Peirce's Realistic Approach to Mathematics l 35
the "Platonistic world of pure forms" (.CP 4.1 18), or that "Inner World" (CP 41.161) in which these eternal, abstract, and "airy nothings" (C!' 6.455) are not "absolutely created by the mathematician" (CP 4.161).2 In that respect, we would not be very far from h'rege's universe of true "thoughts," from "laws of Pure Being," or thoughts independent of the senses and of the empirical world. Indeed, it is in a way that reminds us very much of F'rege inasmuch as the development of the whole theory of numbers is described by Peirce as arising from a small number of first and primitive propositions (CP 2.361; 6.595). For the same reasons, we could understand some of Peirce's assertions as signifying that the truth conditions of any mathematical proposition are transcendent in respect to their conditions of verification and that arithmetic, at least, is an independent domain of entities (cf. CP 4.114).
The Real Nature of Peirce's
Realistic Solution in Mathematics
I would like to claim, however, that these inconsistencies between conceptualistic and Platonistic elements are a wrong way to look at the matter. In fact, they hide the real nature of Peirce's way of dealing with the problem of universals in mathematics and the typical realistic solution he proposes. Let us start with the problem of conceptualistic conventionalism.
Hypotheses and Definitions
First of all, it is clear enough that Peirce's conventionalism is never so absolute as that of, say, a Poincaré, (who is at times criticized and viewed harshly, though wrongly, as such). No doubt, Peirce would not be ready to reduce .mathematics to an ideal science of hypotheses that would consist in nothing more than a simple game of abstract, arbitrary, and convenient formulas.
If mathematics is a matter of creations, these are never totally arbitrary (NEM 4:xii), first, because most often the mathematician's hypoth-
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esis is provoked by some real problem which arises in other sciences (NEM 4:xv).3 This is why the distinction between pure and applied mathematics is not so decisive after all (NEM 2:vi)j.
But the arbitrariness of hypotheses is weakened in another way. Contrary to the poet's hypotheses (Cl' 4.238; NEM 4:68) the mathematician's hypotheses, which indeed require the highest intellectual capacity to be found, are subject to the rules of deduction. A mathematician is interested in hypotheses only for the forms of inference that can be drawn from them (NEM 4:268). Before going any further, let me add that it is surely not enough to invoke the ideal character of hypotheses, especially in geometry, to solve entirely the question of the nature of their content: namely, that topics is the sole part of geometry that is really pure and ideal, because it deals more with pure continua. A lot could be said here about the interface between mathematics and straightforward metaphysics (cf. Engel-Tiercelin 1986 and 1990).
Rules of Deduction and Habits of Reasoning: Our "Hereditary Metaphysics"
If one wishes to define mathematics as a system that is ideal; and yet not arbitrary, one must first be able to verify whether its creations are relevant or just simply not meaningless. This explains why we do not describe surfaces or lines as mere ad hoc constructions. "We here use the traditional phraseology which speaks as if lines and surfaces were something we make. This is not strictly correct. The lines and surfaces are places which are there, whether we think of them or not. They are quite ideal, it is true. But they are there, in this sense that we can think of them as being there without being drawn into any absurdity" (NEM 2:387).
But the lack of arbitrariness is also due, in good part, to the character of logical necessity, that is, to the fact that the rules we follow in our demonstrations do nothing except exhibit the habits we have acquired through reasoning (CP 5.367). The leading principles of inference are but the linguistically codified formulation (logica docens) (CP 1.417) of these habits of reasoning under the form of habits of inference, which
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must not be confused with some psychological constraint (in the mode of Sigwart or Schröder (CP 2.52, 209; 3.432). Even if it is true, in the end,that it is not the constraint exerted by the rule but rather the fact that the conclusion is true when the rules of transformation and substitution are correctly applied (CP 2.153; 5.365) that determines the objective validity of the reasoning, one must not underrate the important role played in inference by habits. It explains why logical necessity is something like a fact which is felt as such and which, for that reason, hardly needs any justification. Here Peirce is very near Wittgenstein: necessity is present in our acts and practices; we stumble against something whose explanation could hardly he anything other than "we use it so. "4
Because logical necessity exerts itself as a constraint difficult to explain and to justify. the teacher in mathematics often has difficulties in trying "to make another person feel the force of that demonstration who does not do so already" (NEM 4:xiv). Peirce's realism comes to the fore here: logical necessity is one of those habit‑facts which are absolutely real and yet irreducibly vague, before .which no further explanation seems appropriate or even required (CP 2.173). Something like a sort of constraint of necessity should be granted here. One can wonder if it is enough to warrant the self‑evident character of mathematics (CP 2.191). But obviously, Peirce would have answered that in such a case, it was precisely the hopeless attempt at asking for some kind of justification that would need to be justified.
But such habits of reasoning also have the strength of general rules (and this is part of Peirce's realism too)‑on the one hand, because they express collective and not subjective practices and presuppose some agreement among the members of the community (which is a decisive constituent of the kind of exactitude that can be reached in mathematics) (CP 5.577) and, on the other hand, because they result from a controlled use ruled by intelligence (or thirdness). Here lies all the difference there is between the mere thinking in images (which is more often a handicap than a help) and the experimenting on images, that is, icons or abstract schemes. Because intelligence rules practice, it is of no consequence in mathematics if one uses formulas or notations which are mere flatus vocis. In fact, the more they are so, the better it is, for it will facilitate experimentation. Hence, the utility of children's rhymes like "eeny,
meeny, miney‑mo" in which one counts words, not things. What is essential is what one does with them, the way one thinks in this notation, for "one secret of the art of reasoning is to think" (NEM 1:136).5
Peirce's realism insists as much on the irreducible vagueness of our habits of reasoning as on the fact that thirdness is the category of generalization, of abstraction, of all the operations of reasoning for which self-control or criticism is always on the lookout. In that sense, mathematics does not constitute an exception in principle to fallibilism. Rather it is, most often, an exception in fact. But this again does not warrant any absolute or fundamental value being assigned to mathematical necessity: "Mathematical certainty is not absolute certainty. For the greatest mathematicians sometimes blunder, and therefore it is possible‑barely possible‑that all have blundered every time they added two and two" (CP 4.478).
Realism as an Alternative to Platonism in Arithmetic
What about Peirce's Platonistic claims in arithmetic? I think they are counterbalanced by other kinds of claims. First, paradoxically, such concepts as "number," "zero," and "successor" are counted not as primitive concepts (Russell) but as mere variables (Peano) (Murphey 1961:238ff.). Peirce tried to construct several systems of pure number (see CP.4.160ff., 677-81; 3.562ff.), giving only an implicit definition of its primitive terms, thus allowing a perfectly formalistic interpretation of his system (Murphey 1961:244‑45).
Second, the system of pure number was for him but a~particular case of ordinals, 6 which were the primitive pure numbers (CP 3.628ff.; 4.332, 657‑59, 673ff., etc.) not only because they expressed a relative place but because they illustrated it, so that Peirce believed they exemplified the pure serial relation which was instantiated in all the series (Murphey 1961:273‑74): "But the highest and last lesson which the numbers whisper in our ear is that of the supremacy of the forms of relation for which their tawdry outside is the mere shell of the casket" (CP 4.681).
But there are reasons other than Peirce's relational realism that ex
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plain his preferences for ordinals, and these are taken either. from sensible experience (CP 4.154) or from straightforward pragmatical considerations. 7
It would be as wrong to overrate any so‑called supremacy of arithmetic above the other sciences for it is as vain to hold to the idea of some compartmentalization of mathematics (CP 4.247) as to underrate the pragmatic considerations that are part of Peirce's way of dealing with numbers: This is true, even though he was careful to distinguish between pure, or "scientific,". arithmetic (CP 3. 562A), which "considers only the numbers themselves and not the application of them to counting," and practical arithmetic (CP 4.163). Peirce spent a lot of time achieving a whole pedagogy for arithmetic. What does this signify, except that in arithmetic, too, what is important is not only the type of objects or propositions8 but consideration of the system, in which the learning of the rules is decisive?
The learning of arithmetic, and the role played in it by iconic representations, was taken by Peirce to be essential and yet responsible for so many errors that he kept reconstructing the steps that ought to be performed in the techniques of education arid of learning.y This is not to say that Peirce would reduce meaning to use (this is why, for example, instinct should not decide for the right interpretation of a system). One should manage to teach this at the pedagogical level. But we can go further than this:. "Some children learn by first acquiring the use of a word, or phrase, and then, long after, getting some glimmer of what it means" (NEM 1:213‑14). So it is surely true that a distinction should be maintained between pure arithmetic, which is "the knowledge of numbers," and practical arithmetic, which is "the knowledge of how to use numbers" (NEM 1:107). But.how are we to understand, in the writings of this so‑called avowed Platonist in arithmetic, such claims as the following one: "The way to teach a child what number means is to teach him to count. It is by studying the counting process that the philosopher must learn what the essence of number is" (NEM 1:214)? Does this not say that, in a certain way, the reality of numbers is to be found as much in the rules determining their meaning as in any primitive prédeterminate meaning which would only need to be discovered in some Realm of Ideas?
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First, it is not easy to determine what Peirce's final position is concerning the mathematical aspects of the problem of universals. However, I have tried to show that we can reconcile the obvious conceptualistic arid conventionalistic claims with an equally obvious Platonism in other respects.
I do not think that Peirce's waverings express merely an ill‑assumed tension between the metaphysically inclined logician, more eager to respect the principles and conclusions of his categorical and realistic analysis, and a kind of natural reflex of the mathematical practitioner who is easily led toward Platonism. Of course, we could always say, to the detriment of Peirce, that Platonism is not easy to avoid, even if, as B. Van Fraassen says (1975:40), Platonists win Pyrrhic victories because the conventionalists fail to provide the good arguments.
But such a conclusion is unsatisfactory. In fact, as I have shown, I am not at all convinced about the seriousness of Peirce's Platonism, which, should we he content with it, would constitute a severe objection to his whole theoretical project.
Second, a number of Peirce's vacillations merely,‑reflect the difficulty of a coherent position on the subject; they.do not always result in confusions but, on the contrary, succeed in stressing the immense complexity of the problem. In that respect, one of the paradoxes and characteristics of Peirce's reflections on mathematics is, by his wide definition of the domain of mathematics, to have pointed out its specificity, namely, that in mathematics, what is most difficult is not the solution of problems but the fact that there are problems to be solved and that consequently acute analysis is required as much~on the methods and on reasoning as on the nature of rnathernatical objects and propositions.
Third, even if Peirce did nut always avoid inconsistencies, because he openly adopted positions which came closer either to conventionalism or to Platonism, he was also perfectly aware of the impossible alternative that would consist in believing that one accounts for the specificity of the mathematician's work and of mathematical invention in terms of an opposition between a mathematics of discovery and a mathematics of invention, and he tried to find a third way to understand why there can be real
Peirce's Realistic Approach to Mathematics l 4
problems in mathematics to be solved. I have tried to show that Peirce's idiosyncratic realism is such a third way, a very interesting attempt to answer the obvious major difficulty for a solution to the problem of universals in its mathematical form. A problém which he himself formulated quite clearly is, how is one to explain that "although mathematics deals with ideas and not with the world of sensible experience, its discoveries are not arbitrary dreams but something to which our minds are forced and which were unforeseen" (NEM 2:346)?
Fourth, Peirce obviously felt himself facing the following difficulty: on the one hand, his anti‑Platonism, which was a natural consequence of his belief in the realism of indeterminacy, forced him to consider mathematics as simple meanings to be displayed or to "use" in practice, and not as truths to be discovered. But on the other hand, realism, or simply good sense, forbade him to adopt some strict form of verificationistic constructivism and to take mathematical demonstrations as pure determinations of meaning, free products of arbitrary creations and constructions.
The stress put on the ideal and hypothetical character of mathematics, the definition of it as the science of pure reasoning, and the fact that questions of method, practice, procedure, and demonstration are at least as important as questions bearing on the nature of objects or propositions all tend to show Peirce's awareness of the fact that a purely realistic answer, in the Platonist sense, cannot constitute a sufficient warrant for the necessity and objectivity of mathematics. For that reason, it is clear that Peirce's view cannot be compared to Frege's type of solution.
Again, if one of the Platonist's arguments consists in assuming a universe of objects, entities, and truths not only independent of, but transcending, our capacity to recognize them, Peirce's version cannot be reduced to such a position. First, because mathematics was not for him a science of truths (for reasons different from Wittgenstein, who, in somewhat related terms, thought that mathematics consisted not of true or false propositions but of autonomous rules of grammar), Peirce did not want mathematical statements, which are hypotheses to be taken as true or false, as describing any kinds of facts whatsoever. Second, Peirce's view cannot be reduced to Platonism because he thought that the meaning of mathematical statements could not be given independently of any
demonstration: in that sense, although Peirce's pragmatistic realism of indeterminacy prevented him from reducing the meaning of a proposition to its conditions of verification, or reducing meaning to use, Peirce never separated the meaning of any mathematical proposition from its conditions of assertibility.
Such a view prevails, even when Peirce seems to show some form of Platonism. For example, although he admits that mathematics has a certain autonomy as an ideal system and goes so far as to talk of it as a universe ruled by dichotomies and truth, whose reality ~lies in its entities subsisting, even when no one is thinking about them or trying to know them, he also adds that if they can be called real ideas, it is because, one day or another, they will be "capable of getting thought," and that is but a question of time (CP 3.527; 6.455). We are far from the Platonist definition of a completely independent and transcendent universe. Faithful to the principle of the impossibility of incognizables, Peirce never defined mathematics as a universe totally independent of our possibility of knowing it, nor did he assume some completely given and prerstablishcd meaning of mathematical statements which was waiting to be discovered, without any construction.
Indeed, Peirce was so interested in "pure numbers" that he tried to construct several systems of pure number. The parallel with the intuitionists seems obvious. In that respect, Murphey has shown (1961:286-87) that the way Brouwer defines a set‑as a law according to which the elements of the set may be constructed and which is not a finished totality of any particular element of it necessarily a finished totality‑is very near to Peirce's analysis, by its generating relation, of an infinite collection. But we have here all the differences between the antimetaphysical verificationism of the intuitionists and Peirce's realism. For example, Peirce would admit with Heyting (1966:15) that mathematical objects must have a consistency which in a way renders them independent of the acts of thought which aim at them, and at the same time he would consider that it makes no sense to think of an existence of these objects independently of any relation to human thought (Heyting 1964:42). But Peirce would find truistic the intuitionist's confusion between the fact of an object to be actually thought by an individual and the fact of an object to be dependent upon some form of general or possible thought, which is
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Peirce's definition of reality. For Peirce, then, intuitionism would be a form of nominalistic Platonism, born from a misunderstanding of the real issues in the problem of universals. No wonder if the intuitionists finally drew anti metaphysical conclusions, for fear of the Platonism in which they fell after all (Heyting 1966:3), and decided to adopt a purely historical and constructivist definition of mathematics by considering only actual mental constructions (ibid.:8).
Even if Peirce used intuitionistic distinctions (e.g., between enumerate, denumerable, and nondenumerable collections), since his realism forced him to admit that one could, without contradiction, talk about possibilities, he could not be identified with intuitionism, for at least two reasons. First, we do not have to adopt strict verificationism (CP 6.455, 4.114) or "actually construct the correspondences" (CP 4.178); second, we can always treat possibilities as forming collections and extend the operations of classical logic (including the law of excluded middle) to such collections (CP 6.185ff. ), even if Peirce also considered borderline cases and multivalued systems. Thus, we can think about the infinite, and Peirce believed that it was by calling up collections of possibilities that the paradoxes of the theory of sets could be avoided. As Murphey said (1961:287), nothing could be more opposed to intuitionism.
Fifth, Peirce has also shown, in defining numbers, collections, and multitudes as entia rationis, that for. him they were inseparable from their conditions of assertability. In that respect, stressing as he did the operations of mathematical reasoning, Peirce also understood that it is necessary to remove the problems from the metaphysical ground and to concentrate first on their elaboration, on the problem of their meaning conditions. As Dummett wrote (1987:2), there is perhaps no hope of settling the argument between those who favor mathematical realism, who hold that we discover mathematical objects, and those who favor an idealist position, for whom mathematical objects are creations or conventions of our mind. The only way to clarify the issue is therefore to place it first at the level of meaning: Peirce's way of dealing with problems in mathematics seems perfectly appropriate to that kind of recommendation.
Finally, it is nonetheless true that Peirce would never dream of renouncing the metaphysical ground either. In that respect his treatment of
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the continuum problem is typical. If there is any reality in mathematics, it is perhaps much less in some particular nature of its objects than in that fundamental idea of the synechistic metaphysics, according to which there is, after all, no difference in nature between the "Inner Universe" (the universe of our representations) and the "Outer Universe" (or reality). One may consider either that such a position is fatal to any serious treatment of the mathematical aspects of the problem of universals or that a Dummettian way of dealing with realism leaves the ontological questions undecided, which in some cases may also be fatal to a grasp of some mathematical issues (think of the debates over the continuum problem). Even if we do not decide in favor of the second solution, I think Peirce's idiosyncratic realism is full of insights that may help clarify the issues and contribute to their ultimate solution.
1. It is indeed easier for arithmetic to reach such a formal ideal of purity, but even if it is difficult, as space is very often a matter of experience (NEM 4:xv), it is necessary (as Riemann showed) to come to .a purely formal conception ofgeometry too (namely, one dealing only with pure continua), so that no distinction could finally be made between geometry and pure arithmetic, both containing analytical propositions, that is, deductions derivable from definitions in apurely logical way with no consideration of their possible empirical validity.Hence the question of some real correspondence between mathematical and real space or between the hypotheses or axioms of geometry and their empirical validity has no real value (NEM 2:251‑52). The reason is that, even if it existed, it could not be demonstrated; real space is a ding an sich; we cannot apply our cognitive devices to it. And even if it were meaningful to say that we do apply them, we should have to consider their fallible character and possible margin oferror. All of this explains why we are justified in adopting constructions that areextremely far from the properties of real space, insofar as they are practical and convenient.
2. As Murphey (1961: 239) rightly points out, we can see how faithful Peirce was to Platonism even After 1885, since he titled the fourth volume of his "Principles of Philosophy" (a project in twelve volumes) "Plato's World: An Elucidation of the Ideas of Modern Mathematics" (1893).
3. It is, indeed, because we find ourselves in such complicated situations
Peirce's Realistic Approach to Mathematics 1 45
that it is impossiblé to determine with exactitude what the consequences could be that one calls for the help of the mathematician (NEM 2:9). So it is most often from such a practical suggestion that the mathematician will "frame a supposition of an ideal state of things," then "study that ideal state of things and find out what would be true in such a case," before generalizing to a third stage from that state of things, namely, by "considering other ideal states of things differing in definite respects from the first" (NEM 2:10). In so doing, "he not only finds out, but also produces a rule by which other similar questions may be answered" (ibid.).
4. Compare Wittgenstein 1974:2:74 and Peirce NEM 4:59: "It is idle to seek any justification of what is evident. It cannot be rendered more than evident."
5. This is why "the student must learn to use notations to think in, but he must not try to make the notation think for him, if he wishes to push his reasonings far. Thinking is done by experimenting in the imagination. Notations arc excellent things to experiment with; but still experimentation requires intelligent supervision to come to much" (in Eisele 1979:186).
6. Contrary to most authors (among whom is Cantor), Peirce thought that ordinal and not cardinal numbers were the primitive pure numbers. A cardinal number, "though confounded with multitude by Cantor, is in fact one of a series of vocables the prime purpose of which, quite unlike any other words, is to serve as an instrument in the performance of the‑experiment of counting" (CP 3.628). In consequence of which, "The doctrine of the so‑called ordinal numbers is a doctrine of pure mathematics; the doctrine of cardinal numbers, or rather, of multitude, is a doctrine of mathematics applied to logic" (CP 3.630). For that reason, Peirce thought that Dedekind could have gone even further when he and others considered "the pure abstract integers to be ordinal . . . . [They] might extend the assertion to all real numbers" (CP 4.633). Ordinals express a relative position in a simple sparse sequence (CP 4.337) and not, as in Cantor, the sequences themselves (Murphey 1961:247, 255). Ordinals only name "places" which are relative characters, which determine classes of members of these sequences. Hence, being classes, ordinals are more general than their members, among which are the collections to which multitudes are attributed.
7. Indeed, if ordinals are more primitive than cardinals, it is also because "the essence of anything lies in what it is intended to do." Now what are numbers? "Simply vocables used in counting. In order to subserve that purpose best, their sequence should stick in the memory, while the less signification they carry the better." The children are quite right in counting with nonsense rhymes, but these are always purely ordinals. Second, "The ultimate utility of counting is to aid reasoning. In order to do that, it must carry a form akin to that
46 / Engel‑Tiercelin
of reasoning. Now the inseparable form of reasoning is that of proceeding from a starting‑point through something else, to a result. This is an ordinal, not a collective idea" (CP 4.658‑59).
8. For example, Peirce was not at all convinced by the superiority in arithmetic of any one system above another; in particular, he did not believe that the decimal system should be more "natural" than, say, the "secundal," or binary, numerical system which he proposed. He went so far as to say that if the ten fingers account for the almost universal use of base ten among all races of humankind, then the decimal system is a monument to human stupidity (NEM 1:237, 241). On the contrary, base six would seem extremely advantageous, even for counting on fingers and toes, although he thought that much the prettiest of the Aryan systems is the secundal. It is, to say the least, extremely convenient in logic, especially in the logic of denumerable and abnumeral series (MS 1121), its major merit lying in "its having several different methods of performing each operation, from which one can at sight select the one most convenient for the case in hand." But, as Carolyn Eisele has pointed out, although aware of its merits, Peirce did not believe "that any propaganda would ever move the world, because there is nothing in secundals to excite the emotional nature" (MS 1, in Eisele 1979:205).
9. See "Teaching Numeration" (MS 179; NEM 1:212ff. ). "Imagination, concentration, generalization"‑such are the qualities that have to be educated. In a seventeen‑page manuscript entitled "Practical Arithmetic" (NEM 1:107ff.), we have a sketch of the maxims of a work to help acquire exactitude and agility in the use of numbers. Most individuals cannot help thinking about an abstract number without accompanying it with colors and forms having no intrinsic connection with the number. One has to take account of such "phantasms" and try to prevent "the formation of associations so unfavorable to arithmetical facility" (NEM 1:213). Again, "The teacher must not fail in his teaching to show the child, at once, how numbers can serve his immediate wishes. The schoolroom cluck should strike; and he must count the strokes to know when he will be free. He should count all stairs he goes up. In school recess, playthings should be counted out to him; and the number required of him. This is to teach the ethical side of arithmetic" (ibid.).
Dummett, M., 1987. (An interview on) Philosophical Doubts and Religious Certaientics. Cogito 2 (1):1-3.
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Eisele, Carolyn. 1979. Studies in the Scietific and Mathematical Philosophy of Charles J . Peirce. Ed. R. M. Martin. The Hague: Mouton.
Engel‑Tiercelin, C. 1986. Le vague est‑il réel? Sur le realisme de C. S. Peirce.
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Haack, Susan. 1979. F'allibilism and Necessity. Synthese 41:37‑63.
Heyting, A. 1964. The Intuitionist Foundations of Mathematics. 1931. In Philosophy of Mathematics: Selected Readings, ed. P. Benaeerraf and H. Putnam, 52‑61. Oxford: B. Blackwell. 2d ed., 1983.
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Van Fraassen, Bas C. 1975. Platonism's Pyrrhic Victory. In The Logical Enterprise, ed. A. R. Anderson, M. R. Barcan, and R. M. Martin, 38‑47. New Haven: Yale University Press.
Wiggins, David. 1980. Sameness arid Substance. Oxford: B. Blackwell.
Wittgenstein, L. 1974; Bermerkungen über die Grundlagen der Mathematik. Frankfurt: Suhrkamp.
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