2010. Peirce on Mathematical Objects and Mathematical Objectivity.

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4 Peirce on Mathematical Objects and Mathematical Objectivity

Claudine Tiercelin

For quite some time now, Platonism has been the target of numerous attacks evenfrom realists of all stripes. Indeed, realism in mathematics is usually taken, as anontological thesis, as the view that mathematics is the scientific study of objectivelyexisting mathematical entities just as physics is the study of physical entities, and asan epistemological and semantic thesis, as the view that the statements of mathe-matics are true or false depending on the properties of those entities, independentof our ability, or lack thereof, to determine which. But Platonism seems to involvemore than this. Not only does it assume that mathematical entities are abstract, out-side of physical space, eternal and unchanging, necessarily existing, regardless of thedetails of the contingent make-up of the world, but it most often takes the knowledgeof such entities to be a priori, certain, as distinguished from fallible scientific knowl-edge. And this is why, even more than for realists, a Platonistic account of mathe-matical reality makes the question of “how we humans come to know the requisite apriori certainties, painfully acute” and produces another mystery concerning the suc-cessful application of the non-spatio-temporal realm to the ordinary physical thingsof the world we live in (Maddy 1992:21).

In particular, if mathematics consists of generalizations of objects such as arenamed, for example by “[t]he words ‘five” and ‘twelve’ . . . which are sizes of sets ofapples and the like” (Quine 1981) is there any justification for positing the existenceof such intangible objects (epistemological question)?1 And if, as has been in theforeground of discussion since the publication of Paul Benacerraf ’s famous paper“Mathematical truth” (Benacerraf 1973),2 reference to objects presupposes thatsuch objects are those that we can observe or otherwise causally interact with, or atleast describe in terms of properties and objects with which we can causally interact,since we do not have physical interactions with numbers or mathematical objects, ifthere be such, which are causally inert, how can we so much as refer to them(semantic question) (Putnam 2001:150)? Following the terms in which Benacerrafposed the dilemma:

64 Claudine Tiercelin

1. There are abstract mathematical entities, independent of language,with no spatio-temporal location nor any contact with the humanmind;

2. Knowledge is defined as a form of causal contact with the knownentities;

3. Therefore, there cannot be any knowledge of abstract entities inmathematics.

Hence it looks as if it were impossible to reconcile the Platonist ontology of abstractentities with its epistemology. “If Platonism is true, we can have no mathematicalknowledge. Assuming that we do have such knowledge, Platonism must be false”(Maddy 1992:37). So it seems that the only way to reconcile ontology and epistemol-ogy is (1) either to resort to a mysterious faculty of intuition in order to explain howsuch a knowledge is possible (Gödel’s strategy is along such lines); or (2) to replythat the causal theories are irrelevant to mathematics, because they are theories of aposteriori, contingent knowledge, while mathematical knowledge is a priori and neces-sary—however, this sort of response is of no use to the “compromise Platonist”who following Quine or Putnam’s revised realism, is ready to question these distinc-tions (Maddy 1992:41); or (3) or simply to get rid of the Platonist ontology: mathe-matical entities are not abstract entities—rather we should conceive them either asforms of mental constructions (along intuitionistic or formalistic or finitistic lines),or as physical entities, or as symbols of a language and adopt some form of nominal-ism.3 Most of the time, the rejection of Platonism has implied the adoption of somekind of anti-realism.

Indeed, such anti-realistic programs are appealing. For example, in Field’snominalistic version, which attempts to undermine the Quine/Putnam indispens-ability arguments by showing how mathematics could be useful in applications with-out being true, it would allow us to say, if it worked, that, strictly speaking, there is nomathematical knowledge, although the use of mathematics in science is neverthelessjustified. The advantage of this picture over the Platonist’s would then be that itdoesn’t leave a puzzling open question in the psychological part of our theory abouthow people come to have reliable beliefs about platonic entities (Maddy 1992:47).But are we ready to pay the cost of it: to give up the use of being clearer about thenature of what knowledge, in general, and mathematical knowledge, in particular,really amount to?

Suppose we are not and that, instead, we try to answer Benacerraf ’s dilemmathrough another route. Is it possible to stick to some kind of realistic approach tomathematics without condemning it to the apparently unsuperable difficulties whichPlatonism seems to face? In recent years, there have been several proposals in thatdirection, and among the most fruitful ones, Maddy’s naturalized version of Pla-

Peirce on Mathematical Objects and Mathematical Objectivity 65

tonism and more recently, Putnam’s latest realistic approach. Although both havegreat merits, which I shall underline as we proceed, they also have some lacunae, andthis is why it might be worth taking a closer look at some earlier suggestions madealong such lines by Charles Peirce. Indeed not only did Peirce identify the majorproblems at issue: how is one to explain that “although mathematics deals with ideasand not with the world of sensible experience, its discoveries are not arbitrarydreams but something to which our minds are forced and which were unforeseen”(Peirce 1894:N2.346)? But he offered a realistic though non-Platonist solution tothis which we might be well inspired to reconsider.4

1. Why Is It So Important to Have Our Ideas Clear about Mathematical Knowledge?

Although some might not take knowledge to be such a fundamental issue to beinvestigated when we deal with the philosophy of mathematics, Peirce was of atotally opposed view. As is shown by his classification of the sciences, mathematicsis in the first position: it is the one discipline that depends on no other, which stands“in no need of Ethics” and “has no need of any appeal to logic” (Peirce1902e:C4.241,242).5

This foundational or architectonic role played by mathematics is related to twomain features it has, in the mind of the son of the mathematician Benjamin Peirce:first, it neither refers to a special academic discipline, nor covers a special domain ofentities; second, it is basically a science of reasoning, more specifically “the sciencewhich draws necessary conclusions” (Peirce 1898d:C3.558, 1902e:C4.229). Indeed itis not defined by the specificity of its objects (space, time, quantity) or by the natureof its propositions (analytical, a priori), or by the kinds of truths it can exhibit.Against Hamilton and De Morgan, Peirce denied any dependence of mathematicson space, time or any form of “intuition” (Peirce 1898d:C3.556). As to the analyticor synthetic nature of mathematical propositions, Peirce said almost nothing, con-vinced that the real issues were elsewhere and that they had to be thought throughand formulated in new terms (and especially through the distinction of corollarialand theorematic forms of deductive reasoning). If mathematics has nothing to sayabout truth, it is because it is not—as opposed to logic—a science of facts, but a sci-ence of hypotheses and abstractions. Such an attitude has two important conse-quences.

First of all, as Hookway rightly noted, “the special foundational role of mathe-matics means that we are free to rely upon its techniques in advance of developing


the substantive notion of truth in the normative sciences . . . Peirce would haveendorsed Wittgenstein’s dictum that mathematics is antecedent to truth. Our expla-nation of the validity of mathematical reasoning must not consist simply in showingthat it is a means to discovering truths: rather, our understanding of truth and realityreflects a prior grasp of mathematical necessities” (Hookway 1985:183). On theother hand, if mathematics is very generally defined as a science of reasoning, itmeans that “all a priori reasoning, all the thinking that we do in paper or in our heads,counts as part of mathematics.” This is why Peirce “wants to offer a uniformaccount of mathematical practice which incorporates both our everyday practice of‘necessary reasoning’ and the more rigorous practice of professional mathemati-cians. Any proposition, he thinks, can be looked upon as a mathematical theory andused as a starting point for mathematical reasoning” (Hookway 1985:182). And thisexplains why, in particular, Peirce thought that his “special business [was] to bring . .. modern mathematical exactitude into philosophy, and to apply the ideas of mathe-matics in philosophy” (Peirce 1887-1909:Ms-L387) (quoted in (Eisele 1979b:277)).

In the second place, since not only is all mathematical reasoning diagrammatic,but all necessary reasoning is mathematical reasoning, no matter how simple it maybe (Peirce 1902a:N4.47), when Peirce affirmed the fundamentally iconic, observa-tional, and experimental character of deduction, he not only defined mathematicaldeduction as such, but accounted for all kinds of deduction, thus reviving the wholeconception of logical necessity.

At the outset, such a position faces obvious difficulties, at least when we try tothink of it in terms of our initial Benacerrafian dilemma. First of all, Peirce’s denialthat mathematics depends upon logic obliges him to some kind of rejection of (atleast straightforward) Platonism. Mathematics does not attempt to discover a special(Fregean) range of facts or truths, or to reveal to us a particular part of reality; butthen how are we to account for the “objective validity” of its statements and results?The situation gets worse when we consider another element of Peirce’s position:indeed, such a rejection of Platonism often goes hand in hand with attributing theuncontroversial character of mathematical results to the fact that they lack content.They simply reflect arbitrary conventions, and are tautologous or analytic proposi-tions, true in virtue of the meanings of expressions contained in them. However, it isnot clear at all that Peirce’s rejection of Platonism drives him to a straightforwardconceptualistic conventionalism nor to the view that mathematical statements aredevoid of content. On the contrary, his conception of (both corollarial and, evenmore, theorematic) deduction seems to go against such a view. This might explainwhy, in Peirce’s own formulation of the difficulties to be met, he insists so much onthe fact that we should be able to explain also how it is that we should come tounforeseen discoveries in mathematics, in other words, that mathematics should


present “as rich and apparently unending a series of surprising discoveries as anyobservational science” (Peirce 1885:C3.363). As Hookway notes: “If mathematicsdoes not study a range of distinctive facts, how can it provide useful and surprisingresults?” (Hookway 1985:184). In other words, how is one to adjust the idea ofmathematics as a purely formal and ideal system to realism? What is the status ofthese entia rationis? Are they pure conventions, arbitrarily chosen, which never referto reality? Are they simple tautological or analytical statements, incapable of beingqualified as true except as concerns the meaning of the expressions they involve?Then why insist (as Quine and Putnam will also do in reply to all kinds of anti-real-isms), on the practical side of mathematics? Why take so seriously the problem of itsapplication? For obviously, its results can be applied in thinking about problems thatarise in other areas of inquiry: we can use mathematics to help with difficulties inphysics, psychology or philosophy. If one accepts the notion of applied mathematicsas something which is needed by all sciences, what is to warrant that such idealiza-tions have the objective validity which justifies their being used by these other sci-ences (Tiercelin 1993:31)?

But Peirce faces another problem. If his realism is a realism of indeterminacy(as a consequence of his scholastic and Scotistic reformulation of realism),6 whichimplies, on the one hand, that it should be possible to think without contradictionnot only about reals but also about possibles (hence also about individuals that arenot perfectly determinate in all respects (Eisele 1976:Vol. 4, p. xiii), and about theinfinite) and, on the other hand, that indeterminacy renounces any idea of absoluteor infallible necessity and exactitude—how is such indeterminacy going to be takencare of? Clearly enough, for Peirce, mathematical necessity was perfectly compatiblewith the notion of an ideal system in which one reasons not merely about what isactually the case but about “a whole general range of possibility” (Peirce1902e:C4.232), about possibles and not about real cases. And he holds that we canknow with certainty that mathematical results hold in all possible worlds, not just inthe actual one. We can achieve a certainty which seems to go beyond what is foundin the natural sciences (Peirce 1902e:C4.237). According to Hookway

Our knowledge is a priori and requires no observation of external objects; and it isuncontroversial—we are practically infallible for ‘only blundering can introduce error inmathematics’ (Eisele 1976:Vol. 4, p. xv). Consensus on the correct answer to a problemwill be held up only by stupidity, a slip, or a failure to understand the terms of a problem.There is no reason here to talk of a long run. (Hookway 1985:184)

However, things are not so clearcut, and one can measure the importance of theissue by noticing Peirce’s many hesitations about answering the question of whetheror not fallibilism should finally be extended to mathematics. For Haack (Haack


1979:37), Peirce never really made up his mind, at times declaring that the necessityof mathematics prevents our being wrong about our mathematical beliefs and thatwe are only fallible as far as our factual beliefs are concerned (Peirce 1893c:C1.149);at other times he emphasized that fallibilism does extend to mathematics (Peirce1902b:C1.248, 1910(?):C7.108) and that mathematical inferences are only probableafter all. In that sense, the basic principles of his realism that are attached to indeter-minacy (possibility and generality, or firstness and thirdness) would be obeyed. Butcan one be satisfied with such a definition of necessity (Tiercelin 1993:32)?

Finally, and this has to do with the epistemological worry embedded in Benac-erraf ’s second premise, since Peirce makes reference to abstract objects—sets, num-bers, structures—which are not existing individuals, how is it that we come to knowabout them, since it looks as if we can neither perceive nor causally interact withthem? Or is it finally the case that we do in fact perceive them, and causally interactwith them? And if this is the case, how?7

So to a certain extent, not only does Peirce realize the acuteness of the issuesunderlined by Benacerraf ’s dilemma, but he adds more worries to it: not only isthere a possible inconsistency between our ontology and our epistemology if westick to Platonism and adopt the prerequisite of some causal account of the way werelate to the mathematical realm, at least if we are (as we should be) interested inunderstanding what mathematical knowledge amounts to—but even if we give up aPlatonistic ontology, we should be able not only to understand how we have contactwith such an ideal domain (in particular since we have to account for its usefulnessin other domains in realistic terms), but also to explain why so many new and“unforeseen” things happen in it . How is Peirce going to handle all these problems?

2. Peirce’s qualified rejection of Platonism: where does the “essence” of number lie?

Before getting into the details of Peirce’s solution, it might be well to understandwhy and, to begin with, to what extent exactly, Peirce did (or did not) reject Pla-tonism in mathematics. Indeed, such a claim might be misleading. Undoubtedly,Peirce’s position is far from clear on all these points. Indeed, some of his analyses,mostly in arithmetic but not only there, sound realistic, even in the most traditionalor 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 “apriori,” which “involves the mistaken notion that the operations of demonstrativereasoning are nothing but applications of plain rules to plain cases” (Peirce


1893d:C4.92). It is, of course, in arithmetic that the temptation of Platonism is thestrongest, numbers being at times qualified as “ideas” belonging as such to a differ-ent “universe of experience” from that of facts and laws; the “Platonistic world ofpure forms” (Peirce 1893d:C4.118) or that “Inner World” (Peirce 1897d:C4.161) inwhich these eternal, abstract, and “airy nothings” (Peirce 1908c:C6.455) are not“absolutely created by the mathematician” (Peirce 1897d:C4.161).8

In that respect, we would not be so far from Frege’s universe of true“thoughts,” from “laws of Pure Being,” or thoughts independent of the senses andof the empirical world. Indeed, it is in a way that reminds us very much of Frege thatthe development of the whole theory of numbers is described by Peirce as arisingfrom a small number of first and primitive propositions (Peirce 1893f:C6.595,1901a:C2.361). For the same reasons, we could understand some of Peirce’s asser-tions as signifying that the truth conditions of any mathematical proposition aretranscendent in respect to their conditions of verification and that arithmetic, atleast, is the study of an independent domain of entities (Peirce 1893d:C4.114).

However, such a Platonism, even in arithmetic, needs a lot of qualifications(Tiercelin 1993:39ff). First, and paradoxically, such concepts as “number,” “zero,”and “successor” are counted not as primitive concepts (Russell) but as mere vari-ables (Peano) (Murphey 1961:238ff). Peirce tried to construct several systems ofpure number (Peirce 1897d:C4.160ff, 1898a:C6.77-81, 1898d:C3.562ff), giving onlyan implicit definition of its primitive terms, thus allowing a perfectly formalisticinterpretation of his system (Murphey 1961:244-245). Second, and importantly, thesystem of pure number was for him but a particular case of ordinals which were theprimitive pure numbers (Peirce 1901a:C3.628, 1904a:C4.332, 1908a:C4.657-659,673ff,etc.) not only because they expressed a relative place but because theyillustrated it, so that Peirce believed they exemplified the pure serial relation whichwas instantiated in all the series (Murphey 1961:273-274):9 “But the highest and lastlesson which the numbers whisper in our ear is that of the supremacy of the formsof relation for which their tawdry outside is the mere shell of the casket” (Peirce1908a:C4.681). Indeed, and contrary to most authors (among whom is Cantor),Peirce thought that ordinal and not cardinal numbers were the primitive pure num-bers. A cardinal number, “though confounded with multitude by Cantor, is in factone of a series of vocables the prime purpose of which, quite unlike any otherwords, is to serve as an instrument in the performance of the experiment of count-ing” (Peirce 1901a:C3.628). In consequence of which “the doctrine of the socalledordinal numbers is a doctrine of pure mathematics; the doctrine of cardinal num-bers, or rather, of multitude, is a doctrine of mathematics applied to logic” (Peirce1901a:C3.630). 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” (Peirce 1908d:C4.633).

But there are other interesting reasons besides his relational realism that mightexplain Peirce’s preferences for ordinals, and these are taken either from sensibleexperience (Peirce 1897d:C4.154) or from straightforward pragmatical consider-ations. Indeed, if ordinals are more primitive than cardinals, it is also because “theessence of anything lies in what it is intended to do.” Now what are numbers? “Sim-ply vocables used in counting. In order to subserve that purpose best, their sequenceshould stick in the memory, while the less signification they carry the better.” The chil-dren are quite right in counting with nonsense rhymes, but these are always purelyordinals. Besides, “the ultimate utility of counting is to aid reasoning. In order to dothat, it must carry a form akin to that of reasoning. Now the inseparable form ofreasoning is that of proceeding from a starting-point through something else, to aresult. This is an ordinal, not a collective idea” (Peirce 1908a:C4.658-659). We havehere two important features to reflect upon: it would be as wrong to overrate anysocalled supremacy of arithmetic above the other sciences for, it is as vain to hold tothe idea of some compartmentalization of mathematics (Peirce 1902e:C4.247) (asQuine and Putnam will later insist on),10 as it is to underrate the pragmatic consider-ations that should be part of our way of dealing with numbers: this explains in par-ticular why even if Peirce was careful to distinguish between pure, or “scientific,”arithmetic (Peirce 1898d:C3.562A), which “considers only the numbers themselvesand not the application of them to counting,” and practical arithmetic (Peirce1897d:C4.163), he also 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 onlythe type of objects or propositions but consideration of the system, in which the learningof the rules is decisive?

As far as the first aspect is concerned, Peirce was not at all convinced, forexample, by the superiority in arithmetic of any one system above another; in partic-ular, 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 tosay that if the ten fingers account for the almost universal use of base ten among allraces of humankind, then the decimal system is a monument to human stupidity(Peirce 1912(?):N4.241, n.d.-r:N4.237). On the contrary, base six would seemextremely advantageous, even for counting on fingers and toes, although he thoughtthat 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 abnumeralseries (Peirce n.d.-r), its major merit lying in “its having several different methods ofperforming each operation, from which one can at sight select the one most conve-nient 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 theworld, because there is nothing in secundals to excite the emotional nature” ((Peirce1903(?)-a), cited in (Eisele 1979d:205)).

As far as the second aspect is concerned, the learning of arithmetic, and the roleplayed in it by iconic representations, was taken by Peirce to be essential and yetresponsible for so many errors that he kept reconstructing the steps that ought to beperformed in the techniques of education and of learning: in an important text, enti-tled “Teaching Numeration” (Peirce 1893e:N1.212ff), Peirce emphasizes such quali-ties as “Imagination, concentration, generalization” which he thinks have to beeducated. In a seventeenpage manuscript entitled “Practical Arithmetic” (Peircen.d.-o:N1.107ff), we have a sketch of the maxims of a work to help acquire exacti-tude and agility in the use of numbers. Most individuals cannot help thinking aboutan abstract number without accompanying it with colors and forms having nointrinsic connection with the number. One has to take account of such “phantasms”and try to prevent “the formation of associations so unfavorable to arithmeticalfacility” (Peirce 1893e:N1.213). Again, “the teacher must not fail in his teaching toshow the child, at once, how numbers can serve his immediate wishes. The school-room clock should strike; and he must count the strokes to know when he will befree. He should count all stairs he goes up. In school recess, playthings should becounted out to him; and the number required of him. This is to teach the ethical sideof arithmetic” (Peirce 1893e:N1.213).

This is not to say that Peirce would reduce meaning to use (this is why, forexample, it is not instinct that should settle the right interpretation of a system). Oneshould manage to teach this at the pedagogical level. But we can go further thanthis: “Some children learn by first acquiring the use of a word, or phrase, and then,long after, getting some glimmer of what it means” (Peirce 1893e:N1.213-214). So itis surely true that a distinction should be maintained between pure arithmetic, whichis “the knowledge of numbers,” and practical arithmetic, which is “the knowledge ofhow to use numbers” (Peirce n.d.-o:N1.107). But how are we to understand, in thewritings of this, in some respects, avowed Platonist in arithmetic, such claims as thefollowing one: “The way to teach a child what number means is to teach him tocount. It is by studying the counting process that the philosopher must learn whatthe essence of number is” (Peirce 1893e:N1.214)? Is it not clear enough then that,for the most part, the reality or essence of numbers is to be found, at least as much inthe rules determining their meaning as in any primitive predeterminate meaningwhich would only need to be discovered in some Platonic or Fregean Realm ofIdeas?


3. Mathematical Objectivity and the Temptation of Conventionalism

Before trying to make sense of Peirce’s non-platonistic realism in mathematics, we havefirst to ask whether it makes sense to speak of Peirce’s realism at all. Indeed, a num-ber of instances seem to suggest that he adopted a conceptualistic, or even nominal-istic, position in which pure mathematics was viewed by him as the domain of purehypotheses and ideal creations. Moreover, its necessity is apparently not due to anycharacteristics of its objects or to any particular nature of its propositions that wouldafford a specific objectivity. “All necessary reasoning is reasoning from pure hypoth-esis in this sense that if the premise has any truth for the real world, that is an acci-dent totally irrelevant to the relation of the conclusion to the premise, while in thekinds of reasoning that are more peculiarly topics of logical [rather than mathemati-cal] discussion, it has all the relevancy in the world” (Peirce 1903n:N4.164) (cf.(Peirce 1895(?)-b:N4.270, 1902e:C4.233).

If mathematics is “the study of pure hypothesis regardless of any analogies theymay have in our universe” (Peirce 1898d:C3.560, 1903n:N4.149), which is particu-larly clear in the case of arithmetic (Eisele 1976:Vol. 4, pp. xv-xvi), and if “it certainlynever would do to embrace pragmatism in any sense in which it should conflict withthis great fact” (Peirce 1903n:N4.157), it would seem that mathematical necessitywas derived, not from some necessity in things, but merely from the link of logicalconsequence between premises and conclusion (Peirce 1902e:C4.232) and from thehypotheses, conventions, and rules which the mathematician has chosen to adopt(cf. Defs. 32 and 33 in (Peirce 1894:N2.251)). If that is the case, Peirce would be veryclose to what is usually referred today as an “if-thenist” approach.11

So mathematical systems are purely formal. The meaning of the terms appear-ing in the postulates, hypotheses, and theorems is totally irrelevant as such: “A prop-osition is not a statement of perfectly pure mathematics until it is devoid of alldefinite meaning, and comes to this—that a property of a certain icon is pointed outand is declared to belong to anything like it, of which instances are given” (Peirce1901a:C5.567). For example, the only definitions that have to be retained as con-forming to the “dignified meaninglessness of pure algebra” (Peirce 1902e:C4.314)are those implicit definitions that postulates impose on their terms (see (Peirce1867:C3.20)). In their turn, the postulates will be considered as implicitly definingthe objects to which they apply, in the exact sense in which Riemann declared thatthe axioms of geometry provide a definition of space (see (Murphey 1961:235)).

No doubt all this contributes to qualifying Peirce’s position as anti-realistic in thesense given by Dummett (Dummett 1987), that mathematical propositions have


apparently no predeterminate meaning or truth, or such that we just needed to dis-cover them. Their meaning amounts to what we postulate, then demonstrate. Inmathematics, there are no propositions which could benefit from truth conditionsthat were utterly independent of our capacity to recognize them as such, or whosetruth conditions could be realized without our being able to recognize that they areso. In other words, one should always be able to define the truth conditions of anymathematical statement in terms of its conditions of assertibility. This is clearlystated in Ms 94:

The meaning of any speech, writing, or other sign is its translation into a sign more con-venient for the purposes of thought; for all thinking is in signs. The meaning of a math-ematical term or sign is its expression in the kind of signs in the imaginary or othermanifestation of which the mathematical reasoning consists. For geometry, this (expres-sion) is [in] a geometrical diagram. (Peirce 1894:N2.251)

This is also why one must ordinarily attach great importance to the mathematicalprocedures of demonstration, to the modus operandi (Peirce 1895(?)-a:N2.10-11,1903b:C4.429). That is, meaning is not given from the start but, on the contrary, isdetermined by the demonstration. To reason is not to use meanings; it is to con-struct them, to manipulate them in order to determine them. And the analysis of theway these constructions work may help in clarifying, if not in constituting, such adetermination of meaning (Peirce 1885:C3.363). Hence, as we shall see, the greatimportance of the iconicity of reasoning (Peirce 1895:C2.279, 1902a:N4.47-48) butnot only of that (Tiercelin 1991); for indeed, there are two other essential proce-dures in the determination of the meaning of mathematical statements. These areabstraction and generalization, both of which allow a better grasp of the status ofthe entia rationis the mathematician works upon. Hypostatic abstraction has a decisiverole (Peirce 1902a:N4.49) because it is that operation by which something “denotedby a noun substantive, something having a name,” which belongs to the category ofsubstance as such, is transformed into an assertion, and the reality of which “canmean nothing except the truth of statements in which the real thing is asserted”((Peirce 1903n:N4.161-162); see (Peirce 1902e:C4.234)). Thus, to say that numbersare real is not to reduce them to some singular existing entities but merely to indi-cate that there are statements in which numerical expressions are used to describeclasses adjectively (Peirce 1897d:C4.154-155). Thanks to such abstractions as num-bers, lines, or collections, “it thus becomes possible to study their relations and toapply to these relations discoveries already made respecting analogous relations”(Peirce 1901a:C3.642). It thus becomes unnecessary to assume some kind of objectspre-existing in some kind of mathematical universe.


All this tends to suggest the image of a conceptualist and conventionalist whois more eager to present mathematics, not as a realm of objects to be discovered orof independent truths describing already given facts and transcending all possibilityof verification or refutation, but as a body of rules, practices, mental constructions,procedures of decision, and methods of demonstration from which mathematicsderives its instrumental value and the ground of its necessity.

If that is the case, nothing would differentiate Peirce from many anti-realisticaccounts that have been given, as a reaction to the problems linked with Platonism,either through some version of (Brouwerian or Dummettian) intuitionism or verifi-cationism or constructivism, or some kind of (Hilbertian) formalism, Fregean logi-cism, Carnapian conventionalism, or even straightforward nominalism, simplyholding that there are no mathematical entities.

Such was, for example, Carnap’s position: there are no logical objects of anykind, and the laws of logic and mathematics are true only by arbitrary conventions.Thus mathematics is not, as the Platonist insists, an objective science. The advantageof this otherwise counterintuitive view is that mathematical knowledge is easilyexplicable: it arises from human decisions. Question: Why are the axioms of Zer-melo-Fraenkel set theory true? Answer: Because they are part of the language we’veadopted for using the word ‘set’. It is indeed one easy way to solve Benacerraf ’s puz-zle.

An important problem with such an approach, as Quine was to note, is that itpresupposes the possible separation of the mathematical from the physical language.Now, how are we to separate the conventionally adopted mathematical part of thelanguage from the factually true physical hypotheses? Quine argues that it isn’tenough to say that the scientific claims, not the mathematical ones, are supported byempirical data:

The semblance of a difference in this respect is largely due to overemphasis of depart-mental boundaries. For a self-contained theory which we can check with experienceincludes, in point of fact, not only its various theoretical hypotheses of so called naturalscience but also such portions of logic and mathematics as it makes use of. (Quine1954:367)

Mathematics is part of the theory we test against experience, and a successful testsupports the mathematics as much as the science (Maddy 1992:27).

More generally, the adoption of a kind of “if-thenism” in mathematics isplagued by a number of difficulties, and among them: Which logical language isappropriate for the statement of premises and conclusions? Which premises are tobe presupposed in cases like number theory, where assumptions are usually leftimplicit? From among the vast range of arbitrary possibilities, why do mathemati-


cians choose the particular axiom systems they do to study? What were historicalmathematicians doing before their subjects were axiomatized? What are they doingwhen they propose new axioms? And so on (Maddy 1992:27). And even more seri-ously:

How can the fact that one mathematical statement follows from another be correctlyused in our investigation of the physical world? The general thrust of the if-thenist’sreply seems to be that the antecedent of a mathematical if-then statement is treated as anidealisation of some physical statement. The scientist then draws as a conclusion thephysical statement that is the unidealization of the consequent. Notice that on this pic-ture, the physical statement must be entirely mathematics-free; the only mathematicsinvolved is that used in moving between them. Unfortunately, many of the statements ofphysical science seem inextricably mathematical. (Maddy 1992:27)

In other words, “the if-thenist account of applied mathematics requires that naturalscience be wholly non-mathematical but it seems unlikely that science can be sopurified (Maddy 1992:27). And this is at the heart of Quine’s and Putnam’s indis-pensability argument; as Putnam notes:

“Mathematics and physics are integrated in such a way that it is not possible to be a real-ist with respect to physical theory and a nominalist with respect to mathematical theory”(Putnam 1975:74). From which he concludes that talk about “mathematical entities isindispensable for science . . . therefore we should accept such [talk]; but this commits usto accepting the existence of the mathematical entities in question. This type of argu-ment stems, of course, from Quine, who has for years stressed both the indispensabilityof [talk about] mathematical entities and the intellectual dishonesty of denying the exist-ence of what one daily presupposes” (Putnam 1971:347).

In other words, it looks as if we were, after all, committed to the existence ofmathematical objects because they are indispensable to our best theory of the world,and we accept that theory. Now it may well be that Peirce was already aware of theacuteness of the problem and that it is the reason why his conventionalism was, inits turn too, somewhat qualified.


4. Three Peircian Keys to Account for Our Knowledge of (and Causal Contact with) Mathematical Objects: Perceptual Abduction, Iconic Observation and Experimentation, Normative Habit-rules of Reasoning

First of all, it should be noted that Peirce’s conceptualism always has realism in itsbackground. Thus, after defining a pure diagram as that which is “designed to repre-sent and to render intelligible, the Form of Relation, merely” and asserting that “anintelligible relation, that is a relation for thought, is created only by the act of repre-senting it” (Peirce 1906(?):N4.316, n1), Peirce added that if we should some day findout the metaphysical nature of relation, that would not mean that it would thereby becreated, for the intelligible relation surely existed before, in thought or in the way Godrepresented the universe.12 More precisely, there are at least three ways in whichPeirce’s conventionalist conceptualism is counterbalanced by strong realistic ele-ments, two of which have to do with his understanding of the concept of “hypothe-sis” itself. It is clear enough that Peirce’s conventionalism is never so absolute as thatof, say, a Poincaré (who is at times criticized and viewed harshly, though wrongly, assuch). No doubt, Peirce would not be ready to reduce mathematics to an ideal sci-ence of hypotheses that would consist in nothing more than a simple game ofabstract, arbitrary, and convenient formulas.

Secondly, if mathematics is a matter of creations, these are never totally arbi-trary (Eisele 1976:Vol. 4, p. xii), first, because most often the mathematician’shypothesis is provoked by some real problem which arises in other sciences (Eisele1976:Vol. 4, p. xv). It is, indeed, because we find ourselves in such complicated situ-ations that it is impossible to determine with exactitude what the consequencescould be that one calls for the help of the mathematician (Peirce 1895(?)-a:N2.9). Soit is most often from such a practical suggestion that the mathematician will “framea supposition of an ideal state of things,” then “study that ideal state of things andfind out what would be true in such a case,” before generalizing to a third stage fromthat state of things, namely, by “considering other ideal states of things differing indefinite respects from the first” (Peirce 1895(?)-a:N2.10). In so doing, “he not onlyfinds out, but also produces a rule by which other similar questions may beanswered” (Peirce 1895(?)-a:N2.210). This is why the distinction between pure andapplied mathematics is not so decisive after all (Eisele 1976:Vol. 2, p. vi).

(1) But the arbitrariness of hypotheses is weakened in another way, which canonly be understood if one sees the very strong and original links Peirce drawsbetween an abductive origination of innovative hypotheses and perception itself,13 and we surely


have here an extremely original account of the reasons why and the way in whichour mind comes into causal connection with mathematics:

It is the truth that the nodus of any mathematical proof consist precisely in a judgmentin every respect similar to the perceptual judgment except only that instead of referringto a percept forced upon a perception, it refers to an imagination of our creation. Thereis no more why or wherefore about it than about the perceptual judgment, “This whichis before my eyes looks yellow.” (Peirce 1903u:C7.659).

So there is a compulsive ingredient (Secondness) in mathematical proof: however,Peirce’s view is that it never arises as such, but already as the experience of generalityor continuity (or Thirdness); indeed, it is so true that “general principles are reallyoperative in Nature” and that Thirdness “pours in upon us in our very perceptualjudgments, and all reasonings, so far as it depends on necessary reasoning, that is tosay, mathematical reasoning, turns upon the perception of generality and continuityat every step” (Peirce 1903p:C5.150).14

How does such a causal though at once theorized encounter take place at the levelof perception? Briefly (i) “Nothing is in the intellect that was not first in the senses,”so that a perceptual judgement is indeed the starting point of all reasoning andknowledge (Peirce 1903r:E2.227); but it is just as crucial—in order to avoid mereSensationalism (Peirce 1903q:E2.223)—to demonstrate that (ii) perceptual judg-ments contain general elements (Peirce 1903q:E2.223-224) and that (iii) “abductiveinference shades into perceptual judgments without any sharp line of demarcationbetween them” so that “our first premisses, the perceptual judgments are to beregarded as an extreme case of abductive inferences” (Peirce 1903q:E2.223-224), orthat, in other words, “the abductive faculty, by which we divine the secrets of nature,is, as we may say, a shading off, a gradation of that which in its highest perfection wecall perception” (Peirce 1903q:E2.224).

Such true connecting links between abductions and perceptions, mid-waybetween a seeing and a thinking, are illustrated via several examples and experiencessuch as optical illusions (the serpentine line or the Schroeder staircase) in which “weseem at first to be looking at the steps from above; but some unconscious part of the mindseems to tire of putting that construction upon it and suddenly we seem to see thesteps from below, and so the perceptive judgment and the percept itself seem tokeep shifting from one general aspect to the other and back again” (Peirce1903r:E2.228) (my emphasis). In such cases, “the point is that there are two ways ofconceving the matter,” both being “general ways of classing the line, general classes underwhich the line is subsumed. But the very decided preference of our perception forone mode of classing the percept shows that this classification is contained in theperceptual judgment” (Peirce 1903r:E2.228).


But Peirce alludes to a second slightly different class of phenomena, not per-taining to mere optical illusions but which seem to involve both our constitution asa natural tendency to interpret and some intentional characteristics of the objects them-selves.15 Finally, there is a third category of experiences which takes us even furtherinto the “thinking” rather than the “seeing” part of the scale of experience: forexample, “we can repeat the sense of a conversation but we are often quite mistakenas to what words were uttered. Some politicians think it a clever thing to convey anidea which they carefully abstain from stating in words. The result is that a reporteris ready to swear quite sincerely that a politician said something to him which thepolitician was most careful not to say” (Peirce 1903r:E2.229).

In all three cases, “if the percept or perceptual judgment were of a natureentirely unrelated to abduction, one would expect that the percept would be entirelyfree from any characters that are proper to interpretations, while it can hardly fail tohave such characters if it be merely a continuous series of what discretely and con-sciously performed would be abductions.” From this, it is easy to conclude that “acertain theory of interpretation of the figure has all the appearance of being given inperception” (Peirce 1903r:E2.228) and that “nothing is more familiar (especially toevery psychology student) as the interpretativeness of the perceptive judgment. It isplainly nothing but the extremest case of Abductive Judgment” (Peirce1903r:E2.229).

Hence we better understand what such a compulsive or causal perception of gener-ality consists in: (i) It must not be wholly general but must take singularity intoaccount, and “the singular is that which reacts” (Peirce 1903q:E2.209). Now itshould be clear that “reaction is existence” and that “the perceptual judgment is thecognitive product of a reaction” (Peirce 1903q:E2.210) (my emphasis). (ii) Singularity itselfinvolves some generality and/or vagueness: singularity must not be thought of astotally opposed to generality, since “the being of a singular may consist in the beingof other singulars which are its parts” (Peirce 1903q:E2.208).16 (iii) While being, inits first ordinary meaning, of the nature of predication and representation, generalityis better understood as continuity or Thirdness (Peirce 1903o:E2.184): it is because, asis the case with general laws of nature, it can “produce physical effects” (Peirce1903o:E2.184), but also because it “involves . . . possibilities absolutely beyond allmultitude” (Peirce 1903o:E2.184), which is another word for continuity. To see how“Thirdness ‘pours upon’ us at every avenue of sense,” it suffices, for example toexperience that “whatever the psychical process may be, we seem to perceive a gen-uine flow of time, such that instants melt into one another without separate individ-uality” (Peirce 1903r:E2.238). (iv) The perception of generality is a perception lessof general elements or characters than of general classifications or forms: it is not only“every general element of every hypothesis” which is initially “given somewhere in


perception”—we have to admit that “every general form of putting conceptstogether is, in its elements, given in perception” (Peirce 1903r:E2.229; Turrisi1990:486). Hence we literally perceive generals or universals and not mere propertiesof things: applied to what happens at the level of our mathematical perception, itmeans that nothing prevents us from literally dealing with universals (a viewdefended, for example, in (Bigelow 1988)).

In order to have a clear grasp of Peirce’s causal account and of his explanationof the way in which we get “surprising” and “unforeseen” results, it is important toget the exact meaning of such an abductive logic of perception. Peirce is clear that, inclaiming that perception contains general elements, he “certainly never intended tobe understood as enunciating any proposition in psychology”: he intended to stayon logical ground (Peirce 1903q:E2.210). And from a such a standpoint, to say that“perceptual judgements contain general elements” can only mean that it is possibleto show that “universal propositions are deducible from them in the manner inwhich the logic of relations shows that particular propositions usually, not to sayinvariably, allow universal propositions to be necessarily inferred from them” (Peirce1903r:E2.227). Indeed, “among those opinions which [he has] constantly main-tained, is this, that while abductive and inductive reasoning are utterly irreducible,either to the other or to deduction, or deduction to either of them, yet the only ratio-nale of these methods is essentially deductive or necessary” (Peirce 1903p E2.206).Another constantly maintained opinion is that of inference, as something which iscontrolled and critical (Peirce 1903o:E2.188), in which one belief not only followsafter another, but follows from it” (Peirce 1893a:C4.53) (Peirce’s emphasis), whichproduces an “acceptance” of the conclusion in the mind of the reasoner (Peirce1902f:C2.148), and whose validity (however weak it may be) implies that whateverappears as a result in the conclusion must already be present in the premises. As allcommentators have noted (Frankfurt, Anderson, Roth, Kapitan, Hintikka), this isone of the tricky aspects in Peirce’s account of the logic of abduction: how can onemaintain that, in the inference, the abductive result, unless loosing a “perfectly defi-nite logical form,” should not contain elements foreign to its premises (Peirce1903r:E2.231) and yet ensure the specific function (or justification) of abduction,i.e., to be “the only logical mechanism which introduces any new idea” (Kapitan1990:499ff; Peirce 1903p:E2.205, 1903q:E2.216). Besides, since a novel hypothesis issaid to “result from” an abductive argument (Peirce 1902c:C2.96), how can such anhypothesis first emerge in the conclusion of an abduction without the reasoneralready assuming that the hypothesis has explanatory merit? The only way to breakthe circle is to provide an explanation which, in some way or other, analyzes abduc-tion as emerging in the course of reasoning itself (Kapitan 1990:505). Peirce’s claim


is that this happens precisely as soon as one analyzes what goes on as a kind of seeingor perceiving. Let us recall the structure of abductive inference:

The surprising fact, C, is observed;But if A were true, C would be a matter of course.Hence, there is reason to suspect that A is true. (Peirce 1903r:E2.231)

There is a crucial objection to dismiss: “It may be said that granting that the abduc-tive conclusion ‘A is true’ rests upon the premiss, ‘If A is true, C is a matter ofcourse’, still it would be contrary to common knowledge to assert that the anteced-ents of all conditional judgments are given in perception, and thus it remains almostcertain that some concepts have a different origin.” To this, Peirce has an answer:

The entire logical matter of a conclusion must come from the uncontrolled part of themind. But self-control is the character which distinguishes reasonings from the pro-cesses by which perceptual judgments are formed, and self-control of any kind is purelyinhibitory. It originates nothing. Therefore, it cannot be in the act of adoption of an infer-ence, in the pronouncing of it to be reasonable, that the formal conceptions in questioncan first emerge. It must be in the first perceiving that so one might conceivably reason. And whatis the nature of that? I see that I have distinctively described the phenomenon as a “per-ceiving.” I do not wish to argue from words, but a word may furnish a valuable sugges-tion. What can our first acquaintance with an inference, when it is not yet adopted, bebut a perception of the world of ideas? In the first suggestion of it, the inference must bethought of as an inference, because when it is adopted there is always the thought that soone might reason in a whole class of cases. But the mere act of inhibition cannot intro-duce this conception. The inference must, then, be thought of as an inference in the firstsuggestion of it. (Peirce 1903r:E2.233) [All emphases added, except the first]

Any inference involves three steps: (a) colligation (or conjoining of distinctpropositions into a whole and itself asserting a result, itself a type of deductive rea-soning (Kapitan 1990:500; Peirce 1893g:C2.442-443, 1898b:C5.579); (b) observation“the most essential part of reasoning” (Peirce 1901a:C2.605); and (c) judgment,which includes an acceptance that what is observed in the premises yields, by fol-lowing a rule, that conclusion, and thereby, an acceptance of the conclusion itself(Peirce 1893(?):C7.459, 1897(?)-c:C2.444). Hence it is in the second part of the infer-ential process, i.e., the contemplating or observational part, which is uncontrollableor irresistible (Peirce 1893g:C7.555), that abduction suggests itself under the formof the seeing not so much of a general character or feature as of a generalizingthought (Peirce 1892a:C6.146). The reasoner has “the thought that the inferred con-clusion is true because in an analogous case an analogous conclusion would be true”(Peirce 1903p:C5.130), which has all the appearance of a general rule. This is what


happens in Peirce’s favorite illustration of the process, viz., Kepler’s discovery ofelliptical orbits (Peirce 1902c:C2.96). As Kapitan rightly notes:

what one observes is not the hypothesis by itself, nor the bare conditional, but rather,that one may reason from such and such background information to the said conditional.So observing is not itself inferential; no control is exercised in the “having” of an insightthat one might reason in a given manner—e.g., that given the background information, ifMars moves in an elliptical path then it would have such and such longitudes—for this isthe novel uncontrolled insight into the “world of ideas,” into Thirdness as given in per-ception (Peirce 1901c:C7.198, 1903p:C7.198, 1903q:C5.160,173, 1903r:C5.209,212).Instead control enters when one decides to “adopt” the inference, thereby judging theconclusion to be true insofar as it is a consequence of information one began with. Thecreative insight, recorded in the conditional premiss, occurs when one initially observesthat one can reason in a certain way. So while the novel hypothesis is generated within anepisode of reasoning, it does not itself occur as the result of a controlled inferential step.(Kapitan 1990:506)

But one must also remember that “the perceptive judgement is the result of aprocess, although of a process not sufficiently conscious to be controlled, or to stateit more truly, not controllable and therefore not fully conscious” (Peirce1903q:E2.211), and that the observation phase itself should rather be viewed asimplying some subconscious, continuous, not fully controllable inferential pro-cesses; this is why it neither amounts to some “immediate consciousnesss of gener-ality, direct experience of generality” (Peirce 1903p:E2.207) nor to “an infinite seriesof acts of criticism each of which must require a distinct effort” such as are repre-sented in the sophism of Achilles and the tortoise (Peirce 1903r:E2.227). If Peircenotes that “Generality, Thirdness, pours upon us in our very perceptual judgments,and all reasoning, so far as it depends on necessary reasoning, that is to say, mathe-matical reasoning, turns upon the perception of generality and continuity at everystep” (Peirce 1903p:E2.207), it is because the abductive suggestion is more correctlydescribed (i) on the one hand as “coming to us like a flash”, as “an act of insight,although of extremely fallible insight” (Peirce 1903r:E2.227): “It is true that the dif-ferent elements of the hypothesis were in our minds before; but it is the idea of put-ting together what we had never dreamed of putting together which flashes the newsuggestion before our contemplation.” (Peirce 1903r:E2.227), and (ii) on the otherhand, as the construction of an icon or a diagram of the hypothetical (or may-be)state of things (Peirce 1903p:C5.148), “the observation of which leads us to suspectthat something is true, which we may or may not be able to formulate with preci-sion, and we proceed to inquire whether this is true or not” (Peirce 1903q:E2.212).As Turrisi has noted, just as in mathematics where the reasoner is supposed to “see”the fact that the argumentation of the single diagram is not merely single but of a


general nature, perception functions as “an abstractive observation” of the general-ity of a necessity in the relation of certain features of a diagrammatical icon (Peirce1893a:C7.467, 1897(?)-c:C2.227, 1903p E2.206). In the case of abduction there is“an abstractive perception of what seems to be a really operative generality (a gen-eral law of nature; a necessity) in the relation of certain features of the perceptualfacts, and a perception of the generality of the possibility, or, more exactly, the plausi-bility of the supposition that this real generality is the explanation of the relation ofthese features” (Turrisi 1990:484).

Such a suggestive power of the Icons which “merely suggest the possibility ofwhat they represent, being percepts minus the insistency and percussivity of per-cepts,” since “in themselves, they are mere Semes, predicating of nothing, not somuch as interrogatively” may sound mysterious. It is indeed “a very extraordinaryfeature of Diagrams that they show—as literally as a Percept shows the PerceptualJudgment to be true,—that a consequence does follow, and more marvellously yet,that it would follow under all varieties of circumstances accompanying the premises”(Peirce 1906(?):N4.317-318). To talk of a “perceiving” may sound a bit odd, Peirceadmits: “But what better can we do in logic?” (Peirce 1903q:E2.213) (see (Tiercelin2005a:400)).

And to say the least, it is much clearer and more precise than the fuzzy descrip-tion that we might find in Gödel’s account of our “intuition” of mathematical enti-ties.

(2) Now, Peirce provides a second level of explanation which confirms the viewthat, although we are dealing with an abstract and ideal realm of hypotheses, ourminds are still solidly “hooked” on the world, and this is related to his analysis oficonic observation and experimentation in mathematical deduction (see (Tiercelin1991:197-206, 1995:61-65)). Peirce’s central idea about necessary deductive reason-ing is indeed that it proceeds by construction of diagrams which are a species oficons, the essential feature of which is to be able to represent the formal sides ofthings, so that they have less a function of resemblance to their objects than ofexemplification or exhibition. Also they are formal and not mere empirical images,and it requires some efforts of hypostatic abstraction for such “skeletons” to be rep-resented (Peirce 1896e:C3.434), hence to feel the difference between the thing andits copy. Why are icons so useful in deductive reasoning? First, because it is a directconsequence of the fundamental pragmatist principle according to which all thoughtis in signs, but is constituted by signs; and second, because Peirce is convinced of theinability of mere symbols to convey any information unless they be accompanied byindices and icons (Peirce 1893d:C4.127, 1895:C2.278). Moreover, one of the distinc-tive traits of icons is their capacity to show a necessity, a would-be (Peirce1906b:C4.532). Hence their decisive role in the conviction we get of the necessary


character of our inferences (Peirce 1906(?):N4.318). Such an iconic capacity isreflected by diagrams as well as by algebraic formulas, which we are more prone toconsider as mere conventional compounded signs (Peirce 1895:C2.279). On thecontrary, Peirce argues, though it is true that icons are rendered such through therules of commutation, association, and attribution of symbols, and that their like-ness is aided by conventional rules, the symbolic character is in them secondarycompared with the iconic one (Peirce 1885:C3.363). This is mainly because of oneremarkable property of the icon: namely, that “by the direct observation of it, othertruths concerning its object can be discovered than those which suffice to determineits construction’” (Peirce 1895:C2.279) (cf. (Peirce 1885:C3.363)).

An important thesis follows from this: it is unnecessary to draw any distinction,under such heading at least, between the exact sciences and the physical sciences; asin any science, in logic and mathematics, one makes observations, experiments, con-firms hypotheses. The only difference is that observations do not arise from exter-nal objects but from models or diagrams constructed by our imagination (Peirce1895(?)-a). The iconic experimentation warrants a sort of accord between the modeland the original. Just as in the chemist’s case, we consider that the model does notallow us to grasp the object itself (which is always general, since the chemist is notinterested in the sample, but in the molecular structure), in the same way the icongives us a hold on reality. Very importantly, this confirms that mathematical reason-ing thus exhibits some form of a relation (Peirce 1906b:C4.530). What finally justifiesthe interpretation of mathematical reasoning is, Peirce argues, such an iconic charac-ter, which shows the form of a relation achieving a sort of isomorphism between thetheory and the reality to which it applies. As Peirce stresses, “the icon does not standunequivocally for this or that existing thing; its Object may be a pure fiction, as to itsexistence, much less is its Object necesarily a thing of a sort habitually met with. Butthere is one assurance that the icon does afford in the highest degree. Namely thatwhich is displayed before the mind’s gaze—the Form of the Icon, which is also itsobject—must be logically possible.” Accordingly “icons are specially requisite forreasoning’” since it “has to make its conclusion manifest’” (Peirce 1893d:C4.127,1906b:C4.531) (cf. (Peirce 1895:C2.278)).

How does such ideal experimentation work in deductive reasoning? Peircethinks that on this point he has made an important discovery, viz., that “all mathe-matical reasoning is diagrammatic and that all necessary reasoning is mathematicalreasoning” (Peirce 1902a:N4.47-48). A reasoning is diagrammatic, when it “con-structs a diagram according to a precept expressed in general terms, performsexperiments upon the diagram, notes their results, and expresses them in generalterms” (Peirce 1902a:N4.47-48). How then does the procedure work? First themathematician has to


state his hypothesis in general terms; second to construct a diagram, whether an array ofletters and symbols with which conventional “rules” or permissions to transform, areassociated, or a geometrical figure, which not only secures him against any confusion ofall and some, but puts before him an icon by the observation of which he detects relationsbetween the parts of the diagram other than those which were used in its construction.This observation is the third step. The fourth step is to assure himself that the relationwould be found in every iconic representation of the hypothesis. The fifth and final step,is to state the matter in general terms. (Peirce 1900(?):N3.749)

As can be seen, reasoning combines symbolic and iconic procedures. But Peirce doesnot only underline the observational and experimental character of the procedure, but healso notes that the diagram is not limited to a construction from data alreadyinscribed in the premises, and that it would suffice to idealize. He considers thatproperly speaking, there is no demonstration, unless one modifies or adds somethingto the initial diagram (this is why he finally judges that a machine is unable to go sofar as a demonstration).

From this, Peirce draws a new distinction, between two possible forms ofdeduction, corollarial and theorematic: his first real discovery, as he calls it (Peirce1902a:N4.49). (See (Tiercelin 1991:200ff).) Corollarial deduction is such that “it isonly necessary to imagine any case in which the premisses are true in order to per-ceive that the conclusion holds in that case” (Peirce 1902a:N4.38). The corollary is“deduced directly from propositions already established without the use of anyother construction than one necessarily suggested in apprehending the enunciationof the proposition” (Peirce 1903(?)-b:N4.288). Theorematic reasoning on the otherhand, yields surprises: “it is necessary to experiment in imagination upon the image ofthe premiss in order from the result of such experiment to make corollarial deduc-tions to the truth of the conclusion” (Peirce 1902a:N4.38). Thus a theorem can onlybe demonstrated from previously established propositions if “we imagine somethingmore than what the condition (indicated in the premisses) supposes to exist” (Peirce1903(?)-b:N4.288).

Yet, and even if the aim of the distinction is to stress that the real inventive kindof mathematical reasoning is performed by theorematic deduction, insofar as itobviously calls for imagination, for invention in experimenting upon the icon, andfor widening the context of our hypotheses by supposing more than is strictlyrequired, the basic lesson to be learned from this is that even in the simplest corol-larial deduction (even the simplest syllogism), something like an iconic representa-tion is required—rather than a strict distinction between three kinds of deduction(syllogism, corollarial deduction), we only have a hierarchy of degrees in iconicity.17

(3) Finally there is at least a third and extremely important element in the expla-nation Peirce provides of the type of close connection we have with mathematics,


and this has to do with the way he understands the rules/habits of deduction thatwe follow, in both naturalistic and normative terms, as being part of our “HereditaryMetaphysics.” Indeed, this is invoked as one of the reasons why the mathematician’shypotheses differ from the poet’s hypotheses (Peirce 1902e:C4.238): they are subjectto the rules of deduction. A mathematician is interested in hypotheses only for theforms of inference that can be drawn from them (Peirce 1895(?)-b:N4.268).18 So thelack 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 exceptexhibit the habits we have acquired through reasoning (Peirce 1877:C5.367). Theleading principles of inference are but the linguistically codified formulation (logicadocens) (Peirce 1896b:C1.417) of these habits of reasoning under the form of habitsof inference, which must not be confused with some psychological constraint (in themode of Sigwart or Schröder (Peirce 1896e:C3.432, 1901a:C2.209, 1902c:C2.52).Even if it is true, in the end, that it is not the constraint exerted by the rule but ratherthe fact that the conclusion is true when the rules of transformation and substitu-tion are correctly applied (Peirce 1877:C5.365, 1902f:C2.153) that determines theobjective validity of the reasoning, one must not underrate the important role playedin inference by habits. It explains why logical necessity is something like a fact whichis felt as such and which, for that reason, hardly needs any justification. Here Peirceis very near Wittgenstein: necessity is present in our acts and practices; we stumbleagainst something whose explanation could hardly he anything other than “we use itso.”19

Because logical necessity exerts itself as a constraint difficult to explain and tojustify, the teacher in mathematics often has difficulties in trying “to make anotherperson feel the force of that demonstration who does not do so already” (Eisele1976:Vol. 4, p. xiv). Peirce’s scholastic realism comes to the fore: logical necessity isone of those habit-facts which are absolutely real and yet irreducibly vague, beforewhich no further explanation seems appropriate or even required (Peirce1902f:C2.173). 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(Peirce 1902f:C2.191). But, at least to some extent,20 Peirce’s answer consists in say-ing that it is precisely the hopeless attempt to ask for some kind of justification thatwould need to be justified.


Conclusions. Mathematics without Ontology?

As is well known, the two major realistic reactions to the various brands of anti-real-ism generated by the difficulties met by Platonism have shown their own limits. Onthe one hand, the brand that arose from the Quine/Putnam’s indispensabilityarguments21 was accused of being unable to account for unapplied mathematics22

and, in making mathematics enter only at fairly theoretical levels, of “leaving unac-counted for precisely the obviousness of elementary mathematics” (Parsons 1979/80:151); hence a major defect in the Quine/Putnam realistic compromise: its unfaith-fulness to mathematical practice.

However the other most famous brand of realism, instantiated by Gödel,23

although more in keeping with the actual experience of doing mathematics, insofaras it seems to be a better account of the reasons the most elementary axioms of settheory are obvious: the view that they “force themselves upon us as being true”(Gödel 1947/1964:484) involves positing a faculty of mathematical intuition thatplays a role in mathematics analogous to that of sense perception in the physical sci-ences, but which remains somewhat mysterious. Hence a current objection toGödelism, namely, “its lack of a straightforward argument for the truth of mathe-matics” (Maddy 1992:35).24 Should we then conclude with Putnam that “nothingworks”?25

I would like to suggest that Peirce’s realistic though non-Platonistic approachmay be more convincing than those that have been currently proposed and, basi-cally, for the following reasons.

To be sure, it is not always easy to reconcile Peirce’s obvious conceptualistic andconventionalistic claims with an equally obvious Platonism in other respects. How-ever, such waverings may not merely express an ill-assumed tension between themetaphysically inclined logician, more eager to respect the principles and conclu-sions of his categorical and realistic analysis, and a kind of natural reflex of themathematical practitioner who is easily led toward Platonism. Of course, we couldalways say, to the detriment of Peirce, that Platonism is not easy to avoid, even if, asB. van Fraassen says (van Fraassen 1975:40), Platonists win Pyrrhic victories becausethe conventionalists fail to provide the good arguments. But such a conclusion isunsatisfactory. Indeed a number of Peirce’s vacillations may also be viewed as merelyreflecting the difficulty of a coherent position on the subject—as I already pointedout in (Tiercelin 1993)—as well as the great complexity of the problems at hand. Inthat respect, one of the paradoxes and characteristics of Peirce’s reflections on math-ematics is, by his wide definition of the domain of mathematics, to have pointed outits 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,an acute analysis is required as much of the methods and of reasoning as of the natureof mathematical objects and propositions. In other words, there is always a strong epis-temological worry in Peirce’s attitude: he is concerned about the type of knowledge wedeal with and the way we reach it in mathematics. In particular, he was aware of theimpossible alternative that would consist in believing that one accounts for the spec-ificity of the mathematician’s work and of mathematical invention in terms of anopposition between a mathematics of discovery and a mathematics of invention, and triedto find a third way to understand why there can be real problems in mathematics tobe solved, and in his view, the basic ones were a combination of the difficulties high-lighted by Benacerraf ’s dilemma, to which one should add the following (andimportant) one: How is one to explain that “although mathematics deals with ideasand not with the world of sensible experience, its discoveries are not arbitrary dreamsbut something to which our minds are forced and which were unforeseen” (Peirce1894:N2.346) (my emphasis)? I think these additions to our original problems mayexplain what may at first sight look like a too complex attitude both towards Pla-tonism and towards Conventionalism.

Peirce obviously felt himself facing the following difficulty: (1) On the one hand, his anti-Platonism, which was a natural consequence of

his belief in the realism of indeterminacy embedded in his scholastic realism, forcedhim to consider mathematics as simple meanings to be displayed or to “use” in prac-tice, and not as truths to be discovered. The stress put on the ideal and hypotheticalcharacter of mathematics, the definition of it as the science of pure reasoning, andthe fact that questions of method, practice, procedure, and demonstration are atleast as important as questions bearing on the nature of objects or propositions, alltend to show Peirce’s awareness of the fact that a purely realistic answer, in the Pla-tonist sense, cannot constitute a sufficient warrant for the necessity and objectivityof mathematics, and his 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, enti-ties, and truths not only independent of, but transcending, our capacity to recognizethem, Peirce’s version cannot be reduced to such a position. First, because mathe-matics was not for him a science of truths (for reasons different from Wittgenstein’s,who, in somewhat related terms, thought that mathematics consisted not of true orfalse propositions but of autonomous rules of grammar), Peirce did not want math-ematical statements, which are hypotheses to be taken as true or false, to be con-strued as describing any kinds of facts whatsoever. Second, Peirce’s view cannot bereduced to Platonism because he thought that the meaning of mathematical state-ments could not be given independently of any demonstration; in that sense,although Peirce’s pragmatistic realism about indeterminacy prevented him from


reducing the meaning of a proposition to its conditions of verification, or reducingmeaning to use, Peirce never separated the meaning of any mathematical proposi-tion from its conditions of assertibility. In that respect, stressing as he did the opera-tions of mathematical reasoning, Peirce also understood, in a very modern way, thatit is necessary to remove the problems from the metaphysical ground and to concen-trate first on their elaboration, on the problem of their meaning conditions. As Dum-mett wrote (Dummett 1987:2), there is perhaps no hope of settling the argumentbetween those who favor mathematical realism, who hold that we discover mathe-matical objects, and those who favor an idealist position, for whom mathematicalobjects are creations or conventions of our mind. The only way to clarify the issue istherefore to place it first at the level of meaning: Peirce’s way of dealing with prob-lems in mathematics seems perfectly appropriate to that kind of recommendation.Such a view prevails, even when he seems to show some form of Platonism. Forexample, although he admits that mathematics has a certain autonomy as an idealsystem 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 themor trying to know them, he also adds that if they can be called real ideas, it isbecause, one day or another, they will be “capable of getting thought,” and that isbut a question of time (Peirce 1897b:C3.527, 1908c:C6.455). We are far from thePlatonist definition of a completely independent and transcendent universe. Faithfulto the principle of the impossibility of incognizables, Peirce never defined mathe-matics as a universe totally independent of our possibility of knowing it, nor did heassume some completely given and pre-established meaning of mathematical state-ments which was waiting to be discovered, without any construction.26

(2) But on the other hand, realism, or simply good sense, forbade him to adoptsome strict form of verificationistic constructivism and to take mathematical dem-onstrations as pure determinations of meaning, free products of arbitrary creationsand constructions. Indeed, how could one understand, in such a fashion, the con-straint exerted on us by mathematical necessity, or again explain the simple origina-tion of hypotheses and the occurrence of so many right results in mathematics?Because of such plain facts, he did not flatly reject the intuitions present in Pla-tonism, which any mathematician encounters in his experience: the impression thatwe deal with an ideal realm, and at the same time that we have a kind of perceptionof it, in terms of a sort of a priority and necessity, which gives it a foundational sta-tus. Now this was not simply, on Peirce’s part, something like a mathematician’sreflex or a mysterious intuition: as we saw, he really tried to give a precise anddetailed—in the mode of a non-reductionistic or normative naturalism—account ofthe cognitive mechanisms involved in such an access at the level of perception, abduc-tion, iconic observation and experimentation, and rule-following.27


(3) But it is also important to realize that part of Peirce’s solution never impliesrenouncing the metaphysical ground either or to pronounce, as the latest Putnam(Putnam 2004) is now doing, the “obituary of ontology.” In that respect, his treat-ment of the problem of continuity is typical. If there is any reality in mathematics, itis perhaps to be found much less in some particular nature of its objects than in thatfundamental idea of the synechistic metaphysics, according to which there is, afterall, no difference in nature between the “Inner Universe” (the universe of our repre-sentations) and the “Outer Universe” (or reality). And this is why the view that“Thirdness (or continuity) pours upon us” is part and parcel not only of Peirce’ssynechistic or relational ontological realism, but of the kind of epistemological explana-tion we need in order to provide a genuine account of our causal interaction with themathematical realm.

These three major features of Peirce’s solution explain why, although it hassome links with two recent and interesting realistic approaches, namely Putnam’sand Maddy’s, it remains different from both of them.

(1) Peirce’s view has much in common with Putnam’s approach, both with thePutnam of the seventies (close to Quine’s position), and with the latest more “prag-matist” or “Wittgenstein-inspired” Putnam. With the first one, Peirce would haveagreed about the indispensability argument, and with the stress laid on the necessitynot to view the problems in the philosophy of mathematics as sui generis, not to raisetoo many frontiers between mathematics and physics, nor to forget about the appli-cability of mathematics to the empirical world. The view that mathematics has toaccount for our best theory of the world is a view which Peirce would have no doubtendorsed, even if he insisted on the ideal character of the mathematical realm.Again, he would have sympathized with many aspects of Putnam’s late criticism ofthe dispensability of such platonistic “mysterious,” “supersensible” and “idle” con-cepts as “object” or “existence” in mathematics (which Putnam thinks Quine stillstayed in the grips of (Putnam 2004:81)), which imply that the function of mathe-matical sentences is “to describe reality,” or is supposed to lie “behind our languagegames,” in favour of some form of “pragmatic” or “wittgensteinian” “pluralism,”viewing mathematics more as a realm of “conventions,” “optional languages” oreven corrigible “conceptual truths,”28 in which following rules is not so much a mat-ter of “conforming to the standards of a community” (along a Kripkean scepticaltype of reading) (Putnam 2001:144) as (along a realistic and pragmatist reading) adeeply normative practice, which does not presuppose, for all that, any mentalistic orontological “super mechanism” (Putnam 2001:145-147).

However, there are at least two respects in which Peirce would have disagreedwith Putnam (and, also, with Wittgenstein), one of which is the following: even ifPeirce insists on the conventionalistic side of mathematics, he would not be ready to


go too far in that direction. This may be seen not only in the limitations he sees in apurely “if-thenistic” account of mathematical reasoning,29 but also in the way hewould not be willing to reduce conventions to mere façons de parler (Putnam 2004:43),which would imply our being mainly attentive to the understanding of “the life welive with our concepts” in the domain of mathematics (Putnam 1996:263-264).

Indeed if “conceptual truths” are supposed to mean something close to whatPutnam aims at, namely to be sensitive to the fact that “conceptual truth and empir-ical description interpenetrate,” that such truths are neither trifling nor unrevisableand involve some corrigible activity (Putnam 2004:61) and that “we learn whatmathematical truth is by learning the practices and standards of mathematics itself,including the practices of applying mathematics” more than by “supposing that math-ematical truths are ‘made true’ by some set of objects” (Putnam 2004:66), Peircewould to a certain extent agree. His realism insists as much on the irreducible vague-ness of our habits of reasoning as on the fact that thirdness is the category of gener-alization, of abstraction, of all the operations of reasoning for which self-control orcriticism is always on the lookout. In that sense, mathematics does not constitute anexception in principle to fallibilism. Rather it is, most often, an exception in fact. Butthis again does not warrant any absolute or fundamental value being assigned tomathematical necessity or to mathematical certainty: “Mathematical certainty is notabsolute certainty. For the greatest mathematicians sometimes blunder, and there-fore it is possible—barely possible—that all have blundered every time they addedtwo and two” (Peirce 1903b:C4.478).

But this is not to say that mathematics is a pure matter of conventions or arbi-trary notations. This is why Peirce observes that the student must learn to use nota-tions to think in, but he must not try to make the notation think for him, if hewishes to push his reasonings far. Thinking is done by experimenting in the imagina-tion. Notations are excellent things to experiment with; but still experimentationrequires intelligent supervision to come to much (in (Eisele 1979d:186)). Rules resultfrom a controlled use ruled by intelligence (or thirdness). Here lies all the differencethere is between the mere thinking in images (which is more often a handicap than ahelp) and the experimenting on images, that is, icons or abstract schemes. Becauseintelligence rules practice, it is of no consequence in mathematics if one uses formu-las or notations which are mere flatus vocis. In fact, the more they are so, the better itis, 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 iswhat one does with them, the way one thinks in this notation, for “one secret of theart of reasoning is to think” (Peirce 1890a:N1.136). This is another way of sayingthat the rules we follow when we reason are not only more than standards fixed bythe community—although Peirce agrees that they express collective and not subjec-


tive practices and presuppose some agreement among the members of the commu-nity (which is a decisive constituent of the kind of exactitude that can be reached inmathematics) (Peirce 1898b:C5.577)—but also more than the mere exercise of apractice, however regulated and normative it may be (Putnam 2001:141-147). Con-trary to Wittgenstein (who criticized such a “super” mentalistic or platonisticapproach in Ramsey), Peirce’s account of what is involved in our following rules(viewed basically as dispositional intelligent, partly uncontrolled and partly con-trolled habits)—implying, as he says, our “hereditary metaphysics”—is not at allhostile to, but on the contrary, recommends a cognitive or even physiological studyof its mechanisms, and in particular of the way in which their normativity may emergefrom nature itself.

(2) This may explain why, in many ways, Peirce’s approach is close to the “natu-ralized Platonism” proposed a few years ago by Penelope Maddy, whose explicitplan, in an effort to try and find some reconciliation of both premises of Benacer-raf ’s dilemma, was “to reject the traditional characterization of mathematical objectsand to bring them into the world we know and into contact with our familiar cogni-tive apparatus through an account of our ‘perception-like’ connections” (Maddy1992:48).30 Obviously Peirce would for the most part have applauded the programsince, like Maddy, he tried to make sense of a form of realism which would keepwhat is sound in the Platonistic intuitions, while being clearer—both at the level ofthe kind of perception involved in our access to the mathematical realm and at thelevel of the type of epistemic justification we might give of such an access—in a waywhich, though insisting on the causal aspect of the justification, also underlines itsinsufficiencies as a full-fledged account of the kind of mathematical knowledge onethus gains.31 Like Maddy also, and along lines already developed by Piaget,32 Peirceemphasizes, as we noted, learning and pedagogy in our acquisition of numbers, thussuggesting that it might be worth looking into some kind of (either inneist or empir-ical) cognitive story of their acquisition.33 Finally, it is part of the set theoretic realismMaddy develops (which she does not find so different in fact from structuralistaccounts),34 that some sets are really localized where their members are and are notless observable than their members, so that we acquire direct beliefs about setsthrough perception (Maddy 1992:87). However, such a view presents obvious diffi-culties: the fact that sets of physical objects have a spatio-temporal localization doesnot transform them into physical objects. Besides, how can we be sure, as Fregealready objected to Mill’s suggestion, that granted we have some “numerical capac-ity” or sensitivity to “numerosity,” this can be equated with a sensitivity to numbersthemselves, rather than to physical aggregates for example (to which numbers can inno way be reduced); surely identity and quantification require more than this:between the quantifier “there exists an x” and the numerical quantifier “there exists


exactly n x”, there is a gap—in other words it is hard to claim that the concept ofnumerosity is perfectly identical with the concept of numericity, or to the knowledge ofnumbers.

To a certain extent, Peirce’s waverings about the real “essence” of numbersbetray some similar difficulties. But undoubtedly, for him, the only way to have achance to come close to a solution to this involves keeping firm on the type of enti-ties we are dealing with when we are reasoning in and doing mathematics: for him, aswe often noted, the key answer lay in the focus he put on the centrality of continuity,viewed as the universal par excellence. Maybe, as commentors have often remarked,there is as much unclarity in such a concept (at least as Peirce viewed it) as there maybe in such concepts as “object,” “existence,” “set,” “class” or “structure,” but atleast one thing Peirce should be praised for (although this is no doubt a second andimportant point of departure from Putnam),35 was his acute awareness of the factthat, unless the difficulties highlighted by such a dilemma as Benacerraf ’s were takencare of—not merely at the epistemological level but also at the ontological level, whichfor him did not mean: by getting rid, once for all, of ontology but, on the contrary,by trying to erect a satisfactory scientific and realistic metaphysics—the prospectswe might have of solving them would be extremely limited indeed.

Notes

1. Quine’s answer, as is well known, along the lines of an “indispensability argu-ment,” also endorsed by Putnam (Putnam 1971:346-347) is that the justification isanalogous to the justification for positing the existence of electrons and other suchunobservable entities in physics (‘posits’ is the terminology used in (Quine 1948/1980)).

2. The sense that there is a problem goes back to Plato himself, but the modernform comes from (Benacerraf 1973). Since then, it has become commonplace tobegin by dismissing Platonism on the basis of Benacerraf ’s argument, although, asMaddy notes (Maddy 1992:36), Benacerraf himself draws no such dogmatic conclu-sion, even if his successors, including those with generally realistic leanings, havescorned Platonism (Field 1980; Hellman 1989; Kitcher 1983; Resnik 1981).

3. Such is the anti-platonist moral defended, for instance, by Hartry Field (Field1989b:25-27). But it is also defended by some contemporary neuro-scientists whoclaim that there are some neural bases of our mathematical knowledge. Mathematicswould thus result, for the most part, from the capacity of our brain to “invent” newrules and languages, and to explore the logical consequences of such rules. There-


fore one should view mathematics as an ever growing and evolving activity of man’sbrain rather than as a pre-established world which mathematicians would just, asexplorers, come to discover (see (Changeux and Connes 1989)).

4. I have already given some reasons for this in (Tiercelin 1993), but although Istill agree with what I defended then, the paper was written before new and impor-tant realistic attempts were made to answer Benacerraf ’s dilemma. In view of suchattempts, I think we are in an even better position to see what were the defects but,mostly, the great merits of Peirce’s original approach.

5. On this, see (Haack 1993; Houser 1993; Tiercelin 1991). It might be worthnoting that this does not imply an attitude of contempt towards logic proper; to acertain extent it is even the opposite: mathematics is in a way “superficial.” Briefly,Peirce did not understand logic in a narrow and in a wider sense, according to whichformal logic, in the narrow sense, would be constituted by the deductive part oflogic, whereas the wide sense would be covered by the theory of logic as semiotic:namely, the general theory of signs, or the study of anything whose function is torepresent something (Peirce 1901a:C4.373). The basic distinction to be made israther between logic and mathematics. For example, the calculus of classes as a for-mal, deductive, symbolic system is a piece of mathematics, not logic: “Formal logicis nothing but mathematics applied to logic” (Peirce 1902e:C4.263) (cf. (Peirce1901a:C3.615)). Nevertheless, Peirce stresses the differences between them ratherthan the similarities: from the start, as a follower of Aristotle and Kant, he findsthere is something more to mathematical logic than mathematics. That somethingmore is precisely logic, which is always viewed from a philosophical and ontologicalperspective. Hence an opposition between logic and mathematics; but it is notgrounded on a distinction between two specific domains, since mathematics is notdefined by its objects but widely as the science of necessary reasoning. Hence thereal opposition between logic and mathematics lies between the theoretical or obser-vational aspect of inference on the one hand, and its practical or operational part onthe other. The mathematician practices deduction (Peirce 1893b:C2.532,1893d:C4.124, 1902e:C4.242), reasons deductively, whereas the logician studiesdeductive reasonings and arguments. According to a dictum of his father, Peircecharacterises mathematics as “the science which draws necessary conclusions”(Peirce 1898d:C3.558, 1902e:C4.229); logic, by contrast, is “the science of drawingnecessary conclusions.” This makes mathematics a “prelogical science” which is inno need of logic, for a theory of the validity of its arguments: those are acritical andevident, “more evident than any such (logical) theory could be” (Peirce1902f:C2.120) . Peirce’s apparent antilogicism should rather be interpreted as a dif-ference of attitude according to the position that is being adopted, the respectiveaims and methods of logic and mathematics being different. From the mathemati-


cian’s standpoint, the instrumental value of the calculus is decisive because he is onlyinterested in finding the simplest and shortest way to get to the result (Peirce1902e:C4.239). Logical constructions are superfluous here (Peirce 1880:C3.222). Butfrom the logician’s point of view, it should be clear that his end “is simply and solelythe investigation of the theory of logic, and not at all the construction of a calculusto aid the drawing of inferences” (Peirce 1901a:C4.373). Therefore, the mathemati-cian’s and the logician’s purposes are incompatible: “The system devised for theinvestigation of logic should be as analytical as possible, breaking up inferences intothe greatest possible number of steps, and exhibiting them under the most generalcategories possible; while a calculus would aim, on the contrary, to reduce the num-ber of processes as much as possible, and to specialize the symbols so as to adaptthem to special kinds of inference” (Peirce 1901a:C4.373). Thus the calculus is animportant tool of reasoning, but it is only a tool (Peirce 1885:C3.364, 1903b:C4.424,1903s:C3.322, 1906b:C4.553), or a “special system of symbols” for treating deduc-tive logic. When Peirce criticises logicians like Boole and Schröder for being exces-sively mathematical his objection is that they attempt to draw metaphysicalconclusions from logical calculi whose merit consists in the ease with which calcula-tions can be performed with them, rather than in their ability perspicuously to revealthe semantic structure of arguments and propositions. But he is also convinced ofthe possible plurality of symbolic systems, applied to deduction itself, and above allof the superiority of his logical graphs compared to an algebra of logic (Peirce1908d:C4.617). The logician is not interested in reaching conclusions, but in theoriesabout their relations to premisses (Peirce 1902e:C4.239, 1903b:C4.481,1903l:C4.370, 1906b:C4.533). Hence the natural purpose of logic is “to analyze rea-soning and see what it consists in” (Peirce 1893b:C2.532). (1) Since the business oflogic is “analysis and theory of reasoning” (Peirce 1893d:C4.134)—cf. (Peirce1896b:C1.417, 1901a:C4.373, 1902e:C4.242))—its domain is widened so as to covernot only deductive reasonings but inductive and abductive seasonings as well; this iscrucial for understanding Peirce’s conception of logical and scientific inquiry. (2) Italso means that the aim of analysis as opposed to that of a calculus will be guidednot by simplicity, but on the contrary by complexity (as may be seen from thenumerous steps involved in the graphic presentation), in order to reach the mostbasic and irreducible elements. It is precisely here that semiotic gets into the picture;indeed, the business of semiotic is to explain “the gist” (Peirce 1893b:C2.532) or the“essence of reasoning,” through the various functions exhibited by different signs,in order to discover the nature of arguments (Peirce 1903a:C1.575, 1903b:C4.425)(3) That logic should be concerned with reasoning makes it a normative science(Peirce 1903a:C1.577), and even a branch of ethics (Peirce 1903a:C1.575,1903c:C1.611, 1903p:C5.130, 1905b:C1.573, 1905g:C5.535), for every reasoning is


the product of a deliberate and self-controlled thought (Peirce 1903c:C1.606,1903p:C5.130) “with a view to making it conform to a purpose or ideal” (Peirce1905b:C1.573). Logical criticism should apply to that type of reasoning alone; this isalso a consequence of the principles of pragmatism, according to which all thinkingis a kind of conduct (Peirce 1905g:C5.534), so that reasoning is a kind of deliberateconduct, for which a man can be held responsible. Such a normative conception ofreasoning is particularly decisive for the understanding of the basic features ofPeirce’s theory of assertion, as well as the principles governing inductive and abduc-tive reasonings in the methods of scientific inquiry (Peirce 1903c:C1.615). (4) Finally,such a definition of logic as a science of reasoning implies an appeal to some sort topsychology, but not in the sense that logic should be founded on it (as is the casewith the German introspective tradition, against which Peirce fights). Peirce’s antip-sychologism never goes so far as to deny certain facts of psychology: namely,doubts, beliefs, etc. On the contrary, since logic is a positive science—as opposed tomathematics, which is a science of pure hypotheses—it may, or even must, take intoaccount certain facts or certain indubitable observations concerning mind: “Formallogic must not be too purely formal; it must represent a fact of psychology, or else itis in danger of degenerating into a mathematical recreation” (Peirce 1883b:C2.710).

6. I cannot go into the details, within the scope of this paper, of such a sophisti-cated position, which I have analyzed elsewhere: see (Tiercelin 1992).

7. Here I disagree with Hookway, who does not seem to think that there isindeed, for Peirce (in terms close to Maddy’s own approach), an important causaland perceptual (though not intuitive) aspect in our encounters with the mathemati-cal realm (Hookway 1985:185).

8. As Murphey (Murphey 1961:239) rightly points out, we can see how faithfulPeirce was to Platonism even after 1885, since he titled the fourth volume of his“Principles of Philosophy” (a project in twelve volumes, 1893) Plato’s World: An Elu-cidation of the Ideas of Modern Mathematics.

9. Ordinals express a relative position in a simple sparse sequence (Peirce1904a:C4.337) and not, as in Cantor, the sequences themselves (Murphey1961:247,255). Ordinals only name “places” which are relative characters, whichdetermine classes of members of these sequences. Hence, being classes, ordinals aremore general than their members, among which are the collections to which multi-tudes are attributed.

10. In fact, as we shall see below, this is one argument Putnam and Quine use inorder to refute a Carnapian type of conventionalism.

11. It is indeed easier for arithmetic to reach such a formal ideal of purity, buteven if it is difficult, as space is very often a matter of experience (Eisele 1976:Vol. 4,p. xv), it is necessary (as Riemann showed) to come to a purely formal conception of


geometry too (namely, one dealing only with pure continua), so that no distinctioncould finally be made between geometry and pure arithmetic, both containing ana-lytical propositions, that is, deductions derivable from definitions in a purely logicalway with no consideration of their possible empirical validity. Hence the question ofsome real correspondence between mathematical and real space or between thehypotheses or axioms of geometry and their empirical validity has no real value(Peirce 1894:N2.251-252). The reason is that, even if it existed, it could not be dem-onstrated; real space is a ding an sich; we cannot apply our cognitive devices to it. Andeven if it were meaningful to say that we do apply them, we should have to considertheir fallible character and possible margin of error. All of this explains why we arejustified in adopting constructions that are extremely far from the properties of realspace, insofar as they are practical and convenient.

12. However, as David Wiggins (Wiggins 1980:139) notes, this does not conflictnecessarily with conceptualism: “Conceptualism properly conceived must not entailthat before we got for ourselves these concepts, their extensions could not existautonomously, i.e., independently of whether or not the concepts were destined tobe fashioned and their compliants to be discovered. What conceptualism entails, isonly that, although horses, leaves, sun and stars are not inventions or artifacts, still,in order to single out these things, we have to deploy upon experience a conceptualscheme which has itself been fashioned or formed in such away as to make it possibleto single them out.”

13. On this, see (Turrisi 1990), (Hull 1994) and (Tiercelin 2005a:395-400).14. And through every avenue of sense, hence already at the level of Firstness

and Secondness. This is one of the reasons why, for example, odors have “a remark-able power of calling to mind mental and spiritual qualities” (Peirce 1905c:C1.313)or why we cannot spend five minutes of our waking life without making some kindof prediction (Peirce 1903f:C1.26).

15. Such are “the whole series of hypnotic phenomena, of which so many fallwithin the realm of ordinary everyday observation—such as our waking up at thehour we wish to wake much nearer than our waking selves could guess it, involve thefact that we perceive what we are adjusted for interpreting though it be far less perceptiblethan any express effort could enable us to perceive; while that to the interpretation ofwhich our adjustments are not fitted, we fail to perceive, although it exceed in intensitywhat we should perceive with the utmost ease if we cared at all for its interpretation.It is a marvel to me that my clock in my study strikes every half hour in the mostaudible manner, and yet I never hear it. I should not know at all whether the strikingpart were going, unless it is out of order and strikes the wrong hour. If it does that, Iam pretty sure to hear it. Another familiar fact is that we perceive, or seem to per-ceive, objects differently from how they really are, accommodating them to their manifest


intention. Proofreaders get high salaries because ordinary people miss seeing mis-prints, their eyes correcting them” (Peirce 1903r:E2.229) (my emphasis).

16. This is due to the fact that singulars are always pseudo-singulars (Peirce1870b:C3.93). Even such names as ‘Theodore Roosevelt’, which it is convenient toregard as singulars are a mixture of “hazy recollections” of all kinds of perceptualexperiences which have “led me semi-instinctively to suppose that one person pre-serving an identity through the continuity of space, time, character, memory, etc. hasbeen one singular connected with all these phenomena; and though I have not madeany formal induction to test this theory, yet my impression is that I am in possessionof [an] abundance of facts that would support such an induction quite irresistibly . .. . The notion that all those reacting singulars were in the relation of personal iden-tity to one another, and that their separate singularities consist in a connection to asingular, the collection of them all, this notion is an element of Thirdness abductively con-nected with them . . .” (Peirce 1903q:E2.222) (my emphasis). Indeed, for Peirce, it isplain that our knowledge of the majority of general conceptions comes about in amanner altogether analogous to our knowledge of an individual person (such as, e.g.,the general idea of dog, a combination of associations of perceptual experiencescommon to me and to others, which “I have generalized by abduction chiefly, withsmall doses of induction,” acquiring thus “some general ideas of dogs’ ways, of thelaws of caninity, some of them invariable so far as I have observed, such as his fre-quent napping, others merely usual, such as his way of circling when he is preparingto take a nap” and which make me finally encounter the word dog more like a classthan like an individual (Peirce 1903q:E2.221-222). “These are laws of perceptualjudgments, and so beyond all doubt, are the great majority of our general notions”(Peirce 1903q:E2.223).

17. Hence it is not at all certain that Peirce wants to make a specific logicalpoint by the stress he puts on theorematic reasoning; rather it seems to me to be anepistemic point, first, because he once conjectures that the need for theorematic rea-soning reflects the current state of mathematical ignorance—which means that it isonly a question of time before all theorematic reasonings are reduced to corollarialreasonings (Peirce 1903(?)-b N4.289, n.d.-d:N4.81). “Perhaps, when any branch ofmathematics is worked up into its most perfect forms, all its theorems will be con-verted into corollaries.” He also expresses his puzzlement whether this distinction is“inherently impossible in some cases” (Peirce 1903(?)-b:N4.290), or the reflection ofsome psychological convenience rather than of some logical truth (Peirce1903n:N4.158). Indeed theorematic reasoning is important as an epistemic procedure;this is owing to the fact that the main “business of the mathematician is to discovernew theorems,” while “leaving the grinding of them down into corollaries to thelogician” (Peirce 1903(?)-b:N4.289). So Peirce can hold both that, even in the face of


counterexamples, his distinction still remains valid (Peirce 1903(?)-b:N4.290) andalso that, to a certain extent, it is logically dispensable (Peirce 1907a:N3.491). But ifthe distinction tends to evaporate from a logical point of view, this is preciselybecause grinding down theorems to corollaries does not imply at all that the obser-vational, experimental or iconic part is absent. Peirce’s point is that, even in the sim-plest syllogistic form, something like an iconic representation is required, so that thedevelopment introduced by the logic of relations is no departure from that view butonly a generalisation. On this, see (Tiercelin 1991:197-206, 1995:61-65).

18. We could also add that it is surely not enough to invoke the ideal character ofhypotheses, especially in geometry, to solve entirely the question of the nature oftheir content: namely, that topics is the sole part of geometry that is really pure andideal, because it deals more with pure continua. A lot could be said here about theinterface between mathematics and straightforward metaphysics. If one wishes todefine mathematics as a system that is ideal, and yet not arbitrary, one must first beable to verify whether its creations are relevant or just simply not meaningless. Thisexplains why we do not describe surfaces or lines as mere ad hoc constructions. “Wehere use the traditional phraseology which speaks as if lines and surfaces were some-thing we make. This is not strictly correct. The lines and surfaces are places which arethere, whether we think of them or not. They are quite ideal, it is true. But they arethere, in this sense that we can think of them as being there without being drawninto any absurdity” (Peirce 1894:N2.387).

19. Compare (Wittgenstein 1967:II, §74) and Peirce (Peirce 1902a:N4.59): “It isidle to seek any justification of what is evident. It cannot be rendered more than evi-dent.”

20. These qualifications are indeed in order, since one should not underestimatethe fact that although there are many common elements between Peirce and Wit-tgenstein, the fact remains that Wittgenstein has a basically neo-pyrrhonianapproach to knowledge and justification, whereas Peirce, among other things, isdeeply committed to Critical Commensism and a radical fallibilism. See (Tiercelin2005b:63-108,252-274).

21. Which, incidentally, as noted by Maddy, had some revolutionary features, incontrast with traditional Platonism, among them its view of mathematics as part ofour best theory of the world, hence as liable to revision, so that mathematical knowl-edge is neither a priori nor certain (Maddy 1992:30-31).

22. “Unapplied mathematics is completely without justification on the Quine/Putnam model; it plays no indispensable role in our best theory, so it need not beaccepted (Putnam 1971:346-347). Now mathematicians are not apt to think that thejustification for their claims waits on the activities in the physics labs. Rather mathe-maticians have a whole range of justificatory practices of their own, ranging from


proofs and intuitive evidence, to plausibility arguments and defences in terms ofconsequences. From the perspective of a pure indispensability defence, this is all justso much talk: what matters is the application” (Maddy 1992:30-31).

23. Although it would be unfair to Gödel to limit his views to this. As Maddynotes, Gödel’s platonistic epistemology is two-tiered, involving also the view (closeto Quine and Putnam) that, our belief in some “unobservable” facts is justified by theirrole in our theory, by their explanatory power, their predictive success, their fruitfulinterconnections with other well-confirmed theories, and so on. So the simpler con-cepts and axioms are justified intrinsically by their intuitiveness; but more theoreticalhypotheses are justified extrinsically, by their consequences. Extrinsically justifiedhypotheses are not certain, and not a priori either. It remains true that, in contrastwith Quine and Putnam, Gödel gives full credit to purely mathematical forms of jus-tification—intuitive self-evidence, proofs, and extrinsic justifications within mathe-matics—and the faculty of intuition does justice to the obviousness of elementarymathematics (Maddy 1992:32-33).

24. It should be noted that some mathematicians have tended to adopt a kindof Gödelian strategy which, incidentally also drives them to refuse Benacerraf ’sdilemma as such: thus, R. Penrose claims that we have a direct access, through a fac-ulty of intuition, which is non-algorithmic but yet perfectly natural, because thephysics that is needed in order to understand consciousness and the mind must beanother physics than the one considered by the neo-mechanists. According to him itshould be possible to accept the two following theses: (1) Brain processes causeconsciousness, but no algorithmic process can simulate such processes. (2)Although the Platonic universe cannot be reduced to our imperfect mental con-structions, our mind can nonetheless have a direct access to it, thanks toan “immediate knowledge” of the mathematical forms and a reasoning capacity toreason upon such forms. See (Penrose 1989, 1994). As (Engel 2000) has shown, thisis also an interesting combination (or “platonist naturalism”), which consists inclaiming that both premises of Benacerraf ’s dilemma can be sustained. Immediateknowledge cannot be explained by any scientific knowledge: it is a natural process,which can nevertheless be explained by a physics of the mind, which, however, mustbe conceived in an entirely different way (not yet available to us) if we want tounderstand the natural basis of the mind. In other words, mathematical Platonismand the epistemology according to which we have an access to mathematical entitiesare viewed as perfectly compatible views.

25. “There is an attitude towards the philosophy of mathematics which iswidely shared, and to which I myself contributed, according to which the problemsin question are so hard that one should despair of seeing any way at all of resolvingthem” (Putnam 2001:142).


26. Indeed, Peirce was so interested in “pure numbers” that he tried to con-struct several systems of pure number. The parallel with the intuitionists seemsobvious. In that respect, Murphey has shown (Murphey 1961:286-287) that the wayBrouwer defines a set—as a law according to which the elements of the set may beconstructed and which is not a finished totality, nor need any particular element of itbe a finished totality—is very near to Peirce’s analysis, by its generating relation, ofan infinite collection. But we have here all the differences between the antimeta-physical verificationism of the intuitionists and Peirce’s realism. For example, Peircewould admit with Heyting (Heyting 1966:15) that mathematical objects must have aconsistency which in a way renders them independent of the acts of thought whichaim at them, and at the same time he would consider that it makes no sense to thinkof an existence of these objects independently of any relation to human thought(Heyting 1931:42). But Peirce would find truistic the intuitionist’s confusionbetween the fact of an object to be actually thought by an individual and the fact ofan object to be dependent upon some form of general or possible thought, which isPeirce’s definition of reality. For Peirce, then, intuitionism would be a form of nomi-nalistic Platonism, born from a misunderstanding of the real issues in the problemof universals. No wonder the intuitionists finally drew anti-metaphysical conclu-sions, for fear of the Platonism into which they fell after all (Heyting 1966:3), anddecided to adopt a purely historical and constructivist definition of mathematics byconsidering only actual mental constructions (Heyting 1966:8). Even if Peirce usedintuitionistic distinctions (e.g., between enumerate, denumerable, and nondenumera-ble collections), since his realism forced him to admit that one could, without con-tradiction, talk about possibilities, he could not be identified with intuitionism, for atleast two reasons. First, we do not have to adopt strict verificationism (Peirce1893d:C4.114, 1908c:C6.455) or “actually construct the correspondences” (Peirce1897c:C4.178); and second, we can always treat possibilities as forming collectionsand extend the operations of classical logic (including the law of excluded middle) tosuch collections (Peirce 1898c:C6.185ff), even if Peirce also considered borderlinecases and multivalued systems. Thus, we can think about the infinite, and Peircebelieved that it was by calling up collections of possibilities that the paradoxes of thetheory of sets could be avoided. As Murphey says (Murphey 1961:287), nothingcould be more opposed to intuitionism.

27. On the subtle balance of normative and naturalistic elements in his neo-emergentist explanation of the constraint and normativity of rule following in infer-ence, see (Tiercelin 1997).

28. See in particular (Putnam 2001:151-153, 2004:37,43,46-47,54,60-61,66-67).See also Conant’s interpretation of Putnam (Conant 1997) and Putnam’s comments(Putnam 2004:55).


29. To be compared and opposed here with (Putnam 2004:55-60) and his ten-dency to put logical truth and mathematical truth on the same footing.

30. Indeed, for Maddy “[w]e can and do perceive sets and our ability to do sodevelops in much the same way as our ability to see physical objects. Consider thefollowing case: Steve needs two eggs for a certain recipe. The egg carton he takesfrom the refrigerator feels ominously light. He opens the carton and sees, to hisrelief, three eggs there. My claim is that Steve has perceived a set of three eggs. Bythe account of perception just canvassed, this requires there be a set of three eggs inthe carton, that Steve acquire perceptual beliefs about it, and that the set of eggsparticipate in the generation of these perceptual beliefs in the same way that myhand participates in the generation of my belief that there is a hand before me whenI look at it in good light” (Maddy 1992:58). To the question whether Steve “sees” aset, Maddy’s anwer is “Yes”: “First, I contend that the numerical belief—there arethree eggs in the carton—is perceptual, that is, that it looks to Steve, in a non-meta-phorical sense, as if there are three eggs there. There is empirical evidence, based onreaction times, that such beliefs about small numbers are non-inferential. Further-more, this belief about the number of eggs can non-inferentially influence and beinfluenced by other clearly perceptual beliefs acquired on this occasion; for example,the welcome fact that there are enough eggs for the recipe can make the eggs them-selves look larger. This particular belief about the number of eggs is thus part of arich collection of perceptual beliefs acquired on this occasion, beliefs about the sizeand colour of the eggs, the fact that two eggs can be selected from the three in vari-ous ways, the locations of the eggs in the nearly empty carton, and so on” (Maddy1992:59-60). From this, she concludes that we perceive a structure of a general setassembly which is responsible for various intuitive beliefs about sets, for examplethat they have number properties, that those number properties don’t change whenthe elements are moved; that they have subsets, that they can be combined, and doon. And these intuitions underlie the most basic axioms of our scientific theory ofsets (Maddy 1992:70).

31. I cannot dwell more, within the scope of this paper, on the elements Peircedeems important in a correct account of justification and knowledge, generallyspeaking, and not only as far as mathematical knowledge is concerned (see (Tiercelin2005b), in particular pp. 267-272). Let me just point out that Peirce would not besatisfied with a merely causal theory of knowledge, nor even with a merely reliabilistaccount of it. Although such externalist components are important elements, oneshould not underestimate some internalist aspects of knowledge, mostly related tothe self-control and self-criticism implied in Peirce’s critical commonsensism andfallibilism, together with his subtle position toward scepticism, both critical—likeWittgenstein—of radical or “paper” doubt, yet attentive to the genuine reasons we


might have to doubt, which are part and parcel of the way we (as members of thescientific community) conduct our investigations, and aim at truth (or warrantedassertibility) at the ideal limit of our inquiries.

32. See (Piaget 1937) and (Piaget and Inhelder 1948). “The behavioural evi-dence of Piaget and his colleagues suggests that the ability to gain perceptual beliefsabout sets develops in a series of stages parallel to those for perceptual beliefs aboutphysical objects, though at a somewhat later age. At the beginning of this period, achild may be able to classify objects into groups in a consistent way—say triangleswith triangles, squares with squares—but she does not correctly grasp the inclusionrelation—in a group of two black squares and five black circles, the child thinksthere are more round things than black. For the younger child, a set ceases to existwhen its subsets are attended to” (Maddy 1992:63). This acquisition of set conceptscan be developed in the complete absence of language. Maddy, like Peirce (andPiaget), is very sensitive to the development of the child who gradually develops theideas of inclusion, collection and numeration leading to the discrete set concepts,just as he develops parallel ideas based on the part/whole and enclosure relationsrather than inclusion, proximity and distance rather than collection, measurementrather than number. “These developments, beginning, as their set theoretic counter-parts do, in perception and action, lead eventually to the perception of lines—edges,intersections of planes, trajectories—as continuous structures. By the age of twelve,the child expresses intuitive beliefs about geometric figures that reveal a primitivenotion of continuity” (Maddy 1992:74-75). Just as Peirce would have done, in hisexperimental and laboratory way of conducting inquiry, Maddy suggests that “allthese hypotheses however demand theoretical and extrinsic support as in natural sci-ence, in terms of verifiable consequences, lack of disconfirmation, breadth andexplanatory power, intertheoretic connections, simplicity, elegance, and so on”(Maddy 1992:74-75).

33. Contra Putnam: “What neural process, after all, could be described as theperception of a mathematical object?” (Putnam 1980:430). In that respect, theexperimentally inclined Peirce, as far as psychology is concerned, would surely havebeen interested in the works led by some cognitive psychologists and neuro-scien-tists who suggest that a certain number of animal species, superior primates andchildren have a sensitivity to numbers, which is based in the structures of brain orga-nization. Hence, according to Karen Wynn (Wynn 1992) five-month-old childrenare able to calculate precise results of simple additions and subtractions, and themental symbols on which they operate have a structure which allow them to abstractinformation from the numerical relations between the numerosities. This suggestssome sort of innate representation of numbers, which might serve as a basis, then,for the development of mathematics (Wynn 1992:317); from which she concludes


that a conception according to which mathematical knowledge is of an essentiallyempirical nature, as had been suggested by Mill and by more contemporary writerssuch as Philip Kitcher (Kitcher 1983), might well be the right one. This view has alsobeen defended by the neuro-scientist Changeux and the mathematician Dehaene(Changeux and Dehaene 1993), who hold that neural models of elementary numeri-cal development can be formulated which indicate that the concept of numberclearly has a psychological reality, and that the human brain possesses one or several“numerical organs.” (This has been observed and commented upon by (Engel2000)).

34. Maddy (Maddy 1992:170ff). As Resnik presents the structuralist position:“In mathematics . . . we do not have objects with an ‘internal’ composition arangedin structures, we have only structures. The objects of mathematics, that is, the enti-ties which our mathematical constants and quantifiers denote, are structurelesspoints or positions in structures. As positions in structures they have no identity orfeatures outside of a structure” (Resnik 1981:530).

35. I have explained my reservations about Putnam’s condemnation of ontol-ogy in (Tiercelin 2006).


2010. Peirce on Mathematical Objects and Mathematical Objectivity.

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