**Peirce's Realistic Approach
to Mathematics: Or, Can One Be a Realist without Being a Platonist?**

*Claudine Engel‑
Tiercelin, *Université de Paris

(Pantheon‑Sorbonne)

(*Charles
S. Peirce and the philosophy of science* (papers from the Harvard
sesquicentennial Congress), ed. by Edward C. Moore, The University of Alabama
Press, Tuscalosa and London, 1993 p.30-48)

Peirce's
position on the problem of universals is that of a sophisticated realism
inherited from the Avicennian‑Scotistic tradition,which may he briefly
characterized by its opposition to metaphysical realism (or Platonism) and
various forms of nominalism. Against the idea that there could be either a
reality totally independent of our ways of knowing or verifying (an
incognizable *ding an sich*) or realities that could be reduced to
singular existents (sense‑data, concepts, determinate individuals),
Peirce's realism is a unique and ambitious analysis of all the aspects‑logical,
physical, and metaphysical‑of the problem of universals, combining
scholastic and categorical elements with scientific, pragmatistic, and
straightforward metaphysical concerns.

In
this chapter I consider how Peirce's realism fits his approach to mathematics,
which is often presented as a somewhat incoherent mixture of Platonistic and
conceptualistic elements. Without denying these, I shall attempt to claim that
the subtlety of Peirce's position not only helps to clear up some of these so‑called
inconsistencies but offers many insights for contemporary ways of dealing with
the mathematical aspects of the problem of universals.

**Peirce's
Definition of Mathematics:**

**Or, How to Apply Realism to
Mathematics**

Briefly** **one could say
that Peirce's definition of mathematics has two main characteristics. First, it
does not cover a special domain of entities.

It
is not defined by the specificity of its objects (space, time, quantity) or by
the nature of its propositions (analytical, a priori) or by the kinds of truths
it can exhibit. Against Hamilton and De Morgan, Peirce denied any dependence of
mathematics on space, time, or any form of "intuition" (CP 3.556). As
to the analytic or synthetic nature of mathematical propositions, he said
almost nothing, convinced that the real issues were elsewhere and that they had
to be thought through and formulated in new terms (and especially through the
distinction of corollarial and theorematic forms of deductive reasoning). If
mathematics has nothing to say about truth, it is because it is not‑contrary
to logic‑a science of facts but a science of hypotheses and abstractions.

But
for the son of Benjamin Peirce, a second characteristic of mathematic, was that
it was also a science of reasoning, more specifically, "the science which
draws necessary conclusions" (C/' 3.558; 4.229). This is a very wide
definition of mathematics, since not only is all mathematical reasoning
diagrammatic, but all necessary reasoning is mathematical reasoning, no matter
how simple it may be *(NEM *4:47). In
that sense, when Peirce affirmed the fundamentally iconic, observational, and
experimental character of deduction, he not only defined mathematical deduction
as such, he accounted for all kinds of deduction, thus reviving the whole
conception of logical necessity.

Such
a definition presents obvious difficulties for any kind of realism. First, how
is one to adjust the idea of mathematics as a purely formal and ideal system to
realism? What is the status of these *entia
rationis? *Are they pure conventions, arbitrarily chosen, which never refer
to reality? Are they simple tautological or analytical statements, incapable of
being qualified as true except as concerns the meaning of the expressions they
involve? Then why insist on the practical side of mathematics? Why take so
seriously the problem of its application? If one accepts the notion of applied
mathematics as something which is needed by all sciences, what is to warrant
that such idealizations have the objective validity which justifies their being
used by these other sciences?

Second,
if Peirce's realism is a realism of indeterminacy‑which implies, on the
one hand, that it should be possible to think without contradiction not only
about reals but also about possibles and, on the other hand, that indeterminacy
renounces any idea of absolute or infallible

Peirce's
Realistic Approach to Mathematics / 31

necessity and
exactitude‑how is such indeterminacy going to be taken care of by
mathematics? According to Susan Haack, one can measure the importance of the
issue by noticing Peirce's many hesitations about answering the question of
whether or not fallibilism should be extended to mathematics. Fo .Haack
(1979:37), Peirce never really made up his mind, at times declaring that the
necessity of mathematics prevents our being wrong about our mathematical
beliefs and that we are only fallible as far as our factual beliefs are
concerned (CP 1.149, written c. 1897); at other times he emphasized that
fallibilism does extend to mathematics (CP 7.108, 1892; 1.248,** **1902) and
that mathematical inferences are only probable after all. Clearly enough, for
Peirce, mathematical necessity was perfectly compatible with the notion of an
ideal system in which one reasons. about possibles (hence indeterminates) and
not about real cases. Not only does mathematics allow for individuals that are
not perfectly determinate in all respects *(NEM
*4:xiii), but also one can think about the infinite. In that sense, the
basic principles of Peirce's realism that are attached to indeterminacy
(possibility and generality, or firstness and thirdness) seem to be obeyed. But
can one be satisfied with such a definition of necessity?

**Peirce's Conventionalism and Conceptualism**

Before
trying to make sense of Peirce's realism in mathematics, we have to ask if it
makes sense to speak of Peirce's realism at all. Indeed, a number of instances
seem to suggest that he adopted a conceptualistic, or nominalistic, position in
which pure mathematics is the domain of hypotheses and ideal creations.
Moreover, its necessity is apparently not due to any characteristics of its
objects or to any particular nature of,its propositions that would afford a
specific objectivity.

Conceptualism
always has realism in its background. Thus, after defining a pure diagram as
that which is "designed to represent and to render intelligible, the Form
of Relation, merely" and asserting that "an intelligible relation,
that is a relation for thought, is created only by the act of representing
it" *(NEM 4:316 *n. 1), Peirce
added that if we should some day find
out the** **metaphysical nature
of relation, that would not

32 **Engel‑Tiercelin**

mean that it
would thereby be created, for the intelligible relation surely existed before,
in thought or in the way God represented the universe. But this does not
necessarily conflict with conceptualism. As David Wiggins (1980:139) says:

Conceptualism
properly conceived must not entail that before we got for ourselves these
concepts, their extensions could not exist autonomously, i.e., independently of
whether or not the concepts were destined to be fashioned and their compliants
to be discovered. What conceptualism entail, is only that, although horses,
leaves, sun and stars are not inventions or artifact,, still, in order to
single. out these things, we have to deploy upon experience a conceptual scheme
which has itself been fashioned or formed in such away as to make it *possible *to single them out.

And
this is, indeed, the spirit that animates such Peircean assertions as the
following: "All necessary reasoning is reasoning from pure hypothesis in
this sense; that if the premise has any truth for the real world, that is an
accident totally irrelevant to. the relation of the conclusion to the premise,
while in the kinds of reasoning that are more peculiarly topics of logical
[rather than mathematical) discussion, it has all the relevancy in the world” (*NEM*:4:164;
cf. CP 4.233, *NEM 4:270).*

If
rnathematics, is "the study of pure hypothesis regardless of any analogies
they may have in our universe" *(NEM
4:149; *CP 3.560), which is particularly clear in the case of arithmetic *(NEM 4:xv‑xvi), *and if "it
certainly never would do to embrace pragmatism in any sense in which it should
conflict with this great fact" *(NEM
4:157), *it would seem that mathematical necessity was derived, not from
some necessity in things, but merely from the link of logical consequence
between premises and conclusion (CP 4.232) and from the hypotheses,
conventions, and rules which the mathematician has chosen to adopt (cf. defs.
32 and 33 in MS *94 [NEM *2:251]). ^{1}

So
mathematical systems are purely formal. The meaning of the terms appearing in
the postulates, hypotheses, and theorems is totally irrelevant as such: "A
proposition is not a statement of perfectly pure mathematics until it is devoid
of all definite meaning, and comes to this‑that a property of a certain
icon is pointed out and is declared to belong to anything like it, of which
instances are given" (CP 5.567). For example,

Peirce's
Realistic Approach to Mathematics l 33

the only
definitions that have to be retained as conforming to the "dignified
meaninglessness of pure algebra" (CP 4.314) are those implicit definitions
that postulates impose on their terms (see CP 3.20). In their turn, the
postulates will be considered as implicitly defining the objects) to which they
apply, in the exact sense in which Riemann declared that the axioms of geometry
provide a definition of space (see Murphey 1961:235).

No
doubt all this contributes to qualifying Peirce's position as *ant-realistic*
in the sense given by Durnmett (1987) that mathematical propositions have
apparently no predeterminate meaning or truth, or such that it should suffice
to discover them. Their meaning is such as is postulated, then demonstrated. In
mathematics, there are no propositions which could benefit from truth
conditions that were utterly independent of our capacity to recognize them as
such, or whose truth conditions could be realized without our being able to
recognize that they are so. In other words, one should always be able to define
the truth conditions of any mathematical statement in terms of its conditions
of assertibility. This is clearly stated in MS 94: "The *meaning *of
any speech, writing, or other sign is its translation into a sign more
convenient for the purposes of thought; for all thinking is in signs. The
meaning of a mathematical term or sign is its expression in the kind of signs
in the imaginary or other manifestation of which the mathematical reasoning
consists. For geometry, this (expression) is [in] a geometrical diagram" *(NEM *2:251).

This
is also why one must ordinarily attach great importance to the mathematical
procedures of demonstration, to the modus operandi (CP 4.429; *NEM *2:10‑11). That is, meaning is
not given from the start but,** **on the contrary, is determined by the
demonstration. To reason is not to use meanings; it is to construct them, to
manipulate them in order to determine them. And the analysis of the way these
constructions work may help in clarifying, if not in constituting, such a
determination of meaning (CP 3.363). Hence the great importance of the
iconicity of reasoning (CP 2.279; *NEM *4:47‑48)
but not only of that (Engel‑Tiercelin 1991), for indeed, there are two
other essential procedures in the determination of the meaning of mathematical
statements. These are abstraction and generalization, both of which allow a
better grasp of the status of the entice rationis the mathematician works upon.
Hypostatic

**34 / Engel‑Tiercelin**

abstraction has a
decisive role *(NEM 4:49) *because it
is that operation by which something "denoted by a noun substantive,
something having a name," which belongs to the category of substance as
such, is transformed into an assertion, and the reality of which "can mean
nothing except the truth of statements in which the real thing is
asserted" *(NEM *4:161‑62;
see CP 4.234). Thus, to say that numbers are real is not to reduce them to some
singular existing entities but merely to indicate that there are statements in
which numerical expressions are used to describe classes adjectivrly (CP 4. 154‑55).
Thanks to such abstractions as numbers, lines, or collections, "it thus
becomes possible to study their relations and to apply to these relations
discoveries already made respecting analogous relations" (CP *3.642). *It thus becomes unnecessary to
assume some kind of objects preexisting in some kind of mathematical universe.

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

**Arithmetic and the
Temptation of Platonism**

**It **would be an
exaggeration, however, to say that Peirce's position is utterly clear on all
these points. It seems indeed difficult to deny that some of his analyses,
mostly in arithmetic but not only there, do not sound conceptualistic at all,
but realistic, in the most traditional or Platonistic sense of the word.

Thus
Peirce did not hesitate to speak of the "innate" propositions of
mathematics, preferring that rather than such a term as "a priori,"
which "involves the mistaken notion that the operations of demonstrative
reasoning are nothing but applications of plain rules to plain cases" (CP 4.92*). *It is, of course, in arithmetic that the temptation of Platonism
is the strongest, numbers being at times qualified as "ideas"
belonging as such to a different "universe of experience" from that
of facts and laws;

Peirce's
Realistic Approach to Mathematics l 35

the
"Platonistic world of pure forms" (.CP 4.1 18), or that "Inner
World" (CP 41.161) in which these eternal, abstract, and "airy
nothings" (C!' 6.455) are not "absolutely created by the
mathematician" (CP 4.161).^{2} In that respect, we would not be
very far from h'rege's universe of true "thoughts," from "laws
of Pure Being," or thoughts independent of the senses and of the empirical
world. Indeed, it is in a way that reminds us very much of F'rege inasmuch as
the development of the whole theory of numbers is described by Peirce as
arising from a small number of first and primitive propositions (CP 2.361;
6.595). For the same reasons, we could understand some of Peirce's assertions
as signifying that the truth conditions of any mathematical proposition are
transcendent in respect to their conditions of verification and that
arithmetic, at least, is an independent domain of entities (cf. CP 4.114).

** **

**The Real Nature of Peirce's**

**Realistic Solution in Mathematics**

I
would like to claim, however, that these inconsistencies between
conceptualistic and Platonistic elements are a wrong way to look at the matter.
In fact, they hide the real nature of Peirce's way of dealing with the problem
of universals in mathematics and the typical realistic solution he proposes.
Let us start with the problem of conceptualistic conventionalism.

**Hypotheses
and Definitions**

First
of all, it is clear enough that Peirce's conventionalism is never so absolute
as that of, say, a Poincaré, (who is at times criticized and viewed harshly,
though wrongly, as such). No doubt, Peirce would not be ready to reduce
.mathematics to an ideal science of hypotheses that would consist in nothing
more than a simple game of abstract, arbitrary, and convenient formulas.

If
mathematics is a matter of creations, these are never totally arbitrary *(NEM *4:xii), first, because most often
the mathematician's hypoth-

**36 / Engel‑Tiercelin**

esis is provoked
by some real problem which arises in other sciences *(NEM *4:xv).^{3 }This is why the distinction between pure
and applied mathematics is not so decisive after all *(NEM 2:vi)j.*

But
the arbitrariness of hypotheses is weakened in another way. Contrary to the
poet's hypotheses (Cl' 4.238; *NEM *4:68)
the mathematician's hypotheses, which indeed require the highest intellectual
capacity to be found, are subject to the rules of deduction. A mathematician is
interested in hypotheses only for the forms of inference that can be drawn from
them *(NEM 4:268). *Before going any
further, let me add that it is surely not enough to invoke the ideal character
of hypotheses, especially in geometry, to solve entirely the question of the
nature of their content: namely, that topics is the sole part of geometry that
is really pure and ideal, because it deals more with pure continua. A lot could
be said here about the interface between mathematics and straightforward
metaphysics (cf. Engel-Tiercelin 1986 and 1990).

**Rules
of Deduction and Habits of Reasoning: Our "Hereditary Metaphysics"**

If
one wishes to define mathematics as a system that is ideal; and yet not
arbitrary, one must first be able to *verify
*whether its creations are relevant or just simply not meaningless. This
explains why we do not describe surfaces or lines as mere ad hoc constructions.
"We here use the traditional phraseology which speaks as if lines and
surfaces were something we make. This is not strictly correct. The lines and
surfaces are *places *which are *there, *whether we think of them or not.
They are quite ideal, it is true. But they are there, in this sense that we can
think of them as being there without being drawn into any absurdity" *(NEM *2:387).

But
the lack of arbitrariness is also due, in good part, to the character of
logical necessity, that is, to the fact that the rules we follow in our
demonstrations do nothing except exhibit the habits we have acquired through
reasoning (CP 5.367). The leading principles of inference are but the
linguistically codified formulation *(logica
docens) *(CP 1.417) of* *these habits of reasoning under the
form of habits of inference, which

Peirce's
Realistic Approach to Mathematics / 37

must not be
confused with some psychological constraint (in the mode of Sigwart or Schröder
(CP 2.52, 209; 3.432). Even if it is true, in the end,that it is not the
constraint exerted by the rule but rather the fact that the conclusion is true
when the rules of transformation and substitution are correctly applied (CP
2.153; 5.365) that determines the objective validity of the reasoning, one must
not underrate the important role played in inference by habits. It explains why
logical necessity is something like a fact which is felt as such and which, for
that reason, hardly needs any justification. Here Peirce is very near
Wittgenstein: necessity is present in our acts and practices; we stumble
against something whose explanation could hardly he anything other than
"we use it so. "^{4}

Because
logical necessity exerts itself as a constraint difficult to explain and to
justify. the teacher in mathematics often has difficulties in trying "to
make another person feel the force of that demonstration who does not do so
already" *(NEM *4:xiv). Peirce's
realism comes to the fore here: logical necessity is one of those habit‑facts
which are absolutely real and yet irreducibly vague, before .which no further
explanation seems appropriate or even required (CP 2.173). Something like a
sort of constraint of necessity should be granted here. One can wonder if it is
enough to warrant the self‑evident character of mathematics (CP 2.191).
But obviously, Peirce would have answered that in such a case, it was precisely
the hopeless attempt at asking for some kind of justification that would need
to be justified.

But
such habits of reasoning also have the strength of general rules (and this is
part of Peirce's realism too)‑on the one hand, because they express
collective and not subjective practices and presuppose some agreement among the
members of the community (which is a decisive constituent of the kind of
exactitude that can be reached in mathematics) (CP 5.577) and, on the other
hand, because they result from a controlled use ruled by intelligence (or thirdness).
Here lies all the difference there is between the mere thinking *in *images (which is more often a
handicap than a help) and the experimenting *on
*images, that is, icons or abstract schemes. Because intelligence rules
practice, it is of no consequence in mathematics if one uses formulas or
notations which are mere flatus vocis. In fact, the more they are so, the
better it is, for it will facilitate experimentation. Hence, the utility of
children's rhymes like "eeny,

**38 Engel-Tiercelin**

meeny, miney‑mo"
in which one counts words, not things. What is essential is what one does with
them, the way one thinks in this notation, for "one secret of the art of
reasoning is to *think" (NEM 1:136).5*

Peirce's
realism insists as much on the irreducible vagueness of our habits of reasoning
as on the fact that thirdness is the category of generalization, of
abstraction, of all the operations of reasoning for which self-control or
criticism is always on the lookout. In that sense, mathematics does not
constitute an exception in principle to fallibilism. Rather it is, most often,
an exception in fact. But this again does not warrant any absolute or
fundamental value being assigned to mathematical necessity: "Mathematical
certainty is not absolute certainty. For the greatest mathematicians sometimes
blunder, and therefore it is possible‑barely possible‑that all have
blundered every time they added two and two" *(CP 4.478).*

**Realism
as an Alternative to Platonism in Arithmetic**

What
about Peirce's Platonistic claims in arithmetic? I think they are
counterbalanced by other kinds of claims. First, paradoxically, such concepts
as "number," "zero," and "successor" are counted
not as primitive concepts (Russell) but as mere variables (Peano) (Murphey
1961:238ff.).* *Peirce tried to construct
several systems of pure number (see CP.4.160ff., 677-81; 3.562ff.), giving only an implicit definition of its
primitive terms, thus allowing a perfectly formalistic interpretation of
his system (Murphey 1961:244‑45).

Second,
the system of pure number was for him but a~particular case of ordinals, ^{6
}which were the primitive pure numbers (CP 3.628ff.; 4.332, 657‑59,
673ff., etc.) not only because they expressed a relative place but because they
illustrated it, so that Peirce believed they exemplified the pure serial
relation which was instantiated in all the series (Murphey 1961:273‑74):* *"But the highest and last lesson
which the numbers whisper in our ear is that of the supremacy of the forms of
relation for which their tawdry outside is the mere shell of the casket" *(*CP
4.681).

But
there are reasons other than Peirce's relational realism that ex

Peirce's
Realistic Approach to Mathematics / 39

plain his
preferences for ordinals, and these are taken either. from sensible experience
(CP 4.154) or from straightforward pragmatical considerations.^{ 7}

It
would be as wrong to overrate any so‑called supremacy of arithmetic above
the other sciences for it is as vain to hold to the idea of some
compartmentalization of mathematics (CP 4.247) as to underrate the pragmatic
considerations that are part of Peirce's way of dealing with numbers: This is
true, even though he was careful to distinguish between pure, or
"scientific,". arithmetic (CP 3. 562A), which "considers only
the numbers themselves and not the application of them to counting," and
practical arithmetic (CP 4.163). Peirce spent a lot of time achieving a whole
pedagogy for arithmetic. What does this signify, except that in arithmetic,
too, what is important is not only the type of objects or propositions8 but
consideration of the system, in which the learning of the rules is decisive?

The
learning of arithmetic, and the role played in it by iconic representations,
was taken by Peirce to be essential and yet responsible for so many errors that
he kept reconstructing the steps that ought to be performed in the techniques
of education arid of learning.y This is not to say that Peirce would reduce
meaning to use (this is why, for example, instinct should not decide for the
right interpretation of a system). One should manage to teach this at the
pedagogical level. But we can go further than this:. "Some children learn
by first acquiring the use of a word, or phrase, and then, long after, getting
some glimmer of what it means" *(NEM
1:213‑14). *So it is surely true that a distinction should be
maintained between pure arithmetic, which is "the knowledge of
numbers," and practical arithmetic, which is "the knowledge of how to
use numbers" *(NEM *1:107). But.how are we to understand,
in the writings of this so‑called avowed Platonist in arithmetic, such
claims as the following one: "The way to teach a child what number means
is to teach him to count. It is by studying the counting process that the
philosopher must learn what the essence of number is" *(NEM *1:214*)? *Does this not say that, in a certain
way, the reality of numbers is to be found as much in the rules determining
their meaning as in any primitive prédeterminate meaning which would only need
to be discovered in some Realm of Ideas?

40 (
Engel‑Tiercelin

**Conclusions **

First,
it is not easy to determine what Peirce's final position is concerning the
mathematical aspects of the problem of universals. However, I have tried to
show that we can reconcile the obvious conceptualistic arid conventionalistic
claims with an equally obvious Platonism in other respects.

I do
not think that Peirce's waverings express merely an ill‑assumed tension
between the metaphysically inclined logician, more eager to respect the
principles and conclusions of his categorical and realistic analysis, and a
kind of natural reflex of the mathematical practitioner who is easily led
toward Platonism. Of course, we could always say, to the detriment of Peirce,
that Platonism is not easy to avoid, even if, as B. Van Fraassen says (1975:40),
Platonists win Pyrrhic victories because the conventionalists fail to provide
the good arguments.

But
such a conclusion is unsatisfactory. In fact, as I have shown, I am not at all
convinced about the seriousness of Peirce's Platonism, which, should we he
content with it, would constitute a severe objection to his whole theoretical
project.

Second,
a number of Peirce's vacillations merely,‑reflect the difficulty of a
coherent position on the subject; they.do not always result in confusions but,
on the contrary, succeed in stressing the immense complexity of the problem. In
that respect, one of the paradoxes and characteristics of Peirce's reflections
on mathematics is, by his wide definition of the domain of mathematics, to have
pointed out its specificity, namely, that in mathematics, what is most
difficult is not the solution of problems but the fact that there are problems
to be solved and that consequently acute analysis is required as much~on the
methods and on reasoning as on the nature of rnathernatical objects and
propositions.

Third,
even if Peirce did nut always avoid inconsistencies, because he openly adopted
positions which came closer either to conventionalism or to Platonism, he was
also perfectly aware of the impossible alternative that would consist in
believing that one accounts for the specificity of the mathematician's work and
of mathematical invention in terms of an opposition between a mathematics of
discovery and a mathematics of invention, and he tried to find a third way to
understand why there can be real

Peirce's
Realistic Approach to Mathematics l 4

problems in
mathematics to be solved. I have tried to show that Peirce's idiosyncratic
realism is such a third way, a very interesting attempt to answer the obvious
major difficulty for a solution to the problem of universals in its
mathematical form. A problém which he himself formulated quite clearly is, how
is one to explain that "although mathematics deals with ideas and not with
the world of sensible experience, its discoveries are not arbitrary dreams but
something to which our minds are forced and which were unforeseen" *(NEM *2:346*)?*

Fourth,
Peirce obviously felt himself facing the following difficulty: on the one hand,
his anti‑Platonism, which was a natural consequence of his belief in the
realism of indeterminacy, forced him to consider mathematics as simple meanings
to be displayed or to "use" in practice, and not as truths to be
discovered. But on the other hand, realism, or simply good sense, forbade him
to adopt some strict form of verificationistic constructivism and to take
mathematical demonstrations as pure determinations of meaning, free products of
arbitrary creations and constructions.

The
stress put on the ideal and hypothetical character of mathematics, the
definition of it as the science of pure reasoning, and the fact that questions
of method, practice, procedure, and demonstration are at least as important as
questions bearing on the nature of objects or propositions all tend to show
Peirce's awareness of the fact that a purely realistic answer, in the Platonist
sense, cannot constitute a sufficient warrant for the necessity and objectivity
of mathematics. For that reason, it is clear that Peirce's view cannot be
compared to Frege's type of solution.

Again,
if one of the Platonist's arguments consists in assuming a universe of objects,
entities, and truths not only independent of, but transcending, our capacity to
recognize them, Peirce's version cannot be reduced to such a position. First,
because mathematics was not for him a science of truths (for reasons different
from Wittgenstein, who, in somewhat related terms, thought that mathematics
consisted not of true or false propositions but of autonomous rules of
grammar), Peirce did not want mathematical statements, which are hypotheses to
be taken as true or false, as describing any kinds of facts whatsoever. Second,
Peirce's view cannot be reduced to Platonism because he thought that the
meaning of mathematical statements could not be given independently of any

43/
Engel‑Tiercelin

demonstration: in
that sense, although Peirce's pragmatistic realism of indeterminacy prevented
him from reducing the meaning of a proposition to its conditions of
verification, or reducing meaning to use, Peirce never separated the meaning of
any mathematical proposition from its conditions of assertibility.

Such
a view prevails, even when Peirce seems to show some form of Platonism. For
example, although he admits that mathematics has a certain autonomy as an ideal
system and goes so far as to talk of it as a universe ruled by dichotomies and
truth, whose reality ~lies in its entities subsisting, even when no one is
thinking about them or trying to know them, he also adds that if they can be
called real ideas, it is because, one day or another, they will be
"capable of getting thought," and that is but a question of time (CP
3.527; 6.455). We are far from the Platonist definition of a completely
independent and transcendent universe. Faithful to the principle of the
impossibility of incognizables, Peirce never defined mathematics as a universe
totally independent of our possibility of knowing it, nor did he assume some
completely given and prerstablishcd meaning of mathematical statements which
was waiting to be discovered, without any construction.

Indeed,
Peirce was so interested in "pure numbers" that he tried to construct
several systems of pure number. The parallel with the intuitionists seems
obvious. In that respect, Murphey has shown (1961:286-87) that the way Brouwer
defines a set‑as a law according to which the elements of the set may be
constructed and which is not a finished totality of any particular element of
it necessarily a finished totality‑is very near to Peirce's analysis, by
its generating relation, of an infinite collection. But we have here all the differences** **between the antimetaphysical verificationism of the intuitionists
and Peirce's realism. For example, Peirce would admit with Heyting (1966:15)
that mathematical objects must have a consistency which in a way renders them
independent of the acts of thought which aim at them, and at the same time he
would consider that it makes no sense to think of an existence of these objects
independently of any relation to human thought (Heyting 1964:42). But Peirce
would find truistic the intuitionist's confusion between the fact of an object
to be actually thought by an individual and the fact of an object to be
dependent upon some form of general or possible thought, which is

Peirce's
Realistic Approach to Mathematics / 43

Peirce's
definition of reality. For Peirce, then, intuitionism would be a form of
nominalistic Platonism, born from a misunderstanding of the real issues in the
problem of universals. No wonder if the intuitionists finally drew anti
metaphysical conclusions, for fear of the Platonism in which they fell after
all (Heyting 1966:3), and decided to adopt a purely historical and
constructivist definition of mathematics by considering only ac*tual *mental constructions (ibid.:8).

Even
if Peirce used intuitionistic distinctions (e.g., between enumerate,
denumerable, and nondenumerable collections), since his realism forced him to
admit that one could, without contradiction, talk about possibilities, he could
not be identified with intuitionism, for at least two reasons. First, we do not
have to adopt strict verificationism (CP 6.455, 4.114) or "actually
construct the correspondences" (CP 4.178); second, we can always treat
possibilities as forming collections and extend the operations of classical
logic (including the law of excluded middle) to such collections (CP 6.185ff.
), even if Peirce also considered borderline cases and multivalued** **systems.
Thus, we can think about the infinite,
and** **Peirce believed that it
was by calling up collections of possibilities that the paradoxes of the theory
of sets could be avoided. As Murphey said (1961:287), nothing could be more
opposed to intuitionism.

Fifth,
Peirce has also shown, in defining numbers, collections, and multitudes as *entia rationis, *that for. him they were
inseparable from their conditions of assertability. In that respect, stressing
as he did the operations of mathematical reasoning, Peirce also understood that
it is necessary to remove the problems from the metaphysical ground and to
concentrate first on their elaboration, on the problem of their meaning
conditions. As Dummett wrote (1987:2), there is perhaps no hope of settling the
argument between those who favor mathematical realism, who hold that we
discover mathematical objects, and those who favor an idealist position, for
whom mathematical objects are creations or conventions of our mind. The only
way to clarify the issue is therefore to place it first at the level of
meaning: Peirce's way of dealing with problems in mathematics seems perfectly
appropriate to that kind of recommendation.

Finally,
it is nonetheless true that Peirce would never dream of renouncing the
metaphysical ground either. In that respect his treatment of

44 !
**Engel‑Tiercelin**

the continuum
problem is typical. If there is any reality in mathematics, it is perhaps much
less in some particular nature of its objects than in that fundamental idea of
the synechistic metaphysics, according to which there is, after all, no
difference in nature between the "Inner Universe" (the universe of
our representations) and the "Outer Universe" (or reality). One may
consider either that such a position is fatal to any serious treatment of the
mathematical aspects of the problem of universals or that a Dummettian way of
dealing with realism leaves the ontological questions undecided, which in some
cases may also be fatal to a grasp of some mathematical issues (think of the
debates over the continuum problem). Even if we do not decide in favor of the
second solution, I think Peirce's idiosyncratic realism is full of insights
that may help clarify the issues and
contribute** **to their ultimate
solution.

**Notes**

1. It is indeed easier for arithmetic to reach
such a formal ideal of purity, but even if it is difficult, as space is very
often a matter of experience *(NEM 4:*xv*),
*it is necessary (as Riemann showed) to come to .a purely formal conception
ofgeometry too (namely, one dealing only with pure continua), so that no
distinction could finally be made between geometry and pure arithmetic, both
containing analytical propositions, that is, deductions derivable from
definitions in apurely logical way with no consideration of their possible
empirical validity.Hence the question of some real correspondence between
mathematical and real space or between the hypotheses or axioms of geometry and
their empirical validity has no real value *(NEM
2:251‑52). *The reason is that, even if it existed, it could not be
demonstrated; real space is a *ding an
sich; *we cannot apply our cognitive devices to it. And even if it were meaningful
to say that we do apply them, we should have to consider their fallible
character and possible margin oferror. All of this explains why we are
justified in adopting constructions that areextremely far from the properties
of real space, insofar as they are practical and convenient.

2.
As Murphey *(1961: 239) *rightly points
out, we can see how faithful Peirce was to Platonism even After *1885, *since he titled the fourth volume
of his "Principles of Philosophy" (a project in twelve volumes)
"Plato's World: An Elucidation of the Ideas of Modern Mathematics" *(1893).*

*3. *It is, indeed, because we
find ourselves in such complicated situations

Peirce's
Realistic Approach to Mathematics 1 45

that it is
impossiblé to determine with exactitude what the consequences could be that one
calls for the help** **of the mathematician *(NEM *2:9*). *So it is most
often from such a practical suggestion that the mathematician will "frame
a supposition of an ideal state of things," then "study that ideal
state of things and find out what would be true in such a case," before
generalizing to a third stage from that state of things, namely, by
"considering other ideal states of things differing in definite respects
from the first" *(NEM *2:10*). *In so doing, "he not only finds out, but also produces a
rule by which other similar questions may be answered" (ibid.).

*4. *Compare Wittgenstein *1974:2:74 *and Peirce *NEM *4:59*: *"It is idle
to seek any justification of what is evident. It cannot be rendered more than
evident."

*5. *This is why "the student
must learn to use notations to think in, but he must not try to make the
notation think for him, if he wishes to push his reasonings far. Thinking is
done by experimenting in the imagination. Notations arc excellent things to
experiment with; but still experimentation requires intelligent supervision to
come to much" (in Eisele 1979:186*).*

6.
Contrary to most authors (among whom is Cantor), Peirce thought that ordinal
and not cardinal numbers were the primitive pure numbers. A cardinal number,
"though confounded with multitude by Cantor, is in fact one of a series of
vocables the prime purpose of which, quite unlike any other words, is to serve
as an instrument in the performance of the‑experiment of counting"
(CP 3.628*). *In consequence of which, "The doctrine of the so‑called
ordinal numbers is a doctrine of pure mathematics; the doctrine of cardinal
numbers, or rather, of multitude, is a doctrine of mathematics applied to
logic" (CP 3.630*). *For that reason, Peirce thought that
Dedekind could have gone even further when he and others considered "the
pure abstract integers to be ordinal . . . . [They] might extend the assertion
to all real numbers" (CP 4.633*). *Ordinals express a relative position
in a simple sparse sequence (CP 4.337*) *and not, as in Cantor, the sequences
themselves (Murphey 1961:247, 255*). *Ordinals only name "places"
which are relative characters, which determine classes of members of these
sequences. Hence, being classes, ordinals are more general than their members,
among which are the collections to which multitudes are attributed.

*7. *Indeed, if ordinals are more
primitive than cardinals, it is also because "the essence of anything lies
in what it is intended to do." Now what are numbers? "Simply vocables
used in counting. In order to subserve that purpose best, their sequence should
stick in the memory, while the less *signification
*they carry the better." The children are quite right in counting with
nonsense rhymes, but these are always purely ordinals. Second, "The
ultimate utility of counting is to aid reasoning. In order to do that, it must
carry a form akin to that

46 /
Engel‑Tiercelin

of reasoning. Now
the inseparable form of reasoning is that of proceeding from a starting‑point
*through *something else, *to *a result. This is an ordinal, not a
collective idea" (CP 4.658‑59).

8.
For example, Peirce was not at all convinced by the superiority in arithmetic
of any one system above another; in particular, he did not believe that the
decimal system should be more "natural" than, say, the
"secundal," or binary, numerical system which he proposed. He went so
far as to say that if the ten fingers account for the almost universal use of base ten among all races of
humankind, then the decimal system is a monument to human stupidity *(NEM *1:237, 241). On the contrary, base
six would seem extremely advantageous, even for counting on fingers and toes,
although he thought that much the prettiest of the Aryan systems is the
secundal. It is, to say the least, extremely convenient in logic, especially in
the logic of denumerable and abnumeral series (MS 1121), its major merit lying
in "its having several different methods of performing each operation,
from which one can at sight select the one most convenient for the case in
hand." But, as Carolyn Eisele has pointed out, although aware of its
merits, Peirce did not believe "that any propaganda would ever move the
world, because there is nothing in secundals to excite the emotional
nature" (MS 1, in Eisele 1979:205).

9.
See "Teaching Numeration" (MS 179; *NEM *1:212ff. ). "Imagination, concentration,
generalization"‑such are the qualities that have to be educated. In
a seventeen‑page manuscript entitled "Practical Arithmetic" *(NEM *1:107ff.), we have a sketch of the
maxims of a work to help acquire exactitude and agility in the use of numbers.
Most individuals cannot help thinking about an abstract number without
accompanying it with colors and forms having no intrinsic connection with the
number. One has to take account of such "phantasms" and try to
prevent "the formation of associations so unfavorable to arithmetical
facility" *(NEM *1:213). Again,
"The teacher must not fail in his teaching to show the child, at once, how
numbers can serve his immediate wishes. The schoolroom cluck should strike; and
he must count the strokes to know when he will be free. He should count all
stairs he goes up. In school recess, playthings should be counted out to him;
and the number required of him. This is to teach the ethical side of
arithmetic" (ibid.).

**References**

Dummett,
M., 1987. (An interview on) Philosophical Doubts and Religious Certaientics*.
Cogito* 2 (1):1-3.

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Eisele,
Carolyn. 1979. *Studies in the Scietific
and Mathematical Philosophy *of *Charles
J . Peirce. *Ed. R. M. Martin. The Hague: Mouton.

Engel‑Tiercelin,
C. 1986. Le vague est‑il réel? Sur le realisme de C. S. Peirce.

*Philosophie
*10:69‑96.

.
1990. *Le problème des universaux chez C. S. Peirce.* Doctoral diss.,
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.
1991. The Semiotic Version of the Semantic Tradition in Formal Logic. In *New Inquiries into Meaning and Truth, *ed.
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Haack, Susan.
1979. F'allibilism and Necessity. *Synthese
*41:37‑63.

Heyting,
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Readings, *ed. P. Benaeerraf and H. Putnam, 52‑61. Oxford: B.
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1966.
*Intuitionism: An Introduction. *2d ed.
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Murphey,
Murray G. 1961. *The Development of
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Van
Fraassen, Bas C. 1975. Platonism's Pyrrhic Victory. In *The Logical Enterprise, *ed. A. R. Anderson, M. R. Barcan, and R. M.
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Wiggins,
David. 1980. *Sameness arid **Substance.
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Wittgenstein,
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48
1 Engel‑Tiercelin