arithmetic for the millian

P H I L I P K I T C H E R

A R I T H M E T I C F O R T H E M I L L I A N

(Received 10 April, 1979;in revised form 5 July, 1979)

John Stuart Mill is probably the most famous champion of the thesis that

mathematical knowledge is empirical. Yet Mill's views on mathematics are not usually taken seriously. Since Frege ridiculed what he took to be Mill's position, philosophers have been inclined to dismiss Mill as a man who

proposed too crude a solution to the problem of accommodating mathematics within an empiricist epistemology. 1 Even those who are sympathetic to Mill's

epistemological thesis that arithmetical knowledge is empirical do not turn to his writings in the expectation of finding insightful defenses of that thesis .2

I hope to show that Mill's views about arithmetic can be developed into a

satisfactory theory of arithmetical truth and arithmetical knowledge. Despite

the fact that Mill is sometimes confused and sometimes inconsistent, he presents some important insights which have been ignored by those who have

viewed him as a man whose mind was "essentially illogical ''3 . I shall begin by

trying to clear away the confusions and to arrive at Mill's optimal position.

Mill concedes that most arithmetical knowledge is obtained by deduction

from first principles, and focusses his attention on the status of the first

principles. He aims to show that arithmetical first principles include statements reporting 'matters of fact', which therefore deserve the title of axioms. He completes his argument for the thesis that arithmetic is empirical by proposing that an empiricist account of our knowledge of the axioms can be given, and that this account will either explain, or else explain away, the apparent characteristics of arithmetical knowledge.

This strategy divides into two parts. Mill first attempts to salvage what he takes to be the valid core of the idea that the first principles of arithmetic are definitions. He then uses his conclusion to develop a positive account of arithmetical knowledge. It is impossible to understand the latter unless we

Philosophical Studies 37 (1980) 215-236. 0031-8116/80/0373-0215502.20 Copyright �9 1980 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

216 P H I L I P K I T C H E R

have come to terms with Mill's treatment of the thesis that the first principles

of arithmetic are definitions, and, in particular, with the dark utterance that

they are definitions which "covertly assert a matter of fact ' '4 . So I shall begin

with Mill's views on definition.

Mill regards each meaningful expression as having two semantic functions:

the functions of denotation and connotation. Proper names denote their

bearers. Mill thinks of predicates as general names which denote each of the

objects to which the predicate applies; in other words, a predicate denotes

each of the objects in its extension, not its extension. The notion of conno-

tation is less familiar. General names (predicates) connote an attribute which

is shared by each of the objects they denote. Ordinary proper names, by con-

trast, are taken to be non-connotative. Since connotation is closer to our pre-

theoretic concept of meaning, Mill identifies connotation with meaning.

Accordingly, he declares that proper names are meaningless, s

The purpose of definition is to fix meaning, that is, connotation (p. 87).

It follows that proper names are strictly indefinable, although we can provide

something analogous to a definition, namely a specification o f the denotation of the name. Mill's semantics thus compels him to diverge from Frege's treat-

ment of arithmetic in at least one way: he must either hold that numerals

are not proper names or he must contend that they are not definable.

Mill distinguishes two ways in which terms can be defined. To define a

name 'directly' we use the form ' _ _ i s a name connoting the attributes...'

where the blanks are Idled with a name of the name to be defined and names

o f the attributes which it connotes. However, it is more concise to proceed 'indirectly': we "predicate of [the name to be defined] another name or

names of known signification, which connote the same aggregation of

attributes" (p. 87). Instead o f saying '"Bachelor" is a name connoting the

attributes masculinity and unmarriedness', we usually introduce the term

'bachelor' by declaring that bachelors are unmarried males. Definitions of the

indirect kind are a subspecies of what Mill calls 'merely verbal propositions'

and, as he notes (p. 74 fn. 2), of Kant's 'analytic judgments' . Apart from two muddled passages, in which he tries to show that defini-

tions do not have truth values at all, 6 Mill's central thesis is that the defini-

tions on which mathematics is based covertly assert a matter of fact because

they imply the (actual or possible) existence of objects denoted by the name

defined. The types o f definitions with which we have so far been concemed,

which present the connotation of an expression either directly or indirectly,

A R I T H M E T I C F O R T H E M I L L I A N 217

Mill calls logical definitions. He envisages a different sort of definition -

which, for ease of reference, we may call factual definitions - and proposes

that, although the first principles of mathematics appear to be indirect logical

definitions, they are really factual definitions. The notion of a factual definition is not as clear as it might be. Factual definitions are supposed to do more

than merely present the connotation of a name. Mill wavers between saying

that factual definitions imply the existence of things denoted by the name

defined, and saying that they only imply the possible existence of such things. At one point, he summarizes what he has accomplished as follows:

"...we have pointed out that ...what apparently follows from a definition,

follows in reality from an implied assumption that there exists a real thing conformable thereto" (p. 147). Elsewhere he is more cautious: "[A defini-

tion] affirms the actual or possible existence of things possessing the combination of attributes set forth in the definition" (p. 94).

Unfortunately, Mill's most prominent arguments for the thesis that

mathematical definitions are factual and not logical are seriously flawed. Mill thinks that he can show that the first principles of geometry must be factual definitions, and, more generally, that no logical definition can be a premise

in valid reasoning concerning matters of fact. The argument about geometry begins with the observation that geometrical demonstrations frequently use a

premise to the effect that a particular type of figure (for example, a circle)

can be constructed and that no logical definition of the name of the figure in question can assure the existence of such a figure (p. 95). (Logical

definitions merely present connotation.) This observation is perfectly correct, but Mill seems to overlook the possibility that the existence assumptions

which are found among the axioms of Euclidean geometry can be viewed as

implicitly characterizing the concept of Euclidean space. Instead of supposing

that the Euclidean axiom 'Circles may be drawn with any centre and radius' follows from the (logical) definition of a circle as the locus of points

equidistant from a fixed point, a wily defender of the thesis that geometry begins from logical definitions will suppose that axioms and theorems of

Euclidean geometry involve tacit relativization to Euclidean space. In more explicit form, the axiom states that in a Euclidean space circles may be drawn with any center and radius, and this, it is alleged, forms part of an indirect logical definition of 'Euclidean space'.

Mill's more general argument proceeds as follows (pp. 95-6) . Suppose

that we define a term which does not denote any actual object. To use Mill's


example, assume that we take a dragon to be a serpent which breathes flame. Then, exploiting the fact that Aristotelian logic lets us reformulate the defini-

tion as two premises, ('A dragon is a thing which breathes flame', 'A dragon is a serpent'), we can infer to the false conclusion that some serpents breathe

flame. Mill concludes that logical definitions cannot be used in valid reasoning

about matters of fact. Clearly, the root of the problem is his acceptance of the rule which permits the inference of FSomething is both F and G 7 from

FAll F 's are G -q. However, granted that he accepts this rule, Mill should not

suppose that logical definitions can be presented in the form F-All A's are

B's -q. The argument depends on equivocation. Mill cannot both suppose that

his cited definition of 'dragon' is a correct logical definition and also claim

that the syllogism he produces is valid]

So far we have found no good reason to adopt Mill's claim that the first

principles of arithmetic are not logical definitions. Without considering the criticisms which Frege levelled against Mill's proposals, we have found

ample evidence of confusion. Ironically, by considering Frege's attack, we can isolate a more interesting, though less obvious, strand in Mill's thought.

II

In The Foundations of Arithmetic Frege points out that there is an important lacuna in the proofs of numerical identities favored by some of his predecessors. 8 For example, Leibniz had proposed to prove that 2 + 2 -- 4 by using

the definitions (a) 2 = 1 + 1, (b) 3 = 2 + 1, (c) 4 = 3 + 1. His reasoning runs thus: 2 + 2 = 2 + 1 + 1 (by (a)) = 3 + 1 (by (b)) =4 (by (c)). Frege notes

that the argument depends on tacitly assuming the associativity of addition, but that, since Mill does not see this gap, he accepts the argument without demur. Frege now poses an awkward question for Mill: where in the proof do we need to use any assumption of existence or matter of fact? Having thus begun to reinterpret Mill's thesis in epistemological terms, Frege presses the point. If we suppose that justification for the use of an arithmetical definition depends on observation of the associated matter of fact, then it

seems that we shall be committed to implausible claims about our observa- tional accomplishments. The proof that 23671 + 49823 = 73494 will require extremely recherch6 observations! 9

The epistemological reinterpretation is appropriate. At the heart of Mill's enterprise is the thesis that arithmetic (as well as geometry) is an empirical


science. To achieve his goal he must expose the errors of theories which

construe arithmetical knowledge as a priori. In particular, he tries to block

the approach to mathematical knowledge which regards the first principles of mathematics as definitions which can be known a priori through knowledge of the language. I suggest that we read him as offering an epistem-

ological thesis about definitions: to be justified in accepting the definitions on which arithmetic rests we must have empirical evidence that those definitions are applicable. I shall try to show that this thesis is defensible, that it accords with Mill's aims, that he sometimes considers it and that it can be developed to meet Frege's criticisms.

Mill would allow that certain sentences of our language are true in virtue

of the connotations of the expressions they contain, and that we can defend our assertion of these sentences by citing our understanding of the language.

However, he would deny that assertion of sentences on this basis constitutes a priori knowledge of the propositions they express. Our defense is adequate

only so long as our right to use our language is not called into question. In

particular, if experience gives us evidence that certain concepts are not well- adapted to the description of reality our assertion of sentences involving

those concepts is no longer justified, even if the truth of those sentences stems from the connotations of the terms they contain.

The point may be illustrated by a historical example which Mill cites (p.

91). Chemists of the early nineteenth century fixed the connotation of 'acid' as including the property of containing oxygen. Hence they could assert the

sentence 'All acids contain oxygen' solely on the basis of their understanding

of the terms, and, if they had appealed to that understanding to support their assertion, the defense would have been adequate. After the discovery that

what they had called 'muriatic acid' (hydrochloric acid) consists only of hydrogen and chlorine, this defense was undermined. Experimental research showed that the old definition of 'acid' was inadequate to the compendious

description of chemical phenomena. If some traditionalist had continued to insist that acids contain oxygen, and had appealed to his understanding of

'acid' to support his claim, it would not be correct to say that he knew that acids contain oxygen. Rather, in the light of the experimental evidence available to him, we should say that the continued use of the old language is unjustified and his belief no longer warranted. To put the point another way, his assertion commits him to linguistic practices which he should not rationally adopt. This example shows that, while the appeal to understanding of our


language often serves as a local justification for our beliefs, empirical discovery is relevant to the continued success of that justification. Hence our practice of justification by appeal to understanding does not provide justification which is independent of experience.

This thesis is not new with Mill. Locke's discussion of 'real knowledge '1~ already suggests that the beliefs we form on the basis of acknowledgment of

relations among our ideas will be justified only if we are entitled to belief that our ideas conform to reality. Moreover, in replying to the suggestion that

the judgments of arithmetic can be made analytic, Kant argues that the required enrichment of the subject-concepts of arithmetical propositions

would set us the task of proving the objective validity of those conceptsJ 1

Yet the thesis plays a more important role in Mill's epistemology than it does in the doctrines of his predecessors. For it finally provides a clear sense for the

claim that the definitions used in arithmetic presuppose an observed matter of fact.

Unfortunately, the thesis can be developed in an atomistic or holistic way, and Mill's predilection for the former approach involves him in unnecessary

difficulties. On several occasions, he suggests that our use of particular

arithmetical definitions presupposes a related, observed matter of fact. So,

for example, he takes the definition '3 = 2 + 1' to presuppose the fact that

'collections of objects exist, which while they impress the senses thus, o~

may be separated into two parts, thus, oo o' (p. 169). Frege fastened on this

remark with glee, and chided Mill for not having drawn attention to the facts presupposed by the definitions of 777864 and 0.12 Now if Mill were to

propose that our use of individual arithmetical definitions is justified by our observation of associated facts, then he would face troubles in accounting

for our reasonings about large numbers. However, he can avoid this embarrassing consequence by taking a holistic line: use of the system of arithmetical definitions can be seen as justified by the applicability of that system to our experience. The holistic approach also enables Mill to answer Frege's question concerning the role of experience in the Leibnizian proof of '2 + 2 = 4'. Mill can reply that the argument only counts as a proof if the employment of the arithmetical definitions is justified, and that the justification for the employment of these definitions derives from the continued success of the system of arithmetical definitions as a whole. (Of course, it would still be correct to point out that there is a gap in the proof.)

The interpretation which I have offered thus ascribes to Mill a position which accords with his basic goals. At some points he considers it explicitly:

ARITHMETIC FOR THE MILLIAN 221

To save the credit of the doctrine that definitions are the premises of scientific knowledge, the proviso is sometimes added, that they are so only under a certain condi- tion, namely that they be framed conformable to the phenomena of nature; that is that they ascribe such meanings to terms as shall suit objects actually existing (p. 94).

However, Mill immediately tries to strengthen this claim, launching into the

confused discussion o f the Euclidean axiom about the construction ofcficles

which we considered in Section I. Interestingly enough, when he recognizes

that the thesis that gemetfical definitions imply the existence of objects

answering to them would commit him to the existence of ideal geometrical

objects, he becomes more circumspect, retreating to the type of posit ion I

have at t r ibuted to him.

We might suppose an imaginary animal, and work out by deduction, from the known laws of physiology, its natural history; or an imaginary commonwealth and from the elements composing it might argue what would be its fate. And the conclusions which we might thus draw from purely arbitrary hypotheses might form a highly useful intellectual exercise: but as they could only teach us what would be the properties of objects which do not really exist, they would not constitute any addition to our knowledge of nature: while, on the contrary, if the hypothesis merely divests a real object of some portion of its properties without clothing it in false ones, the conclusions will still express, under known liability to correction, actual truth (p. 150).

At this point, Mill continues to maintain that our definitions should define

concepts which are applicable to experience, but he gives up the demand that

those concepts must be instantiated. Instead, as the case of geometry teaches

us, we can sometimes employ concepts in the study of real things, which do

not fit these concepts exactly but which can be treated, in practice, as i f they

do instantiate them (see pp. 148-9 ) .

Mill's posit ion is a compromise between the view that there are analytic

truths which we know a priori and the view that there are no analytic truths

and no a priori knowledge. He maintains that our knowledge of analytic

truths is a posteriori. Despite appearances, this posit ion is not absurd. On

Mill's conception, elementary analytic sentences would be those sentences

which users of the language would defend by appeal to their understanding of

the language; the analytic sentences would be the logical consequences of the

elementary analytic sentences. Mill would deny that defense by appeal to

language constitutes a justification for the assertion of those sentences which

is independent of experience. Empirical evidence can undermine one's right

to use the language by showing that the meanings of our terms do not 'suit

objects actually existing'. I believe that these claims represent what is

defensible in Mill's thesis that the definitions used in arithmetic presuppose

matters of fact.


III

At this point we are ready to consider the second part of Mill's theory, his

account of the meaning of arithmetical statements. Mill is forced to consider

this issue because, given the approach to arithmetical knowledge just

attributed to him, he needs an explanation of how we are justified in accepting the definitions of arithmetical expressions. As we have just seen, his boldest claim about the acceptability of definitions is that we are justified in adopting a definition only if we have observed that the term defined is instantiated. The more cautious criterion, to which Mill is driven by the case of geometry, is that the defined term must "merely divest a real object of

some portion of its properties" (p. 159). Use of either of these criteria would make it difficult for Mill to accept a standard account of the logical form of arithmetical sentences. As I noted above, if he were to regard numerals as

proper names he would be committed to holding them to be indefinable. But,

in any case, any theory about numerals will raise awkward questions about the objects to which they apply. At the very least, Mill is committed to showing how the denotata of the numerals abstract from the properties

of real entities. In general, Mill is unhappy with discourse which is apparently about

abstract objects. He declares forthrightly that "All numbers must be numbers of something: there are no such things as numbers in the abstract" (p. 167). He cites as one of the evil consequences of taking mathematics

to be a priori the supposition that mathematical truths "...relate to, and express the properties of purely imaginary objects" (p. 147; see also p. 148). Although he talks about classes, Mill is also explicitly suspicious

about the notion of a class. He remarks that the usual method of explaining

the distinction between proper names and general names by reference to classes is faulty because "...it explains the clearer of two things by the more obscure" (p. 17; see also p. 60), and he suggests that talk of classes can be

reformulated in more enlightening ways:

For a class is abso lu t e ly n o t h i n g b u t an inde f in i t e m u l t i t u d e of ind iv idua l s d e n o t e d by a general name. The n a m e given to t h e m in c o m m o n is w h a t makes t h e m a class. To refer a n y t h i n g to a class, there fore , is to l o o k u p o n i t as one of the th ings w h i c h are to be cal led by tha t c o m m o n name. To exc lude i t f rom a class is to say t h a t the c o m m o n

n a m e is no t app l icab le to i t (p. 60) .

Mill's problem is to offer an account of the logical form of arithmetical statements which will avoid the repugnant idea that we are justified in


accepting arithmetical definitions by apprehending the properties of

abstract objects. His desire to construct an empiricist epistemology guides him to a solution of the problem. He wants to claim that our justification for accepting arithmetical definitions comes from observation of ordinary physical objects. Prominent among such observations are experiences of separating, combining and arranging physical objects. Mill's idea is to provide a view of arithmetical truth which will exhibit directly the relevance of such

observations to our acceptance of the basic laws of arithmetic. He often expresses his conclusion as the thesis that numerals connote

physical properties of phenomena (p.400). Numerals are general names,

and Mill explains their denotation by saying that "Two ...denotes all pairs of things, and twelve all dozens of things" (p. 400). He tackles the question

of the connotation of numerals as follows:

What, then is tha t which is connoted by a name of number? Of course, some property belonging to the agglomeration o f things which we call by the name; and that property is the characteristic manne r in which the agglomeration is made up of, and may be separated into, parts . . . . What the name of number connotes is the manner in which single objects of the given kind mus t be pu t together, in order to produce that particular aggregate (p. 400).

Unfortunately, this passage, in which he addresses the problem of arithmetical

truth directly, appears to conflict with some of Mill's views. For it suggests

that numerals denote 'agglomerations' or 'aggregates', and the most obvious way to read the latter term is to take Mill to be talking about sets. His

position would thus be a variant of Platonism, regarding arithmetic as

concerned with abstract objects which are classified by our use of numerals.

This would be quite incompatible with his hostility to abstract objects.

Fortunately, the slogan that numerals connote properties of aggregates (or

agglomerations) masks a more interesting view. As a first approximation, we might take Mill to be an early mereologist

and take agglomerations as sums. In the Foundations of Arithmetic, Frege attributes this position to Mill, and argues that, since agglomerations can be

regarded as composed in different ways, we cannot objectively speak of an agglomeration as having a particular number as a property. These arguments can be circumvented by supposing that numbers are relational properties which hold between sums and particular parts, or between sums and

principles of decomposition. We might attempt to work out a Millian theory of arithmetic along these lines. ~3 However, I think that there is a simpler approach, which avoids the references to sums, and which makes sense of

Mill's continued emphasis on 'modes of formation' of numbers.


We have seen that Mill suggests that apparent references to classes can be

parsed away. His idea is that to say that objects belong to a class is to assert that we regard those objects as associated (we fashion a 'general name' for

them). Thus the root notion in Mill's ontology is that of a collecting, an

activity of ours, rather than that of a collect ion, an abstract object. There is already a hint of this in the passage quoted above, where Mill takes the

connotation of number words to be connected with the ways in which objects can be put together. Perhaps we can regard statements of arithmetic not as expressing the properties of types of collections but as concerned with the properties of types of collectings. At times, Mill seems to come

very close to an explicit proposal of this kind. He interprets the sentence 'Two pebbles and one pebble are equal to three pebbles' as affirming that

"...if we put one pebble to two pebbles, those very pebbles are three" (p. 168). Similarly, he asserts that "Every arithmetical proposition, every statement of the result of an arithmetical operation, is a statement of one of

the modes of formation of a given number" (p. 400).

We can begin to lessen some of the obscurity of the proposal by considering the ways in which Mill takes us to learn arithmetic. Children

come to learn the meanings of 'set', 'number', 'addition' and so forth by

engaging in activities of collecting and segregating. Experiences of this type, on Mill's theory, provide justification for the acceptance of arithmetical

definitions. But what do children learn by having such experiences?

Presumably, that particular types of collective operations have particular

properties: they recognize, for example, that if one performs the collective

operation called 'making two', then performs on different objects the collective operation called 'making three', then performs the collective operation of combining, the total operation is an operation of 'making five'. Knowledge

of such properties of such operations is relevant to arithmetic because arithmetic is concerned with collective operations.

My interpretation so far might suggest that Mill is a peculiar type ofcon- structivist, one who holds that arithmetic describes constructions which are public rather than nebulous transactions in some mental intuition. However, it would be more accurate to regard him as concerned less with what we

do to the world than with what the world will let us do to it. To coin a Millian phrase, arithmetic is about 'permanent possibilities of manipula- tion'. More straightforwardly, arithmetic describes those structural features of the world in virtue of which we are able to segregate and


recombine objects, features which become manifest to us in our actual

operations of segregation and recombination. Another qualification is in order. I am not claiming that the only way to

interpret arithmetical statements is to regard the numerals as predicates

which apply solely to operations of heaping up small objects. Neither did

Mill. Arithmetic studies the properties of types of collective operation, and some operations of these types are the operations of physical collection

and segregation in which we sometimes engage and which are important in teaching arithmetic to children. It is helpful to think of this physical activity as a crude paradigm of the kind of activity which is of interest to

arithmetic. To collect all the red objects on a table is, at first, to assemble the

red objects and to segregate them from other objects on the table. Yet, reliance on physical activity of assembly and segregation is quickly

diminished; we soon become able to collect objects by 'drawing a mental

line around them' (as we might say) or by 'looking at them as things which are to be called by a common name' (in Mill's phrase).

At times, Mill courts misinterpretation, suggesting that arithmetic is

concerned with particular physical operations (see p. 169). Frege pointed out, correctly, that Mill's discussion of the definition of '3 ' will not allow us to

speak of three strokes of a clock or three methods of solving an equat ion) 4

However, some of Mill's own remarks suggest that he is sensitive to the Fregean point. Three paragraphs before the passage which Frege cites, Mill

points out that algebra is applicable to all things, and, in an earlier chapter, he is even more explicit:

The properties of number, None among all known phenomena are, in the most rigorous sense, properties of all things whatever. All things are not coloured, or ponderable, or even extended; but all things are numerable (p. 146). is

My proposal to construe numerals as predicates of collective operations, of which physical collectings furnish one example, can thus be seen as according with Mill's most enlightened aims.

Why, then, does Mill suggest that arithmetical definitions presuppose facts about physical operations which produce spatial configurations? The answer is straightforward: Mill wants to claim that arithmetical definitions presuppose facts about collective operations and, in explaining the notion of a collective operation, he naturally turns to those physical operations which are the most comprehensible examples. These particular operations warrant us in accepting the definitions, and Mill mistakes their epistemological


relevance, concluding that the definitions of the numerals presuppose facts

about spatial configurations. This narrow view of a collective operation is belied by his more insightful remarks concerning the universal scope of

arithmetic.

IV

[ have sketched what I take to be Mill's optimal view about the logical form

of arithmetical statements. Our next task is to see if these hints can be developed into a workable account.

The primitive notions to be used in reformulating arithmetic are those of

a one-operation, 16 of one operation being a successor to another operation,

of an operation being an addition on other operations, and of the matchability

of operations. (I shall ignore multiplication for purposes of simplicity and

deal solely with elementary additive arithmetic; it is easy to extend the account to multiplicative arithmetic.) These notions are readily comprehensible,

either in terms of our crude physical paradigm or in more abstract terms. We perform a one-operation when we perform a segregative operation in which a single object is segregated. An operation is the successor of another operation if we perform the former by segregating all of the objects

segregated in performing the latter, together with a single extra object. When we combine the objects collected in two segregative operations on distinct objects we perform an addition on those operations.

We now turn to the important notion of matchability. This notion will play in our theory a central role akin to that of identity in the standard

presentation of arithmetic. Individual operations are of little interest to us. We are concerned with types of operations, where operations which are

matchable belong to the same type. Two segregative operations will be said to be matchable if the objects they segregate can be made to correspond with one another. (The notion of matchability will thus be an equivalence relation.) Arithmetic is concerned with those properties of segregative operations which are invariant under matchability.

It should be clear that the arithmetical notions taken as primitive here can

be related to more general notions - the notions of a collective operation and of a correlative operation - which we may consider to be the concern of a reformulated set theory. Integrating arithmetic within a revised set theory


would take us too far from the project of reconstructing Mill's views, so I

shall not pursue the issue here. 1 ~ Instead, I shall develop a formal system of arithmetic which will

recapitulate the work of the standard systems of first-order arithmetic. We

take a first-order theory with identity with (nonlogical) primitive predicates

'Ux', 'Sxy', 'Pxyz', 'Mxy', ('x is a one-operation', 'x is a successor operation

of y ' , 'x is an addition on y and z ' , 'x and y are matchable'). Clearly, we need

to provide some axioms about matchability. Prominent among these will be

assertions that matchability is reflexive, symmetric and transitive. But we

know much more than this about our intended concept of matchability.

Anything matchable with a one-operation is a one-operation and, conversely,

any two one-operations are matchable. If two operations are successors of

matchable operations then they are matchable. If an operation a is matchable

with a successor of some operation b then there is an operation matchable

with b of which a is a successor. So we already arrive at the following axioms

of Mill Arithmetic:

(1) (x)Mxx (2) (x) (y) (Mxy -, Myx) (3) (x)(y)(z)(Mxy -+ (Myz ~ Mxz)) (4) (x)(y)((Ux &Mxy) -+ Uy) (5) (x)OO((Wx & wy) ~ Mxy) (6) (x)(y)(z)(w)((Sxy & Szw &Myw) -+Mxz) (7) (x)(y)(z)((Sxy &Mxz -+ (3w)(Myw & Szw))

The first-order Peano postulates need to be embodied within our system.

The principle that no two distinct numbers have the same successor will be reformulated as the statement that if two operations are successor operations

and are matchable then the operations of which they are successors are mat-

chable. The statement that one is not the successor of any number is analyzed

as the claim that no one-operation is a successor operation. Finally, the induc-

tion principle is glossed as the assertion that whatever property is shared by

all one-operations and which is such that if an operation has the property

then all successor operations of that operation have the property is a property which holds universally. So we add to our axioms:

(8) (x)(y)(z)(w)((Sxy & Szw & Mxz) ~Myw)


(9) (x)(y)~(Ux & Sxy) (10) ((x)(Ux "+ (bx) & (x)(y)((~y & Sxy) ~ (bx)) -+ (x)~x

for all open sentences 'cbx' of the language. The recursive definition of addition is often introduced by adding two

further axioms: the result of adding one to the number n is the successor of

n; and the result of adding the successor of rn to n is the successor of the result of adding m to n. These specifications can easily be reformulated in

our language. We add the axioms

(11) (x)(y)(z)(w)((Pxyz & Uz & Swy) -+Mxw) (12) (x)O,)(z)(u)(v)(w)((Pxyz &Szu &Svw &Pwyu) ~Mxv)

Do these axioms suffice for the development of arithmetic in the usual

way? Unfortunately not. The reason is not hard to discern. A standard

system of first-order arithmetic must be interpreted by assigning an object

to '1', and functions to the symbols for successor and addition. These

constraints, together with those imposed by the Peano postulates, require us to choose an infinite domain of interpretation. Mill arithmetic, as

presented so far, is much more liberal. The twelve axioms given allow for

interpretations in finite domains. As an example, let D be a set of arbitrary

finite cardinality; assign to 'U' the entire set D, to 'M' the Cartesian product

D xD, to 'S' and to 'P ' the null set. The inadequacy of the reformulation appears as soon as we try to prove

some of the usual theorems of elementary arithmetic. Standard proofs of such results as the commutativity of addition require us to assume the existence of numbers, which we can establish by using the fact that successor and addition are functions. (Thus we can assume that each number has a unique successor and that there is a unique number which is the sum of two other numbers.) In Mill arithmetic, as presented so far, these proofs break down because we have no right to the existence assumptions on which they depend. (For an example see the Appendix.)

We can easily remedy the situation by adding principles which ensure the existence of enough operations. The following naturally suggest themselves:

(13) (3x)Ux (14) (x)(3y)Syx (15) (x)(y)(3z)ezxy


These principles are obvious analogues of the commitments involved in introducing '1' as a constant and the symbols for successor and addition as

function symbols. By adding them, we can replicate in Mill arithmetic the standard development of elementary arithmetic.

However, it may seem that, in exposing the need for such existence assumptions, we have uncovered the Achilles heel of the suggested treatment.

We have used the crude physical paradigm of operations of spatial segregation as a means of explaining the primitive predicates. The axioms (1)-(12) seem

to accord with this schema of interpretation. Yet when we try to construe the existence assumptions (13)-(15) by taking the predicates to apply to

physical operations we run into trouble. For when the predicates are read in this way, (14), for example, is straightforwardly false. Given that the existence of an operation consists in its performance, it is just not true that

for any physical operation of segregation there are successor operations: Mill arithmetic seems to founder on the same difficulty which besets tradi-

tional nominalist proposals. For example, if one regards arithmetic as having

an ontology of stroke-symbols, construed as concrete physical inscriptions,

there is the familiar problem that there are not enough physical inscriptions to do duty for the natural numbers.

Mill arithmetic has an advantage over standard nominalist programs in that

it offers a direct explanation of the applicability of arithmetic. Arithmetical

truths are useful because they describe operations which we can perform on any objects. The classical nominalist, by contrast, faces the puzzle of why

studying the properties of physical inscriptions should be of interest and of service to us in coping with nature (or, alternatively, the puzzle of why the

features of :extralinguistic entities are reflected in the properties of signs). 18 This advantage can be turned to good account. Suppose that we concede that the existence assumptions (13)-(15), when interpreted by means of our

crude physical paradigm, are false. This means that when we interpret Mill arithmetic according to the crude physical paradigm Mill arithmetic will be false. But the fact that Mill arithmetic cannot accurately be applied to the

description o f the physical operations o f segregation, spatial rearrangement, and so forth, is not fatal to the applicability o f Mill arithmetic. Mill arithmetic would be accurately applicable to a world in which there were

more physical operations than there are in ours, enough to satisfy the existence assumptions when they are construed in physical terms. Thus, in applying the

principles of Mill arithmetic to our world we regard our world as if it were an


ideal world in which the existence assumptions are true. In Mill's terms, we

'divest' our world 'o f some portion of its properties' . Mill arithmetic is

applicable to the physical operations which we perform, only if we idealize.

Here we must be careful. One natural way to view an idealizing theory is to interpret it as descriptive of a close possible world, one which is very

similar to the actual world but has the advantage of lending itself to simpler description. The trouble with this picture is that it may cause epistemological

worries. When we begin to think of possible worlds as planets in space, some

of which are relatively close to the actual world (the home planet) and others of which are remote, then we may wonder how experience which is tied to

one planet can give us knowledge of others. Aficionados of possible worlds naturally protest that the picture misleads us and that epistemological

anxieties are unfounded. However, since I want to defend the epistemological adequacy of Mill's theory, I shall try to present the thesis that arithmetic provides an idealized description of our physical operations in a way which avoids the unfortunate suggestions of the possible worlds picture.

Let us focus on the example of tile successor operation. As I have already

remarked, when we interpret Mill arithmetic by means of our crude physical paradigm, (14) is false. Sometimes we perform a collective operation but do

not perform a successor to it. However, equally obviously, when we try to perform a successor operation, we always succeed. Observing this elementary fact, we project that trial would have been followed by success, even in '

those cases where we did not try. As I pointed out above, Mill's thesis about arithmetic is that arithmetic records 'permanent possibilities of manipula- tion'. So we might gloss the thesis that trial results in success, by concluding

that those features of the world to which arithmetic ultimately owes its truth always allow for the performance of successor operations, and that the falsity of (14), when physicalistically construed, exposes the imperfection of the

manifestation of these features. Continuing, we project that our ability to perform successor operations is unlimited, and hence that it is possible that (14) be satisified, even under its physicalistic construal. Proceeding analogously, we arrive at the conclusion that it is possible that (1) (15) should describe our physical operations, regarding this possibility as making manifest the way in which the world of objects permits our rearrangement of it. 19

At this stage, we may introduce, without risk of confusion, the terminology of possible worlds. To say that it is possible that (1) - (15) should be physicalistically satified is simply to say that there is a possible world in which they


are physicalistically satisfied. Our evidence for the latter claim is the evidence

for the former, evidence adduced in the last paragraph. We can now, finally,

state a Millian theory of arithmetic.

Recall that in Section II we interpreted Mill as advocating the view that

the first principles of arithmetic have the character of definitions and that

definitions must be justified by means of their applicability. Suppose that we

now regard the principles (1)-(12) as (partial) implicit definitions of the

primitive notions of Mill arithmetic. These principles pin down the concepts of one-operation, successor operation, addition and matchability. Once these concepts have been fixed we can consider possible worlds in which there exist a stock of operations which fall under them, a stock which is rich enough to

satisfy the principles (13)-(15). Call such worlds M-worlds. Let the operations which fall under the concepts of one-operation, successor operation, and addition in an M-world be the arithmetical operations of that M-world. The

hypothesis that Mill arithmetic is applicable in the study of our world, in its

most straightforward version, is the thesis 'that our world is an M-world and that the physical operations of segregation which we perform are among its arithmetical operations. In this version the hypothesis is false. We do not perform enough such physical operations to make our world an M-world with

respect to these operations.

What is at fault is not the hypothesis that Mill arithmetic is applicable to

the study of our physical operations but the interpretation which we have given to that hypothesis. As we saw above, in demanding that our definitions

be applicable to the world, Mill wavered between the claim that objects

corresponding to the term defined exist and the claim that such objects be

possible. Our discussion helps to expose the criterion towards which Mill was

groping. We accept idealizing theories when we believe these theories describe the world, not as it is, but as it would be if accidental or compli-

cating features were removed. Mill arithmetic applies directly to M-worlds, that

is, to possibilities which we recognize on the basis of our experience and which we see as making manifest features which are partially revealed in our accidentally limited manipulations. In describing possible worlds which exhibit more clearly some features of the actual world, Mill arithmetic may reasonably be said to apply to the study of our world, and we are thus justified in accepting it.

The point will be reinforced by comparing arithmetic and geometry. The first principles of geometry can be formulated as a set of definitions (the


definitions of the names of various types of figures) supplemented with a set

of existence assumptions (assertions that circles exist with any center and

radius (etc.). As we saw in Section I, the existence assumptions can

themselves be regarded as characterizing the notion of a Euclidean space. We

may then reformulate the standard theorems of geometry as statements

which are implicitly relativized to the notion of a Euclidean space. The

relativization can be made explicit: so, for example, we can rewrite the usual

claim that all triangles have the property that the sum of their angles is 180 ~

as the sentence 'In any Euclidean space, the sum of the angles of any triangle

is 180 ~ When they are rewritten in this way, the theorems of geometry are

logical consequences of the definitions of the terms which they contain.

Geometry explores the consequences of a system of definitions. We are

justified in accepting those definitions because we can regard objects in our

world as conforming to the definitions of the names of geometrical figures (we can regard rigid rods as covering straight paths in space, coins as cover-

ing circular regions etc.), and, having made this idealization, we can regard

our world as if it were a Euclidean space.

Similarly, if we take (1) - (12) to provide (partial) definitions of the

notions of the various arithmetical operations and (13)- (15) to characterize

the notion of an M-world, then the usual theorems of arithmetic can be

reinterpreted as sentences which are implicitly relativized to the notion of

an M-world. The analogues of statements of ordinary arithmetic will be

sentences describing the properties of operations in M-worlds. ( '2 + 2 = 4'

will be translated as 'In any M-world if x is a 2-operation and y is a 2-opera-

tion and z is an addition on x a n d y then z is a 4-operation.') These sentences

will be logical consequences of the definitions of the terms which they

contain. Arithmetic explores the consequences of a system of definitions. We

are justified in accepting those definitions because we can regard operations

which we perform (for example, operations of physical segregation as well as

less crude physical operations) as if they conformed to the notions of the

arithmetical operations as they are defined, and, having made this idealiza-

tion, we can regard our world as if it were an M-world. In Mill's treatment,

arithmetic and geometry are strikingly parallel. 2~

V

I hope to have shown that Mill's approach to arithmetic is not as bad as it has


been taken to be. If Mill's inconsistencies and his enthusiasm for any

considerations which conflict with rival doctrine are once cleared away, his position can be seen to rest on two interesting theses. He claims that the

principles of arithmetic are analytic a posteriori and that our physical

operations of segregation and combination provide an approximate model for

those principles. His theses dovetail to yield an account of arithmetical

knowledge which is prima facie defensible. Which of his empiricist successors has done as well? 21

A P P E N D I X

To see the inadequacy of (1)- (12) , we need only consider the standard proof

of the elementary theorem (x)Cv)(z)(x = y -+x + z = y + z), which would be

reformulated in Mill arithmetic as (x)(y)(z)(u)(v)((Puxz & Pvyz &Mxy)-+

Muv). The standard formal proof formalizes the following simple argument:

(i) I f z = O , t h e n x + z = x , y + z = y . So, i f x = y , x + z = y + z .

(ii) Suppose that, i f x = y then x + n = y + n . Let n' be the succes-

sor of n. By the recursion axiom for addition, we know that

x + n' = (x + n)', y + n' = (y + n) '. So, i f x = y, x + n = y + n, and

hence x + n' = (x + n) ' = (y + n) ' = y + n' .

There is no difficulty in adapting (i) to Mill arithmetic. Trouble arises when

we try to reconstruct (ii). To complete the induction step, we need to show

that the supposition that if Puxz & Pvyz & Mxy then Muv enables us to

conclude that, if z ' is a successor of z then, i fPu 'xz ' & Pv'yz' & Mxy then

Mu'v'. We can attempt to do this as follows:

(a) . . . . . Suppose Pu xz , Pv yz , Mxy, Sz z, Puxz, Pvyz.

(b) Let u* be a successor of u, v* be a successor for v. (i.e. Su*u, Sv*v).

(c) By (12), Mu'u*, My'v*. Given the supposition that i fPuxz & Pvyz & Mxy then Muv, we can conclude, by (a), that Muv. So,

by (b), Mu*v*. Using (2) and (3) we can now obtain the desired conclusion that Mu'v'.

This argument recapitulates the proof idea deployed in the standard treat-

ment. However, given (1)- (12) , we have no warrant for (b). Only when we

add (14) are we entitled to assume that u, v have successors. Hence, Mill

234 PHILIP KITCHER

ar i thmet ic without exis tence assumpt ions ( tha t is, ( 1 ) - ( 1 2 ) ) does n o t suffice

for the derivat ion o f the analogues o f s tandard theorems , bu t I th ink it is no t

ha rd to see tha t the addi t ion o f ( 1 3 ) - ( 1 5 ) repairs the def ic iency.

University o f Vermont

NOTES

i For Frege's criticisms, see: The Foundations of Arithmetic, pp. 12-17, 29-33. Examples of more recent criticisms can be found in A.J. Ayer, 'The a priori' (in P. Benacerraf and H. Putnam (eds.), Philosophy of Mathematics, Prentice-Hall, 1964, pp. 289-301), pp. 291-2, in Rudolf Carnap, 'Intellectual autobiography', (in P. Schilpp (ed.), The Philosophy of Rudolf Carnap, Open Court, 1963), pp. 47, 65, and in C. G. Hempel, 'On the nature of mathematical truth' (in Benacerraf and Putnam op. cit., pp. 366-381), pp. 367-8.

See, for example, W. V. Quine, Philosophy of Logic (Prentice-Hall, 1970) p. 100. 3 The phrase is that of Jevons. In a series of articles, Jevons tried to show that Mill's views on logic and mathematics are hopelessly muddled. A lucid rebuttal of Jevons' critique of Mill's views on geometry has been given by Reginald Jackson in 'Mill's treatment of geometry' (J. B. Schneewind (ed.), Mill, Doubleday, 1968, pp. 84-110). 4 j .S . Mill, A System of Logic (Longmans, 1970). Future references to this work will be given parenthetically in the text. s Obviously, there are similarities between Mill's distinction of connotation and denotation and Frege's distinction of sense and reference. The idea that expressions usually play two semantic roles is shared by both writers. Moreover, Frege's principle that the sense of an expression determines the referent of the expression finds its analogue in Mill's claim that "connotation precedes denotation". Yet it would be wrong to indentify the notions of sense and connotation. Mill explicitly claims that connotations are attributes, a view which seems quite foreign to Frege's ideas about sense. Nor is Mill concerned with the ways in which the connotations of complex expressions result from the connotations of their constituents. Thus his semantic theory lacks the systematic character which is found in Frege's. For a good discussion of some aspects of Mill's semantic theory, see M. Lockwood, 'On predicating proper names', Philosophical Review LXXXIV (1975), pp. 471-498. 6 See pp. 70 and 94. I conjecture that Mill finds the doctrine that definitions lack truth value enticing because it can be used in a reductio of the thesis that mathematical first principles are definitions: one can't squeeze mathematical truth out of premises which are neither true nor false! Mill's enthusiasm for the doctrine leads him into a number of confusions, and, in particular, he tends to obliterate his distinction between direct and indirect definitions. I shall ignore these passages because they are at odds with the main tenets of his position.

In any case, this argument would prove too much. If it were cogent, it would show that definitions which do not imply the actual existence of objects conforming to the definition cannot be used as premises in arguments which are not solely concerned with words. As a result, Mill would be committed to the thesis that if definitions of the names of geometrical figures can be used in proofs then there must be objects which answer to those definitions. Since he denies that objects found in nature exemplify geometrical definitions, he would be forced to conclude that there are ideal objects which do exemplify the definitions, a conclusion which he strenuously rejects (pp. 97, 147-150;

ARITHMETIC FOR THE MILLIAN 235

in 'Mill's treatment of geometry' Jackson gives a clear presentation of Mill's views on this issue). As we shall see below, cases like that of geometry incline Mill to the view that mathematical definitions imply only the possible existence of objects conforming to them.

Frege, op. cir., pp. 7-8 . 9 [bid. ,p . 9. 10 j. Locke, Essay Concerning Human Understanding (ed. by A.C. Fraser, Dover, 1959), Vol. 2, pp. 226-9. 11 See Kant's reply to Eberhard, quoted in L. W. Beck, 'Can Kant's synthetic judgments be made analytic?' (in R. P. Wolff (ed.), Kant, Doubleday, 1967, pp. 3-22), pp. 13-14. Quine appears to make a similar point in 'Truth by convention' (in: The Ways of Paradox, Random House, 1966, pp. 70-99). 12 Frege, op. cit., p. 9. 13 After arriving at the ideas presented here, I discovered that, i n an unpublished paper, Glenn Kessler develops a reconstruction of Mill's arithmetic in which numbers are relational properties of aggregates and aggregates are construed as sums. In Kessier's treatment, numbers relate aggregates to properties, and the formalization consequently involves quantification over properties. There are interesting parallels between my approach and that which Kessler adopts, but I believe that my reconstruction is not only simplelc but also retains more of the epistemological benefits which Mill hoped to achieve. 14 Frege, op. cit., pp. 20-21, 30-32. Quine seems to recapitulate these charges in: Philosophy of Logic, p. 100. ~s See also p. 401 (and compare Frege, op. cir., p. 21). Of course, much depends on how we read Mill's term 'thing'. At the very least, however, Frege's interpretation is un- charitable. 16 This corresponds to standard presentations of arithmetic in which one is taken as the first number. Alternatively, but less naturally, we could begin with the notion of a zero- collection. The relative unnaturalness of the latter notion pays some tribute to the Fregean point about the nonobservability of the facts presupposed in the definition of ~0'. 17 Just as collective operations do duty for sets, so too correlative operations do duty for relations. For suggestions about the integration of arithmetic within set theory, see the final section of my paper 'The plight of the Platonist' (Nofis XII (1978), pp. 119- 136). The present paper articulates further the approach to arithmetic which I proposed there. A program which is, in some ways, similar was suggested be Dedekind (see 'The nature and meaning of numbers', in W. Beman (ed. and trans.), Dedekind's Essays on the Theory of Numbers (Dover, 1963)). is The most sophisticated nominalist treatment of arithmetic of which I am aware is that given by Dale Gottlieb in 'The truth about arithmetic', American Philosophical Quarterly 15 (1978), pp. 81-90. Gottlieb's proposal has four disadvantages compared with that given here. Firstly, in using substitutional quantification, Gottlieb is apparently committed to divorcing the semantics for arithmetic from that for the rest of the language. Secondly, his approach provides no explanation of the global applicability of arithmetic. Thkdly, as he notes, his own account provides only a reduction of epistemological questions about arithmetic to epistemological questions about logic, not a full-fledged theory of arithmetical knowledge. Finally, there seem to be legitimate worries about extending the account to deal with real numbers. (See Gottlieb, p. 89, for a candid discussion of this). On my approach, given the integration of arithmetic within set theory indicated in 'The plight of the Platonist', an analogue of real arithmetic should be obtainable in familiar ways. 19 Obviously this paragraph owes a considerable debt to Chapter II of Nelson Goodman's: Fact, Fiction and Forecast (Bobbs-Merrill, 1965). My purpose is to block worries about our knowledge of possibility by assimilating that knowledge to knowledge we obtain

236 PHILIP KITCHER

through induction. I also sense agreement with the attitude towards modality presented in F. Mondadori and A. Morton 'Modal realism, The poisoned pawn' (Philosophical Review LXXXV (1976), pp. 3-20). The central thesis of Mondadori and A. Morton's paper is that modal statements describe properties which we can only pick out in certain special ways. Analogously, we might say that, on the Millian account given here, arithmetic is true in virtue of properties of the world which we can only pick out in special ways, viz. through our operations on objects. We arrive at full recognition of the properties by induction from observations of the operations we perform. 2o My concern here has been with a Millian reconstruction of arithmetic, but it is worth pointing out that there are easy ways to give accounts of ascriptions of cardinality within the framework I provide. Let us introduce the two place predicate 'Cxy' to mean that x is a collective operation which collects (possibly among other things) the object y. This two place predicate is the analogue of the membership relation in standard systems of set theory. Then, using the predicate 'Nx' to mean that x is an n-operation (such predicates being definable from the notions of unitary operation and successor in obvious ways), we could write the statement 'There are n F ' s ' as:

( 3 x ) ( ( y ) ( F y ~ Cxy) & Nx) .

Alternatively, using the counterfactual conditional '[] ~ ' (where ~ [] ~q-q means that if it were the case that p it would be the case that q), we could write

( x ) ( ( y ) ( C x y .~ Fy) [] -+ Nx) .

Either of these approaches can be integrated within our framework in a way which avoids Frege's criticisms of Mill's account of cardinality and which offers a parallel treatment of cardinality to that which Frege provided. 2a In writing this paper, I have been helped by reading an unpublished manuscript by Michael Resnik which deals, in part, with Mill's theory of arithmetic. (Resnik's conclusions about the viability of Mill's approach are, however, somewhat different from my own.) I am very grateful to Resnik and to Leslie Tharp for constructive criticisms of an earlier draft, and I would like to thank David Fair, Patricia Kitcher and George Sher for advice and encouragement.

arithmetic for the millian

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