intuition and reflection in arithmetic: michael potter

INTUITION AND REFLECTION INARITHMETIC

Michael Potter and Bob Hale

I—Michael Potter

ABSTRACT If arithmetic is not analytic in Kant’s sense, what is its subject mat-ter? Answers to this question can be classified into four sorts according as theyposit logic, experience, thought or the world as the source, but in each case weneed to appeal to some further process if we are to generate a structure richenough to represent arithmetic as standardly practised. I speculate that thisfurther process is our reflection on the subject matter already obtained. Thissuggestion seems problematic, however, since it seems to rest on a confusionbetween the empirical and the metaphysical self.

Two questions. Can we give an account of arithmetic whichdoes not make it depend for its truth on the way the world is?

And if so, what constrains the world to conform to arithmetic?In trying to answer these questions philosophers have repeat-

edly been drawn to the idea that arithmetic is in some sensedevoid of content. Just which sense has varied according to thephilosopher: some (for example, Leibniz) have thought it analyticin something like Kant’s sense, some have tried to extend thenotion of the analytic more widely; some (for example, Ramsey)have tried to make arithmetic tautologous in the sense of theTractatus. But none of these attempts has been altogether suc-cessful. It seems impossible to justify arithmetic without appeal-ing to some subject matter which it is ineliminably about. But ifthere is now widespread agreement on this, there is certainly nosuch agreement on the nature of the ineliminable subject matterin question. My principal aim here will be the relatively modestone of recommending a classification of the subject matters thathave been proposed into four sorts, although I shall concluderather more ambitiously by suggesting that this classificationenables us to see all of them as deficient in essentially the samerespect.

We can agree straightaway with Kant that arithmetic cannotbe analytic in his narrow sense. Aristotelian logic, strictly under-stood, recognizes no means of distinguishing between objects

I—MICHAEL POTTER64

independent of the concepts they are supposed to satisfy. Objectsfunction merely as faceless placeholders for the application ofconcepts. In such a logic we cannot even make sense of the ideaof counting. Since the extent of the Kantian analytic coincideswith what can be proved in this restricted logic, arithmetic is notanalytic in Kant’s sense. To go beyond this restricted logic weneed the general notion of an object. But do we have access toany such general notion independent of any particular structurewithin which we can conceive of the object as occurring? Canwe, in other words, perceive an object without perceiving it insome particular way? It was essential to the coherence of Kant’ssystem as a whole to say that we cannot, but to insist that theonly way we can be affected by an object is via the sensibility,which imposes a structure to which we conceive of the objectas conforming. The difficulty with this is that it makes not justarithmetic but quantified logic depend—implausibly—on thespatio-temporal structure of the world as we experience it.

Treating the sensibility as schematic, so that quantified logic iswhat all beings with any sort of sensibility have in common, helpsonly until its incoherence is pointed out: viewed in this way therewould not really be anything the beings had in common. Thealternative is to deduce the notion of an object from that of aconcept: purely conceptual thought is impossible since the notionof a concept presupposes that of an object to which it may ormay not apply. If we adopt this idea, quantified logic will beapplicable to objects just because objects are the subject matterof propositions.

Once we have quantified logic, we can at least count: we can,for instance express that there are two Fs by the proposition that

(∃x, y)(Fx · Fy · x≠y · (z)(zGx∨zGy∨∼Fz)).

Indeed we can generalize this by writing

0(F )GDf ∼ (∃x)Fx,

s(n)(F )GDf (∃y)(Fy · n(λx · x≠y)),

and defining ‘m is a number’ to mean

(F )((F0 · (n)(Fn⊃F(s(n))))⊃Fm).

On this account numbers are not objects but second level con-cepts. We should not let this deter us too quickly since we have

INTUITION AND REFLECTION IN ARITHMETIC 65

not yet been given any strong reason to think that numbers areobjects. What is more worrying is that although we can countobjects, we cannot count anything else—concepts for instance, ornumbers. Moreover, we cannot generate arithmetic as standardlypractised unless we assume that there are infinitely many objects,and the supposition that thought is by its nature propositional isnot in itself sufficient to deliver that conclusion. So this accountfalls to the same difficulty that led us to reject empiricism andpsychologism, namely that it grounds only strict finitistarithmetic.

If we wish to extend arithmetic any further we must appeal tosomething else to provide it with the necessary content. But oursearch for a source of the content of arithmetic is heavily con-strained by our insistence that we should aim to explain theapplicability of arithmetic to the world.

Each of us has experiences, both of what we take to be anexternal world and of our own inner mental life. These experi-ences do not occur in us merely as an unstructured stream: wehave thoughts involving them, and those thoughts can by the actof thinking become the objects of further experiences. Experi-ence, though, is essentially private: my experience is intrinsicallymine and I cannot share it with you, however hard I try. Butalthough we have no experiences in common, our sharedhumanity somehow permits us to communicate about them toeach other: I do not thereby have your experiences, but I docome to know something about them. This is achieved throughthe medium of language, not just written language but other sortstoo—speech, gesture, music, flags.

One has only to reflect on it to realize that this link betweenlanguage, thought, experience and the world, which is at the verycentre of what it is to be human, is truly remarkable. It is indeed,as Schlick remarked,

astonishing that by hearing certain sounds issuing from the mouthof a person, or by looking at a few black marks on a piece ofpaper I can become aware of the fact that a volcano on a distantisland has had an eruption [...] The marks on the piece of paperand the eruption of the volcano are two entirely distinct and differ-ent facts, there is apparently no similarity between them, and yetknowledge of the one conveys to me knowledge of the other.1

1. Form and content. An introduction to philosophical thinking’, p. 286.


The challenge of accounting for the applicability of arithmetic tothe world evidently participates in this wider puzzle of explainingthe link between experience, language, thought and the world.We can distinguish accounts that look to each of these to supplythe content we require: those that involve the structure of ourexperience; those that explicitly involve our grasp of a ‘thirdrealm’ of abstract objects distinct from the concrete objects ofthe empirical world and the ideas of my private Gedankenwelt;those that appeal to something non-physical that is neverthelessan aspect of reality in harmony with which the physical aspectof the world is configured; and those that involve only our graspof language.

The first sort of account gives the sensibility the originalKantian role of structuring experience. In Kant’s conception thisstructure is spatio-temporal. Whether Kant saw arithmetic asdependent on the temporal part of this structure alone or on bothspace and time is largely irrelevant since the principal difficultywith Kant’s account is that it seems implausible for arithmetic todepend on either space or time. But already we begin to see—only dimly in Kant; rather more clearly, perhaps, in Schultz’scommentary on the Critique2—a distinction between the initialinput from sensibility, which supplies the subject matter to get amathematical argument started, and the subsequent role of sensi-bility in regulating the progress of the proof. Kant evidentlythought that the spatial structure of sensibility plays a part inboth aspects in the case of geometry, but there is no firm evidencethat he thought the same about arithmetic. He may well havethought that there intuition is required only in order to ensurethe existence of enough objects and not in order to constrain theproof procedures thereafter.

Hilbert’s alternative conception of the sensibility as providinga way of experiencing the world as constituted by finite arrays ofconcrete objects is at first sight more promising since it does notcommit us to supposing that the particular structure our currentphysical theories ascribe to the spatio-temporal world is a priori.But even if it delivers arbitrarily large finite numbers, and henceall true numerical equations and inequations, the sensibility soconceived does not on its own ground generalizations about

2. Schultz, Prufung der Kantischen Critik der reinen Vernunft.


numbers. In order to generate even Primitive Recursive Arith-metic (which is still in some ways rather weak) we need to beable to intuit not only particular arrangements of objects but thestructure to which they are all subject: we need, to put the matterin Kantian terms, a pure intuition of the form of finiteness.

Even then, however, we do not have the whole of arithmetic.The surprising fact demonstrated by Godel is that we need toattend not only to the objectual content of arithmetic but to theorder of quantification over that content. If we wish to groundPeano Arithmetic on Hilbertian lines we need intuitions not onlyof finite arrangements of things and the structure to which thesearrangements are subject but also of properties of arrangements,and properties of such properties, ad infinitum.

Frege’s account of arithmetic was of the second sort. Hederived arithmetic from the numerical equivalence

NxFxGNxGx ≡ F∼G

and hoped to show that this was a purely logical principle. Hethought this could be done indirectly via an explicit definition ofnumbers as extensions of concepts, but that depended on BasicLaw V, which cannot be a logical law since it is contradictory.To do it directly instead we should have to establish a notion ofcontent for which the left hand side of the numerical equivalencecould be seen as doing no more than recarving the contentexpressed by the right hand side. Frege did not supply any suchnotion of content, and the difficulty involved in doing so is severesince we know that the syntactically similar Basic Law V cannotbe viewed in terms of such a recarving.

Frege’s appeal to the notion of content at this point indicatesthat he had failed to justify the numerical equivalence as purelylogical. In the same way Dedekind’s construction of the naturalnumbers appeals to a domain of thoughts which we can come toknow by means of the intellect. Of course, for Dedekind’s proofthat there are infinitely many objects to be valid ‘thoughts’ heremust mean thoughts available to me rather than thoughts I shallactually have. What Dedekind does not supply, however, is anargument to show that our experience must conform to the struc-ture of the domain of thoughts so conceived. Kant, of course,never considered this matter because he thought intellectualintuitions of the kind Dedekind appeals to were possible only for


God and not for finite beings such as ourselves. In any case thestep in Dedekind’s treatment that is quite unmotivated is the for-mation of a system with all these thoughts as members. (Thislacuna is understandable: Dedekind was writing without anyawareness of the paradoxes with a conception of systems asfusions, not as classes.)

In any case the position is analogous to that faced by Hilbert’saccount: the appeal to a realm of thoughts may supply us withinfinitely many objects, but this still does not deliver any morethan the quantifier-free arithmetic of equations and inequationsunless we can take the further step of regarding the numbers —whether obtained by Frege’s route via the numerical equivalenceor Dedekind’s via intellectual abstraction—as lying in thedomain of the quantifiers. And as soon as we take this step, ouraccount is evidently impredicative. We have two choices, both ofwhich seem unattractive: if we say that thoughts occupy a realmquite distinct from the empirical things we normally talk about,we need to explain why what we say about one realm can beused in reasoning about the other; if we say instead that theyoccupy the same realm, of which we have a single, unified grasp,we seem to have lapsed into idealism, since there is no reason tosuppose that anything real belongs to this realm at all.

One way out of this impasse might be to follow Russell inlooking to the world itself for the content of arithmetic, sincethis makes the applicability of arithmetic to the world whollyunproblematic. Alone among the accounts we have been con-sidering it makes arithmetic truly independent not only of mebut of all of us; but it is hard to see how the world could supplythe infinite content we require unless there are in fact infinitelymany things. The trouble with this is that it is an empiricalhypothesis which there seems no good reason to believe. Andeven if we can bring ourselves to believe the axiom of infinity asa claim about the world, there is a further difficulty with thelogic. For even if the physical world supplies enough objects forarithmetic, Godel’s theorems show that it cannot supply on itsown a structure rich enough to ground the unrestricted inductionscheme any more than Hilbert’s concrete intuition could. Inorder to make up this deficit Russell had to posit other entities—universals of some kind or other—which participate in the worldand lend it structure; and what he had to assume about these


other entities was the axiom of reducibility. But it is hard to seewhat possible way there could be of telling whether this is trueor not. Russell advocated it on the ground that it has so farenabled us to prove mathematical propositions we alreadybelieved for other reasons to be true (e.g. 2C2G4) and not oneswe already believed to be false (e.g. 2C2G5). But this merelythrows the question back onto the ‘other reasons’ we had forbelieving that 2C2 is 4 and not 5.

The fourth sort of account gives the role of sensibility to ourgeneral grasp of language. This is what Wittgenstein appealed toin the Tractatus to explain arithmetic. He thought that byextending the account based on numerically definite quantifiersso that it applied to operations in general he could get a treat-ment of arithmetic adequate for practical purposes. In order todo so he had to invoke another layer of content beyond whatcan be obtained from individual propositions, namely our aware-ness of the general concept of proposition. If we allow operationsto apply at the level of symbols, thus in effect ascribing to our-selves an explicit awareness of the structure of our own language,we can ground a limited portion of arithmetic, rather as in theHilbertian account, but we do not obtain a theory of classes oreven the whole of Peano Arithmetic.

Wittgenstein was not troubled by the fact that his account didnot ground the whole of mathematics. ‘In life it is never a math-ematical proposition which we need’,3 he said. This seems hardto maintain. Although it is true that nothing in life seems to turnon the truth or falsity of Goldbach’s conjecture, we frequentlyact on beliefs dependent on arithmetical generalizations, quiteapart from the myriad applications of higher mathematics. Whatwe need if we are to pursue the idea of placing language in theKantian role of structuring experience is an extension of Wittgen-stein’s system that will ground higher mathematics.

In an attempt to achieve just this Ramsey abolished symbolsfrom the account and used instead an informal metalanguagewithin which the meanings we attach to the signs of the languagecould be described, but he was then forced to introduce a hier-archy of meanings of the word meaning in order to avoid theheterological paradox. Ramsey obscured the difficulty this causes

3. TLP, 6.211.


by keeping his metalanguage informal, but in fact he had beenled to exactly what Russell proposed in his Introduction to theTractatus, namely a hierarchy of metalanguages to which thereis no limit. In order to avoid a recurrence of the paradox in themetalanguage, we are forced, as Russell predicted, to deny thatthis hierarchy of metalanguages has a union. This is whyassuming that we can reflect on our symbolism forces us to con-clude that our grasp of language is fundamentally incomplete (inDummett’s terms ‘indefinitely extensible’).

The second part of Ramsey’s account provides for a definitionof identity, and hence permits a theory of classes, by means ofthe notion of a propositional function in extension, i.e. any map-ping from objects to propositions. Ramsey was concerned princi-pally with avoiding contradictions: his notion achieved this aimbecause the domains of objects and propositions have been delin-eated in advance and so the introduction of propositional func-tions in extension cannot lead to any explosion in these domains.Moreover, if we combine this notion with the simple theory oftypes already considered we do obtain a system that is adequatefor mathematics. But the difficulty now is that of explaining ourgrasp of the notion of a propositional function in extension. Ifwe try to explain this notion on anything like the model of ourprevious understanding of propositional functions as extractedfrom propositions by replacing constant constituents of themwith variables, we run into total perplexity as Wittgenstein real-ized. If we simply appeal to a primitive grasp of the notion, asRamsey was inclined to do, this primitive grasp seems to be anidle wheel that cannot have anything to do with the applicationof mathematics to the world.

Carnap’s account stands in the same Wittgensteinian traditionof making the sensibility linguistic, but viewed in this light thePrinciple of Tolerance amounts to a denial that our sensibilityimposes any particular structure at all. It is the structure oursensibility imposes that determines the sort of beings we are. Ifwe are, as Carnap claimed, free to adopt any structure at all, itis hard to see what connection there can be between the self thatadopts one structure and the self that adopts another.

This point corresponds precisely to the objection we made earl-ier to treating the sensibility as schematic in a Kantian accountof polyadic logic. Any account that attempts to base arithmetic


on a subject matter treated as schematic, abstracting from theparticular features of our sensibility or our language loosens thebinding between us that explains how we communicate with oneanother, and ends up in solipsism.

What conclusions can we draw, then, from this brief survey ofattempts to base arithmetic on some aspect of our relationshipwith the world? What is clear, I think, is that arithmetic is notlogic, if by logic is meant the laws to which our thought shouldconform simply by virtue of being propositional. Arithmetic hasa content that logic, so conceived, cannot supply. It is senselessto look for that content in the world, for we, being finite, areincapable of experiencing the world directly as infinite, whetherit is so or not. But the world and our configuration of it seem tobe the only sources of content available to us, so we are drivento look for a feature of the structure of our experience of theworld that can provide us not only with the infinite, which theaccounts we have considered already can supply, but with anability to impose on the infinite a further structure not implicitin the notion itself.

Where is such a further structure to be sought? I am not,unfortunately, in a position to provide conclusive proof that itcannot be derived directly from the structure of any of the foursources—experience, thought, the world, and language—con-sidered so far, but I hope what I have said has made that con-clusion seem plausible at least. In particular, I find highlysuggestive the similarities that have emerged between the scopesof the various approaches.

If this is right, what it shows, I believe, is that in order tograsp the part of arithmetic that goes beyond the mechanicalcalculations of numerical sums we have to appeal to higher orderconcepts, i.e. what Godel describes as concepts

which do not have as their content properties or relations ofconcrete objects (such as combinations of symbols), but rather ofthought structures or thought contents (e.g. proofs, meaningfulpropositions, and so on), where in the proofs of propositionsabout these mental objects insights are needed which are notderived from a reflection upon the combinatorial (space-time)properties of the symbols representing them, but rather from areflection upon the meanings involved.4

4. Collected Works, vol. II, pp. 272–3.


If we are to flesh out the idea with any more detail than this, weshall evidently have to return to the fourfold narrative adoptedearlier and consider each of the accounts separately. As aninstance let us consider here only the linguistic case. The issuehere is one that has already arisen in our discussion of Ramsey’saccount of the theory of types. We saw that in order to avoidparadox he had to adopt a hierarchy of languages. The danger,of course, is that if my self is constituted by my capacity to investthe language I speak with meaning then a hierarchy of languageswith different meaning relations is really a hierarchy of selves. Inorder to avoid this fragmentation of the self I need to assume afurther relation that binds together this hierarchy of meaningrelations into one. What could that further relation be? If Iregard it as yet another meaning relation, I am simply iteratingthe process into the transfinite—I am treating meaning as whatMichael Dummett calls an indefinitely extensible concept—but Iam only postponing the difficulty, not resolving it. (If it is poss-ible to form the union of the hierarchy, the paradox forces meto treat the meaning relation for that union as part of a furtherlanguage not yet included; if it is not possible, I have a frag-mented self again.)

The only way out that I can see is to treat the self as a distinctitem bearing to each of these languages a relation that makes itmy language. But in saying this I am of course at (or beyond)the very limits of sense, for the relation that the self bears toeach of these languages cannot be expressed in language; nor—crucially—can I simply adopt the Tractarian solution of treatingthe relation as shown by my successful use of the language eventhough it cannot be expressed within it. That way works for asingle language but makes no sense for an infinite number sinceI cannot as a finite being show my successful grasp of infinitelymany different languages at once.

So the way out that I am recommending seems unavoidably toinvolve repeated appeal to the very confusion that Wittgenstein(following Kant) warned us against, namely that between the selfof empirical psychology and the self as metaphysical subject. Ihave to conceive of myself empirically as an object in the worldinsofar as I conceive of myself as bearing determinate relationsto the languages in the hierarchy. But I must also conceive ofmyself metaphysically insofar as these relations are ultimatelyinexpressible.


This is an uncomfortable conclusion to reach: it would indeedbe alarming if the whole of higher mathematics depended on sucha confusion. But I cannot for the moment think what else to say.

REFERENCES

Godel, K., Collected Works, 4 vols (Oxford University Press, 1986–).Schlick, M., ‘Form and content. An introduction to philosophical thinking’, in

Philosophical Papers, 2 vols (Dordrecht: Reidel, 1978–9), pp. 285–369.Schultz, J., Prufung der Kantischen Critik der reinen Vernunft, 2 vols (Konigs-

berg, 1789–92).Wittgenstein, Ludwig, Tractatus Logico-Philosophicus, trans. by D. F. Pears and

B. F. McGuinness (London: Routledge & Kegan Paul, 1961).

intuition and reflection in arithmetic: michael potter

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