UDIEN ER , BRD), D. F0LLES- R. HILPINEN (Turku, ~), R. KAMITZ (Graz, lRNER (Bristol, Great ~. LAMBERT (Irvine, SA), J.L. MACKIE t aly), J. MORA VCSIK , G. PATZIG (Gottin- W. ROD (Innsbruek, . VUILLEMIN (Paris inen zweimal jlihrlieh in :ellungen sind an den )pi B.V., Keizersgraeht uzosiseher Spraehe, in t DISKETTEN im MS- lELLE KORRESPON- uden an "Grazer Philo- Hit Graz, A-SOW Graz, 1 zur Verfiigung. Seitens nem Mitglied aus dem :elegt; der HerausgebeI \.nnahme des Artikels. tigem Zustand und in tze diirfen den Umfang m nieht iibersehreiten. lmen. Handsehriftliehe ;kripten zu vermeiden. .gt werden. Die Anmer- rennt dem Text beige- lige Zusammenfassung er wie fUr die Verfasser ~anuseripts" naeh dem 1 I THE INTERNALIEXTERNAL QUESTION Philip HUGLY & Charles SAYWARD

University of NebraskalLincoln

Introduction Rudolph Carnap expresses his famous distinction between the in- ternal question and external question thus: What is now the nature of the philosophical question concerning the existence or reality of numbers? To begin with, there is the internal question which, together with the affirmative answer can be formulated ... by "There are numbers" ... This statement follows from the analytic statement "five is a number" and is therefore itself analytic. Moreover it is rather trivial ... therefore nobody who meant the question "Are there numbers?" in the internal sense would either assert or seriously consider a negative answer. This makes it plausible to assume that those philosophers who treat the question of the existence of numbers as a serious philosophical problem and offer lengthy arguments on either side, do not have in mind the internal question ... Unfortunately, these philosophers have so far not given a formulation of their question in terms of the common scientific language. Therefore our judgement must be that they have not in giving the external question and to the possible answers any cognitive content. l As a question internal to arithmetic, the question 'Are there num- bers?' has a trivial affirmative answer. As a question external to arithmetic it has no cognitive content. A fundamental question in the philosophy of mathematics is whether there are numbers. Realists produce 'There are numbers'; anti-realists produce 'There aren't any numbers'. These philosoI. Rudolph Carnap, "Empiricism, Semantics and Ontology". Reprinted from Revue Internationale de Philosophie, 4 (1950), in Irving M. Copi and James A. Gould (editors), Contemporary Philosophical Logic (New York, 1978), p. 153.

phers take themselves to be thereby making assertions on which they differ one asserting what the other denies. For Carnap it is not the case that there is either assertion or denial in such exchanges, although, doubtlessly, it feels like assertion and denial. This essay defends Camap's view of the matter. We differ on details but agree on the bottom line that either 'There are numbers' has a content unsuitable for a philosophy of mathematics or it has no content at all. 2. A realist objection A realist objection to Camap goes like this: Carnap admits that 'There are numbers' is a true sentence of arithmetic. So it is true that there are numbers. This is the fundamental thesis of realism. So what Carnap says hardly is an objection to realism. Now it is true that Camap also says that what 'There are numbers' says within arithmetic differs from what it says when asserted by a mathematical realist. But this is just obscure too obscure to bother with. Carnap says that '5 is a number' is analytic, and, since 'There are numbers' follows from it, Carnap holds that 'There are numbers' is also analytic. So Camap might have responded to part of this realist objection by saying that, since 'There are numbers' is analytic, it is not suitable for a metaphysical claim. This is not a persuasive reply to the objection. The thesis that the theorems of arithmetic are analytic is implausible, and it is not a point on which we would wish to defend Camap. 3. 'Number'within mathematics Both Camap and his realist opponent take it as obvious that 'There are numbers' is a sentence of arithmetic. But this is not obvious. After all 'There are numbers' contains no mathematical signs. In developing what we call natural number theory we use the signs for addition, multiplication, equality and the numerals, along with such letters as 'n' and 'm' together with signs for generality and sentential composition. 32 thereby making assertions on which hat the other denies. For Carnap it is assertion or denial in such exchanges, like assertion and denial. p's view of the matter. We differ on m line that either 'There are numbers' :1 philosophy of mathematics or it has tp goes like this: Carnap admits that e sentence of arithmetic. So it is true ; is the fundamental thesis of realism. , is an objection to realism. Now it is It what 'There are numbers' says within it says when asserted by a mathematical re too obscure to bother with. Camap nalytic, and, since 'There are numbers' )lds that 'There are numbers' is also have responded to part of this realist ce 'There are numbers' is analytic, it is ::al claim. This is not a persuasive reply s that the theorems of arithmetic are it is not a point on which we would wish talics opponent take it as obvious that 'There of arithmetic. But this is not obvious. 3rs' contains no mathematical signs. In tural number theory we use the signs for uality and the numerals, along with such ~r with signs for generality and sentential

Among these signs is not to be found a sign corresponding to the word 'number' in the kind of grammatical application it is given in 'There are numbers'. The mathematical sign 'number' is the word together with its use in numerical quantifiers (as in 'For every number n there is some number m such that m is greater than n'). That sign is absent from 'There are numbers'. And, apart from its use in numerical quanti- fiers, the word plays no role in arithmetic. We need to distinguish between the primitive vocabulary of arithmetic and the defined vocabulary. The primitive vocabulary is very sparse, consisting of the signs for addition, multiplication, equality, the numerals, the signs for generality and sentential com- position. (It can be rendered even more sparse with the numerals giving way to two signs: one for zero and one for the successor function.) But, in addition, there are signs which are defined in terms ofthe primitive vocabulary. For example, to say n is even is to say that, for some m, n =2 x m. So there is the option of introducing into arithmetic a predicative use of 'number' . All that is required is that we select some numerical formula Fn which is correct no matter what numeral we put for 'n'; for example, for some m, m =n + 1 will serve as a definition of n is a number thus conferring a predicative use of 'number' within arithmetic. But if 'n is a number' is to be taken as short for 'for some m, m =n + l' then the philosopher's 'There are numbers' comes to nothing other than For some nand m, m =n + 1 in its mathematical sense. Now ask yourself whether the content of mathematical realism

j is entirely mathematical in nature so that the propositions of mathei matical realism are nothing other than the theorems of arithmetic.1 This picture of a person who from time to time and perhaps itt, books and articles writes down various well-known arithmetical I quantifications and calls it a philosophy of mathematics fits nothing' with which we are familiar. ' Someone says 'There are no numbers'. The Zen master repliesj with a theorem of arithmetic, and perhaps lays out its proof. H~ meets every such challenge in just this way. He might also, from' time to time, apply the conclusicns of certain proofs in somd practical way. One might call him a philosopher, but one would not; say that what he says constitutes a philosophy of mathematics. . In any case, the mathematical realist is no Zen master. The realist wants to add something namely, that these theorems and proofsl are not about nothing that there are things (that there are thingsy; which they are about numbers. I But to add this is to utter a sentence in addition to those of! mathematics, and so to utter a sentence not a part of mathematics,~ and thus not a sentence the sense of which is secured by its place! within mathematics. I Realists may present various proofs within mathematics and j make various assertions within mathematics. But then they want tal add something, and try to do so with such words as 'There arel numbers'. This is something they can't so much as try to do if they] stick with the sentences of mathematics. So, we cannot show thad the realist's sentence makes sense by noting that it is just another! ~ mathematical sentence for which it is unproblematic that it makes! , sense. j (And what holds for the realist holds as well for the antirealist I who disputes realism by negation.) ! If 'number' is short for some mathematical formula, then 'There I are numbers' is just another quite ordinary mathematical sentence, l ) one of a kind which goes virtually undisputed. And if 'number' is I not short for some mathematical formula, then one certainly cannot I show that 'There are rumbers' has a sense by showing that it is a ! sentence of mathematics! 4. Model the A realist obje assertion 'Th is an assertiol del theory sh4 ::In n = 0 is true only if 'n' ranges i: are numbers. Modal the, idea is to defi then to define The functi variable v of second, f(,O" any terms t a Truth is t1 (t=r)istrw formulas A ~ only if A is 1 only if A is 1 f::lvA' is true except at me Within th '::In n = 0 n, n = 0 are derivabl theory it is 1 But, sam 'exists', bUI 'some'. The war, 'there exist: 34 II in nature so that the propositions of mathe- :hing other than the theorems of arithmetic. 'son who from time to time and perhaps i rites down various well-known arithmetica .s it a philosophy of mathematics fits nothin iliar. re are no numbers'. The Zen master replies hmetic, and perhaps lays out its proof. He lenge in just this way. He might also, fro he conc1usicns of certain proofs in som ht call him a philosopher, but one would no onstitutes a philosophy of mathematics. lematical realist is no Zen master. The realist g namely, that these theorems and proofs . that there are things (that there are things) numbers. o utter a sentence in addition to those 0 utter a sentence not a part of mathematics, ~ the sense of which is secured by its plac~ It various proofs within mathematics and ; within mathematics. But then they want to 1 to do so with such words as 'There are thing they can't so much as try to do if they ~ of mathematics. So, we cannot show that akes sense by noting that it is just another for which it is unproblematic that it makes the realist holds as well for the antirealist I negation.) )f some mathematical formula, then 'There ther quite ordinary mathematical sentence, ;!s virtually undisputed. And if 'number' is ~matical formula, then one certainly cannot mbers' has a sense by showing that it is a s!

. Model theory A realist objection to all of this goes as follows: Even if the realist assertion 'There are numbers' is not a sentence of mathematics, it is an assertion that is justified by thinking about mathematics. Mo- del theory shows us that a simple arithmetic quantification such as 3n n= 0 is true only if the class of numbers the class over which the variable 'n' ranges is non-empty, and that class is non-empty only if there are numbers. Modal theory for arithmetic is done in different ways. The key idea is to define a function on the terms of the object language, and then to define truth relative to this function. The function, call it 'f', satisfies these conditions: First, for any variable v of the object language, f(v) = n, for some number n; second, f('O') = 0; third, for any term t, fest') = f(t) + 1; fourth, for any terms t and r, f(t+r) = f(t) + fer), and f(txr) = f(t)xf(r) Truth is then defined relative to f. First, for any terms t and r, (t = r') is true relative to f if and only if f(t) = fer). Second, for any formulas A and B, and variable v, r-A' is true relative to f if and only if A is not true relative to f; r AvB' is true relative to f if and only if A is true relative to for B is true relative to f; and, finally, r3vA' is true relative to f if and only if, for some function g like f except at most that f(v) -:f. g(v),A is true relative to g. Within this framework such familiar biconditionals as '3n n = 0' is true relative to f if, and only if, if for some number n, n=O are derivable. What has been set out is a model theory and in this theory it is nowhere said that numbers exist. But, someone might object, we have managed to avoid the word 'exists', but not the concept of existence for we used the word 'some'. The word 'some' is used. But we could just as well have used 'there exists'. Nothing turns on using one phrase or another. No one

thinks that arithmetic changes if we everywhere use 'some' instead jl For every x, if x is ( of 'exists' or vice versa. It makes no difference whether we use '::3 n', 'for some n', 'for Iwe will not want to in1 some number n' or 'there exists a number n such that'. Unless there I is a predicative use of 'number' in the language of the model theory j If red is odd, then rf that is to say, ordinary mathematical English we cannot infer I 'There exists a number' from 'There exists a number n such that ... land to bar this inferenc n ..:. ~nstead An examination of the model theory for arithmetic set out above I reveals no sign corresponding to the word 'number' in the kind of! For every x, if x is a grammatical application it is given in 'There are numbers' or in 1 'There exists a number'. The sign 'number' is used in numerical IBut if we do, we can tt quantifiers. But apart from this use the sign plays no role in model i theory. ! If red is a number tb !which is equally unwal i5. 'Number' outside mathematics I 1 For every number x, Perhaps it is of no significance that we can do mathematics without 11 the predicate 'number'. Perhaps we actually use the word predicwhich yields tively outside of mathematics. ! That appears to be how we use it when, for example, we dis tinI If 3 is odd, then 3 is guish between colors and numbers. We say that three is a number, I but that red isn't. !but not Or just imagine the use of mathematics in a physical theory. We I there can use 'number' in a predicative manner so as to distinguish, I If red is odd, then re e.g., particles from numbers. I It is like the case with sets. Pure set theory does without any sign ISo what is needed to for sets but only because in pure set theory our domain consists! 'number', but the use 0 of sets alone. But the domains of those languages in which set theory for letters used to expr finds an application are not thus limited-and within them a predicais the practice of repla( tive sign for sets finds a use. And so once again we c So we now need to consider such sentences as 'Colors are not !a distinction among thi numbers', 'Red is not a number', 'Particles aren't numbers', and i The fundamental poi 'Tables aren't sets'; (for color and number de Suppose that the following is a truth formulable in the language iword won't in general of some theory of color including at least elementary arithmetic: !the use of number word ! of 3 and 2' nor '3 is brig 36 s if we everywhere use 'some' instead ltics e that we can do mathematics without aps we actually use the word predic- ~ use it when, for example, we distin- nbers. We say that three is a number, mathematics in a physical theory. We edicative manner so as to distinguish, Pure set theory does without any sign l pure set theory our domain consists of those languages in which set theory us limited-and within them a predica- er such sentences as 'Colors are not ber', 'Particles aren't numbers', and is a truth formulable in the language iing at least elementary arithmetic: 37 For every x, if x is odd, x is not even If red is a number then if red is odd, then red is not even which is equally unwanted. What we really need is For every number x, if x is odd, then x is not even which yields If 3 is odd, then 3 is not even but not If red is odd, then red is not even. So what is needed to guard against nonsense is not a predicate 'number' , but the use of that word to delimit acceptable substitutes for letters used to express generalities. And what is essential here is the practice of replacing certain letters only by numerical terms. And so once again we do not need the word as a predicate to draw a distinction among things. The fundamental point would seem to be that the language games for color and number don't' intersect' . A term with the use of a color word won't in general yield a sense when it replaces a term with the use ofnumber word, and conversely. Neither 'Red is the product of 3 and 2' nor'3 is brighter than pink' make sense. We can ofcourse l ; 38 j count each ~ot true, and in that sense false. But in that sense a c~ says anything. In that sense opener also IS false. founded on nothing but the ob: . Just as 'number' serves as an index to generalization, and thus q Objection 2. You need to re dIspe~sable, so also for 'color'. Suppose we lacked this word. Wjt may help if you look at thl yet mIght. say that Joseph had a coat of many colors by using loncept of number, and so SUff: sentence lIke . nder it. Reply. There is no doubt tll For many f, Joseph's coat was f put what kind of concept is it rere are concepts of that kind where it was our practice to recognize as instances of this general! Not every concept is a co ization only such sentences as ~xistence (as expressed in a Sl f. concept of that kind, as is st Joseph's coat was red his concept is a signfor gener Joseph's coat was blue erm (e.g., a noun). It may well be that the cor I and the like that is, as we would say, to recognize as instances ohrithmeticallanguage. But wi this generalization only sentences formed with words for colors 10ncept of number? Might it r words with that kind of use. We construct formulas using by numerals and carry out cert rom these by replacing letter 6. Further objections and replies Won't it be (roughly) our ~ rasp of the concept number Objection 1. This whole involuted enquiry is predicated on thtfnathematical predicate. (Ane supposition that there is a doubt whether 'There are numbers' makepumbers' is the kind of conce sense. But there is no plausibility in this so-called doubt. The words Remember: Even if we I at issue are plain English. You may claim not to understand thf,bnguage of natural number tt sentence 'There are numbers', but you understand it nonetheless ctual use of 'number' is for This doubt is just a pretense, and the enquiry to which it has led ha s like that of a subscript wh no point. ~lo go in for the letters. Reply. The sentence 'There are numbers' has a normal gramm Objection 3. You asked fo and its words are familiar ones. Does that show that it has a sensei, hat 'There are numbers' mak If it did, it would show as well that the sentence 'Three is red' hãomething which supports tt a sense. And it isn't obvious that it has a sense. rens~? And if you couldn't, We grant that it may be that everybody does perfectly wel~that It does? understand this sentence, that it does make sense and that everybodyi That we are not sure how grasps the sense it makes. It may be that it is only a false philosophyfW reason whãsoever for do which keeps us from seeing clearly that this sentence makes a sensd ReplY* ConsIder the case 0 we grasp. But for whatever reason it yet is not clear to us thaition using 'beet' , and then as t. I i 38 39 n that sense false. But in that sense a c says anything. In that sense our doubt is a real one, even if it is . ounded on nothing but the obscurities in our own thought. ; as an index to generalization, and thus Objection 2. You need to relax and learn to accept the obvious. ;olor'. Suppose we lacked this word. t may help if you look at the matter this way: We all grasp the 1 had a coat of many colors by using oncept of number, and so surely can conceive that something falls nder it. Reply. There is no doubt that we grasp the concept of number. Jat was f ut what kind of concept is it? Is it a concept of a kind such that ere are concepts of that kind under which things fall? to recognize as instances of this genera Not every concept is a concept of that kind. The concept of ~s as . xistence (as expressed in a sentence like There are lions') is not concept of that kind, as is shown by the fact that what expresses his concept is a signfor generality (e.g., a quantifier), not a general erm (e.g., a noun). It may well be that the concept of number is expressed by our e would say, to recognize as instances ithmeticallanguage. But what about that language expresses the ~ntences formed with words for colors oncept of number? Might it not be the letters 'm', 'n' and the like? ;e. e construct formulas using these letters and replace these letters y numerals and carry out certain inferences withJormulas resulting rom these by replacing letters by numerals. , replies Won't it be (roughly) our grasp of all this which constitutes our rasp of the concept number not our mastery of one or another involuted enquiry is predicated on th athematical predicate. (And what we would need for 'There are loubt whether 'There are numbers' make umbers' is the kind of concept expressed by a predicate.) ;ibility in this so-called doubt. The word Remember: Even if we had 'number' as a predicate in the . You may claim not to understand th anguage of natural number theory, it would be entirely useless. Our ers', but you understand it nonetheles ctual use of 'number' is forthe expression of generality. It's use e, and the enquiry to which it has led ha s like that of a subscript which reminds us which expressions are o go in for the letters. lere are numbers' has a normal gramm Objection 3. You asked for something which supports the claim ::mes. Does that show that it has a sense hat 'There are numbers' makes sense. But why? Could you produce well that the sentence 'Three is red' ha omething which supports the claim that 'There are beets' makes IS that it has a sense. ense? And if you couldn't, would that at all sap your confidence be that everybody does perfectly weI hat it does? tat it does make sense and that everybod That we are not sure how to show that a sentence makes sense is t may be that it is only a false philosoph o reason whatsoever for doubting that it does make sense. ~ clearly that this sentence makes a sens Reply. Consider the case of beets. Someone might read a descrip- ier reason it yet is not clear to us tha ion using 'beet', and then ask to be shown that there are such roots. 40 1 We know how to respond to this request. We bring various roots a The doubt whether anythiñ see whether any fits the description, and find one does. We th is matched by the doubt whet! agree: Yes, there are beets. timeless' and the like. 1There is something analogous for 'square of 27'. An easy calcq

lation shows that 729 is a square of 27. Having carried it out, WI'

will then agree that, yes, there are squares of 27. 7. Final remarks

But the case for number doesn't fit this familiar pattern. Wh, description do we have for 'number', so that we can decide wheth9 Both Carnap and his (imagina 27 fits that description? Shall we say that a number is a timele 'number' within mathematics. entity? By what method might we find out that 27 fits that descri numbers' is a theorem of aritl tion? Or, what calculation shows that 27 is a number? No calculatio arithmetic is actually set out r shows any such thing. no predicative use within aril A calculation shows e.g., that 27 is 13 + 4 + 10. That 27 is between odds and evens, for e number is not something which can be brought out within math And how these distinctions get matics.. up predicate like 'number'. Someone unfamiliar with our notation might ask whether 27 ia It is true that one could co a number. We could then exhibit our use of that sign. They wOUlj infinitely many degree-one for then be satisfied. '27 is a number' can be used to express a recognh tion about the use of a sign. Fnj Objection 4. This is just so much palaver. The key point gets 10 in all this talk. It is clear that we all realize that three is a number which become true sentences and that re~, for example, isn:t. And. this re.cognition can easily ~~ 'n'. The formula expressed III words as easily as It has Just been expressed Id words! ~ n =n Reply. If there is a recognition here, then it might be lacking. S! let us suppose that someone failed to recognize that three is is an example. If 'number' is ( number. What would they have missed? And how might their failur we say, with Carnap, that it is irr be remedied? of mathematical realism com We here imagine a person who counts, adds, mUltiplies, applie sentences as ':3n n = n' or thai the results of adding and multiplying, etc. We imagine a perso down to the denial of such sen' reasonably competent in the empirical application of mathematica Carnap was on target with hi: terms, and in the arithmetic of those terms. This is enough to hav~ numbers from a point ~f vi.ew e) him be one who grasps the idea of three. But he is supposed to fai~ numbers such as that fIve IS .om to recoqnize that three is a number. J don't exist are unmathematical So now we tell him something about numbers. Our hope is thaI! A final point Carnap made once he gets this information he'll recognize that three is a number~ unmathematical claims lack c( What do we tell him? Shall we say that a number is a timeless,! maties, 'number' has no clear placeless entity? I matics; rather, it is a sign for g 1 40 s request We bring various roots a 'iption, and find one does. We th* us for 'square of 27'. An easy calc are of 27. Having carried it out, are squares of 27. )esn't fit this familiar pattern. Wh nber', so that we can decide wheth we say that a number is a timele we find out that 27 fits that descri 41 The doubt whether anything is asserted by 'There are numbers' is matched by the doubt whether anything is asserted by 'Three is timeless' and the like. 7. Final remarks Both Carnap and his (imaginary) realist critic had it wrong about 'number' within mathematics. They each thought that 'There are numbers' is a theorem of arithmetic. But an examination of how :hat 27 is 13 + 4 + 10. That 27 is :h can be brought out within math* ur notation might ask whether 27 bit our use of that sign. They woul ,er' can be used to express a recog nuch palaver. The key point gets 10 we all realize that three is a numb t. And this recognition can easily b ly as it has just been expressed i on here, then it might be lacking. S failed to recognize that three is missed? And how might their failul1 vho counts, adds, multiplies, applie ltiplying, etc. We imagine a perso npirical application of mathematic those terms. This is enough to hav a of three. But he is supposed to fa nber. ing about numbers. Our hope is tha e'lI recognize that three is a numbe we say that a number is a timeless 's that 27 is a number? No calculatio arithmetic is actually set out reveals no such thing. 'Number' has no predicative use within arithmetic. Distinctions such as those between odds and evens, for example, get made in familiar ways. And how these distinctions get made does not draw on any cooked up predicate like 'number'. It is true that one could cook up such a predicate. There are infinitely many degree-one formulas Fn which become true sentences no matter which numeral is put for 'n'. The formula n=n is an example. If 'number' is defined by some such formula, then we say, with Carnap, that it is implausible to suppose that the content of mathematical realism comes down to the assertion of such sentences as '3n n =n' or that the content of anti-realism comes down to the denial of such sentences. Carnap was on target with his point that philosophers speak about numbers from a point ofview external to mathematics. Claims about numbers such as that five is one of them, that they exist, or that they don't exist are unmathematical claims. A final point Carnap made with which we agree is that these unmathematical claims lack content. As is the case inside mathe- matics, 'number' has no clear use as a predicate outside of mathe- matics; rather, it is a sign for generality.