Wittgenstein on Irretionals and Algorithmic Decidability

VICTOR RODYCH

WITTGENSTEIN ON IRRATIONALS AND ALGORITHMIC DECIDABILITY

Taken in isolation, Wittgenstein’s views on irrational numbers are exceedingly difficult to understand. Even good commentators have failed to fully appreciate Wittgenstein’s unorthodox rejection of certain constructive, rulegoverned “real numbers”.1 The situation improves considerably, however, when we place Wittgenstein views on irrationals admidst his finitism, his crucial distinction between intensions and extensions, and his claim that mathematics is invented bit-by-little-bit. By doing so, we find that Wittgenstein’s remarks on irrationals are a piece with the central tenets of his philosophy of mathematics, and that in particular, they go hand-in-hand with his anti-foundationalism, and especially his opposition to a “gapless mathematical continuum” and a comprehensive theory of the real numbers. Thus, the first article of business here is to present Wittgenstein’s views on irrationals in context, by elucidating Wittgenstein’s conception of “irrationals as rules” and his “comparability with the rationals” criterion. Next I shall present and elaborate Jairo Jose da Silva’s “explicit reference” criterion, and proceed to show that Wittgenstein holds two additional necessary conditions, namely that genuine irrationals must be baseindependent and that genuine reals are essentially measuring numbers in the same system. Third, I will endeavour to show that one of the reasons for Wittgenstein’s constructive presentation of the “diagonal procedure” at (RFM II, 1) is to demonstrate that even a constructive diagonal number is a homeless pseudo-irrational, without any genuine utility. Fourth, utilizing Wittgenstein’s criterion of mathematical meaningfulness, namely algorithmic decidability, I shall show why Wittgenstein denies that expressions such as “There are three consecutive 7’s in the decimal expansion of π ” (PIC) are meaningful mathematical propositions, while admitting that analogous statements about the decimal expansions of rationals are meaningful mathematical propositions. Lastly, I will argue that Wittgenstein’s rejection of lawless and pseudo-irrationals is inextricably tied to his anti-foundationalism and, in particular, his attack on set theory.

Synthese 118: 279–304, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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1. FINITISM AND THE EXTENSIONAL - INTENSIONAL DISTINCTION

According to Wittgenstein, there is no such thing as an infinite mathematical set in extension. What we call an “infinite class” is represented by a recursive rule, or what Wittgenstein sometimes calls “an induction”. But although “[a]n induction has a great deal in common with the multiplicity of a class (a finite class, of course)”, “it isn’t one, and now it is called an infinite class” (PR §158). An inductive rule is used to generate finite extensions – it is only ‘infinite’ in the sense that it represents an unlimited technique, which “does not mean that it goes on without ever stopping – that it increases immeasurably; but that it lacks the institution of the end, that it is not finished off” (RFM II, 45). When we speak, for instance, of the “infinite set of natural numbers”, this indicates “only the infinite possibility of finite series of numbers”, for “[i]t is senseless to speak of the whole infinite number series, as if it, too, were an extension” (PR §144). We erroneously think that there are infinite sets and sequences in extension for two closely related reasons. First, “[w]e mistakenly treat the word ‘infinite’ as if it were a number word, because in everyday speech both are given as answers to the question ‘how many?’ ” (PG 463; cf. PR §142). But “ ‘[i]nfinite’ is not a quantity”, Wittgenstein insists (WVC 228); “[t]he word ‘infinite’ has a different syntax from a number word”. When we recognize this we see that “[t]he words ‘finite’ and ‘infinite’. . . are not adjectives” which “signify a supplementary determination regarding ‘class’ ” (WVC 102). Thus, when we say, e.g., that “there are infinitely many even numbers”, we are not saying “there are an infinite number of even numbers” in the same sense as we can say “there are 13 people in this room”. “Where the nonsense starts”, says Wittgenstein (PR §138), “is with out habit of thinking of a large number as closer to infinity than a small one”, or when we “interpret the not-limited as length which reaches further than any given length” (RFM VII, 59), or when we construe “and so on ad inf.” as “some gigantic extension” (RFM V, 19). The infinite is understood rightly when it is understood as an “infinite possibility that isn’t exhausted by a small number any more than by a large. . . because it isn’t itself a quantity” (PR §138). “[T]he infinite doesn’t rival the finite” – “[t]he infinite is that whose essence is to exclude nothing finite” (PR §138).2 We make this (first) mistake of grammar or syntax because of a (second) more fundamental mistake: we conflate intensions and extensions, erroneously thinking that there is “a dualism” of “the law and the infinite series obeying it” (PR §180). For instance, we think that because a real number “endlessly yields the places of a decimal fraction” (PR §186), it is “a totality” (WVC 81–82, Ft. §1), when, in reality, “[a]n irrational number

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isn’t the extension of an infinite decimal fraction,. . . it’s a law” (PR §181) which “yields extensions” (PR §186).3 It is ‘nonsense’ to say “we cannot enumerate all the numbers of a set, but we can give a description”, for one “cannot give a description instead of an enumeration” because “[t]he one is not a substitute for the other” (WVC 102). “A law is not another method of giving what a list gives”, and a “list cannot give what the law gives” (WVC 102–103). Indeed, “the mistake in the set-theoretical approach consists time and again in treating laws and enumerations (lists) as essentially the same kind of thing” (PG 461). For Wittgenstein, this conflation reaches its pinnacle of absurdity when, noting that the rationals are enumerable while the irrationals are not, we say that the “infinite set of irrationals” is greater or larger than the “infinite set of rationals”. What we should say, instead, is that “[t]here is no system [“infinite set”] of irrational numbers – but also no super-system, no ‘set of irrational numbers’ of higher-order infinity” (RFM II, 33).

2. IRRATIONALS AS RULES

From this it follows that an irrational number cannot be a unique infinite expansion simply because there is no such thing as an infinite (mathematical) extension. Since, on Wittgenstein’s terms, mathematics consists exclusively of extensions and intensions (i.e., ‘rules’ or ‘laws’), an irrational √ is only an extension insofar as it is a sign (i.e., a ‘numeral’, such as ‘ 2’ or ‘π ’). On Wittgenstein’s account, however, numerals do not refer to numbers – “the signs themselves do mathematics, they don’t describe it” (PR §157). Put even more accurately, we do, make, or invent mathematics by operating with individual signs, concatenations of√ signs, and rules.4 An irrational number is, therefore, both a sign, such as ‘ 2’, and the unique rule for constructing finite, rational ‘approximations’ in some base or other. √ The rule for working out places of 2 is itself the numeral for the irrational number; and the reason I here speak of a ‘number’ is that I can calculate with these signs (certain rules for the construction of rational numbers) just as I can with rational numbers themselves. (PG 484)

√ In saying that we can calculate with signs such as ‘ 2’, Wittgenstein √ means not only that we can calculate with it (i.e., by, e.g., squaring 2 to get 2) just as we can calculate with ‘5’ and ‘2/7’, but also, as he says at (PG 484), that we can calculate successively longer rational expansions (rational ‘approximations’) whose squares converge to 2.5 Thus, the √ sign ‘ 2’ is both numeral and number, but its principal function in our arithmetic calculus is as a rule for calculating expansions.6

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√ The idea behind 2 is this: we look for a rational number which, multiplied by itself, yields 2. There isn’t one. But there are those which in this way come close to 2 and there are always some which approach 2 more closely still. There is a procedure permitting me to approach 2 indefinitely closely. This procedure is itself something. And I call it a real number. It finds expression in the fact that it yields places of a decimal fraction lying ever further to the right. (PR §183)

The second important component to Wittgenstein’s theory of irrationals is his “comparability thesis”, which is articulated in a variety of ways. It seems to be a good rule that what I will call a number is that which can be compared with any rational number taken at random. That is to say, that for which it can be established whether it is greater than, less than, or equal to a rational number.

... A real number is what can be compared with the rationals. When I say I call irrational numbers only what can be compared with the rationals, I am not seeking to place too much weight on the mere stipulation of a name. I want to say that this is precisely what has been meant or looked for under the name ‘irrational number’. (PR §191)

Comparability with the rationals is the pivotal criterion in Wittgenstein’s treatment of irrationals. Indeed, in response to the question why is an irrational ‘a number’, he answers (PR §191) “because there is a definite way for comparing it with the rational numbers”.

3. ADDITIONAL CRITERIA FOR GENUINE I RRATIONALS

The foregoing account of Wittgenstein’s views on irrationals is very similar to accounts given independently by Mathieu Marion and Jairo Jose da Silva. On Marion’s characterization, Wittgenstein restricts genuine irrationals to rules that are (1) recursive and (2) “effectively comparable with every rational number” (Marion 1995, 162, 164). Similarly, da Silva says that, for Wittgenstein, “a cut can be admitted as determining a real number only if it is specified by an arithmetical law, and this means a principle of classification of rational numbers into two classes, which also provides a procedure of decision for the question to which class an arbitrary given rational number belongs” (da Silva 1993, 93).1 The problem with both of these characterizations is that they are incomplete. Thinking that (1) 1 See also p. 94: “No real number independently of a rule, no rule which cannot provide

a method to compare the number with any rational number arbitrarily given”.

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and (2) are exhaustive for Wittgenstein, Marion incorrectly attributes the “recursive continuum” to Wittgenstein,7 providing no real explanation for √ . Inwhy Wittgenstein rejects certain “constructive numbers”, such as 7→3 2 deed, when he discusses such “computable numbers”, Marion first says, erroneously, that they “are not real numbers because one is simply not given a way of constructing them” (Marion 1965, 164), and then, in Endnote §56, he is puzzled because, as he correctly notes, the operation involved in 7→3 √ “seems perfectly effective” (Marion 1965, 164 and 173, respectively). 2 Although da Silva similarly emphasizes (1) and (2), in discussing “con√ , he correctly notes an additional criterion structive numbers” such as 7→3 2 that Wittgenstein uses to further restrict the domain of genuine irrationals. In his very succinct exposition of Wittgenstein’s views, da Silva correctly links this third criterion to (2) (i.e., the comparability criterion). In Section 3.2, I shall elaborate what I believe da Silva has in mind, and then show that he overlooks other important criteria which enable Wittgenstein to further restrict the domain of genuine irrationals. As a preliminary, however, let us first see how Wittgenstein uses (1) and (2) to eliminate “lawless irrationals”. 3.1. Lawless Irrationals Wittgenstein’s rejection of so-called “lawless irrationals” has two components, one for each type of lawless irrational that has been proferred. On the first conception, a lawless irrational is a non-rule-governed, non-periodic, infinite expansion in some base. According to Wittgenstein, however, “we cannot say that the decimal fractions developed in accordance with a law still need supplementing by an infinite set of irregular infinite decimal fractions that would be ‘brushed under the carpet’ if we were to restrict ourselves to those generated by a law”, for “[w]here is there such an infinite decimal that is generated by no law” “[a]nd how would we notice that it was missing?” (PR §181; cf. PG 473) [Y]ou can only put finite series alongside one another and in that way compare them; there’s no point in putting dots after these finite stretches (as signs that the series goes on to infinity). Furthermore, you can compare a law with a law, but not a law with no law. (PR §181)

In short, this type of lawless irrational cannot be compared with a finite decimal expansion because there is no point at which the former leaves the latter behind, and it cannot be compared with a rule-governed irrational because one cannot compare “a law with no law”. If, moreover, we place three dots at the end of a finite expansion for a lawless irrational, this is of no help, for such dots are only admissible if they are “dots of laziness”

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(i.e., a symbolic means of representing an extension that could in principle be completely enumerated) (Moore 1959, 298, seee also (PG 451–452)). On the second conception, a lawless irrational is either a “free-choice sequence” or an irregular, infinite expansion in some base that is generated by some other non-mathematical means. At (PG 483) Wittgenstein addresses this conception by asking: “How does an infinitely complicated law differ from the lack of any law?” One can imagine an irregular infinite decimal being constructed by endless dicing, with the number of pips in each case being a decimal place. But, if the dicing goes on for ever, no final result ever comes out.

It is because “no final result ever comes out” (i.e., no expansion is thereby completed), that an “irregular infinite decimal” is not an infinitely complicated law, and hence no law at all.8 Analogously, if we endeavour to determine a point in an interval by tossing a coin and repeatedly dividing the left or right half of the interval, we do not in this way ‘describe’ “the position of a point”, because “the description doesn’t determine any point explicitly” (PG 484). Here we are confusing the recipe for throwing with a mathematical rule like that for pro√ ducing decimal places of 2. Those mathematical rules are the points. That is, you can find relations between those rules that resemble in their grammar the relations “larger” and “smaller” between two lengths, and that is why they are referred to by these words.

The problem is that endless bisection is not a mathematical rule com√ parable to the rule for 2.9 To say “that the recipe for endless bisection according to heads and tails determines a point. . .would have to mean that this recipe could be used as a numeral, i.e. in the same way as other numerals[;] [b]ut [this] of course. . . is not the case” (PG 485). The problem with this recipe, or pseudo-rule, is (PR §186) that it does not “determine a point ever more closely by narrowing down its whereabouts”, for “[a]fter every throw the point is still infinitely indeterminate”. To be able to actually determine a point “we must be able to construct it”, which means that the number must be a constructive rule which utilizes purely arithmetic operations “in their natural home”. Such constructive rules are the points, just as 5 and 1/7 on the number line are points. No point is determined, or ‘exists’, if it is ‘described’ by a non-mathematical prescription. In sum, lawless irrationals are not genuine irrationals because, insofar as they are not recursive, mathematical rules, they fail to satisfy criterion (1), and insofar as they are not comparable to any rational taken at random, they fail to satisfy criterion (2).

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3.2. da Silva on “Explicit Reference” As I have noted, da Silva correctly says that criteria (1) and (2) are not exhaustive for Wittgenstein. To see this clearly, let us consider one of Wittgenstein’s pseudo-irrationals, what Marion calls Wittgenstein’s “bizarre √ ”, which stands for the prescription: “Construct example”, namely “5→3 2 √ the decimal expansion for 2, replacing every occurrence of a ‘5’ with a ‘3’ ”.10 At (PR §183), Wittgenstein contrasts this rule with two similar rules, one that utilizes a rational and one that utilizes an irrational. ˙ How about 17 (5 → 3)? Of course 0.14283 7˙ isn’t an infinite extension but once again an infinite rule, with which an extension can be formed. But it is such a rule as can so to speak digest the (5 → 3). The suffix (1 → 5) so to speak strikes at the heart of the law 0.1010010001.. . . The law talks of a 1 and a 5 is to be substituted for this 1.

√ ˙ Using (1/7)0 to stand for the “infinite rule” 0.14285 7˙ (5 → 3) and ( 2)0 √ ”, da Silva says that “[a]ccording to Wittgenstein (1/7)0 in place of “5→3 2 √ defines a real number, but ( 2)0 does not”, because ‘the law for the decimal expansion of 1/7 makes explicit reference √ to the occurrence of a digit “5” whereas the law for the expansion of 2 does not’. “The√consequence is that (1/7)0 is comparable with any rational number but ( 2)0 is not (Da Silva 1994, 94). For the sake of brevity, apparently, da Silva does not explicate what is meant by “explicit reference”. In this section I shall elaborate this notion and connect it with Wittgenstein’s distinction between calculation and experiment. In Sections 3.3 and 3.4, I shall argue that even if we add da Silva’s “explicit reference” criterion (3) to criteria (1) and (2), this account is still incomplete for Wittgenstein also maintains (4) that genuine irrationals are base-independent, and (5) that genuine irrationals are all measuring numbers in the same system. By “explicit reference” da Silva undoubtedly means, paraphrasing Wittgenstein’s second example in (PR §183) above, that the law (1/7)0 “talks of a 5 and a 3 is to be substituted for this 5”.11 As Wittgenstein puts it at (PR §186), “The sign π 0 (or: 7→3 π ) means nothing, if there isn’t any talk of a 7 in the law for π , which we could then replace by a 3”. Wittgenstein insists on this necessary condition because of his long-held view, first introduced in the Tractatus, that “[c]alculation is not an experiment” (6.2331). This can best be illustrated by considering the following definitions given for the pseudo-irrationals “π 0 ” and P at (PG 475). π 0 is a rule for the formation of decimal fractions: the expansion of π 0 is the same as the expansion of π except where the sequence 777 occurs in the expansion of π; in that case instead of the sequence 777 there occurs the sequence 000. There is no method known to our calculus of discovering where we encounter such a sequence in the expansion of π.

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P is a rule for the construction of binary fractions. At the nth place of the expansion there occurs a 1 or a 0 according to whether n is prime or not.

What Wittgenstein objects to in both of these definitions is the presupposed conception of predeterminateness in mathematics.12 π 0 presupposes that the occurrence of ‘777’ in the infinite decimal expansion of π is completely determined (or fixed) by the rule for generating its expansion. Analogously, P presupposes that “[t]he position of all primes must somehow be predetermined. We work them out only successively, but they are all already determined. God, as it were, knows them all. And yet for all that it seems possible that they are not determined by a law” (PG 481). At (PG 479), Wittgenstein makes this denial more explicit: ‘ “Can God know all the places of the expansion of π ?” would have been a good question for the schoolmen to ask’.13 In this context we keep coming up against something that could be called an ‘arithmetical experiment’. Admittedly the data determine the result, but I can’t see in what way they determine it. (Cf. e.g. the occurrences of 7 in π.) The primes likewise come out from the method for looking for them, as the results of an experiment. To be sure, I can convince myself that 7 is a prime, but I can’t see the connection between it and the condition it satisfies. – I have only found the number, not generated it. I look for it, but I don’t generate it. I can certainly see a law in the rule which tells me how to find the prime, but not in the numbers that result. And so it is unlike the case +1/1!, –1/3!, +1/5!, etc., where I can see a law in the numbers. (PR §190)

Thus, on Wittgenstein’s account, the problem with both π 0 and P is that neither rule “understand[s] itself”, for each “is made up out of two heterogeneous parts” (PR §183). A necessary condition of π 0 and P being well-defined is that the occurrences of ‘777’ in π and the distribution of the primes, respectively, must be generatable by a law. The acid test here, according to Wittgenstein (PR §190), is that “I must be able to write down a part of the series, in such a way that you can recognize the law”.14 Since this condition is not satisfied in either case, in constructing each pseudoirrational I must “look and find” – i.e., conduct an infinite arithmetical experiment – in order to construct the number. Given that this is not possible, such numbers are simply not well-defined; they are not self-contained rules, and in that sense, they do not “understand themselves”. It should be emphasized that, for Wittgenstein, the knowledge of laws and decision procedures is of paramount importance. And if our calculus contains a method, a law, to calculate the position of 777 in the expansion of π, then the law of π includes a mention of 777 and the law can be altered by the substitution of 000 for 777. But in that case π 0 isn’t the same as what I defined above; it has a different grammar from the one I supposed. In our calculus there is no question whether π ≥ π 0 or not, no such equation or inequality. π 0 is not comparable with π. And one can’t

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say “not yet comparable”, because if at some time I construct something similar to π 0 that is comparable to π, then for that very reason it will no longer be π 0 . (PG 475–476)

3.3. Base-Independence: The Invariance of Irrationals As was mentioned in the previous section, criterion (3) is at best an additional necessary condition. The foundation of Wittgenstein’s views on the irrationals is perhaps best stated at (PR §188): Arithmetical operations only use the decimal system as a means to an end; that is, the rules for the operations are of such a kind that they can be translated into the language of any other number system, and do not have any of them as their subject matter.15

For instance, the rule for multiplication in the decimal system is the (one and only) arithmetic rule for multiplication. It and its fellow rules of operation use the decimal system, but they could just as easily use a binary system, or a system in base 7. Put differently, a “general rule of operation” is invariant across different base-notational systems. A general rule of operation gets its generality from the generality of the alteration it effects in the numbers. That is why 7→3 × won’t do as a general rule of operation, since the result 7→3 of a×b doesn’t depend solely on the nature of the numbers a and b; the decimal system also comes in. (PR §188)16

General arithmetic rules of operation partially define an arithmetic calculus because they are invariant across base-notational systems. According to Wittgenstein, the same must be true of any number, and in the case of irrationals, the rule that is a particular irrational. A real number yields extensions, it is not an extension. A real number is: an arithmetical law which endlessly yields the places of a decimal fraction. This law has its position in arithmetical space. Or you might also say: in algebraic space. Whereas π 0 doesn’t use the idioms of arithmetic and so doesn’t assign the law a place in this space. It’s as though what is lacking is the arithmetical creature which produces these excretions. The impossibility of comparing the sizes of π and π 0 ties in with this homelessness of π 0 . (PR §186)

When Wittgenstein says that “π 0 doesn’t use the idioms of arithmetic” he means that it doesn’t use only the invariant arithmetic rules of operation. Whereas π has its place in the natural home partially defined by the general arithmetic rules (i.e., idioms of arithmetic), and hence “has its position in arithmetical [/algebraic] space”, π 0 is ‘homeless’ because it is dependent upon the particular ‘incidental’ notation of a particular system (i.e., in some particular base).17 In just this way 7→3 π makes the decimal system into its subject matter (or would have to do so, if it were genuine), and for that reason it is no longer sufficient that we can use the rule to form the extension. For this application has now ceased to be the criterion for the rule’s

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being in order, since it is not the expression of the arithmetical law at all, but only makes a superficial alteration to the language. (PR §188)

For this reason, π 0 is not a genuine number comparable with the rationals, or even the irrationals for that matter, and hence, it is not a genuine irrational. Even though of π 0 is “as unambiguous √ the “rule for [the] expansion” as that for π or 2. . . that is no proof that π 0 is a real number, if one takes comparability with rational numbers as an essential mark of real numbers” (PG 476).18 Thus, when we endeavour to create a new irrational whose rule is parasitic upon, e.g., π , we must necessarily fail, for in doing so we necessarily create a new system. The expansion of π is admittedly an expression both of the nature of π and of the decimal notation, but our interest is usually restricted exclusively to what is essential to π, and we don’t bother about the latter. That is a servant which we regard merely as a tool and not as an individual in its own right. But if we now regard it as a member of society, then that alters society. (PR §188)

That is to say, if we were to make the decimal expansion of π essential to π as a number, then the decimal system would “stop being a servant” and “join the others at the table, observing all the required forms, and leave off serving, since it can’t do both at once” (PR §188). Wittgenstein’s point about π 0 is that, insofar as it brings the decimal system to the table by explicitly referring to symbols and their relations in its “replacement rule”, it necessarily creates a new calculus.19 This calculus is different from ordinary arithmetic, because in ordinary arithmetic irrationals use the decimal system, just as they could use any other base-notational system. Wittgenstein puts the matter a little differently at (PG 477). “How far must I expand π in order to have some acquaintance with it?” – Of course that is nonsense. We are already acquainted with it without expanding it at all. And in the same sense I might say that I am not acquainted with π 0 at all. Here it is quite clear that π 0 belongs to a different system from π; that is something we recognize if we keep our eyes on the nature of the laws instead of comparing “the expansions” of both.

We are acquainted with π because we know which arithmetic rules are to be applied, step-by-step, before we have constructed its expansion even to one place! Much to the contrary, we are not acquainted with π 0 “at all” because its rules are not system-invariant. We see this if we focus on the nature or essence of the laws instead of trying to compare their respective expansions. When we recognize this we see that π 0 belongs to a different system from π ; or, put more precisely, π belongs to a system in which the rules of operation are invariant and in which no irrational is dependent upon a particular base-notational system. If we speak of the various base-

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notational systems, we might say that π belong to all systems, while π 0 belongs only to one.20 Wittgenstein’s strong stance against base-dependence makes one wonder about the irrational 0.1010010001. . . and the parasitic ‘number’ 0.1010010001. . . (1 → 5) presented at (PR §183), above. As Wittgenstein says, in the latter case, “[t]he law talks of a 1 and a 5 is to be substituted for this 1”. da Silva’s “explicit reference” criterion (3) is fully satisfied, but the resultant ‘number’ is not base-independent. The best explanation I can offer for this confusing situation is that Wittgenstein wishes to emphasize the difference between a number that mentions the digit to be replaced, and which is therefore not experiment-dependent, and one that does not explicitly mention the digit to be replaced. This difference in and of itself shows that pseudo-irrationals fail to satisfy criterion (2). What must be borne in mind is that, although 0.1010010001. . . (1 → 5) does satisfy criterion (3), it still fails to satisfy criterion (2) since, in making the decimal system part of its subject matter, it becomes ‘homeless’, losing its unique ‘position’ in “arithmetical/algebraic space”. 3.4. Real Numbers and Measurement It is well-known that Wittgenstein is an anti-foundationalist in a variety of senses. One of the most important of these is his claim that there is simply no need for a comprehensive theory of the real numbers. This impetus, as Wittgenstein sees it, lies at the very beginning of set theory, when Dedekind cuts are introduced to fill in alleged gaps on the allegedly continuous real number line. When the idea of a cut of the real numbers is now introduced by saying that we simply have to extend the concept of a cut of the rational numbers to the real numbers – all that we need is a property dividing the real numbers into two classes (etc.) – then first of all it is not clear what is meant by such a property, which thus divides all real numbers. Now our attention can be drawn to the fact that any real number can serve this purpose. But that gets us only so far and no further. (RFM V, 34)

The problem is that there is no property, no rule, no systematic means of defining each and every irrational number intensionally (e.g., we cannot recursively enumerate the “recursive√reals”). We can, of course, present a rule-governed irrational, such as 2, and then show (PR §183) that “[t]here is a procedure permitting me to approach 2 indefinitely closely”, that an irrational number “runs through a series of rational approximations”. But the problem is that this irrational number never “leaves this series behind” (PR §181). Thus, there is no criterion “for the irrational numbers being complete”, for even if we assume “we have been given all the irrational numbers that can be represented by law”, and are told “that

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there are yet other irrationals” which can be ‘represented’ by ‘cuts’, it is ‘impossible’ “to tell that this is so”, “since no matter how far I go with my approximations, there will always be a corresponding fraction” (PR §181). The problem “with the concept of a ‘cut’ ”, is that it looks “at the matter now in an intensional, now again in an extensional way”, overlooking the fact that although “any [given] real number” can divide “those that we have”, “we have no system of irrational numbers” (RFM II, 29, 33), no ‘system’ or ‘set’ of “all real numbers”, which shows that “[t]he cut is an extensional image” which cannot possibly define or make sense of “the set of all real numbers” (RFM V, 34). Wittgenstein’s view of real numbers and real number arithmetic is one reason why he says that mathematics does not need a foundation. Insofar as Frege, Russell, Zermelo and set theorists are endeavouring to provide a foundation for the various mathematical calculi, they produce unnecessary aberrations which “add nothing” to these calculi, but instead distort them by attaching homeless fictions to their natural elements (e.g., numbers). Nothing is added to the differential and integral calculi by constructing a ‘complete’ theory of real numbers, which includes pseudo-irrationals and lawless irrationals. We think, erroneously, that these are needed for a theory of the ‘continuum’, when, in fact, the mathematical continuum is a fiction. As Wittgenstein says as late as (RFM V, 32), “[t]he picture of the number line is an absolutely natural one up to a certain point; that is to say so long as it is not used for a general theory of real numbers”. Mathematicians go awry, however, when they misconstrue the nature of the geometrical line, thinking it a continuous collection of points, each with an associated real number. This causes mathematicians to go well beyond the ‘natural’ picture of the number line in search of a “general theory of real numbers”. When, to the contrary, Wittgenstein pronounces that “[t]here are no gaps in mathematics” (WVC 35), this includes the dictum that there are no gaps on the number line.21 Each time that we construct a new irrational we create a new irrational, we do not discover one that was already there without our knowing it. This radical constructivism also finds expression when Wittgenstein says (RFM V, 9): “However queer it sounds, the further expansion of an irrational number is a further expansion of mathematics”. Wittgenstein’s views on genuine versus pseudo-irrationals is perhaps best summed up by his discussion of measurement. “Could we perhaps √ put it”, he asks (PR §183), “by saying 0 2 isn’t a measure until it is in a system?” His answer is ‘Yes’. A number must measure. And this doesn’t mean merely: values in its expansion must measure. For we can’t talk of all values, and that rational numbers (which I have formed in accordance with some rule) measure goes without saying.

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What I mean might be put like this: for a real number, a construction and not merely a process of approximation must be conceivable. – The construction corresponds to the unity of the law. (PR §186) A number must measure in and of itself. It seems to me as though that’s its job. If it doesn’t do that but leaves it to the rationals, we have no need of it. (PR §191)

This, I suggest, is Wittgenstein’s trump card against pseudo-irrationals. In real number arithmetic, both rationals and genuine irrationals can measure; indeed that is their raison d’être. We only calculate with them in order to measure with them. This is their place in their natural home. When we need to measure with an irrational, we generate its expansion as far as we need, and then we use that finite expansion (i.e., a rational) to measure. Quite to the contrary, pseudo-irrationals do not measure, because they are homeless, artificial constructions parasitic upon numbers which have a natural place in a calculus that can be used to measure. In what calculus would we use π 0 to measure? If we incorporate it into our customary calculus, it does not add anything to it, as Wittgenstein stresses time and again. We simply do not need these aberrations, because they are not sufficiently comparable to rationals and genuine irrationals. Wittgenstein’s conclusion is simply that they are not real numbers.

4. WITTGENSTEIN ON “ DIAGONAL NUMBERS ”

In RFM , Wittgenstein’s √ treatment of Cantor’s diagonal has two distinct components: (1) like 0 2, the “diagonal number” is fully constructive (RFM II, 1), but it is not a genuine irrational, and (2) Cantor’s “diagonal proof” does not establish that there exist non-denumerable infinite sets which are greater in cardinality than denumerably infinite sets.22 The second component, which is too large and complex to be treated here, is thoroughly examined in my ‘Wittgenstein’s Rejection of Set Theory’. My aim in the present section is to highlight the similarities and differences between Wittgenstein’s purely constructive RFM discussion of Cantor’s diagonal and his intermediate treatment of pseudo-irrationals. As I said earlier, Wittgenstein’s presentation of the diagonal at (RFM II, 1; 1938) is constructive, and there is, I believe, a very good reason for this. If the diagonal is presented, as it often is, with a non-constructive, ‘random’ listing of a few reals on the right hand side, this makes no sense on Wittgenstein’s terms. First, it presupposes irrationals as infinite extensions, and second, the only sense in which Wittgenstein would admit an infinite sequence given by a finite list and three “dots of laziness” would be if the sequence were given by a rule. Thus, Wittgenstein’s presentation

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at (RFM II, 1) uses a constructive list, wherein the assumed rule is: “For every natural n on the left hand side, construct its square root expansion to the nth place on the right hand side”. This constructive presentation of the diagonal allows Wittgenstein to consider the diagonal number thereby constructed and ask whether it is a genuine irrational (component §1). The contrast between the diagonal number constructed at (RFM II, 1) and the pseudo-irrationals discussed by Wittgenstein in PR and PG is perhaps not immediately evident. The crucial difference is that the diagonal number in Wittgenstein’s presentation is a constructive number, which, moreover, is neither base nor experiment-dependent. If someone says: “Shew me a number different from all these”, and is given the rule of the diagonal for answer, why should he not say: “But I didn’t mean it like that!”? What you have given me is a rule for the step-by-step construction of numbers that are successively different from each of these. “But why aren’t you willing to call this too a method of calculating a number?” – But what is the method of calculating, and what the result, here? You will say that they are one, for it makes sense now to say: the number D is bigger than . . . and smaller than . . . ; it can be squared etc. etc. Is the question not really; What can this number be used for? True, that sounds queer. – But what it means is: what are its mathematical surroundings? (RFM II, 3)

In the first paragraph above, Wittgenstein has his speaker balk at the diagonal construction of a number different from the squares, apparently on the grounds that s/he was not asking for a rule of construction, but rather a number given directly. Let us suppose that someone had been given the task of naming a number different from √ every n; but that he knew nothing of the diagonal procedure and had named the number √ √ [3] 2; and had shewn that it was not a value of n. Or that he had said; assume that √ 2 = 1.4142 . . . and subtract 1 from the first decimal, but have the rest of the places agree √ √ with 2. 1.3142 cannot be a value of n. (RFM II, 1)

This, I think, is what Wittgenstein’s speaker was requesting. In the second paragraph of (RFM II, 3) above, Wittgenstein himself answers the question put. That this is Wittgenstein is made clear in the third paragraph, where Wittgenstein reiterates his intermediate “comparability criterion” and alludes to his “natural home” argument. Wittgenstein does this, I believe, √ because unlike pseudo-irrationals such as 0 2, Wittgenstein’s own diagonal number (hereafter referred to as ‘WD’) is constructive in his own restrictive sense. The rule for constructing WD says simply: “For each √ n, (e.g.) add one to the numeral in the nth place in its expansion”. Unlike 0√ 2, this rule does not mention a particular numeral (or digit) to replace, nor does it stipulate a particular numeral to put in its place, and so it is not bound to any particular base-notational system; it tells us what to do,

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and where to do it. Thus, WD is not experiment-dependent, using as it does only general arithmetic rules of operation, and being as it is systeminvariant. This means that the only (intermediate) grounds Wittgenstein could have for not admitting WD as a genuine irrational are those based on the comparability criterion and the natural home argument. It is not surprising, therefore, that these are precisely the grounds from√which he rejects WD as a genuine irrational. WD is just as homeless as 0 2 since it serves no natural purpose in our calculus of real number arithmetic (i.e., “What can this number be used for?”). This shows not only that systeminvariance and the comparability criterion are both necessary conditions for the existence of a genuine irrational, but also that Wittgenstein has not changed his position on this in RFM.

5. ALGORITHMIC DECIDABILITY AND PIC

Having attained a clear understanding of Wittgenstein’s theory of irrational numbers, we are now in a position to understand what Wittgenstein is trying to do when, from 1929 through 1944, he repeatedly wrestles with the question “Are there three consecutive 7’s in the decimal expansion of π ?” On Wittgenstein’s account, the corresponding ‘proposition’, namely, “There are three consecutive 7’s in the decimal expansion of π ” (∼PIC) [and its putative negation, namely, “It is not the case that there are three consecutive 7’s in the decimal expansion of π ” (∼PIC)] is not a meaningful mathematical proposition.23 The claim that a number occurs at a certain place [in the expansion of π] is of course an assertion and can as such be in turn negated. Such a negation simply says that at the place in question that number does not occur. [Weyl’s] error arises from regarding an extension as a totality. For it makes good sense to say: If 7 occurs at the 25th place, then 7 occurs between the 20th and the 30th place. But it does not make sense to say: 7 occurs, full-stop. This is not a statement at all. If you answer the question whether the figure 7 occurs in the expansion of π by saying: Yes, it occurs at the 25th place, you have answered only the question whether 7 occurs at the 25th place but not the question whether 7 occurs at all. If the question has a sense, then the answer has a sense too, no matter if it turns out positive or negative. (WVC 81–82: Footnote #1)

To fully understand the import of this and other passages, we must first understand that, for the intermediate Wittgenstein, a mathematical expression is a meaningful mathematical proposition (in a given calculus) iff we know of a decision procedure (DP) by means of which we can decide it. For brevity, I will say that a meaningful mathematical proposition

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is algorithmically decidable. In Wittgenstein’s words: “What ‘mathematical questions’ share with genuine questions is simply that they can be answered” (PR §151). “Where there’s no logical method for finding a solution”, states Wittgenstein (PR §149), “the question doesn’t make sense either”. “We may only put a question in mathematics (or make a conjecture)”, he adds (PR §151), “where the answer runs: ‘I must work it out’ ” “The question ‘How many solutions are there to this equation?’ is the holding in readiness of the general method for solving it” (PR §151; cf. PG 452, 464). At (PG 468) Wittgenstein sums up the importance of algorithms in mathematics: “In mathematics everything is algorithm and nothing is meaning; even when it doesn’t look like that because we seem to be using words to talk about mathematical things. Even these words are used to construct an algorithm”. According to Wittgenstein, Fermat’s Last Theorem and Goldbach’s Conjecture (GC), qua expressions that quantify over an infinite domain, are not meaningful mathematical propositions, because they are not algorithmically decidable.24 At best, such an expression can only stand ‘proxy’ for two meaningful propositions, namely an inductive base and an inductive step, and this only after a proof by mathematical induction (MI) has been executed (and a new calculus has been thereby created). Analogously, there is no logical, or in principle, barrier to (non-algorithmically) ‘refuting’ GC by proving (‘verifying’) a finitistic proposition of the form “∼G(n)”. Thus, if we proved that some even number is not the sum of two primes, we would prove that proposition, a proposition of the form “∼G(n)”, true, and thereby also refute the inductive step of a possible MI proof of GC, since the negation of that proposition follows from “∼G(n)”. Given, however, that this inductive step was not algorithmically decidable before our proof of “∼G(n)”, it was not a meaningful proposition until we proved “∼G(n)”, which means that just as a successful MI proof creates a new calculus and makes the inductive step a meaningful proposition in that new calculus, a proof of “∼G(n)” also must create a new calculus, since the negation of the inductive step is proved, and its negation, namely the inductive step must now be a meaningful proposition, which it was not before the proof of “∼G(n)”. Wittgenstein’s treatment of PIC involves similar considerations, but there is one important difference (to be mentioned shortly). We can only meaningfully state finitistic propositions regarding the expansion of π , such as “There exist three consecutive 7’s in the first 10,000 places of the expansion of π ”.25 There can be no such question as, Do the figures 0, 1, 2. . ., 9 occur in π? I can only ask if they occur at one particular point, or if they occur among the first 10,000 figures. No

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expansion, however far it may go, can refute the statement ‘They do occur’ – therefore this statement cannot be verified either. What is verified is an entirely different assertion, namely that this sequence occurs at this or that point. Hence you cannot affirm or deny such a statement, and therefore you cannot apply the law of the excluded middle to it. (WVC 71)26

Since, according to Wittgenstein (PR §151), “where the law of the excluded middle doesn’t apply, no other law of logic applies either, because in that case we aren’t dealing with propositions of mathematics”, neither GC nor PIC is a meaningful proposition.27 Finitistic variants of both PIC and ∼PIC, however, are algorithmically decidable, just as finitistic variants of GC are algorithmically decidable. The law of the excluded middle applies to both, and so both are meaningful mathematical propositions. The critical difference between GC and PIC is that, whereas GC is in principle provable by MI (i.e., in Wittgenstein’s restricted sense), ∼PIC is not. On Wittgenstein’s account, we have only the rule for generating finite expansions of π . Since we know that there is no periodicity in π ’s expansion, we cannot prove ∼PIC by MI, as we can, in principle, prove GC by MI (or as we have proved Euclid’s Prime Number Theorem [EPNT] by MI). Thus, although Wittgenstein admits the in principle decidability of GC (i.e., with his finitistic restrictions), he does not admit the in principle decidability of PIC. PIC is not decidable in principle, even in Wittgenstein’s finitistic sense, simply because ∼PIC is not decidable by MI, and, it appears, because ∼PIC also cannot be decided by reductio. Thus, there is no ∼PIC analogue to GC (or EPNT). For the most part, Wittgenstein’s position on PIC in RFM remains unchanged. In Wittgenstein’s earliest (1942–44) remarks on PIC, at (RFM V, 9), he adopts a position indiscernible from his intermediate view. We only see how queer the question is whether the pattern φ (a particular arrangement of digits e.g. ‘770’) will occur in the infinite expansion of π, when we try to formulate the question in a quite common or garden way: men have been trained to put down signs according to certain rules. Now they proceed according to this training and we say that it is a problem whether they will ever write down the pattern φ in following the given rule.

In the last sentence of this passage, Wittgenstein is saying, I believe, that we say that it is a mathematical problem (or question) whether we will ever write down “the pattern φ” in writing down π ’s expansion, when in fact this question is at best only a contingent question (RFM V, 23). If it were a mathematical question (or problem), it would be algorithmically decidable, which it is not. As Wittgenstein says in the continuation of this passage: “The question – I want to say – changes its status, when it becomes decidable. For a connexion is made then, which formerly was not there”.28

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Wittgenstein immediately follows this passage by reiterating his intermediate view that PIC is meaningless because it quantifies over an infinite domain. And if you say that the infinite expansion must contain the pattern φ or not contain it, you are so to speak shewing us the picture of an unsurveyable series reaching into the distance. (RFM V, 10)

“To say of an unending series that it does not contain a particular pattern”, Wittgenstein adds (RFM V, 11), “makes sense only under quite special conditions”. These “special conditions” are ones in which the “proposition has been given a sense for certain cases” – “[r]oughly, for those where it is in the rule for this series, not to contain the pattern”.29 What ˙ Wittgenstein has in mind here, I believe, are laws such as 0.14283 7˙ and 0.1010010001. . .(1 → 5) articulated at (PR §183), above. We could say, for instance, of the first (rational) number that it does not contain the pattern ‘429’, and we could say of the second (irrational) number that it does not contain the pattern ‘010101’. But in the ‘normal’ case, where an irrational is given by a recursive rule, we are not in the position to ‘look’ at the rule and see that this or that pattern is precluded. Nor can we say (PR §184) that “π 0 alludes to a law which is as yet unknown (unlike 1/70 )”, for if “the method” for “finding this law” is “unknown, we can’t speak about the law which we don’t yet know, and is bereft of all sense”. For this reason, expressions such as PIC are meaningless because we do not know of a way to decide them.30 At little later, at (RFM V, 19), Wittgenstein once again speaks of π and says: Isn’t it like this? The concepts of infinite decimals in mathematical propositions are not concepts of series, but of the unlimited technique of expansion of series. We learn an endless technique: that is to say, something is done for us first, and then we do it; we are told rules and we do exercises in following them; perhaps some expression like “and so on ad inf.” is also used, but what is in question here is not some gigantic extension. These are the facts. And now what does it mean to say: “φ either occurs in the expansion, or does not occur”?

Here, once again, we have Wittgenstein’s denial that an irrational is an infinite extension. In the last sentence, Wittgenstein seems to intimate that this ‘assertion’ is meaningless, just as he did during the middle period. In RFM, however, Wittgenstein adds some remarks on the Law of the Excluded Middle that have greatly confused many. When someone hammers away at us with the law of excluded middle as something which cannot be gainsaid, it is clear that there is something wrong with his question.

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When someone sets up the law of excluded middle, he is as it were putting two pictures before us to choose from, and saying that one must correspond to the fact. But what if it is questionable whether the pictures can be applied here? (RFM V, 10)

Wittgenstein’s point here is that if someone invokes the Law of the Excluded Middle in an attempt to convince us that PIC is a meaningful mathematical proposition, s/he is simply begging the question. “In the law of excluded middle”, Wittgenstein stresses (RFM V, 12), “we think that we have already got something solid, something that at any rate cannot be called in doubt. Whereas in truth this tautology has just as shaky a sense (if I may put it like that), as the question whether p or ∼p is the case”. Put slightly differently, someone who has doubts about the meaningfulness of PIC will not be swayed by a person who asserts “PIC ∨ ∼PIC”. There are times, however, when Wittgenstein’s position on PIC seems to soften in RFM, such as (RFM VII, 41). A proof that shews that the pattern ‘777’ occurs in the expansion of π, but does not shew where. Well, proved in this way this ‘existential proposition’ would, for certain purposes, not be a rule. But might it not serve e.g. as a means of classifying expansion rules? It would perhaps be proved in an analogous way that ‘777’ does not occur in π 2 but it does occur in π × e, etc. The question would simply be: is it reasonable to say of the proof concerned: it proves the existence of ‘777’ in the expansion? This can be simply misleading. It is in fact the curse of prose, and particularly Russell’s prose, in mathematics.

Wittgenstein here considers a non-constructive existence proof – perhaps a proof by reductio or by some other means. He seems to allow the possibility of such a proof, yet he still maintains that it is ‘misleading’ to think of this in the same terms as a constructive existence proof. Indeed, in the continuation of this passage, Wittgenstein makes one of his strongest claims that the only way to determine whether a particular sequence occurs in the expansion of π is by constructing the expansion.31 What harm is done e.g. by saying that God knows all irrational numbers? Or: that they are already there, even though we only know certain of them? Why are these pictures not harmless? For one thing, they hide certain problems. – Suppose that people go on and on calculating the expansion of π. So God, who knows everything, knows whether they will have reached ‘777’ by the end of the world. But can his omniscience decide whether they would have reached it after the end of the world? It cannot. I want to say: Even God can determine something mathematical only by mathematics. Even for him the mere rule of expansion cannot decide anything that it does not decide for us. We might put it like this: if the rule for the expansion has been given us, a calculation can tell us that there is a ‘2’ at the fifth place. Could God have known this, without the calculation, purely from the rule of expansion? I want to say: No.

The last two sentences in particular state unequivocally that the only way to determine whether a ‘777’ occurs in the expansion of π is to construct

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the expansion and actually find three consecutive 7’s, which, again, would only show that ‘777’ occurs at that particular place in the expansion. Wittgenstein’s numerous remarks on PIC from 1929 through 1944 consistently challenge the meaningfulness of PIC qua mathematical proposition. Only finitistic propositions about the expansion of an irrational are meaningful because only they are algorithmically decidable. If the “mere rule of expansion cannot decide” the question, as it cannot in the case of π , then the ‘question’ is simply not decidable. Again and again ‘we are tricked by the image of an “infinite extension” as an analogue to the familiar “finite” extension’ (PG 481). Once we realize that an irrational is not an infinite extension, but rather a recursive rule (which satisfies certain conditions), we see that certain alleged ‘assertions’ about these so-called “infinite extensions” are meaningless, and hence, neither true nor false. 6. ESSENTIALISM AND WITTGENSTEIN ’ S ANTI - FOUNDATIONALISM

This is Wittgenstein’s position on irrationals, pseudo-irrationals, and constructive versus non-constructive diagonal numbers. As I have said, Wittgenstein’s view of real numbers and real number arithmetic is one reason why he says mathematics does not need a foundation (RFM VII, 16). On his view, we erroneously think that lawless and pseudo-irrationals are needed for a comprehensive theory of the real numbers, when, in fact, the real number arithmetic that we need is an arithmetic for calculating, and this means that general arithmetic rules of operation are used to calculate numbers, both rational and irrational, which can be used to measure. This is the real need that arithmetic addresses – it is the reason we have constructed it. The points that exist on the number line are numbers that we construct. They do not exist there in some completed infinite totality – we must construct them, and only then do they exist. Put differently, the geometrical continuum does not exist, except in the sense of a possibility. There are only ever a finite number of numbers that actually exist. What is perhaps most striking about Wittgenstein’s theory of real numbers is its inherent essentialism. In saying that in real number arithmetic we have no need for a comprehensive theory of all the supposed “real numbers” (i.e., a foundation for the theory of real numbers), because real numbers are essentially constructions that we invent to count and measure, Wittgenstein presupposes that the calculus of real number arithmetic is partially defined by our aims and our interpretation. What, then, are we to make of Wittgenstein’s claim that his “comparability criterion” is not “the mere stipulation of a name”, but rather a condition that captures “precisely what has been meant or looked for under the name ‘irrational number’ ”

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(PR §191)? Is this is an “ordinary language” observation, and if so, how does it wash with Wittgenstein’s (PG 334) claim that “[f]or [him] one calculus is as good as another” and his (PG 309–310) insistence that extrasystemic application does not give formal calculations “substance in some sense which is essential to mathematics”? Wittgenstein’s critic, particularly the foundationalist (e.g., set theorist), will simply say that the term “irrational number” has been extended to lawless and pseudo-irrationals because these “conceivable numbers” are much more like (rule-governed) irrationals than rationals. Wittgenstein can of course rejoin that a calculus that employs lawless and pseudo-irrationals (e.g., ZF) is a new and acceptable mathematical calculus, but it is doubtful that he would actually respond in this fashion. The point of his lengthy ruminations on irrationals is not just that real number arithmetic need not be extended in this way – it is that real number arithmetic should not be extended in this way. We use the word “number” in all sorts of different cases, guided by a certain analogy. We try to talk of very different things by means of the same schema. This is partly a matter of economy; and like primitive peoples, we are much more inclined to say, “All these things, though looking different, are really the same” than we are to say, “All these things, though looking the same, are really different”. Hence I will have to stress the difference between things, where ordinarily the similarities are stressed, although this too, can lead to misunderstandings. (LFM 15)

Admittedly, this is the way in which Wittgenstein viewed his philosophy of mathematics. Where others tend “to assimilate to each other expressions which have very different functions in language” (LFM 15), creating ‘misunderstandings’ by “gradually [distorting]” “the use of words” (LFM 21), Wittgenstein stresses the differences between the various things called “real numbers”. To say, however, that this is all that Wittgenstein is doing is to grossly distort and underestimate his clear and adamant rejection of foundationalism, and especially set theory. “When set theory appeals to the human impossibility of a direct symbolization of the infinite”, states Wittgenstein (PG 469–470), “it brings in the crudest imaginable misinterpretation of its own calculus”, which “is of course [the] very misinterpretation that is responsible for the invention of the calculus”. The ‘misunderstandings’ that Wittgenstein wishes to dissolve “are misunderstandings without which the calculus [set theory] would never have been invented”, since it is “of no other use” (LFM 16–17). Indeed, “the chief reason [set theory] was invented” “may be” to ‘charm’ us – to make our “head[s] whirl” – by “show[ing] there are numbers bigger than the infinite” (LFM 16). This is Wittgenstein’s real reason for labouring so long and hard on irrationals and reals.

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Mathematics is ridden through and through with the pernicious idioms of set theory. One example of this is the way people speak of a line as composed of points. A line is a law and isn’t composed of anything at all. (PR §173)

The ‘misunderstandings’ and ‘misinterpretation’ revolving around irrationals, on Wittgenstein’s view, are the most dangerous misunderstandings inside and outside mathematics. ‘The confusion in the concept of the “actual infinite” arises [italics mine] from the unclear concept of irrational number, that is, from the fact that logically very different things are called “irrational numbers” without any clear limit being given to the concept’ (PG 471). Although some of us construe the ‘set’ of natural numbers as an example of “the actual infinite”, the real reason why many are unwilling to abandon the notion of the actual infinite is the conception of irrational numbers as necessarily infinite extensions. The origins of set theory lie in this “erroneous conception” (RFM V, 16), together with Dedekind’s ‘laughable’ (PG 464) and ‘ridiculous’ (PG 466) definition of an “infinite class” [i.e., the mistaken idea that the “relation m = 2n correlate[s] the class of all numbers with one of its subclasses” (PR §141)] and “Dirichlet’s [mistaken] concept of a function” (WVC 102–103), which conflates “a kind of list” (i.e., ‘correlation’) with a ‘law’. Once this “fictitious symbolism” (PR §174) became well ensconced in mathematics, it was but a short step to Cantor’s “puffed-up [diagonal] proof”, which purports to prove or show more than “its means allow it” (RFM II, 21). With a little Cantorian “hocus pocus”, “one pretends to compare the ‘set’ of real numbers in magnitude with that of cardinal numbers” (RFM II, 22), when, in fact, “[i]t means nothing to say: “Therefore the X numbers are not denumerable” (RFM II, 10). “There is no system of irrational numbers – but also no super-system, no ‘set of irrational numbers’ of higher-order infinity” (RFM II, 33). The innocent and apparently innocuous “concept of irrational number” has picked us up, and, through this series of missteps, plunged us into “[t]he sickness of [our mathematical] time” (RFM II, 23). For Wittgenstein, therefore, there is a great deal at stake in the war over irrationals. It is not as if we can extend real number arithmetic to include lawless and pseudo-irrationals without engendering the mistakes that constitute set theory. According to Wittgenstein, if we wish to steer clear of the absurdities of set theory, we must take great care to distinguish intensions and extensions. When we do so, we see that irrationals are mathematical rules which are born by means of actual mathematical constructions. Pseudo-irrationals are not real numbers despite the fact that they are constructive – we come to understand this when we grant that the so-called “set of recursive real numbers” cannot be recursively given. Lawless irrationals are not real numbers either, because they are not math-

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ematical rules. What remains to be shown is exactly how Wittgenstein uses the intensional-extensional distinction inherent in his conception of real numbers to undermine the claim that the reals are greater in cardinality than the rationals. This I have done in ‘Wittgenstein’s Rejection of Set Theory’.

NOTES 1 See esp. da Silva (1993) and Marion (1995). 2 This means, I believe, that the infinite does not ‘exclude’ anything finite, as the finite

does when we say a basket of 5 apples does not contain (exactly) 4, or 3, or 23 apples. That Wittgenstein is still a finitist in RFM is further indicated at (RFM II, 1, 38, 59). See also (RFM II, 22), (RFM III, 85), and (RFM V, 11, 19, 34–35, 39), where Wittgenstein distinguishes between genuine (and necessarily finite) sets and extensions versus so-called “infinite sets” and “infinite extensions”, and (RFM II, 40–48, 60–61) and (RFM VII, 59), where Wittgenstein argues that we are mistaken in thinking that whatever is infinite in mathematics is like something very ‘big’ or ‘gigantic’ (RFM V, 19). 3 That the later Wittgenstein maintains this view is evidenced by (RFM V, 19): “The concepts of infinite decimals in mathematical propositions are not concepts of series, but of the unlimited technique of expansion of series”. Wittgenstein similarly treats lines and curves in a strongly constructivist manner. “The straight line isn’t composed of points” (PR √ §172) – “[t]hose mathematical rules [e.g., 2] are the points” (PG 484). “[A] curve is not composed of points, it is a law. . . according to which points can be constructed” (PG 463). See also (PR §§173, 181, 183, 191) and (PG 373, 460, 461, 473, 474, 484). 4 See (PR §159): “[W]e can’t describe mathematics, we can only do it”; (WVC 34, Ft. §1): “We make mathematics”; and (LFM 22) and (RFM II, 38; V, 9, 11; Appendix II, 2), where Wittgenstein says that “[t]he mathematician is an inventor, not a discoverer” (RFM I, 168). Cf. (PG 469): “For mathematics is a calculus; and the calculus does not say of any sign that it is merely possible, but is concerned only with the signs with which it actually operates”. 5 See (PR §180): “My feeling here is the following: no matter how the rule is formulated, in every case I still arrive at nothing else but an endless series of rational numbers”. 6 At this point, since Wittgenstein’s views on irrationals and rules do not change in RFM, it seems reasonable to say that Wittgenstein is a revisionist (or, at least, an interventionist) with respect to classical mathematics, the theory of real numbers, and Analysis, independent of his revisionism viz., set theory. That is, although Wittgenstein’s views on irrationals are tied to his views on set theory, it is not as if his construal of, and criticism of, Cantor’s diagonal and non-denumerability is the only revisionist aspect of his later philosophy of mathematics. Additionally, if I am right that he maintains his finitism in RFM, he is still a revisionist regarding elementary number theory. 7 See esp. Marion (1995, 161) Wittgenstein “spoke from the viewpoint of someone who accepted only the recursive continuum”. Although I am not sure what Marion means (or can mean) by “the recursive continuum”, he presumably means at least that Wittgenstein accepts all and only recursive reals as genuine reals. This, however, is false, as we shall soon see in detail in Section 3.3.

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8 What Wittgenstein is here trying to envisage is a law different from a mathematical law

for, say, π. Thus, the only sense that could be given to “an infinitely complicated law” √ would be a law that gave, completely, a result. And unlike the law (or rule) for 2, which does determine a point “every more closely by narrowing down its whereabouts” (see PR §186, below), and hence which is the point itself, an infinitely complicated law does not do that mathematically, and hence does not do it at all. 9 It should come as no surprise that Wittgenstein persistently ridicules the Multiplicative Axiom (Axiom of Choice). In “the case of infinitely many subclasses”, says Wittgenstein, “[i]t’s obvious that. . . I can only know the law for making a selection”, for it is not ‘conceivable’ to “make something like a random selection. . . in the case of an infinite class of classes” (PR §146). We are not even able to ‘apply’ the Multiplicative Axiom within mathematics (RFM V, 25), because the so-called “choice rule” is simply not a mathematical rule. Indeed, if we were to construe the Multiplicative Axiom as an acceptable mathematical means of making selections from the “infinite number” of finite sequences generated by an irrational rule (e.g., π), lawless irrationals could be ‘constructed’ by making “random selections” of single digits from each next larger finite (rational) expansion ad infinitum. That this ‘axiom’ is not mathematical is made clear at (RFM VII, 33), “We might say: if you did not understand any mathematical proposition better than you understand the Multiplicative Axiom, then you would not understand mathematics”. √ 10 The symbol “5→3 √ ” is first presented at (PR §182), but is replaced by “0 2” at (PR 2

√ ” and “3→5 √ ”. §183). Note also that at (PR §186) Wittgenstein also uses “7→3

2 2 11 It should be noted that in the first paragraph quoted above from (PR §183), Wittgenstein

˙ provides 0.14283 7˙ as the “infinite rule” equivalent to “1/7 (5 → 3)”, and then says that this rule “can so to speak digest the (5 → 3)”. Given what he says immediately following this, he probably means that this new rule, which is parasitic upon 1/7, has already digested the (5 → 3). 12 See, for instance, (PG 481): “one cannot discover any connection between parts of mathematics or logic that was already there without one knowing”. 13 See (RFM VII, 41), Section 5, for Wittgenstein’s RFM remarks on God and π. 14 (PR §190) continues: “That is to say, no description is to occur in what is written down, everything must be represented. The approximations must themselves form what is manifestly a series. That is, the approximations themselves must obey a law”. 15 See also (PR §182): “The rules for the digits belong at the beginning, as a preparation for the expression. For the construction of the system in which the law lives out its life”. 16 Wittgenstein defines this new “multiplication operation” at (PR §186): “Suppose someone were to invent a new arithmetical operation, which was normal multiplication, but modified so that every 7 in the product was replaced by a 3. The operation ×0 would have something about it we didn’t understand so long as we lacked a law through which we could understand the occurrence of 7 in the product in general”. 17 See (PR §188), (PR §182), and (PG 475). 18 Although the rule is as “unambiguous as that for π or √2”, we shall shortly see in more detail that it is not an arithmetical rule. See also (PG 478). 19 Cf. (PG 481). 20 Thus, Wittgenstein says (PR §180): “In some sense or other there can’t be irrational numbers of different types”. 21 See (PR §§181, 183, 191) and (PG 373, 460, 461, 473).

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22 One should note that Wittgenstein does not admit the existence of even denumerably

infinite sets (i.e., in extension) – only rules for constructing sequences. For Wittgenstein’s earliest allusion to the diagonal, see (PR §181). 23 See Bernays (1959), for a criticism of Wittgenstein’s treatment of PIC. 24 This issue is discussed in detail in my ‘Wittgenstein, Finitism, and Mathematical Induction’. 25 Wittgenstein’s most explicit articulation of his finitism is made in relation to π and the “extensional conception” of irrational numbers (PR §145): “Let’s imagine a man whose life goes back for an infinite time and who says to us: ‘I’m just writing down the last digit of π , and it’s a 2’. Every day of his life he has written down a digit, without ever having begun; he has just finished. This seems utter nonsense, and a reductio ad absurdum of the concept of an infinite totality”. 26 Cf. (RFM V, 11): “Good, – then we can say: ‘It must either reside in the rule for this series that the pattern occurs, or the opposite’ ”. But is it like that? – “Well, doesn’t the rule of expansion determine the series completely? And if it does so, if it allows of no ambiguity, then it must implicitly determine all questions about the structure of the series”. – Here you are thinking of finite series”. 27 See also (WVC 82), (PG 368), and (PG 400). Although Marion (1999) does not quote (PR §151), he does quote (PR §173), (PR §168) and (PR §189) on pp. 157 and 159, and yet he mistakenly claims that Wittgenstein criticized “the universal validity of the law of excluded middle” in mathematics (141, 156–161, esp. 157, 159). Marion also misses Wittgenstein’s crucial claim that expressions like ‘PIC’ are not genuine, meaningful, mathematical propositions, mistakenly claiming that, on Wittgenstein’s account, one “is not in the position to assert ∃x F(x)” “unless [one] knows already of one such specific number” (Marion 1999, 147; but see, e.g., PR §150) and that “there are claims [e.g., ‘PIC’], that we can’t make about certain decimal expansions, such as that of π, unless the calculation up to the relevant point is already done” (p. 160). For a detailed explanation of the former mistake, see my ‘Wittgenstein, Finitism, and Mathematical Induction’. 28 See also (RFM V, 9): “I want to say: it looks as if a ground for the decision were already there; and it has yet to be invented”. Cf (RFM V, 11): “If you want to know more about the series, you have, so to speak, to get into another dimension (as it were from the line into a surrounding plane). – . . . the mathematics of this further dimension has to be invented just as much as any mathematics”. 29 Cf. (RFM V, 20): “But does this mean that there is no such problem as: “Does the pattern φ occur in this expansion”? – To ask this is to ask for a rule regarding the occurrence of φ. And the alternative of the existence or non-existence of such a rule is at any rate not a mathematical one. Only within a mathematical structure which has yet to be erected does the question allow of a mathematical decision, and at the same time become a demand for such a decision”. 30 Cf. (RFM IV, 42): “That, if you go on dividing 1: 3, you must keep on getting 3 in the result is not known by intuition, any more than that the multiplication 25 × 25 yields the same product every time it is repeated”. See also (RFM IV, 44). 31 Given (RFM V, 10, 19), quoted above, and (RFM V, 11), quoted in Note 32, Wittgenstein must here mean that all that we can thereby determine is that the particular sequence occurs at a particular interval in the expansion (i.e., the truth of a finite disjunction).

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REFERENCES

Ambrose, Alice (ed.): 1979, Wittgenstein’s Lectures, Cambridge 1932–35, Basil Blackwell, Oxford; referred to in the text as AWL. Bernays, Paul: 1959, ‘Comments on Ludwig Wittgenstein’s Remarks on the Foundations of Mathematics’, in Shanker (1986, 165–182). originally published in Ratio 2. Da Silva, Jairo Jose: 1993, ‘Wittgenstein on Irrational Numbers’, in Puhl (1993, 93–99). Marion, Mathieu: 1993, ‘Wittgenstein and the Dark Cellar of Platonism’, in Puhl (1993, 110–118). Marion, Mathieu: 1995, ‘Wittgenstein and Fintism’, Synthese 105, 143–165. Moore, G. E.: 1959, ‘Wittgenstein’s Lectures in 1930–33’, Philosophical Papers, Allen & Unwin, London, pp. 252–324. Puhl, Klaus (ed.): 1993, Wittgenstein’s Philosophy of Mathematics, Verlag Holder-PichlerTempsky, Vienna. Rodych, Victor: 1995, Review Article: ‘Pasquale Frascolla’s Wittgenstein’s Philosophy of Mathematics’, Philosophia Mathematica (3) 3, 271–288. Rodych, Victor: 1997, ‘Wittgenstein on Mathematical Meaningfulness, Decidability, and Application’, Notre Dame Journal of Formal Logic 38(2), 195–224. Rodych, Victor: 1999a, ‘Wittgenstein’s Inversion of Gödel’s Theorem’, Erkenntnis 51, 173–206. Rodych, Victor: 1999b, ‘Wittgenstein’s Critique of Set Theory’ (submitted). Rodych, Victor: 1999c, ‘Wittgenstein, Finitism, and Mathematical Induction’ (submitted). Shanker, Stuart (ed.): 1986, Ludwig Wittgenstein: Critical Assessments, Vol. 3, Croom Helm, London. Waismann, Friedrich: 1979, Wittgenstein and the Vienna Circle, in B. F. McGuinness (ed. and trans.), Basil Blackwell, Oxford; referred to in the text as WVC. Wittgenstein, Ludwig: 1922, Tractatus Logico-Philosophicus, Routledge and Kegan Paul, London. Wittgenstein, Ludwig: 1956, Remarks on the Foundations of Mathematics, 2nd edn., G. H. von Wright, R. Rhees and G. E. M. Anscombe (eds.). Basil Blackwell, Oxford, 1967; referred to in the text as RFM. Wittgenstein, Ludwig: 1974, Philosophical Grammar, Rush Rhees (ed.), Anthony Kenny (trans.). Basil Blackwell, London; referred to in the text as PG. Wittgenstein, Ludwig: 1975, Philosophical Remarks, Basil Blackwell, Oxford; referred to in the text as PR. Wittgenstein, Ludwig: 1976, Lectures on the Foundations of Mathematics, Cora Diamond (ed.). Cornell University Press, Ithaca, NY; referred to in the text as LFM. Department of Philosophy York University 4700 Keele St. Toronto, Ontario, M3J-1P3 Canada