Bolzano's measurable numbers: are they real?
aa r X i v : . [ m a t h . HO ] M a y Bolzano’s measurable numbers: are they real?
Steve Russ and Kateˇrina Trlifajov´a ∗ Abstract
During the early 1830’s Bernard Bolzano, working in Prague, wrote a manuscriptgiving a foundational account of numbers and their properties. In the final sectionof his work he described what he called ‘infinite number expressions’ and ‘measur-able numbers’. This work was evidently an attempt to provide an improved proofof the sufficiency of the criterion usually known as the ‘Cauchy criterion’ for theconvergence of an infinite sequence. Bolzano had in fact published this criterionfour years earlier than Cauchy who, in his work of 1821, made no attempt at aproof. Any such proof required the construction or definition of real numbers andthis, in essence, was what Bolzano achieved in his work on measurable numbers.It therefore pre-dates the well-known constructions of Dedekind, Cantor and manyothers by several decades. Bolzano’s manuscript was partially published in 1962and more fully published in 1976. We give an account of measurable numbers, theproperties Bolzano proved about them, and the controversial reception they haveprompted since their publication.
It is now widely accepted that any logically sound development of the limitingprocesses underlying mathematical analysis, or the calculus, requires the construc-tion, definition or axiomatisation of the domain of real numbers. Or, at least, itrequires some explicit assumption about the completeness of a linearly orderedfield such that we can guarantee the closure of the field under limiting processes.Such recognition has been slow to be achieved. As late as 1908 the first edition ofG.H. Hardy’s classic textbook
Pure Mathematics simply assumed the rational andirrational numbers taken together had suitable algebraic and completeness prop-erties. Only subsequent editions from 1914 gave a detailed account of Dedekindcuts as a construction of reals from the rationals. Both Dedekind and Weierstrass—leading professional mathematicians of their time—explicitly attributed motivationfor their constructions of real numbers to the need, in the context of their teach-ing from the late 1850’s, for a more rigorous basis for the differential and integralcalculus.It is all the more remarkable therefore that a little-known Bohemian priest, whonever did any formal teaching of mathematics, should have not only seen the needfor such a foundation for calculus as early as 1817, but in the early 1830’s had ∗ Steve Russ, Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK, [email protected] , Kateˇrina Trlifajov´a, Faculty of Information Technology, Czech Technical Uni-versity in Prague, Th´akurova 9, 160 00 Prague 6, Czech Republic, katerina.trlifajova@fit.cvut.cz . lready gone a long way towards developing a bold, original, framework for a the-ory of real numbers. Bernard Bolzano’s measurable numbers are developed in thefinal (7 th ) section of his Reine Zahlenlehre ( Pure Theory of Numbers ), here abbre-viated to RZ. It was written in three manuscript versions in the early 1830’s that arebriefly described by the late Jan Berg in (Bolzano 1976) where his transcription ofthe final version is published. It is the final (7 th ) section of RZ that is our primarysource and wherever paragraph numbers (of the form §
1) appear in isolation in thischapter they refer implicitly to this final section of RZ. The purpose of this chapteris to outline the achievements in this part of Bolzano’s work and to identify, asfar as the evidence allows, how Bolzano regarded his measurable numbers. Wealso endeavour to throw light on the somewhat confused reception his theory hasreceived since its publication.Bolzano’s written output of published works and unpublished manuscripts wasprodigious. It is being given a comprehensive publication in the
Bernard BolzanoGesamtausgabe ( Complete Works ) with over 90 volumes of the projected 129 vol-umes having already appeared. We use the abbreviation BGA for this edition. Thebest overview of Bolzano’s life and work in English is (Morscher 2008) which isparticularly good on the work on logic and philosophy but includes useful materialon mathematics too. Some of the most important mathematical works have ap-peared in English translation, with brief commentary, in (Russ 2004). Much moresustained study of his mathematics appears in (Rusnock 2000) and, in French,(Sebestik 1992). Bolzano’s most important philosophical work, in four volumes,is
Wissenschaftslehre ( Theory of Science ) now with a complete English translation(Bolzano 1837/2014).In a deservedly much-admired paper of 1817 Bolzano presented an ingeniousand original proof, by repeated bisection, of the intermediate value theorem, namelythat a continuous real-valued function that changes its sign at the endpoints of theinterval [ a , b ] has a zero somewhere in the open interval ( a , b ) . In § Bolzano-Cauchy convergence criterion . We state this here as follows:If a sequence of terms a , a , a ,... a n ,... a n + r ,... has the propertythat the difference between its n -th term an and every later one a n + r ,however far this latter term may be from the former, remains smallerthan any given quantity if n has been taken large enough, then thereis always a certain constant quantity , and indeed only one , which theterms of this sequence approach and to which they can come as nearas we please if the sequence is continued far enough. (Bolzano 1817, § Bolzano attempted a detailed proof of the sufficiency of this criterion. Cauchysimply assumed the criterion without comment. It is an irony that Bolzano istoday criticised for his ‘flawed’ proof while Cauchy is not criticised for makingthe silent assumption. As pointed out in (Stedall 2008, 496), Bolzano’s was the only attempted proof of the criterion in the early 19 th century. We can put ‘flawed’in inverted commas here because while the result is not true in the field of rationals This is a slightly modified form of the theorem, the exact version is in (Russ 2004, 266). lone, it is true in the field Hardy was assuming and evidence that this was also thefield Bolzano intended comes from his remark near the end of the Preface whenhe summarises the content of the body of his paper saying that,for anyone having a correct concept of quantity the idea of [thelimit] is the idea of a real, i.e. an actual quantity.The main work of his attempted proof is to show that the limit isnot impossible [because] on this assumption it is possible to deter-mine this quantity as accurately as we please.The possibility of some entity was always regarded by Bolzano as a necessary, butnot sufficient condition for existence. In this case the possibility, combined withthe need, led Bolzano to claim that with a correct concept of quantity (or num-ber, the two concepts were not sharply distinguished by Bolzano) the convergencecriterion was indeed sufficient to ensure the existence of a limit. For detailed dis-cussion of this proof see (Kitcher 1975, 247–251), (Rusnock 2000, 69–84) and(Russ 2004, 149).No-one could be in doubt that to maintain a correspondence between numbersand lengths (such as the diagonal of a unit square), or to locate the zeros of somefunctions (such as x − th century was the logical needto give some construction or definition of the irrationals on the basis of the ratio-nals. That was the problem giving rise to the celebrated solutions such as thoseof Dedekind, Weierstrass, M´eray, and Cantor published from 1872 (although orig-inating in the late 1850’s). It was this same problem that Bolzano had alreadyaddressed, largely successfully but unpublished and quite unknown, in his theoryof measurable numbers of the early 1830’s. It is the purpose of this chapter togive to give an outline of Bolzano’s achievement in this work. The clearest sourcefor the later work, which became a standard reference, is (Dedekind 1963). Theoriginal works of the other authors just mentioned can be hard to locate but arewell-described, and referenced, in many works, for example in chapter IV of (Fer-reiros 1999).During the 1840’s Bolzano realised he would not complete the manuscriptsfor RZ and he passed these and many other mathematical papers on to RobertZimmermann, one of his former students and the son of a good friend. Zimmer-mann gained a Chair in Philosophy in Vienna and deposited a locked suitcase withmany of Bolzano’s mathematical papers in the Austrian National Library. Theylanguished there until discovered by Professor M. Jaˇsek from Pilsen in the 1920’s.Some material was published from the 1930’s onwards but it was not until 1962that a partial transcript of RZ was published as (Rychl´ık 1962). This was in factonly a partial transcription of the final (7 th ) section of RZ. A much fuller publi-cation of all sections of RZ including some parts of the 7 th section which wereillegible to Rychl´ık appeared in (Bolzano 1976). The title of this final section of RZ is ‘infinite quantity concepts’ but from hishand-written revisions, and alternative phrasing, it is clear he had difficulty decid-ing between ‘quantity’and ‘number’ on the one hand, and between ‘concept ’ and‘expression’ on the other hand. We shall not adhere exclusively to any of these hrases but try to keep in mind the ambiguity reflected in Bolzano’s usages. Aregular explicit usage was that an infinite quantity concept is represented by aninfinite quantity expression . The following are some of his examples of the latter:1 + + + + ... in inf.12 − + − + ... in inf. ( − )( − )( − )( − ) ... in inf. a + b + + + ... in inf.where a , b is a pair of natural numbers . The crucial property of an infinite quantityexpression is that it should contain infinitely many operations of addition, subtrac-tion, multiplication or division. But Bolzano is at pains to point out that this doesnot mean we have to have the ideas of all the components of such an expression,we could not. He says it is like the way we can describe, and designate, a pocketwatch very simply and without having ideas of the many components inside thewatch. So he says that ‘infinite’is here being used in a figurative way. The numberconcept itself is a single thing arising from the multitude of operations. He saysthat for every number expression S (not only infinite ones),... we determine by approximation, or measure [the number ex-pression S ], if for every positive integer q we determine the integer p that must be chosen so that the two equations S = pq + P and S = p + q − P arise, in which P and P denote a pair of strictly positive numberexpressions (the former possibly being zero). (RZ, § strictly positive number expression is one in which—according to the earliersection 4 of RZ— no subtraction appears . When these two equations are alwayssatisfiable Bolzano says, we determine the number expression S , as precisely aswe please and then he calls such an expression a measurable expression (RZ, § pq is then called the measuring fraction for S .It appears that we can visualise a measurable expression or number by imag-ining a line of length S which is being measured by rulers with units divided into q equal parts, a q -ruler. For a given ruler either the line will match up exactlywith some division marks (and so S is rational), or the line is always strictly in be-tween two divisions. In either case S is measurable. Such a visualisation assumesa close correspondence between a geometric continuum and an arithmetic contin-uum. In fact the theme of continuum—whether geometric, arithmetic, temporalor physical—is a powerful underlying theme in much of Bolzano’s work fromhis first publication with his analysis of straight line in 1804 to the explicit treat-ment of continuum in §
38 of his well-known
Paradoxes of the Infinite published as(Bolzano 1851) shortly after his death. It is clear from Bolzano’s usage in RZ that this is the appropriate translation of ‘wirkliche Zahlen’. his definition of an expression S as measurable implies that for every positiveinteger q there is a p so that pq ≤ S < p + q and it means that Bolzano is explicitly associating a measurable number with in-finitely many approximating rational intervals. It is easy to show by experimentthat these intervals are not nested (readers may like to try this with examples of S = /
3, or S = √ dually directed in the sense that the intersection of any two (sayfor q and q ′ with q ′ > q ) is a proper subset of some interval for q ′′ with q ′′ > q ′ .But the association of a measurable number with an infinite collection of intervalsis significant. The use of infinite collections to define real numbers is a featurecommon to all the major constructions that emerged in the later 19 th century.In RZ §
21 Bolzano introduces the expression S = + + + ... in inf.After showing S is measurable with p =
0, however large q may be taken,Bolzano goes on to conclude that... we are not justified, at least by the concepts so far, in consider-ing the expression S to be equivalent to 0. ... This is an example of an infinitely small positive number.This represents a major change in Bolzano’s view on the infinitely small fromsome 15 years earlier when he declared that... calculus is based on the shakiest foundations ... on the self-contradictory concepts of infinitely small quantities. (Bolzano 1816,Preface)It would be an interesting project to investigate what influenced his change of mindhere. There are many potential sources for this study such as the series
MiscellaneaMathematica of the BGA (Bolzano’s mathematical diaries that he maintained formost of his life), the sustained reflection on infinite collections in (Bolzano 1851),and the correspondence in the series III of the BGA, as well as numerous furtherarchival sources.The possibility of infinitely small numbers that are measurable gives rise toa crucial revision of his definition of the equality of measurable numbers. In thelong discussion of section §
54, and again as an explicit definition in §
55, Bolzanostates that measurable expressions are equal to one another if for every arbitrary q always one and the same p may be found yielding a measuring fraction pq commonto both. Otherwise stated, two numbers were equal ifthey behave in the same way in the process of measuring.However, Bolzano inserted a highly significant revision near the beginning of § − + + + ... in inf. where the former, for every q ,has measuring fraction with p = q , but the latter has p = q −
1. So he revises thedefinition to say that if the pair of numbers A and B have a difference A − B which n the process of measuring always has a measuring fraction of the form 0 / q then A = B . And if the difference is positive then A > B , if it is negative then A < B .Unfortunately the section §
54, and subsequent sections have not, in fact, been re-vised other than by way of this inserted note. In §
56 Bolzano says it is not so muchthe notion of equality which is extended by his definition but rather it is the object(here measurable number) which is affected. The earlier definition discriminatesmore finely than the new one which, in modern terms, is an equivalence relation.We shall speak here of old measurable numbers and new measurable numbers. Forthe old numbers equality was not transitive, for the new numbers equality is tran-sitive (and reflexive and symmetric). Perhaps Bolzano was one of the first to seethe need for an equivalence relation, and the related equivalence classes, or factor-classes. It is interesting to note that in (Klein 1908) two separate themes in thedevelopment of analysis are identified: the Weierstrassian approach (in the contextof an Archimedean continuum), and an approach with indivisibles and/or infinites-imals (in the context of a richer non-Archimedean continuum). This is cited at thebeginning of the paper (Bair et al 2008) which continues with extensive commen-tary but, curiously, no mention of Bolzano whose two definitions of the equality ofmeasurable numbers clearly straddle both approaches described by Klein.Bolzano proves important properties of the ordering of measurable numbers in §§
61 - 79. We summarise these in the following using modern notation and givingthe relevant paragraphs of RZ in parentheses after each result.
Theorem 1
Let A , B , C be measurable numbers.1. Transitivity. (( A > B ) ∧ ( B > C )) ⇒ ( A > C ) . ( §
63 )2. Linearity. ( A = B ∨ A > B ∨ A < B ) . ( §
61, 73)3. Unboundedness. ( ∀ A )( ∃ B )( ∃ C )(( B < A ) ∧ ( A < C )) . ( § ( A < C ) ⇒ ( ∃ B )(( A < B ) ∧ ( B < C )) . ( § ( ∃ n )( An < B < n · A ) . ( § ( A > B ) ⇒ ( A + C ) > ( B + C ) . ( § The next results are about the arithmetic properties of measurable numbers,they are in §§
45, 51, 59, 99-121 and are gathered here as follows.
Theorem 2
Let A , B , C are measurable numbers.1. Closure under addition. A + B is a measurable number. ( § · B is a measurable number. ( § § § = ⇒ AB is a measurable number. ( § . A · = · A = . ( § · ( B · C ) = ( A · B ) · C . ( § · B = B · A . ( § · ( B + C ) = A · B + A · C . ( § ( A = B ∧ C = ) ⇒ AC = BC , B = ⇒ AB · B = A , etc . ( §§
113 - 121) olzano can now finally prove the sufficiency of the Bolzano-Cauchy convergencecriterion in § Bolzano-Cauchytheorem . He stated it already in (Bolzano 1817/2004), but could not prove the ex-istence of the relevant limit. The following much improved formulation and proof(Russ 2004, 412) is in terms of a sequence of measurable numbers: in modernterms it means that the ordered field of measurable numbers is complete.
Theorem 3
Suppose the infinitely many measurable numbers X , X , X , ··· pro-ceed according to such a rule that the difference X n + r − X n , considered in itsabsolute value always remains smaller than a certain fraction N which itself canbecome as small as we please, providing the number n has first been taken largeenough. Then I claim there is always one and only one single measurable numberA, of which it can be said that the terms of our series approach it indefinitely. Bolzano distinguishes three cases: the sequence X , X , X , ··· is non-decreasing,non-increasing or alternating. He begins with non-decreasing sequences. Heproves that if there is a limit of the sequence then for every q there is a p suchthat pq is a measuring fraction of the (conjectural) limit. So we have a complete setof measuring fractions. Bolzano shows that this limit is determined uniquely. Theproof for non-increasing sequences is similar. From alternating sequences we canchoose subsequences which are either non-decreasing or non-increasing and theirlimit is the limit of the whole sequence. The proof is long and not easy to followin some of the details but we support Rusnock’s opinion in concluding positivelyon the logical structure of this proof in (Rusnock 2000, 188).This was a vindication of Bolzano’s belief, announced in the Preface of his(Bolzano 1817/2004), that with a ‘ correct concept of number’ the convergence cri-terion quoted there was indeed sufficient to establish a limit number. That correctconcept can, we propose, be identified with the measurable numbers. This con-cludes the demonstration that, in spite of what we would now regard as some gapsand confusions, Bolzano’s domain of measurable numbers is a complete linearlyordered field and so isomorphic to the real numbers as we know them today.
It is an irony that the late Bob van Rootselaar, who did so much work in carefultranscription and editing of Bolzano’s mathematical diaries, was also one of theseverest critics of Bolzano’s work on real numbers. Very soon after the publica-tion of (Rychl´ık 1962) there appeared (van Rootselaar 1963) in which the authordeclares in the opening two pages that,Bolzano’s elaboration [of measurable numbers] is quite incorrect,and that the more advanced part of Bolzano’s theory is inconsistent.One might have supposed this to be largely due to the fact that it only came to lightin (Bolzano 1976) how some significant improvements in the content of Bolzano’swork were revealed by the more thorough and detailed reading of the manuscriptversion by Berg. For example, the revision to the equality criterion mentionedabove was not legible for Rychl´ık and was omitted, and other parts that weredeleted by Bolzano, were included by Rychl´ık. But this is evidently not the causeof van Rootselaar’s negative views. He refers, long after publication of Berg’stranscription, to the obscure, but interesting, work (Ide 1803) and concludes that oth the theories [of Ide and Bolzano] presuppose the existence ofthe real numbers ... progress is made only with a theory such as thatof Cantor. (van Rootselaar 2003).On the contrary, we wish to support here the claim that Bolzano did not—in con-trast to the situation in 1817—presuppose the existence of the reals at all and that,in fact, his theory has a close resemblance to that of Cantor and to that of Dedekind.Soon after the criticism of van Rootselaar came a strong rejoinder in (Laugwitz1965) pointing out that it only needed a small change in the definition of infinitelysmall quantity in order to rectify many of Bolzano’s proofs and results. After sucha change it can indeed be viewed as a consistent theory of real numbers. Followingthe publication by Berg of the improved reading of Bolzano’s work it was discov-ered that Bolzano had already made the change that Laugwitz recommended andso (Laugwitz 1982) was able to fully endorse Bolzano’s work as a theory of thereal numbers.Subsequently there have been a variety of commentators with a wide spectrumof views on Bolzano’s work on measurable numbers, certainly enough to justifythe remark that,there is perhaps no area of Bolzano’s research about which there isless agreement than his theory of real numbers ... Bolzano’s analyseswere a preamble to his theory of measurable numbers, which is itselfa tangled thicket of issues, much disputed in the literature (Simons2003, 118).We cannot give here any comprehensive study of the debates but should referat least, further to those already mentioned, the detailed studies in (Spalt 1991),(Sebestik 1992) and (Rusnock 2000). The last-cited work sums up the difficultieswell: Bolzano’s theory of measurable numbers as it has come down tous is obviously in fairly rough shape.Then after some valuable detailed discussion of the proof in RZ §
107 he concludes,. . . on the essential point of conceptual structure, Bolzano wasalmost entirely successful in characterizing the reals. (Rusnock 2000,184-188).So much for the commentary in the literature at a high level. We now give someindication of the detailed discussion and interpretation of what Bolzano was doingin this rich final section of RZ and the direction of our own thinking on the mainthemes.The motivation for the title of the final section of RZ as
Infinite Quantity Con-cepts (or also
Infinite Quantity Expressions ) is clear. The previous three sectionsof the work were all concerned explicitly with rational numbers. Number expres-sions with only finitely many arithmetic operations will only yield rational numberresults. So to address the construction of irrational numbers it was essential toconsider expressions with infinitely many operations. Some commentators seemto have been distracted into a focus on how best to interpret such expressions whenthe central, over-riding concept—that of measurable number—is one which ap-plies to both infinite and finite number expressions. And a measurable number isdefined in terms of two equations or, equivalently, by an approximating interval asdescribed above. he problem, and the need for some interpretation, with the notion of infi-nite quantity expression arises because it appears to be very general (just requir-ing infinitely many arithmetic operations) but the examples given by Bolzano arerather simple. It appears to allow for expressions such as continued fractions, com-pounds of multiple continued fractions or any arbitrarily complex expression, evenwhether or not there is any evident rule for continuing the expression. But the mostcomplex one used by Bolzano in RZ is the one occurring near the end of § s ( + + + ··· in inf. ) − qbq ( + + + ··· in inf. ) . For the proof in this paragraph it is only claimed that this is a positive quantity:it does not need to be evaluated. Ladislav Rieger, one of the editors of (Bolzano1962), suggests in his
Vorwort that such infinite number expressions might be in-terpreted as ‘symbols for effectively described, unbounded, computational proce-dures on rational numbers.’ But for what procedure is it a symbol? In this case itmight be ’natural’ to say we obtain a partial n -th value if we sum the first n termsof numerator and the first n terms of the denominator, then divide. But it is easy toconstruct cases where there is no such natural rule. van Rootselaar attributes theidea of interpreting infinite number expressions as infinite sequences of rationalnumbers to Rychl´ık and takes up the idea himself with enthusiasm even declaring,. . . indeed anyone who reads Bolzano’s manuscript is bound toaccept it [Rychl´ık’s interpretation] (van Rootselaar 1963, 169).So the expression b + + + ... in inf. is interpreted as the sequence { bn } , and the ex-pression 1 + + + ... in inf. corresponds to the sequence { n ( n + ) } . It mightbe noted in the former case that bn is not strictly a partial sum of the expression (itis the reciprocal of a partial sum) but it is a ’partial computation’ to use Rieger’sphrase.Van Rootselaar develops his own elaborate sequence interpretation and usesit to give an interpretion of Bolzano’s measurable numbers. Having expressedthe measurable number S in terms of an infinite sequence { s n } each term of thesequence is then assigned an approximating interval using terms involving triplesubscripts and equations of the form s n = p q ( S ) q + P q , , n = p q ( S ) + q − P q , , n where full details are given in (van Rootselaar 1963, 173). Apart from being rathercumbersome a strong argument against carrying the sequence interpretation to suchlengths is the one put forward in (Becker 1988). Here Becker simply pointsout the obvious fact that Bolzano, although being fluent at working with infinitesequences, nowhere suggests that he was himself making use of a sequence inter-pretation for either infinite number expressions or measurable numbers. However,there is an important aspect of what van Rootselaar is doing in the above formu-lation which we shall ourselves shortly be endorsing. That is, he is making avery tight association between a measurable number and an infinite collection ofapproximating intervals. Whether or not this was in Bolzano’s mind was neverexplicitly stated by him. We can only judge from the surrounding context and hisusage. This is an unpublished dissertation which we have not seen but rely on the report of it in (Spalt 1991). eturning to the more limited application of the sequence interpretation—thatfor simple infinite series—there is a well-known problem arising in the case ofan alternating series in which the partial sums are non-monotonic. The problemoccurs already with the expression S = − + − + ... in inf.which Bolzano himself presents as an example of an infinite number expressionin §
2. If we interpret it as the sequence of partial sums we obtain the sequence , , , ,... that converges to . This is a non-monotonic sequence, the termsare sometimes less, and sometimes greater, than . It is impossible to say that for q = p such that the sequence lies within the interval [ p , p + ) . It is similarfor q = n where n is a positive integer. The question is: did Bolzano considerthis expression, and consequently all convergent alternating series, as measurablenumbers? There are several possibilities:1. Bolzano did not regard the expression S as a measurable number because itdoes not satisfy his definition of measurability (Sebestik 1992, 370). Bolzanogenerally did not regard sequences oscillating around a rational number asmeasurable. Then there is a problem. Bolzano had proved in §
45 that mea-surable numbers are closed under addition. But for instance if we take thetwo measurable numbers A = − + − + − + ... in inf. B = − + + + + ... in inf.then their sum A + B = − + − + − ... in inf.is not measurable.2. Bolzano regarded S as measurable. He speaks about measurable numbers asquantities which we can measure up to q for every q . And the expression S does have this property. If we return to the picture of a q -ruler we seethat if we are allowed to shift the ruler then S can be enclosed between twoscale divisions. Otherwise, if we do not shift the ruler, the expression valueoscillates around one division mark, successively occupying two adjacentintervals.If we admit in the definition of measurable numbers that p can be a rationalnumber then the definition would be: S is measurable if for all q there is a rational number r and two positive expressions P , P such that rq + P = S = r + q − P . Or we could repair the definition in this way: S is measurable if for all q there is a rational number r and two positive expressions P , P such that r + P = S = r + q − P . In the both cases A + B would be measurable and generally the sum of twomeasurable numbers would be measurable too. . Bolzano regarded S as measurable but he had a different concept of an in-finite calculation. He considered S as one exactly given quantity which isequal to . He generally considered number expressions with oscillatingvalues and approaching a rational number as being equal to that rationalnumber. There is some indication of this in his proof that the sum of twomeasurable numbers A , B is measurable in §45 . Bolzano analyses severalcases. The last case is about number expressions which could be interpretedas non-monotonic sequences. Bolzano obtains after many equations the ex-pression A + B = p + p + q + P − Ω = p + p + q − P + Ω where by Ω and Ω Bolzano understands a pair of number which can de-crease indefinitely. Therefore Ω + Ω = P + P Because Ω + Ω can decrease indefinitely one can say that also P and P can decrease indefinitely. The sum A + B evidently alternates and approachesa fraction p + p + q . Bolzano, referring to a similar result for rational num-bers ( §
8, 6th Section), says that A + B = p + p + q It is hard to know what was in Bolzano’s mind here. The manuscript thatremains was not a definitive version. In order to deal with the case of oscillatingvalues Laugwitz demonstrated that it suffices to change the equation in Bolzano’sdefinition of measurable numbers to repair the theory (Laugwitz 1982, 407). Wegive his proposal as follows in a slightly modified form and call it the
Laugwitzcondition . Definition 1
An infinite number concept S is a measurable number if for everypositive natural number q there is a natural number p and two positive numberexpressions P and P such that the two following equations are satisfied:p − q + P = S = p + q − P . It is in the spirit of Bolzano’s idea that the infinite number expression S is measur-able if the determination by approximation, or the measurement of S , canbe carried out as precisely as we please. (RZ § Bolzano himself suggested a generalisation of this modi-fication in §
122 (Russ, 2004, 428) where he writes: Another interesting possibility for repairing the definition of measurable numbers is in (Rusnock 2000,185 - 186). erhaps the theory of measurable numbers could be simplified ifwe formulated the definition of them so that A is called measurable ifwe have two equations of the form A = pq + P = p + nq − P , where forthe identical n , q can be increased indefinitely.We return now to the discussion of the sequence interpretation of infinite num-ber concepts which we began earlier in this section (on p.10). It is surely a reason-able proposal that if we associate a number concept P with the sequence { p n } then P ≥ N such that p n ≥ n ≥ N (Rusnock 2000,185). Then adopting the Laugwitz condition as above, and using modern notation,we shall prove the following. Theorem 4
If a rational sequence { a n } represents the infinite number expressionS then { a n } satisfies the Bolzano-Cauchy convergence criterion (we call it a BC-sequence) if and only if S is a measurable number. Proof 1
Let q , m , n , k , p be natural numbers, P , P are strictly positive numberconcepts.By the Laugwitz condition, the infinite number concept S is measurable if ( ∀ q )( ∃ p )( ∃ P )( ∃ P )( p − q + P = S = p + q − P ) . It means in our interpretation that ( ∀ q )( ∃ p )( ∃ m )( ∀ n > m )( p − q < a n < p + q ) . Remember that { a n } with each a n ∈ Q is a BC-sequence iff ( ∀ k )( ∃ m )( ∀ n > m ) | a n − a m | < k . (i) Let S be measurable. We will prove that { a n } is a BC-sequence. Take any k.Let q = k. Then ( ∃ p )( ∃ m )( ∀ n > m )( p − k < a n < p + k ) . Hence ( ∀ n > m ) | a n − a m | < | p + k − p − k | = k . (ii) Conversely, let { a n } be a BC-sequence. We will prove that S is a measurablenumber. Take any q. Let k = q. Then ( ∃ m )( ∀ n > m )( | a n − a m | < q . We know that a m ∈ Q therefore ( ∃ r )( r q ≤ a m < r + q . Hence r − q = r q − q < a n < r + q + q = r + q . If r is even take p such that r = p and if r is odd take psuch that r = p − . In the both casesp − q < a n < p + q. Approximating intervals
The title of the final section of RZ with which we are mainly concerned here is
Infinite Quantity Concepts (or as we have indicated this could be interpreted as
Infinite Number Expressions or similar variants). And it lives up to this title forthe first 5 sections. But thereafter (and there are more than 100 sections in thethereafter)—with the exception of § § S = pq + P S = p + q − P That is, with intervals of the form [ pq , p + q ) for all values of natural numbers q . Itis possible we suggest, that the pre-occupation with the sequence interpretation ofinfinite number expressions has, at least for some commentators, been a distractionfrom Bolzano’s main focus.As far as we know it has not been observed in the previous literature that thereare two rather different ways of deriving infinite rational sequences from Bolzano’sconcept of measurable numbers. The common approach is that of partial compu-tation that depends on the detailed procedural evaluation of an infinite numberexpression: it is what we have called the sequence interpretation . Another ap-proach is to begin from a concept like √
2, or a rational like , for either of whichwe may derive an algorithm, or a decimal expansion, which will allow us to gen-erate approximating intervals. Choosing the left-hand (or right-hand) endpoints ofthese intervals then also generates an infinite rational sequence. In fact both theseapproaches are at least strongly hinted at in the original publication (Rychl´ık 1962)but they are not equally taken up in the subsequent literature. Both views seem tous legitimate and significant though the sequence interpretation would have lim-ited application to the concept of infinite number expressions in general. But it isBolzano’s analogy between an infinite number concept and a pocket watch ( § associated with an infinite collection of approximating intervals.We should therefore like to re-emphasise the approximating intervals view. Itseems in fact to be the dominant view in RZ. Here we shall rely on the researchesof others for some of our argument. In the chapter (Mainzer 1990) it is reportedthat in the work (Bachmann 1892) there is a systematic use of nested intervals tointroduce real numbers. Mainzer indicates in some detail how such an approachmight be developed in modern terms. A rational net is defined as a sequence ofclosed strictly nested intervals on rationals with lengths tending to zero. A net ( J n ) is said to be finer than ( I n ) if J n ⊆ I n for all n . Then two nets ( I n ) and ( I ′ n ) are saidto be equivalent if there is a net ( J n ) finer than each one. He shows how then realnumbers can be defined as equivalence classes of nets. It would be possible to fol-low this up with definitions of arithmetic operations and ordering on these classesand show they form a complete ordered field. Instead Mainzer follows a different,more interesting, strategy. He establishes a direct correspondence between on theone hand the classes of nets and Dedekind cuts, and on the other hand between thenet classes and Cantor’s classes of fundamental sequences. These correspondences an be set up rather simply and reveal a satisfying underlying similarity betweenclasses of approximating intervals and the Dedekind and Cantor constructions. Fulldetails are given in (Mainzer 1990). We have already explained that Bolzano’s ap-proximating intervals are not, themselves, strictly nested, they are dually directed ,but this property allows us in a straightforward fashion to derive a strictly nestedfamily of intervals which could therefore be used in such a construction as Mainzerdescribes. For example, one way to do this for a given measurable number S is totake, instead of the approximating intervals for all values of q , is to take, for agiven fixed value of q , say q , the sub-collection of intervals for all multiples nq for natural numbers n >
1. In this way each approximating interval is a subset ofthe previous one and they do form a nested sub-collection of all the approximatingintervals. The length of these intervals tends to zero and must have the same uniquecommon point as that of the collection of all intervals, namely that correspondingto the number S .Bachmann was not the only mathematician of the late 19C to define real num-bers in terms of nested intervals. The work (Burn 1992) reports that in an appendixto Volume 3 of the Cours d’analyse (Jordan 1887) Jordan gave a construction ofirrational numbers. It was also using nested intervals but in the later, more influen-tial, editions he gave accounts similar to Dedekind cuts.
In (Rychl´ık 1962) Bolzano’s work is hailed in the title as a ‘theory of real num-bers’. We have tried in this paper to focus on this claim and to give greater em-phasis than in previous treatments to the key feature of measurable numbers thatthey are defined in terms of approximating intervals. In doing so we have in-evitably neglected many interesting and relevant themes. We have not dealt prop-erly, for example, with Bolzano’s concepts of infinitely small numbers, infinitelylarge numbers and infinite collections. These are complicated topics, especially ifthey are to be considered—as they must—alongside the context of Bolzano’s lifeand his times.We cannot now know any more than the evidence that remains to us allowsfor how exactly Bolzano regarded his infinite number concepts and the approxi-mating intervals of his measurable numbers. So there is an essential element ofsurmise and speculation in any judgement we make. It has to be acknowledgedthat Bolzano did not explicitly refer to any such sequence interpretation as manycommentators (including ourselves) have constructed. Nor did he explicitly re-late measurable numbers to the collection of their approximating intervals in theway we have supportted. But we have given—at least in broad outline—good ev-idence, albeit with the hindsight of a modern perspective, that Bolzano’s insightsinto a ‘correct concept of number ’ did indeed constitute the core of the real numbersystem as recognised in modern times.Bolzano never did re-write the crucial discussion around §
54 (as he said wasneeded) relating to the two views of the equality of measurable numbers. But wehave highlighted this issue and drawn attention to his premonitions of the idea ofequivalence relation. His careful development of many of the algebraic propertiesof an ordered field is also worthy of further attention.In addition to the uncertainties surrounding work for which there was nevera ‘final version’ we have mentioned also some other specific factors contributing o the controversies in the literature. For example, there is the fact of the twostages (1962 and 1976) in publication of transcriptions of his manuscripts, thedistractions, and difficulties, in the ‘sequence interpretation’ of infinite numberexpressions and the association (misguided in our view) of Bolzano’s infinitelysmall numbers with non-standard analysis.Finally we mention a theme that calls for further investigation in a future work.The Czech dissident Petr Vopˇenka working in Prague developed in (Vopˇenka 1979)an Alternative Set Theory (AST). Although his motivation did not come fromBolzano’s work his ideas seem to us a framework well-suited to Bolzano’s math-ematical work and particularly his work in relation to infinite numbers and collec-tions. Vopˇenka’s AST makes use of so-called semisets and the phenomenologicalnotion of a ‘horizon’separating finite from infinite numbers. There is an extensivetheory of semisets which can be used to support a theory of numbers, and a theoryof the continuum. Some of this work appears to relate quite closely to Bolzano’sideas and results about measurable numbers References [1] Bachmann P (1892) Vorlesungen ¨uber die Theorie der Irrationalzahlen.Leipzig[2] Bair J, Blaszczyk P, Ely R et al (2013) Is Mathematical History Written by theVictors? Notices AMS 60(7):886–904[3] Becker H P (1988) Dr B Bolzanos meßbaren Zahlen. Unpublished Diplomar-beit, Fachbereich Mathematik, Technische Hochschule, Darmstadt[4] Bolzano B (1816/2004) Binomische Lehrsatz und aus Folgerung aus ihm derpolynomische, und die Reihen, die zur Berechnung der Logarithmen und Expo-nentialgr¨oßen dienen, genauer als bisher erwiesen. C.W. Enders, Prague. Trans-lation in (Russ 2004).[5] Bolzano B (1817/2004) Rein analytischer Beweis des Lehrsatzes, daß zwis-chen je zwey Werthen, die ein entgegengesetztes Resultat gew¨ahren, wenigstenseine reelle Wurzel der Gleichung liege. Gottlieb Haase, Prague. Translation in(Russ 2004).[6] Bolzano B (1837/2014) Wissenschaftslehre. Translation in Rusnock P, Rolf G(2014) Theory of Science, Oxford University Press, Oxford[7] Bolzano B (1851/2004) Paradoxien des Unendlichen. CH Reclam, Leipzig.Translation in (Russ 2004).[8] Bolzano B (1976) Reine Zahlenlehre. In Berg J (ed) Bernard Bolzano Gesam-tausgabe Bd. 2A8, Frommann-Holzboog, Stuttgart-Bad Canstatt[9] Burn RP (1992) Irrational Numbers in English Language Textbooks, 1890-1915: Constructions and Postulates for the Completeness of the Real Numbers.Hist. Math. 19(2):158–176[10] Cauchy A (1821/2009) Cauchy’s Cours d’analyse. An Annotated Transla-tion. Tr. Bradley, RE & Sandifer, CE. Springer[11] Ide J (1803) Anfangsgr¨unde der reinen Mathematik. Heinrich Fr¨olich, Berlin
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