A footnote to The crisis in contemporary mathematics
aa r X i v : . [ m a t h . HO ] A p r A FOOTNOTE TO THE CRISIS IN CONTEMPORARYMATHEMATICS
BORIS KATZ, MIKHAIL G. KATZ, AND SAM SANDERS
Abstract.
We examine the preparation and context of the pa-per “The Crisis in Contemporary Mathematics” by Errett Bishop,published 1975 in
Historia Mathematica . Bishop tried to moder-ate the differences between Hilbert and Brouwer with respect tothe interpretation of logical connectives and quantifiers. He alsocommented on Robinson’s
Non-standard Analysis , fearing that itmight lead to what he referred to as ‘a debasement of meaning.’The ‘debasement’ comment can already be found in a draft versionof Bishop’s lecture, but not in the audio file of the actual lectureof 1974. We elucidate the context of the ‘debasement’ commentand its relation to Bishop’s position vis-a-vis the Law of ExcludedMiddle.Keywords: Constructive mathematics; Robinson’s framework;infinitesimal analysis. Introduction
We will compare three extant versions of Errett Bishop’s 1974 lec-ture entitled “The crisis in contemporary mathematics.” Errett Bishop(1928–1983) delivered a plenary lecture in the session on the Founda-tions of Mathematics of the Workshop on the Evolution of ModernMathematics organized by the American Academy of Arts and Sci-ences (AAAS) in 1974.Three versions of the lecture are extant. The first one is a 2-pageinitial draft of the lecture [Bishop 1974a]. The second is an audiorecording [Bishop 1974b] of the lecture delivered on 9 august 1974. Thethird is the published version of the lecture in
Historia Mathematica [Bishop 1975]. 2.
The three versions
Mathematics Subject Classification.
Primary 01A60; Secondary 26E35,03B20, 03F60.
The draft version.
The draft of the lecture already sets out itsmain theme, namely the Brouwer–Hilbert differences on the meaningof logical connectives and quantifiers:What I am recommending, and I do not know whetherthe possibility occured [sic] to Hilbert, is that we acceptBrouwer’s definitions of “or”, “there exists”, and all theother connectives and quantifiers, without damaging theparadise that Hilbert wished to preserve. [Bishop 1974a,p. 1]There follows a paragraph concerning the work of Abraham Robin-son [Robinson 1966] on infinitesimal analysis and of H. Jerome Keisler[Keisler 1971] on infinitesimal calculus:A more recent attempt at mathematics by formal finesseis non-standard analysis. I gather that it has met withsome degree of success, whether at the expense of givingsignificantly less meaningful proofs I do not know. Myinterest in non-standard analysis is that attempts arebeing made to introduce it into calculus courses. It isdifficult to believe that debasement of meaning could becarried so far. [Bishop 1974a, p. 2].Bishop goes on to discuss recursive function theory, and then commentson applications in science:The reason that mathematics is so successful in thephysical sciences is not clear. To Hermann Weyl, theutility of mathematics extended even to that part ofmathematics that was not inherently computational. Al-though I hesitate to disagree with such an authority, myown impression is that the opposite is true. It would beinteresting and worthwhile to settle this point.[Bishop 1974a, p. 2].2.2. The published version.
Bishop’s lecture was published in
His-toria Mathematica in 1975 as part of the proceedings of the AAASworkshop. The 11-page published version [Bishop 1975] contains anexpanded discussion of Brouwer–Hilbert disagreements over connec-tives and quantifiers, followed by a constructive analysis of the classi-cal result that a function of bounded variation is differentiable almosteverywhere. Bishop’s conclusion, echoing the corresponding remarksin the draft version, is the following: For a historical analysis of the genesis of Robinson’s theory see [Dauben 1995].
FOOTNOTE TO THE CRISIS IN CONTEMPORARY MATHEMATICS 3
In a way, the imaginary dialogue that I presented heremight be regarded as a historical investigation if youbelieve as I do that it shows how two titanic figuressuch as these might have reached an accommodationthat would have changed the course of mathematics ina profound way, had they spoken to each other withless emotion and more concern for understanding eachother. Instead, Hilbert tried to show that it was all rightto neglect computational meaning, because it could ul-timately be recovered by an elaborate formal analysis ofthe techniques of proof. This artificial program failed. [Bishop 1975, p. 513].Immediately following, in the published version, is the ‘debasement’passage on Robinson and Keisler already found in the draft version(see Section 2.1). Bishop goes on to make some comments on recur-sive function theory identical to those found in the draft version, andconcludes:That is all I want to say about pure mathematics. Iwould like to consider next another very interesting ques-tion that has occupied many people: what does theconstructivist point of view entail for the applicationsof mathematics to physics? My own feeling is that theonly reason mathematics is applicable is because of itsinherent constructive content. By making that construc-tive content explicit, you can only make mathematicsmore applicable, Hermann Weyl seems to have had anopposite opinion. For him, the utility of mathematicsextended even to that part of mathematics that was notinherently computational. I hesitate to disagree withWeyl, but I do. It is a very serious subject for investi-gation; it would be interesting and worthwhile to settlethis point. [Bishop 1975, p. 514]Following the published version of Bishop’s lecture in [Bishop 1975] isan extended exchange among mathematicians and philosophers presentat the workshop, containing a number of reactions to Bishop’s ideas by Bishop’s negative appraisal notwithstanding, the proof mining program spear-headed by Ulrich Kohlenbach (see e.g., [Kohlenbach 2008]) has been successful atextracting computational content from proofs in classical mathematics, going as faras improving known results in the literature (classical or constructive). The proofmining program is the archetype of an “elaborate formal analysis of the techniquesof proof” which has successfully produced a plethora of natural results.
B. KATZ, M. KATZ, AND S. SANDERS
Moore, Kline, Mackey, Birkhoff, Freudenthal, Dieudonn´e, Abhyankar,Kahane, and Dreben. Their responses don’t include any reaction toBishop’s ‘debasement’ comments on Robinson and Keisler and it willsoon become clear why.2.3.
The audio version.
The recorded lecture lasted 47 minutes (in-cluding an introduction by Birkhoff; not including responses from theaudience). At minutes 43 and 44 one finds the following comments byBishop:In a way the imaginary dialog I presented here mightbe regarded as a historical investigation if you believeas I do that it shows how these titanic figures may havereached an accommodation that would have changed thecourse of mathematics in a somewhat profound way ifthey had spoken to each other at a less emotional level.Now that’s all I want to say about pure mathematics. Iwant to read a little bit if I still have it. I don’t haveit, well, I’ll try to remember it then. [Bishop 1974b,minute 43]Note that Bishop first declared that “that’s all I want to say about puremathematics” and then searched unsuccessfully for his notes to presentthe next segment of his lecture. This indicates that the misplacednotes did not deal with pure mathematics but rather with applicationsto physics, which as he said he delivered from memory.
Immediatelyfollowing the above in the audio version is the following comment:One very interesting question. . . is what this wouldmean, what this would entail for the applications ofmathematics to physics. . . My own feeling is that theonly reason mathematics is applicable is because of itsinherent constructive content. And by making that con-structive context explicit we can only make mathematicsmore applicable. [Bishop 1974b, minute 44]The passage at minute 43 closely parallels the discussion of the Brouwer–Hilbert differences found in both [Bishop 1974a] and [Bishop 1975].The passage at minute 44 closely parallels the discussion of applica-tions in both [Bishop 1974a] and [Bishop 1975].However, the intermediate ‘debasement’ passage that appears inidentical form in the preliminary draft version [Bishop 1974a] and thepublished version [Bishop 1975] is absent from the audio recording ofthe lecture. Thus, Bishop never made those comments in the actual
FOOTNOTE TO THE CRISIS IN CONTEMPORARY MATHEMATICS 5 lecture, though he was apparently planning to present them accordingto [Bishop 1974a], and ultimately did publish them in [Bishop 1975].One can only speculate concerning the reasons that may have ledBishop to suppress the ‘debasement’ comment when faced with an ac-tual audience on 9 august 1974, or what he meant exactly when hedeclared, at the exact spot of the omission, that “that’s all I wantto say about pure mathematics.” However, a reader of the publishedversion who may have been surprised or disappointed not to find anyreaction to the ‘debasement’ comment on the part of the audience thatincluded a number of logicians (see end of Section 2.2), will now havean explanation for their silence.3.
Reception and meaning
The reception of Bishop’s program among mainstream mathemati-cians has ranged from lukewarm to sceptical. Thus, Jean Dieudonn´ewrote: L’Auteur expose une d´efense des points de vue de L. E.J. Brouwer (dont il se d´eclare le disciple) sur la “signi-fication” des math´ematiques et les tabous qui en r´esul-teraient si ces points de vue ´etaient adopt´es. Le termede “crise” qu’il emploie ne semble gu`ere justifi´e, car untel mot d´esigne un conflit ayant un certain caract`ered’acuit´e, et l’Auteur reconnaˆıt que pour la tr`es grandemajorit´e des math´ematiciens les questions soulev´ees parBrouwer n’ont pas d’int´erˆet. [Dieudonn´e 1975]A small group of followers has successfully pursued Bishop’s program ofwhat he defined as constructive mathematics. The most notable of hisdisciples are Douglas Bridges and Fred Richman who have publishedwidely in constructive mathematics; see e.g., [Bridges–Richman 1987].There have also been attempts to bridge a perceived gap between con-structive mathematics and Robinson’s framework for infinitesimals; seee.g., [Schuster et al. 2001].What Bishop meant by ‘meaning’ is well known: meaningful math-ematics has computational content. Namely, according to Bishop tosay a mathematical object exists, is to provide a construction for it. Translation: “The Author presents a defense of the viewpoints of L. E. J. Brouwer(of whom he professes to be the disciple) on the ‘meaning’ of mathematics and thetaboos that would result should these viewpoints be adopted. The term ‘crisis’ heemploys hardly seems justified since such a word refers to a conflict possessing acertain acuteness, and the Author acknowledges that for the very great majority ofmathematicians the issues raised by Brouwer have no interest.”
B. KATZ, M. KATZ, AND S. SANDERS
The other logical symbols have similar computational interpretations,i.e., essentially as in intuitionistic logic as pioneered by Brouwer andHeyting; see [Heyting 1956]. In particular Bishop’s program called forredevelopment of mathematics based on a computational interpretationin which the integers constitute the foundation of everything: Everything attaches itself to number , and every mathe-matical statement ultimately expresses the fact that ifwe perform certain computations within the set of pos-itive integers , we shall get certain results. (A construc-tivist manifesto in [Bishop 1967], emphasis ours)According to Bishop’s reading, the Law of Excluded Middle (LEM) isthen meaningless as it does not have any computational content, andis therefore rejected. LEM is the crucial ingredient in a typical proofby contradiction. Such proofs are ubiquitous in modern mathematicsbased on classical logic. Many mathematicians do recognize that aproof by contradiction sometimes lacks to deliver constructive contentin the sense that the entity whose existence is proved in this way oftenlacks explicit description. Classically trained mathematicians can alsorelate sympathetically to the normative sentiment that an existenceproof is a construction, not the impossibility of non-existence. In this light, Bishop’s use of the phrase “debasement of meaning”should be interpreted as referring to an allegedly fundamental and ir-reparable absence of numerical content, the latter sketched in the abovequote.It seems very difficult to construct e.g., infinitesimals in terms ofordinary integers (but see [Borovik et al. 2012]), i.e., it seems the for-mer cannot be reduced to the latter in any way acceptable to Bishop.Hence, the very core of Robinson’s framework for infinitesimal analysisdeals with objects (seemingly) unacceptable to Bishop, which is whatpresumably led him to the ‘debasement’ comment. Bishop recognized Brouwer’s criticism of classical mathematics, but ultimatelydeemed intuitionism to be an unsatisfactory answer. Bishop felt the same wayabout (constructive) recursive mathematics. Bishop actually formulated an in-formal framework which produces results acceptable in intuitionism, constructiverecursive mathematics, and classical mathematics. The idea of LEM as computationally meaningless is analyzed in [Katz et al. 2014]in the context of a (weak) Brouwerian counterexample to the Extreme Value The-orem. However, Erik Palmgren and others have established constructive NSA; see e.g.,[Palmgren 1998].
FOOTNOTE TO THE CRISIS IN CONTEMPORARY MATHEMATICS 7
Thus, the thrust of Bishop’s critique of Robinson’s framework con-sisted in alleging that the presence of ideal objects (in particular in-finitesimals) in Robinson’s framework entails the absence of meaning(i.e., computational content). Similar sentiments have been expressedby Alain Connes in [Connes et al. 2001]. Recently the Bishop–Connes critique has been challenged. Namely,the presence of ideal objects (in particular infinitesimals) in Robinson’sframework arguably yields the ubiquitous presence of computationalcontent; see e.g., [Sanders 2017], [Sanders 2018].For instance, the various nonstandard definitions (involving the re-lation ≈ of infinite proximity) of continuity, differentiability, Riemannintegration, etc., are actually stand-ins for Bishop’s constructive defini-tions involving moduli. More generally, in Robinson’s classical frame-work for infinitesimal analysis, the quantifier “there exists a standardobject” has a meaning akin to Bishop’s constructive existential quan-tifier, while “there exists an object” has no computational content. Acknowledgments
We are grateful to Michele M. Lavoie, archivist with the AmericanAcademy of Arts and Sciences, for providing access to the two items[Bishop 1974a] and [Bishop 1974b], and granting permission to repro-duce the passages cited in Section 2.3.
References [Bishop 1967] Bishop, E., 1967. Foundations of constructive analysis. McGraw–HillBook, New York–Toronto, Ont.–London.[Bishop 1974a] Bishop, E., 1974a. The crisis in contemporary mathematics(2 pages). Preliminary version of lecture at Workshop on the Evolution ofModern Mathematics. Archives of the American Academy of Arts and Sci-ences, Record Group XXI: Academy Programs and Projects, 1780-present.[Bishop 1974b] Bishop, E., 1974b. The crisis in contemporary mathematics. Audiorecording of lecture delivered on 9 august 1974. Archives of the AmericanAcademy of Arts and Sciences, Record Group XXI: Academy Programs andProjects, 1780–present.[Bishop 1975] Bishop, E., 1975. The crisis in contemporary mathematics(11 pages). Proceedings of the American Academy Workshop on the Evolu-tion of Modern Mathematics (Boston, Mass., 1974). Historia Math. , no. 4,507–517. Connes’s critique is analyzed in [Kanovei et al. 2013] and [Katz–Leichtnam 2013].Further analysis of the Bishop–Connes critique may be found in [Katz–Katz 2011]and [Kanovei et al. 2015].
B. KATZ, M. KATZ, AND S. SANDERS [Borovik et al. 2012] Borovik, A., Jin, R., Katz, M., 2012. An integer constructionof infinitesimals: Toward a theory of Eudoxus hyperreals. Notre Dame Jour-nal of Formal Logic , no. 4, 557–570. See http://dx.doi.org/10.1215/00294527-1722755 and https://arxiv.org/abs/1210.7475 [Bridges–Richman 1987] Bridges, D., Richman, F., 1987. Varieties of constructivemathematics. London Mathematical Society Lecture Note Series, 97. Cam-bridge University Press, Cambridge.[Connes et al. 2001] Connes, A., Lichnerowicz, A., & Sch¨utzenberger, M., 2001.Triangle of thoughts. Translated from the 2000 French original by JenniferGage. American Mathematical Society, Providence, RI.[Dauben 1995] Dauben, J., 1995. Abraham Robinson. The creation of nonstandardanalysis. A personal and mathematical odyssey. With a foreword by BenoitB. Mandelbrot. Princeton University Press, Princeton, NJ.[Dieudonn´e 1975] Dieudonn´e, J., 1975. Review of Bishop [Bishop 1975] for Zentral-blatt. See https://zbmath.org/?q=an:0361.02001 [Heyting 1956] Heyting, A., 1956. Intuitionism. An Introduction. North–HollandPublishing, Amsterdam.[Kanovei et al. 2015] Kanovei, V., Katz, K., Katz, M., Schaps, M., 2015. Proofsand retributions, or: why sarah can’t take limits. Foundations of Science (2015), no. 1, 1–25. See http://dx.doi.org/10.1007/s10699-013-9340-0 [Kanovei et al. 2013] Kanovei, V., Katz, M., Mormann, T., 2013. Tools, objects,and chimeras: Connes on the role of hyperreals in mathematics. Foun-dations of Science , no. 2, 259–296. See http://dx.doi.org/10.1007/s10699-012-9316-5 and https://arxiv.org/abs/1211.0244 [Katz–Katz 2011] Katz, K., Katz, M., 2011. Meaning in classical mathematics: Isit at odds with intuitionism? Intellectica no. 2, 223–302. See https://arxiv.org/abs/1110.5456 [Katz et al. 2014] Katz, K., Katz, M., Kudryk, T., 2014. Toward a clarity of theextreme value theorem. Logica Universalis , no. 2, 193–214. See http://dx.doi.org/10.1007/s11787-014-0102-8 and https://arxiv.org/abs/1404.5658 [Katz–Leichtnam 2013] Katz, M., Leichtnam, E., 2013. Commuting and non-commuting infinitesimals. American Mathematical Monthly , no. 7,631–641. See http://dx.doi.org/10.4169/amer.math.monthly.120.07.631 and https://arxiv.org/abs/1304.0583 [Keisler 1971] Keisler, H. J., 1971. Elementary Calculus: An Approach Using In-finitesimals. Prindle, Weber & Schmidt, Boston.[Kohlenbach 2008] Kohlenbach, U., 2008. Applied proof theory: proof interpreta-tions and their use in mathematics. Springer Monographs in Mathematics.Springer-Verlag, Berlin.[Palmgren 1998] Palmgren, E., 1998. Developments in constructive nonstandardanalysis. Bull. Symbolic Logic , no. 3, 233–272.[Robinson 1966] Robinson, A., 1966. Non-standard Analysis. North–Holland Pub-lishing, Amsterdam.[Sanders 2017] Sanders, S., 2017. Reverse Formalism 16. Synthese. See http://dx.doi.org/10.1007/s11229-017-1322-2 and https://arxiv.org/abs/1701.05066 FOOTNOTE TO THE CRISIS IN CONTEMPORARY MATHEMATICS 9 [Sanders 2018] Sanders, S., 2018. To be or not to be constructive, That is not thequestion. Indag. Math. (N.S.) , no. 1, 313–381.[Schuster et al. 2001] Schuster, P., Berger, U., Osswald, H., Eds., 2001. Reunit-ing the antipodes–constructive and nonstandard views of the continuum.Proceedings of the symposium held in Venice, May 16–22, 1999. SyntheseLibrary, 306. Kluwer Academic Publishers, Dordrecht. Boris Katz is a Principal Research Scientist at MIT CSAIL anda founding member of the Center for Brains, Minds, and Machines,where he serves as a co-leader of the Vision and Language Thrust. Hisresearch interests encompass natural language understanding, knowl-edge representation, and integration of language, vision, and robot-ics. His recent publications include the following article: Katz, B;Borchardt, G; Felshin, S; Mora, F. A Natural Language Interface forMobile Devices. In K. Norman and J. Kirakowski (Eds.), The WileyHandbook of Human Computer Interaction, John Wiley & Sons, 2018.
Mikhail G. Katz (BA Harvard ’80; PhD Columbia ’84) is Profes-sor of Mathematics at Bar Ilan University, Ramat Gan, Israel. He isinterested in Riemannian geometry, differential geometry and topol-ogy, and history of infinitesimals. His monograph Systolic Geometryand Topology was published by the American Mathematical Society in2007. His recent publications include the following article: Nowik, T;Katz, M. Differential geometry via infinitesimal displacements, Jour-nal of Logic and Analysis :5 (2015), 1–44. Available at Sam Sanders studied mathematics at Ghent University (Belgium)and subsequently obtained a doctorate in pure mathematics there un-der the supervision of Chris Impens and Andreas Weiermann. Hisresearch interests are in the (new) connections between computabilitytheory, reverse mathematics, nonstandard analysis, and formal seman-tics. He is currently a member of the Center for Advanced Studies atLMU Munich (Germany). As part of joint work with Dag Normann, new frontiers in mathematical logic are explored in: https://arxiv.org/abs/1711.08939
B. Katz, Computer Science and Artificial Intelligence Laboratory,MIT, Cambridge MA USA
E-mail address : [email protected] M. Katz, Department of Mathematics, Bar Ilan University, RamatGan 5290002 Israel
E-mail address : [email protected] S. Sanders, Center for Advanced Studies, LMU Munich Seestraße13, 80802 M¨unchen, Germany
E-mail address ::