{"id":3050,"date":"2020-02-26T11:56:46","date_gmt":"2020-02-26T16:56:46","guid":{"rendered":"https:\/\/notes.math.ca\/?post_type=article&p=3050"},"modified":"2020-03-17T12:56:55","modified_gmt":"2020-03-17T16:56:55","slug":"the-two-cultures-of-mathematics","status":"publish","type":"article","link":"https:\/\/notes.math.ca\/en\/article\/the-two-cultures-of-mathematics\/","title":{"rendered":"The Two Cultures of Mathematics"},"content":{"rendered":"\t\t
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CSHPM Notes bring scholarly work on the history and philosophy of mathematics to the broader mathematics community. Authors are members of the Canadian Society for History and Philosophy of Mathematics (CSHPM). Comments and suggestions are welcome; they may be directed to either of the column’s co-editors:<\/em><\/p>

Amy Ackerberg-Hastings<\/strong>,\u00a0Independent Scholar (aackerbe@verizon.net)<\/em>
Hardy Grant,\u00a0<\/em><\/strong>York University [retired] (hardygrant@yahoo.com)<\/em><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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Like the two cultures of the sciences and the humanities, as lamented by C. P. Snow in his influential and controversial Rede Lecture [4], there are two cultures within the mathematics community itself. Snow used specific examples of cultural clashes to illuminate his argument, and similarly here I will attempt to shed some light on the different cultures of pure and applied mathematics by recounting a 1950\u201351 conflict involving Harvard mathematician Garrett Birkhoff and J. J. Stoker, one of the founders of the Courant Institute of Mathematical Sciences.<\/span><\/p>

\"Image
Garrett Birkhoff, MacTutor<\/i>.<\/span><\/figcaption><\/figure>

Garrett Birkhoff, as most readers will know, co-authored (with Saunders Mac Lane) the classic text, A Survey of Modern Algebra<\/em>. However, during and after the Second World War, he began working in more applied areas. I first encountered his monograph on fluid dynamics [1] when I was exploring Euler\u2019s role in the history of d\u2019Alembert\u2019s paradox [2]. As an applied mathematician who had done research in fluid flow (and later developed an interest in the history of mechanics), I was struck by the fresh approach and clarity of Birkhoff\u2019s writing on the subject. The entire first chapter [1, pp. 3\u201339] of his monograph is devoted to paradoxes of fluid flow; d\u2019Alembert\u2019s paradox is the first discussed [1, pp. 10\u201313].<\/span><\/p>

As Birkhoff explained, this paradox involves the steady, uniform flow of a non-viscous, incompressible fluid (often called an ideal fluid) past a smooth, finite body (such as a sphere). Flows of this kind can be described by the gradient of a potential function, which implies the flow exerts no drag on the body, a result contradicted by the physical fact of substantial drag exerted by actual flows. For Birkhoff, d\u2019Alembert\u2019s paradox and others are \u201cin part at least, paradoxes of topological oversimplification and symmetry paradoxes\u201d [1, p. 22]. Euler\u2019s resolution of d\u2019Alembert\u2019s paradox challenged the assumption of an incompressible fluid (especially for a ball shot through the air) [2]; two other plausible resolutions are considered below.<\/span><\/p>

One topological oversimplification associated with \u201cthe hypothesis of an \u2018ideal fluid\u2019\u201d is \u201cthat a locally single-valued velocity potential U<\/em> is single-valued in the large\u201d for two-dimensional flow past a body. To avoid symmetry paradoxes, Birkhoff advised the reader to \u201cadmit the possibility that a symmetrically stated problem may not have any stable symmetric solution<\/em>\u201d; in the case of the uniform flow of an ideal fluid past a sphere, for example, a steady, axially-symmetric solution exists mathematically, but \u201cthere is no reason to suppose that any steady flow is stable<\/em>.\u201d An instability makes a steady, mathematical flow physically unrealizable, and \u201cirregular, turbulent \u2018eddies\u2019 … in the \u2018wake\u2019 of an obstacle\u201d might therefore occur in actual flows [1, pp. 20\u201321; italics are Birkhoff’s]. An instability of this kind might thus resolve d\u2019Alembert\u2019s paradox.<\/span><\/p>

Birkhoff opined that theories of fluid dynamics can be learned \u201cmore effectively … by studying the paradoxes\u201d he described. He criticized textbooks that attributed the gap between theory and experiment to the difference between real fluids with \u201csmall but finite viscosity\u201d and ideal fluids of \u201czero viscosity,\u201d and he thought \u201cthat to attribute them all [the paradoxes he describes] to the neglect of viscosity is an unwarranted oversimplification\u201d\u2014the \u201croot lies deeper, in lack of precisely that deductive rigor whose importance is so commonly minimized by physicists and engineers\u201d [1, pp. 3\u20134]. The paradoxes warn against \u201cthe impression … that mathematical deduction should be supplanted by \u2018physical\u2019 reasoning,” which can lead to flawed approximations and oversimplifications, though Birkhoff admitted the usefulness of \u201coversimplifications based on the \u2018right\u2019 approximations.” He continued, \u201cmathematicians can perform a useful service if they will analyze critically these oversimplifications, by the deductive method, and so establish their limitations more clearly\u201d [1, p. 37]. Birkhoff offered his paradoxes to a subject that is primarily the domain of engineers and applied mathematicians (like me, in my past life as a fluid dynamics specialist). His criticisms were severe and, in fact, he named J. J. Stoker as one who \u201ceffectively exploited\u201d an \u201canalogy\u201d between two kinds of waves, even though another paradox (not d\u2019Alembert\u2019s) made one kind \u201cmathematically impossible<\/em>\u201d [1, pp. 22\u201324; italics are Birkhoff\u2019s].<\/span><\/p>

\"Image
J. J. Stoker, ca 1960, Courant Institute<\/span><\/figcaption><\/figure>

As life and luck would have it, J. J. Stoker wrote a review [5] of Birkhoff\u2019s 1950 monograph. Stoker\u2019s assessment of Birkhoff\u2019s Chapters 2 through 5 was balanced, even complimentary in the cases of Chapter 2 (on problems with free boundaries) and Chapter 3 (on modelling and dimensional analysis). Stoker\u2019s review of Chapter 1, however, was withering. He found \u201cit difficult to understand for what class of readers the first chapter was written\u201d; indicated that \u201cthe majority of cases cited as paradoxes\u201d were either \u201cmistakes long since rectified\u201d or \u201cdiscrepancies between theory and experiment the reasons for which are also well understood\u201d; and worried that \u201cthe uninitiated would be very likely to get wrong ideas about some of the important and useful achievements in hydrodynamics from reading this chapter.\u201d Referring to “some general observations regarding the philosophy and correct attitude toward applied mathematics\u201d made by Birkhoff, Stoker allowed that most \u201cworkers in the field would agree quite well with the author\u2019s observations,\u201d but he thought that \u201cthey are perhaps better informed in some cases than the author would seem to imply” [5, pp. 497\u2013498].<\/span>\u00a0<\/span><\/p>

To illustrate this last point, Stoker offered salient mathematical reasoning underlying the generally accepted resolution of d\u2019Alembert\u2019s paradox: “the small coefficients involving viscosity occur in terms containing derivatives of the highest order in the system of differential equations, and thus developments in the neighborhood of zero viscosity involve boundary layer effects because of the loss of order of the differential equations in the limit.”\u00a0Stoker was referring to the fact that the Navier-Stokes equations, which describe viscous, incompressible flow past a body, are second-order partial differential equations that permit the so-called no-slip condition (that the fluid\u2019s velocity vanish on the body\u2019s boundary) to be satisfied; in the case of zero viscosity, the Navier-Stokes equations become Euler\u2019s equations, which are first-order and allow the flow to be described by a potential function but permit the vanishing of only the flow velocity normal to the body\u2019s boundary; the transition from viscous flow (with the no-slip condition on the boundary) to non-viscous flow (with the corresponding loss of second-order terms) farther from the boundary occurs in what is known as a boundary layer, in which the fluid\u2019s velocity is approximated mathematically using matched asymptotic expansions (or computationally using a very fine mesh).<\/p>

Birkhoff had anticipated this persuasive argument from applied mathematics \u201cin support of the view that the paradoxes of fluid mechanics are due to an unjustified neglect of viscosity.\u201d He conceded that the argument had \u201csome merit\u201d but thought that it was \u201cinconclusive.\u201d For him, \u201cthe real question is, why does separation of the boundary layer occur?\u201d This question alluded to observations that the boundary layer adjacent to a body immersed in a flow often separates from that body (downstream from where it begins) to become the border of a turbulent wake behind the body. Birkhoff believed this question \u201cconcerns the stability<\/em> of nearly non-viscous flows\u201d [1, p. 27; italics are Birkhoff’s]. It seems that Stoker\u2019s argument was inconclusive for Birkhoff because it failed to rule out, deductively, the stability question raised by d\u2019Alembert\u2019s paradox and others.<\/span><\/p>

Birkhoff, as one steeped in the culture of pure mathematics, saw his paradoxes as guidelines to sharpen the deductive skills of fluid dynamics researchers. Stoker rejected the applicability of those guidelines. In the second (1960) edition of Birkhoff\u2019s monograph, the first chapter grew to two chapters; these chapters doubled down on paradoxes (one covered those of non-viscous flow, the other viscous flow), but Birkhoff\u2019s earlier criticisms of physicists, engineers, their lack of deductive rigor, and J. J. Stoker were removed. Further, Birkhoff made no suggestion that an instability in the flow of an ideal fluid might resolve d\u2019Alembert\u2019s paradox. I prefer to believe that these omissions are evidence of Birkhoff\u2019s attempt to reconcile with those working within the culture of applied mathematics. Ironically, however, recent (not yet mainstream) research [3] indicates that Birkhoff\u2019s suggested resolution to d\u2019Alembert\u2019s paradox might be the best one.<\/span><\/p>

William Hackborn (who prefers to be addressed simply as “Bill”) is Professor of Mathematics and Computing Science at the Augustana (Camrose) Campus of the University of Alberta. He has dabbled in various mathematical areas over the years, including fluid dynamics, dynamical systems, mathematical biology, and most recently the history of mathematics and physics. Bill anticipates, with mixed emotions, his upcoming retirement on 30 June 2021<\/em>.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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References<\/strong><\/p>

[1] Birkhoff, G. (1950) Hydrodynamics: A Study in Logic, Fact, and Similitude. Princeton University Press.<\/span><\/p>

[2] Hackborn, W. W. (2018) Euler\u2019s Discovery and Resolution of D\u2019Alembert\u2019s Paradox. In M. Zack and D. Schlimm (eds), Research in History and Philosophy of Mathematics, 43\u201357. Proceedings of the Canadian Society for History and Philosophy of Mathematics. Birkh\u00e4user. doi.org\/10.1007\/978-3-319-90983-7_3<\/a>.<\/span><\/p>

[3] Hoffman, J. and C. Johnson. (2010) Resolution of d\u2019Alembert\u2019s Paradox. J. Math. Fluid Mech.12, 321\u2013334. doi.org\/10.1007\/s00021-008-0290-1<\/a>.<\/span><\/p>

[4] Snow, C. P. and S. Collini. (1993) The Rede Lecture (1959). In The Two Cultures, 1\u201352. Cambridge University Press. doi.org\/10.1017\/CBO9780511819940.002<\/a>.<\/span><\/p>

[5] Stoker, J. J. (1951) Review of Hydrodynamics: A Study in Logic, Fact, and Similitude, by Garrett Birkhoff. Bull. Amer. Math. Soc. 57, 497\u2013499. doi.org\/10.1090\/S0002-9904-1951-09552-X<\/a>.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"author":6,"template":"","section":[58],"keyword":[127,128,126],"class_list":["post-3050","article","type-article","status-publish","hentry","section-cshpm-notes","keyword-applied-mathematics","keyword-fluid-dynamics","keyword-pure-mathematics"],"toolset-meta":{"author-4-info":{"author-4-surname":{"type":"textfield","raw":""},"author-4-given-names":{"type":"textfield","raw":""},"author-4-honorific":{"type":"textfield","raw":""},"author-4-institution":{"type":"textfield","raw":""},"author-4-email":{"type":"email","raw":""},"author-4-cms-role":{"type":"textfield","raw":""}},"author-3-info":{"author-3-surname":{"type":"textfield","raw":""},"author-3-given-names":{"type":"textfield","raw":""},"author-3-honorific":{"type":"textfield","raw":""},"author-3-institution":{"type":"textfield","raw":""},"author-3-email":{"type":"email","raw":""},"author-3-cms-role":{"type":"textfield","raw":""}},"author-2-info":{"author-2-surname":{"type":"textfield","raw":""},"author-2-given-names":{"type":"textfield","raw":""},"author-2-honorific":{"type":"textfield","raw":""},"author-2-institution":{"type":"textfield","raw":""},"author-2-email":{"type":"email","raw":""},"author-2-cms-role":{"type":"textfield","raw":""}},"author-info":{"author-surname":{"type":"textfield","raw":"Hackborn"},"author-given-names":{"type":"textfield","raw":"William"},"author-honorific":{"type":"textfield","raw":""},"author-email":{"type":"email","raw":"hackborn@ualberta.ca"},"author-institution":{"type":"textfield","raw":"University of Alberta"},"author-cms-role":{"type":"textfield","raw":""}},"unknown":{"downloadable-pdf":{"type":"file","raw":"https:\/\/notes.math.ca\/wp-content\/uploads\/2020\/02\/The-Two-Cultures-of-Mathematics-CMS-Notes-3.pdf","attachment_id":3904},"article-toc-weight":{"type":"numeric","raw":"50"},"author-surname":{"type":"textfield","raw":"Hackborn"},"author-given-names":{"type":"textfield","raw":"William"}}},"_links":{"self":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/3050","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article"}],"about":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/types\/article"}],"author":[{"embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":44,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/3050\/revisions"}],"predecessor-version":[{"id":3895,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/3050\/revisions\/3895"}],"wp:attachment":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/media?parent=3050"}],"wp:term":[{"taxonomy":"section","embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/section?post=3050"},{"taxonomy":"keyword","embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/keyword?post=3050"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}