Where Is Mathematics? [Point of View]

  • Published on

  • View

  • Download


P O I N T O F V I E WWhere Is Mathematics?BY ROBERT S INCLAIRMathematical Biology UnitOkinawa Institute of Science and Technology Graduate University, JapanA s a mathematician, I read the Point of View article The rea-sonable ineffectiveness of mathematics [1] with great interest.While I do agree with much of what is written there, I wish tocorrect two impressions the article may have left in somereaders minds; first, that the belief that mathematics has its own existencesomehow implies the notion that mathematical forms underpin the physicaluniverse.For the record, let me state that I do believe mathematics has its ownexistence, but I do not see any compelling reason why mathematical formsshould underpin the physical universe. Put simply, this is what I mean by ownexistence. To be more specific, although I understand that there is a school ofthought, inspired by Platos allegory of the cave (Book 7 of Republic [2]), thatthe physical universe is but a shadow of a mathematical world, I do not feelbound to define my own position as either for or against it. I would not evenknow where to start: Which mathematics is the right one? Do we have differentphysical universes for different types of mathematics (with or without theAxiom of Choice, for example, [3])? Let me be clear that, in asking whichmathematics, I am not asking which branch of mathematics. I am actuallyreferring to the fact that we haveseveral different versions of mathe-matics as a whole, each one basedupon different logical foundations[4][6]. Are they really so different?Yes, and the nonstandard versions areoccasionally finding application inengineering contexts [7]. One canget a taste of what I mean by consi-dering the fact that most workingscientists and engineers are justifiablycontent to use only floating-pointarithmetic in their calculations, andyet floating-point number representa-tions are fundamentally differentfrom the real numbers of standardmathematics, starting with the factthat floating point (I am thinking ofthe IEEE 754 standard [8]) often hasseparate encodings for 0 and 0,and for good reasons [9]. Since onecould, in principle, develop a type ofmathematics upon the foundation offloating-point arithmetic [10], do Ihave to ask whether I should belooking for both 0 and 0 in thephysical world? I think not. Fur-thermore, I should clearly state thateven the most commonly acceptedversion of mathematics involves para-doxes, even of a sort that would allowone to cut up the sun and gently putthe pieces back together to makesomething the size of a pea [3], [11].I do not believe that such mathe-matics is the literal foundation of thephysical world I live in, but neitherdo I reject it as mathematics.Most pure mathematicians wouldpoint to proof as the defining charac-teristic of mathematics, and would doso without reference to anything phy-sical. I agree, but I prefer to think ofmathematics in terms of Cantorscomment that can be translated asDigital Object Identifier: 10.1109/JPROC.2013.22912912 Proceedings of the IEEE | Vol. 102, No. 1, January 2014 0018-9219 2013 IEEEthe essence of mathematics lies in itsfreedom [12], and, for me, that alsomeans freedom from having to de-fine mathematics and the physicaluniverse in terms of each other.Instead, I like to think of mathematicsand engineering or basic science asequal partners.The second impression I wouldlike to correct is that mathematicsmay be purely a product or inventionof the human mind.There are two reasons why I cannotsubscribe to such a point of view. First,we already live in an age where com-puters not only generate new mathe-matics, but also, in some cases, judgethe mathematical work of humans.Second, Homo sapiens is not the onlyspecies to have evolved and used ma-thematical descriptions of the world.Mathematics owes a great deal tothe community of electrical engineersand computer scientists, since wewould otherwise not have had reliablecomputing machines. While I haveoften heard fellow mathematiciansstate that computers could never com-plete a proof which involves an infi-nite number of cases, the real issue iswhether the number of steps in aproof, and the resources required toprocess and store them, is finite. Us-ing symbolic methods, even a com-puter can prove, in a finite number ofelementary steps, that 2 x2 is great-er than unity for all real x. There is noneed to evaluate 2 x2 for specificvalues of x at all. At a less trivial level,one can ask whether a computer hasever found a mathematical proof thathumans had failed to. The earliest ex-amples of this that I am aware of dateback to the last century (I consider aresult relating to the Robbins conjec-ture to be a particularly nice one[13]). It would be tempting to retreatto the position that humans remainthe final arbiters or custodians ofmathematical truth, but there is al-ready one journal I know of, Formal-ized Mathematics, published by Walterde Gruyter, Berlin, Germany, whichsubjects submitted articles to auto-mated proof checking [14] before anypeer review can begin.The question, whether mathemat-ics could ever have been the productof any but a human mind, depends inpart on our definition of what itmeans to understand mathematics. Itis no use shouting mathematical ques-tions at an unsuspecting slime mouldand then celebrating its failure toreply in English. Such mistaken logichas all too often been used in attemptsto demonstrate the superiority of onegroup of humans over another. In-stead, it has been suggested that re-peated correct usage is an effectiveindicator of understanding [15]. Itmay come as a surprise to some thatbees have evolved a polar representa-tion of relative position that they notonly use, but use to communicate re-lative positional information from onebee to another [16]. It is the act ofabstract communication which Ibelieve makes this example more re-levant here than the otherwise im-pressive computational/cryptographicpotential of some genetic systems [17]or the bowerbirds use of perspective[18], etc. There is ample evidencethat animals have and use abstractproblem-solving abilities [19], includ-ing game theory as a real battle ofwits, with the outcome determininglife or death [20]. Furthermore, pure-ly genetic explanations appear to beinadequate to explain the observedtransmission of tool use in dolphins[21], meaning that we must take se-riously the possibility that animalsmay be developing and communicat-ing complex skills in ways which aredifficult to distinguish from ways thathumans develop and communicatemathematics. Even the frequentlyexpressed thought, that pure mathe-matics is play, and only humans dothat, since animals only do what theyneed to survive, crumbles when con-fronted with observations of dolphinsbubble ring play [22].All this leads me to the issue of thelesser success of mathematics indescribing biological systems [1]. Inmy honest opinion, it is too early tomake strong statements concerningthe relationship between mathematicsand biology. To write that biology issomehow harder to model thanother natural sciences is to miss animportant point: The relationship be-tween mathematics and biology al-ready goes deeper than modeling.Regarding the idea of informationand its physical embodiment in DNAsequences, the fundamental theorywas formulated by Turing in hisnotion of a universal Turing machineand deployed by von Neumann in histheory of self-reproducing machines[23]. Also, as any engineer knows,biological systems seem to have muchto teach us about the principles ofcontrol theory. This is one of manyareas in which I expect new mathe-matics to eventually emerge, onceagain a reason to see the relationshipbetween mathematics and biology asmore than just modeling.We can, of course, also expect tosee other types of new engineeringprinciples emerge from careful stud-ies of living organisms. I do not wishto go into a list of specific exampleshere, but do want to make one generalcomment at a rather abstract level:Evolution teaches us that optimality isnot always the strategy that wins inthe long run [24]. There are manyreasons for this. One is that optimalityoften implies a loss of flexibility inconstruction, and both parasites andpredators usually target any aspect ofan organism that is unvarying. Anoth-er is that optimality in a given contextoften translates into weakness inanother, and vice versa. Thus, a smallfraction of bacteria may occasionally,without any apparent reason, go into adormant state, only to wake upagain later. The switching is de-termined by a reversible stochasticmechanism which operates in allnormal cells [25]. The growth of thecolony is slightly reduced as a result,but it is these few cells which cansurvive when antibiotics are adminis-tered, and their seemingly ridiculousbehavior is of great long-term benefitfor their lineage.Given the valid points raised byAbbott and others, it is useful to re-examine what it is that makes math-ematics valuable. I wish to proposePoint of ViewVol. 102, No. 1, January 2014 | Proceedings of the IEEE 3that it is precisely its othernesswhich makes mathematics, as abranch of philosophy, valuable toengineering and science, but onlywhere there is real engagement fromboth sides (I have put science andengineering on the same side here,since they both deal more or lessdirectly with the world we live in,whereas mathematics, in my opinion,does not). I am painfully aware of thefailures of parts of the mathematicalcommunity to engage, and see this asone of the major social challenges tobe overcome. I am one of the fortu-nate few who has had the experienceof having done pure mathematicalwork with no application in mind, butthen subsequently to see it being ap-plied [26]. I wish more mathema-ticians could have such experiences.This does not mean that I propose anysort of dilution of mathematics itself.The power of mathematical ap-proaches tends to be in their general-ity. The finite element method isfortunately not tied to a narrow areaof application, for example, but isuseful to many. This is because it hasnot been developed with any overlyspecific or exclusive set of problems inmind, and general mathematical prin-ciples were allowed to shape its foun-dations [27]. I suggest that we canlearn much from this example, in thesense that a combination of a genuineneed for a practical method and agenuine interest in basic mathema-tical issues, even when from a point ofview that is completely divorced fromapplications, is a winwin situation.The only requirement is a willingnessto communicate across traditionalboundaries.Engineering and the basic naturalsciences have always been sources ofinspiration for mathematicians. It isno accident that we have seen, timeand again, new mathematics (and sta-tistics) emerge exactly at the interfacebetween mathematics and a practicalneed for answers. Abbotts Point ofView [1] was an exciting read for mebecause it reminded me that engi-neers do care about the nature ofmathematics, and this can only ben-efit us all. hREF ERENCE S[1] D. Abbott, The reasonable ineffectivenessof mathematics [Point of view], Proc. IEEE,vol. 101, no. 10, pp. 21472153, Oct. 2013.[2] Plato, Republic. Oxford, U.K.: Oxford Univ.Press, 2008.[3] C. K. Gordon, The third great crisis inmathematics, IEEE Spectrum, vol. 5, no. 5,pp. 4752, May 1968.[4] G. Priest, Mathematical pluralism, LogicJ. IGPL, vol. 21, pp. 413, Feb. 2013.[5] C. W. Henson, Foundations of NonstandardAnalysis, in Nonstandard Analysis,L. Arkeryd, N. Cutland, andC. W. Henson, Eds. Amsterdam,The Netherlands: Springer-Verlag, 1997,vol. 493, pp. 149.[6] E. B. Davies, Pluralism in mathematics,Philos. Trans. R. Soc., Math. Phys. Eng. Sci.,vol. 363, pp. 24492458, Oct. 15, 2005.[7] L. Ferrucci, D. Mandrioli, A. Morzenti, andM. Rossi, Modular automated verificationof flexible manufacturing systems withmetric temporal logic and non-standardanalysis, in Formal Methods for IndustrialCritical Systems, M. L. Stoelinga andR. Pinger, Eds. Berlin, Germany:Springer-Verlag, 2012, vol. 7437,pp. 162176.[8] IEEE Standard for Floating-Point Arithmetic,IEEE Std. 754-2008, 2008, pp. 170.[9] W. Kahan, Branch cuts for complexelementary functions, or much ado aboutnothings sign bit, in The State of theArt in Numerical Analysis, A. Iserles andM. J. D. Powell, Eds. Oxford, U.K.:Oxford Univ. Press, 1987, pp. 165211.[10] W. S. Brown, A simple but realistic modelof floating-point computation, ACM Trans.Math. Softw., vol. 7, pp. 445480, 1981.[11] L. M. Wapner, The Pea and the Sun: AMathematical Paradox. Boca Raton, FL,USA: A K Peters/CRC Press, 2007.[12] G. Cantor, Ueber unendliche, linearePunktmannichfaltigkeiten, MathematischeAnnalen, vol. 21, pp. 545591, Dec. 1883.[13] W. McCune, Solution of the Robbinsproblem, J. Autom. Reason., vol. 19,pp. 263276, Dec. 1997.[14] A. Grabowski, A. Kornilowicz, andA. Naumowicz, Mizar in a nutshell,J. Formalized Reason., vol. 3, no. 2,pp. 153245, 2010.[15] L. Wittgenstein, Wittgensteins Lectures onthe Foundations of Mathematics, Cambridge,1939. Chicago, IL, USA: Univ. ChicagoPress, 1989.[16] F. C. Dyer, The biology of the dancelanguage, Annu. Rev. Entomol., vol. 47,pp. 917949, 2002.[17] L. F. Landweber and L. Kari, The evolutionof cellular computing: Natures solutionto a computational problem, Biosystems,vol. 52, pp. 313, Oct. 1999.[18] J. A. Endler, L. C. Endler, and N. R. Doerr,Great bowerbirds create theaterswith forced perspective when seen bytheir audience, Current Biol., vol. 20,pp. 16791684, Sep. 28, 2010.[19] A. H. Taylor, D. Elliffe, G. R. Hunt, andR. D. Gray, Complex cognition andbehavioural innovation in New Caledoniancrows, Proc. R. Soc. B, Biol. Sci., vol. 277,pp. 26372643, Sep. 7, 2010.[20] R. R. Jackson and R. S. Wilcox,Spider-eating spiders, Amer. Sci.vol. 86, pp. 350357, Jul.Aug. 1998.[21] M. Krutzen, J. Mann, M. R. Heithaus,R. C. Connor, L. Bejder, and W. B. Sherwin,Cultural transmission of tool use inbottlenose dolphins, Proc. Nat. Acad. Sci.USA, vol. 102, pp. 89398943, Jun. 21, 2005.[22] B. McCowan, L. Marino, E. Vance,L. Walke, and D. Reiss, Bubble ring playof bottlenose dolphins (Tursiops truncatus):Implications for cognition, J. Comparat.Psychol., vol. 114, pp. 98106, Mar. 2000.[23] S. Brenner, The revolution in the lifesciences, Science, vol. 338, pp. 14271428,Dec. 14, 2012.[24] K. Parvinen and U. Dieckmann,Self-extinction through optimizingselection, J. Theor. Biol., vol. 333, pp. 19,2013.[25] E. Maisonneuve, M. Castro-Camargo, andK. Gerdes, (P)ppGpp controls bacterialpersistence by stochastic induction oftoxin-antitoxin activity, Cell, vol. 154,pp. 11401150, 2013.[26] J. Osborne, G. Hicks, and R. Fuentes,Global analysis of the double-gimbalmechanismVDynamics and control onthe torus, IEEE Control Syst. Mag., vol. 28,no. 4, pp. 4464, Aug. 2008.[27] G. Strang, Piecewise polynomials and thefinite-element method, Bull. Amer. Math.Soc., vol. 79, pp. 11281137, 1973.Point of View4 Proceedings of the IEEE | Vol. 102, No. 1, January 2014 /ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages false /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice


View more >