Zambezia( 1988), XV (ii).THE THRUST OF MATHEMATICS IN DEVELOPMENT:TRENDS AND PROSPECTS*H. A. M. DZINOTYIWEYIDepartment of Mathematics, University of ZimbabweTHROUGHOUT HISTORY, AS mankind grappled with nature in the process ofdevelopment, many new disciplines emerged and greater interdependence amongall activities of developmental significance ensued. As far as we can see, this trendwill continue Š and continue at an increasing pace Š possibly indefinitely. Inmost of these developments the use of mathematics is both apparent andincreasingly necessary for their advancement. In fact, today it is correct to say thatin every activity Š and I must emphasize activity in whatever form Šmathematics is applicable. In support of this assertion let me cite two cases thatare easy to comprehend and have been encountered by most people in one formor other before.Firstly, there is the concept of 'force' which can be mathematicallycharacterized by an equation involving a general function (i.e. a differentialequation). For a specific phenomenon the general function can be explicitlyderived from the conditions peculiar to that phenomenon and thus the equationwould then quantify the force involved. Turning to real life, we know that someform offeree exists in every activity: it may be air pressure; blood pressure; electriccurrent; heat intensity; effect of light or rain on plant growth; the force of inflationon consumer expenditure or on the exchange rate; the force of the public sectorborrowing requirement on interest rates; the force of rural developmentprogrammes on government expenditure Š it may even be the force of deatharising from the disease AIDS. Each of these has a corresponding mathematicalformulation that is a special case of the general equation of force. To put this inother words, one may say that the general equation offeree is a dynamic modelwhose parameters can be found for specific occurrences. The parameters woulddepend on the history of, and other factors contributing to, the state of theoccurrence. In particular, the historical trends of the parameters would constitutea basis for predicting their future form. Let me also add that in some cases suchparameters can be difficult to find, largely because of current human inability tocomprehend fully the factors involved. However, even in such cases, approximatevalues for the parameters can often be found and the equation can be employedfor general guidance.I now turn to a second case illustrating why mathematics is penetrating*An inaugural lecture delivered before the University of Zimbabwe on 1 October 1987.105106 THE THRUST OF MATHEMATICS IN DEVELOPMENTvirtually all disciplines and is closely linked with their development. In appraisingmost activities,'one observes, sometimes collecting a sub-sample of the result andtesting it or performing some other simulation of the experience. Thus empiricalstudies are widely employed in projects and, with the coming of computers,almost every development programme will incorporate such studies in one formor another. Mathematics, or more specifically the arm of mathematics calledstatistics, is the main tool employed in synthesizing the results based on empiricalstudies. Directly and indirectly, other areas of mathematics also contribute (onecan cite an area such as Probability Theory, the applied arm of which is statistics).Undoubtedly, the major developmental thrust ot mathematics falls into thearea of Science and Technology, and a great deal of the stock-in-trade of appliedmathematics awaits delivery to that area. Furthermore, much of the developmentof our time and perspective is closely tied up with Science and Technology. It is,therefore, imperative to focus first on the role of mathematics in Science andTechnology, and then turn to the other complementary areas of development. Butbefore this the role of Science and Technology in development needs clarification.To this end, I prefer to look at Science and Technology as consisting of four mainareas: Basic Sciences Š which include subjects like Biology, Chemistry, Physics,Geology, Mathematics and their offshoots like Biophysics, Theoretical Physicsand Biochemistry; Applied Sciences Š which include Computer Science,Agriculture, Engineering and Medicine; Technology Development Š whichdeals with the adaptation of existing scientific and (imported) technologicalresults for prospective use in industry and may also encompass part of the workdone under Engineering (including Computer Engineering); and, finally,Technology Application Š which deals with the use of developed technologies inindustry and the commercialization of products and other results discovered.In most cases the results of Science and Technology employed in develop-ment have their roots in the Basic Sciences and are gradually turned into somepractical use via the Applied Sciences, through Technology Development, finallyreaching Technology Application. It is also pertinent to note that countries thatare deficient in any one of these four areas often find themselves unable to utilizefully the results of Science and Technology. Indeed, scientists in some developingcountries have made useful discoveries at the levels of the Basic and AppliedSciences Š I have fields like traditional medicine and the biochemistry of variousplants in mind Š but their countries cannot fully utilize the discoveries becausethey do not have the relevant infrastructure needed for the implementationprocess (at the level of Technology Development). Eventually, the results findtheir way to the developed countries, where they are processed and, in most cases,generate enormous profits partly through export to the developing world Š theircountries of origin. My message here is that for a country that is weak at theTechnological (Development and/or Application) level most (Basic andH. A M. DZINOTYIWEYI 107Applied) Scientific Research may not be of direct benefit to the country'sdevelopment.Yet the science of today is the root of the technology of tomorrow. Thus,technological advancement cannot be sustained without recourse to science.Indeed, countries that ignore (Basic and/or Applied) Scientific Research wouldsoon find their long-term development programmes strangled in one form oranother. What is desirable is development in all areas of Science and Technologywith the level of emphasis for each specific area of Science and Technologydetermined by the peculiarities of the particular country. Of course, like all issuesin real life, the dividing line between these areas of Science and Technology isthin.Let me now turn to the thrust of Mathematics in each of the four main areas ofScience and Technology that I have cited. Firstly, I would like to look at the BasicSciences. As Mathematics is itself one of the Basic Sciences, I will start by lookingbriefly at the situation within Mathematics. The branches of Mathematics that amathematician commonly looks at and often treats as distinct fields Š and whichare the ones employed in mathematical reviews Š amount to a total exceedingeighty. However, the more familiar divisions of Mathematics, that should also befamiliar to the general reader, are Pure Mathematics and Applied Mathematics.Pure Mathematics embraces aspects of the theory of mathematics and, at researchlevel, the creation of new mathematics. On the other hand, Applied Mathematics(including Statistics) deals with the extensive and intensive application ofmathematical results to physical and practical situations and the formulation ofthese in mathematical terms. Strictly, the techniques employed are not new butrather are borrowed from Pure Mathematics and adapted accordingly as thephysical situation at hand demands. The dividing line between the two branchesis faint and blurred by various aspects of Mathematics of interest both to Pure andApplied mathematicians. These may be viewed as aspects of Pure Mathematicsundergoing reformulation for absorption into Applied Mathematics.Both Pure and Applied Mathematics have expanded, especially over the lastfew decades. Recent major trends in the composition of the subject include theincreasing interaction between Pure and Applied Mathematics, with some (evenabstract) areas of Pure Mathematics being quickly turned into applicable form,thus increasing the Applied Mathematics stock. Let me underline one feature ofthe pattern of mathematical expansion. As I have indicated earlier, both Pure andApplied Mathematics have expanded to make up more than eighty somewhatdistinct sub-fields of Mathematics. What fascinates me is the way this hashappened: In Pure Mathematics the expansion is largely vertical; althoughvarious sub-branches have emerged and signs of other offshoots to come areapparent, the trend is in the vertical direction. This has serious implications forfuture research in Pure Mathematics, for the top of the tree (of Pure Mathematics)108 THE THRUST OF MATHEMATICS IN DEVELOPMENTis moving further and further away from the ground; even if one tries climbingalong one of the branches, the branches too have their ends vertically directed andare growing fast. This gives pressure for more and more specialization and, ipsofacto, the emergence of more vertically growing sub-branches of Pure Mathe-matics. On the other hand, the expansion of Applied Mathematics has beenlargely horizontal, through interaction with other disciplines. As will be apparentlater, this phenomenon has been brought about mainly by the practical realities oflife.You will recall that at the beginning I mentioned that the aspect of force canbe studied in mathematical terms. But then force is also the starting point ofPhysics, the approach being largely experimental. The alliance of Mathematicsand Physics, therefore, dates back to the birth of Physics. As is the case with manydevelopments, theory leads practice, and in Physics the theory not onlyaccelerated great discoveries, paving the way for experimentation, but did so witha flavour that is heavily mathematical. This led to the spin-off Theoretical Physics,now standing as a distinct discipline. Others, perhaps with a mathematical bias,prefer to call it Mathematical Physics and some even view it as one of the manysub-branches of Applied Mathematics. In a related manner, the interactionbetween Mathematics and Biology has brought about another spin-off, Bio-mathematics. The use of statistics in the classification of species and the rapidgrowth of Mathematical Modelling, which is invading areas such as Ecology,particularly population dynamics, are wetting the appetites of many mathe-maticians. In fact Mathematical Modelling touches on some aspects of virtuallyall the Basic Sciences Š Chemistry, Biochemistry and Geology included Š andhelps to reveal the interdependence among these sciences.It is pertinent to note that new developments among the Basic Sciences makemathematicians very inquisitive not just because such developments expand theterritory for mathematical applications but also because they often contributetowards mathematical innovation Š both Pure and Applied. Thus theinteraction of Mathematics and the other Basic Sciences reveals a two-wayprocess enriching both sides.Such a two-way process seems to be either absent or minimal betweenMathematics and the Applied Sciences, Here Š barring Computer Science,which has only recently emerged, and viewing Applied Sciences as distinct fromTechnology Š the situation seems to be static. Even at university level themathematical input in subjects like Engineering and Agriculture tends to consistof standard results that have been with us for quite some time. (To some extent theinput of Physics in Engineering shows a similar trend.) However, lately thereseems to have emerged a process of change in the approach employed in theApplied Sciences whereby the intrinsic outlook of the division is being replacedby two thrusts: One calls for greater specialization, not just relying on the input ofH. A. M. DZINOTYIWEYI 109i Basic Science but also independently grappling to understand relevant aspects ofnature, thus attaining a form much like that of a Basic Science; such specializationis on the increase in Medicine and Agriculture. The other thrust is that leaningtowards Technology Development; indeed, most universities have now createdTechnology faculties in which the main departments relate to Engineering.Where an Applied Science reveals a leaning towards the Basic Sciences orTechnology Development the mathematical picture in that discipline shows asimilar bias. I shall discuss the situation between Computer Science andMathematics, which I consider one of the rewarding developments of thiscentury, at a convenient point later.Turning to the area of Technology Development, the scenery largely containsa lot of mathematical results awaiting use. I see the role of a mathematician in thisarea largely as formulating practical problems in mathematical terms, then eithersolving the mathematical problems or identifying their solutions among thewidely available mathematical results, and, finally, interpreting the mathematicalresults in terms of the physical situations at hand Š thus enabling the technologistto effect implementation. All this can often be done without calling for originalmathematical research but rather by extracting the relevant results and techniquesfrom the large stock of mathematics already available in the literature. Sometimesspin-offs occur whereby a mathematical solution obtained not only answers thephysical problem at hand but is also widely applicable to other issues of a differentphysical form. In some cases the practical problem may look complicated yet themathematical solution may be readily available. We illustrate such a case inExample 1.It seems quite evident that the solution to this problem could be applied tomany real-life issues including aspects of safety control. (The mathematicalsolution uses ideas commonly employed in the area of Mathematics called GroupTheory and can be studied at university undergraduate level). Other variations,like tying a million rings in such a way that if any two of the rings break theremainder will fall in tied pairs, are also possible.As 1 noted earlier, a peculiar feature of Mathematics for TechnologyDevelopment Š which fascinates many technologists especially now with theemergence of high-technology disciplines like microelectronics Š is the largestock of mathematical results that may be applicable. Thus the thrust ofMathematics in Technology Development is vigorous, not so much because ofthe high turnover in original mathematics but rather because of the increasingrelevance to results already on the market Š sometimes picking very elementarymathematics, sometimes stumbling on theorems that had been shelved in thearchives since the last century and occasionally encountering some recentmathematical theorems, all with concrete applications.Take, for instance, the so-called four-colour conjecture which states that for110THE THRUST OF MATHEMATICS IN DEVELOPMENTExample 1Consider a case where there is a need to tie together many, say one million, rings(made of flexible material) in such a manner that if any one of the rings breaks theremainder will all fall apart. How should the rings be tied?To obtain the mathematical solution we proceed as follows: Let the rings belabelled 1,2,3,... and position them in a horizontal plane. Define the rings rlt r2, r3,... as follows: Let P be a fixed point to be used as a reference point in the sense thatwhen a ring passes through P we say the ring starts and ends at P. Let /Ł, be a ring thatstarts at P and passes through Ring 1 from below coming out above to end at P.Ring /-,-' (called the inverse of r,) is the ring that reverses the process of r,, thatmeans it passes through Ring 1 from above and it starts and ends at P. Similarly onedefines the rings r2, r2"', r3, r3"', . . . with respect to Rings 2, 3, ... .Next we define ring r, r2' : this starts at P, passes through Ring 1 from belowcoming out above and then passes through Ring 2 from above to come out belowand end at P. Similarly we can define r, r2 and, ipso facto, products of any of therings /Ł— r, -' , r2, r2', r3> r3-'Now we are in a position to give the solution to our problem: To tie three rings, 1 23, such that if one breaks the remaining two will fall apart, Ring 3 may beconstructed using the formulat'l. r2] =(3)H. A. M. DZINOTYIWEYI111Example 1 (cont)To tie four rings, 1,2,3,4, such that if one breaks the remaining three will fall apart,Ring 4 may be constructed using the formula= [i-,, r2 ] r3 [/Ł—(4)Proceeding in this manner we see that to tie one million rings, 1,2,3,..., 1 000 000,in such a way that if one breaks the remainder will all fall apart, we may constructthe I 000 000th ring using the formula8], f999 99>)J(1 000 000)which, with the aid of a computer, can be done easily.From the formulae (3), (4),..., (1 000 000) we see that if one removes r,, rf1 orr2, r2 ' or... or nn>> 999, ^99 999 ' Š which corresponds to breaking Ring 1,2,3,...,999 999, respectively Š the formula collapses to 1 Š which corresponds to all theremaining rings falling apart.This completes our solution.112 THE THRUST OF MATHEMATICS IN DEVELOPMENTany real or imaginary map of any size, the minimum number of colours to colourthe countries so that no two adjoining countries have the same colour is four. Theconjecture was well known to mathematicians even during the last century butwas only solved, with the partial aid of a computer, in the USA around 1977. Thisresult should be of great interest to, for instance, designers and artists Š especiallywhere paint may be in short supply Š as a minimum of four different kinds ofpaint can always be employed to give distinct patterns. The result can also beapplied to some military matters. In short, the result simply says that the number 4has a special role in many cases where a minimal number of items is required for adefined purposes and the message is that it be recalled in developmental issues ofthat form.Though the Mathematics employed in Technology Application occasionallygrows like that for Technology Development, especially on aspects of Transpor-tation and Communications Studies the main trend concerns the construction olsimple models and numerical tables for use by the extension services, the public,the commercial sector and other bodies who need not necessarily have themathematical know-how but are engaged in activities where Mathematics isapplicable. For instance, many life assurers only know how to calculate thepremium of a life or endowment policy by using ready-made tables for thepurpose but do not know how the tables are derived; such is also the case withbuilding societies when calculating instalments, for a house mortgage, forexample. Behind these people are mathematicians engaged in the translation ofthe theory on life expectation Š with the desired rates of return on theinvestments and the main risk factors incorporated Š to figures given in the tablesand other simple models. The extent to which such mathematical activity willincrease is largely dependent on the pace of Technology Development; inparticular, the emergence of the computer has inspired the construction of manymathematical models employed in various aspects of Technology Application. Itis also pertinent to note that here one is mostly dealing not only withmathematically well-established results but with results whose relevance forspecific cases has already been noted in the course of Technology Developmentand/or Applied Scientific study.For the time being, this completes my analysis of Mathematics and Scienceand Technology per se, with the exception of the linkage between Mathematicsand Computer Science which I am deferring to a more convenient point.Both from the stand-point of Mathematics and generally, if I were asked tocite in order of priority the three most critical requirements for prospectivedevelopment, I would say that, firstly, it is the ability to predict oncoming naturalphenomena Š like rain, earthquakes, biological phenomena and droughtslong in advance. Secondly, I would say it is again the ability to predict oncomingnatural phenomena long in advance and, thirdly, well, it is still the ability toH. A. M. DZINOTYIWEYI 113predict oncoming natural phenomena long in advance. For instance it is possible,with a good degree of confidence, to predict the weather for the next twelve hoursor so, which seems to be the furthest meteorologists today can take it. Furtherahead, more variables enter the picture and the result becomes increasinglyunpredictable. So far, economists are probably the main professional groupdeserving praise for embarking on long-term forecasts of issues pertinent to theirdiscipline for many decades. However, they are also the best known for erroneouspredictions. This, in my view, makes economics one of the main areas of ourlifetime in need of intensive cultivation both in the theoretical arena and on theapplied side. It is in such a cultivation that I see Mathematics as one of the maintools to be used.The entry of Mathematics into economics received its first celebratedwelcome not so much from Karl Marx's quantification of surplus labour butrather from Leontief s input-output table (1919) which still remains possibly themost widely known and widely employed mathematical model in macro-economic planning. The model is given for a particular year in matrix form, andeach entry has a numerical value indicating the input from one economic sector,say, /', into another, say,/ For convenience let a (i,f) denote such a numericalentry. Using elementary mathematics (on matrix algebra) the model can beprojected to give an estimate of inter-sectoral linkages of a given future year. I seetwo ways in which Mathematics can be further employed to improve thereliability of such a model. Firstly, production functions, incorporating variousbackground factors, can be devised and employed in deriving the numericalentries a (i,j). For instance, in Zimbabwe, where the agricultural sector has oftenbeen depressed by droughts, the amount of rain could be incorporated as one ofthe variables of production functions depicting inputs from the agricultural sector.Secondly, both the form of such production factors and the compilation ofsectoral data will need to be attained in a manner that either does not allowinclusion of structural errors or minimizes such errors with increasing data. Thispoint is illustrated in Example 2.It is apparent that both the idea of 'force' and 'empirical study' cited at thebeginning are incorporated in the construction of production functions. Also themodel can be suitably adjusted and applied to other macro-economic questionsŠ some even apply it in forecasting future energy consumption requirements ofthe various sectors of the national economy.Other models of different mathematical formulation are employed in variousfields of economics. Even institutional investors Š though hestitant Š often get aglimpse of models with a view to selecting investment portfolios that givemaximal returns at certain levels of risk. In the field of financial accounting, ascompanies expand in cross-border activities and currency fluctuations widen,coupled with price inflation of most commodities not falling below a double-digit114 THE THRUST OF MATHEMATICS IN DEVELOPMENTExample 2Suppose we have several squares and for each square we are given estimates, x andy, of the length of a side. If data giving the area of each square, A, is needed, shouldwe use the formula(1)A ŠBoth (1) and (2) are correct formulae for the area of a square; however, formula (1)contains a bigger structural error than formula (2). Thus, formula (2) would give abetter result. The actual mathematical proof comes from an area called ExpectationTheory.* From the latter field we in fact learn that the ideal formula to use isA = xywhich does not contain a structural error. Also, generalizations of(l)and(2)inthecase where n estimates of the length of a square are given give a formulacorresponding to (2), whose structural error tends to zero as n increases, while theformula corresponding to (1) retains a structural error that does not changeregardless of the size of n.Ł With the usual statistical notation for mean, p, and variance, a2, of random variables x andy, and with E denoting expectation, we get= E= A+CT2 giving error of a2E(X) =E ((£12. \) _ E(*2 + f) *= A+ Š giving error of ŠE (A) = E (xy) = n2 = A which has no error term.H. A. M. DZINOTYIWEYI 115annualized percentage, I foresee current cost-accounting practice Š whichrevalues assets, stock, credits and debits in line with price changes Š gainingpopularity, and mathematical models (on current cost accounting) being devisedfor use at least in planning the budget.Lately, mathematical applications have extended even to social sciences suchas sociology and international law, for instance, in analysing internationaldisputes and devising negotiating strategies and in the general aspects ofmanagement decision-making. The techniques commonly employed in theseareas come from a field called Discrete Mathematics including one called GraphTheory. However, I have a feeling that a concrete mathematical breakthrough insuch studies is hampered by the heavily subjective nature of strategies employedwhich often do not agree with mathematical deductive reasoning. Despite this,where the studies focus on prospective events Š I have in mind an area such asmarketing Š some useful guidelines (to the events) can be achieved.In all these models the mathematics used is standard and widely known andthe computer is used in analysing data and testing the models. I now want to lookat the relationship between Mathematics and Computer Science. The latter as adiscipline is relatively new, as I stated before, and its whole existence is centred onthe emergence of the computer. It is, therefore, relevant to look at Mathematicsvis-a-vis the computer (and its capabilities). Many people have often wonderedwhether certain inventions can now be made more easily using a computer inplace of Mathematics and whether the outlook of Mathematics is going to changebecause of computer usage. To answer these speculations let me start byreiterating that Mathematics per se remains solidly a Basic Science subject withtentacles Š namely its applications Š spreading virtually over all the otherdisciplines. Viewed from a mathematical perspective, the emergence of thecomputer enabling the transmission of mathematical results from the BasicScience level to users on the ground is the natural outcome of TechnologyDevelopment. Over the next few years those mathematical results withnumerically based solutions, or with solutions comprehensible in relatednumerical forms, will be delivered to real-life use with the aid of the Mathematicsagency, namely the computer. My guess is that within twenty years, the computerwill be more common all over the world than the hi-fi (music) system is today.Unfortunately, the scope of mathematical results that the computer can perceiveis still very small and hardly covers a significant part of advanced PureMathematics. There are many mathematical results, especially those of thiscentury, that society cannot use in real life simply because there does not yet exista relevant technological capability to usher in their usage. I am optimistic,however, that in the near future Špossibly within five decades Š we will witnessa second stage of higher technology emerging; some kind of super high-technology that would include the invention of super-computers or some other116 THE THRUST OF MATHEMATICS IN DEVELOPMENTrelated technology capable of employing a wide range of abstract mathematicalresults in real-life issues. Such a technology would have a remarkable multipliereffect on the pace of development.For most mathematical (especially pure) research, barring cases of the typediscussed above, the computer is rarely used, and my view is that this situation islikely to remain like that until, possibly, some remarkable development of thetype I have called super high-technology happens. In short, the computer remainsa tool, albeit an intelligent one, to be used in various areas, particularly those inwhich mathematics is applicable.I have discussed how the mathematical perspective differs between the maindivisions of Science and Technology, in particular, that the relationship betweenMathematics and the other Basic Sciences reveals a two-way process with eachside having an input into the other, especially at the level of research. On the otherhand, Mathematics relates to Technology in a largely one-way process, depictingthe application of Mathematics in technological issues. Technology plays acritical role in enabling us to realize the benefits of nature and remains equallycrucial in solving problems of our own making. This reinforces our confidence; asKarl Marx once stated, the only problems which mankind is capable of creatingare those which mankind can solve. Indeed, Mathematics not only provides aconcrete framework for analysing technical issues but even the kind ofmathematical techniques employed are often available within the armoury ofresults established long before.I also reflected on the increasing role of Mathematics in socio-economicstudies. In particular, I have noted the good prospects of greater intimacy betweenMathematics and Economics. In this connection I wish to underline the followingpoint: In my view, the future of Economics as a discipline with particularrelevance to development will depend on its ability to interpret and forecastpractical economic scenarios. In that endeavour I see Mathematics as Economics'key partner, especially in deriving models depicting such scenarios.I also underlined that, equipped with Mathematics, various phenomena canbe understood much better and one tends to face issues from an aggressive ratherthan a defensive stance. As Julius Caesar said, 'The things that frighten us look atour backs but when they see our faces, they vanish'. Indeed, there are severalreal-life problems that, prima facie, are difficult to comprehend, yet, once lookedat squarely and formulated into mathematical terms, are not only more easilyunderstandable, but often their solutions readily follow.I also discussed why and how Mathematics is rapidly expanding Š with thetrend for Pure Mathematics being largely vertical while that for AppliedMathematics is mainly horizontal. Since Pure Mathematics is the pivot of allMathematics, the future of Mathematics will also depend on our ability to deviseH. A. M. DZINOTYIWEYI 117approaches that will enable the human mind to comprehend a lot moremathematical results in line with the (vertical) growth in Pure Mathematics.Finally, I have in some way underscored the following point: the dialectics ofdevelopment reveal that all things and all activities are interlinked and constantlychanging, and what we call development is simply a manifestation of howmankind manages such linkages. From a mathematical perspective, not only dowe find (mathematically) quantifiable phenomena like force apparent in allactivities of whatever form, but even mathematical models derived for specificactivities often turn out to be applicable to many other situations. Mathematics,therefore, provides a way of perceiving the totality of such linkages as a unitedprocess Š a process which mankind will continue to strive to come to grips within its struggle for a better life. Such signifies the Thrust of Mathematics inDevelopment: Its Trends and Prospects.