The Shard and Southwark Cathedral, London. The cathedral is built on the site of 800 years of cathedrals from mid-19th century. The Shard by Renzo Piano built in 2009-2012. Reclaiming Education Renewing Schools and Universities in Contemporary ♦ Western Culture Edited by Catherine A. Runcie & David Brooks W\\'\V.cdwin.hlowcpublishii1g.ct)m DISCIPLINES AND METHODS 148 Mathematics, Core of the Past and Hope of the Future James Franklin i\fathemacics has always been a core pare of western education, from the medieval q11arhivi11111 to the large amount of arithmetic and algebra still compulsory in high schools. It is an essential part. Its commitment to exactitude and to rigid demonstration balances humanist subjects devoted to appreciation and rhetoric as ,,*ell as gidng the lie to postmodernisr insinuations that all "truths" are subject co political negotiation. In recent decades, the character of mathematics has changed--or rather broadened: it has become the enabling science behind the complexiC): of contemporary knowledge, from gene interpretation to bank risk. Mathematical understanding is all the more necessary for future jobs, as well as remaining, as ever, a prophylactic against the more c'orrosive pbjlosophical views emanating from the humaruties. The Ancients and D eductive Proof ln the mid-fifth century BC, the Parthenon was rising over Athens, built accorrung to the best geometrical principles. The constructio n lines are not visible on it, though on some other Greek temples they are. 1 Among the trageruans, artists, sophjsts and merchants creating \'\festern civilization in the city below, a number of visitors from out of town brought an interest in what we would now call scientific questions. According to P armenides of Elea, for example, chinking about the geometry of eclipses is enough for a convincing argument that the earth is round, even for someone fixed in one place on the earth's surface.2 A lesser-known genius who visited Athens was a certain H ippocrates of Chios.3 He came to appreciate that geomeu1* could be orgarused so that the co mplicated and less obvious propositions followed \\~th strict logic from the simple and obvious ones. ffis project was perfected a century or more later in 'Lothar Haselberger, "The construction plans for the Temple of Apollo at Didyma," Scientific American 253, no. 6 (1985): 126-32. ' Otto Neugebauer, A History of Ancient Mathematical Astronomy (Berlin: Springer, 1975), 576. 3 Confusingly, not the same person as Hippocrates of Cos, the medical writer after whom the Hippocratic Oath is called. 149 James Franklin the Ele111mls of Euclid, which was such a success that its predecessors have not survived. But one fragment of Hippocrates's work is left, enough to demonstrate his extraordinary brilliance. Being still at the "bare bands" phase of mathematical development, it can be appreciated by anyone. The fragment concerns the "quadrature of lunes", that is, finding the area of a crescent-shaped figure as shown in the shaded portion at the top left of the figure: E 0 C Fig. 1: The Lune of Hippocrates 4 The lune is bounded by two arcs of circles: the upper one AEB which is half of a circle with diameter ADB, and the lower one AFB which is a quarter of a circle with diameter AOC. Hippocrates proves, amazingly, that the area of the lune is exactly the same as the area of the right-angled triangle AOB (also shaded). Note that there is no n in the answer, as tl1crc is in the formula for the area of a circle: H ippocrares docs not use anything about the area of a circle, but proves directly that the two areas, one curved and one straight-sided, are equal. If the radius AO is one unit of length, both shaded areas are exactly half a square unit. The original text of his proof is given in this footnote in case some readers wish to skip it.5 'Michael Hardy, "Lune of Hippocrates", 7 March 2009,. https :// u pica d. wi kim ed ia. org/wi ki ped ia/ cam mo ns/ e/ e0/Lu ne .s vg s "He started with, and laid down as the first of the theor-ems useful for the purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases. And this he proved by first showing that t he squares on the diameters have the same ratio as the circles. After proving this, he proceeded to show in what way it was possible to square a lune the outer circumference of which is that of a semicircle. This he effected by circumscribing a semicircle about an isosceles right-angled triangle and a segment of a circle similar to those cut off by the sides. Then, since the segment about the base is equal to the sum of those about the sides, it follows that, when the part of the triangle above the segment about the base is added to both alike, the lune will be equal to the triangle." "Text of Simplicius", translated in Thomas L. Heath, A Manual of Greek Mathe150 Mathematics, Core of the Past and Hope of the Future d ! ,he ~re;ks immediately reaLised there was something special about deucti, e ptoo as a way of acquiring knowled e l . r . number of Junes and triangles and fmding that gm* . llt is notl I kl'e measuring a * J * a cases t 1c une equals tl e t:~g :hm a_rca. Pbroof somehow gets to a deeper level of reality. It rcve~s ñt o ) \\ at ,.r so, ut what 11111.rt be so. And the xoof all ' know what must be 50 6 d . 1 O\\ s us not only to , ut to un erstand 111/!Y It must be so. The results are not thereb)' cut off f I . al . b rom P 1ys1c, reaLiry as if tl a out a Platonic realm of ideal forms If real j d . , iey are P t1 , . . * unes an mangles are drawn on aper, ie) are nearly equal m area and the more per*r:ectl tl d ne * * l d h ' 1' Y 1ey are rawn the th:;:õ~eappr~;; s ~pes ;pproximate the exact parts of real space to wl1ich ' an so ie nearer they are equal in area. There are plenty more . d - . . . exact an provable results where that came fr as laid out in the thirreen books of E lid' E' om, I . I b uc s ~ 1e111e11ts and manv thousands of mat:ematica ooks and papers since. The method of deducti,;e . f . ~::~:~::~~ñ:,~!~ :::;. resuJcs. It is just a matter of human interesf ~~~ :r~:r::; A very simple example shows l h . now t e rnitial and most basic results of matl1ematics are known. Fig. 2: Why 2 x 3 = 3 x 2 In1 the figure; th_e six crosses are arranged both as two rows of 3 and as three co umns of-* Smee they are the same six crosses 2 x 3 = 3 x 2 \Y/ li erally see not only !bat 2 x 3 = 3 x 2 b h 2 '3 . . e can t- , ut t at x v111.rt be 3 x 2 6 row;~,~~her,Jwe can easily see that the same reasoning appLies if ~ve add more co umns. If we have 111 columns and " rows, the same reasoning is matics (Mineola NY: Dover, 2003), 121_132. a more . . . ics: A Concise History and Philosophy (New~ k S ~ccess1ble explanation in W.S. Anglin, Mathemat6 C h . or : pnnger 1994) 52 at errne Legg and James Franklin "Pe * * ' ' * (2017), 320-343. , rce1vmg necessity," Pacific Philosophical Quarterly 98, no. 3 James Franklin vali_d. Therefore 111 x JI = JI x 111, for any numbers 111 and 11. So the insight produces an infinite number of truths, all of them understood to be true with certainty. Again, these truths apply directly to the real world of actual crosses written on paper, or of any other objects whatsoever. There is no need for getting out in the wet and observing, but the abstract trutl,s predict what will be observed by anyone who does gee out there.7 Excited by these possibilities, Aristotle proposed in his Poste,ior AJ1n!J•tics that all sciences should follow the model of mathematics, with immediately understood a_•.doms supporting a superstrucrure o f more complex theorems, and proofs explaining why the theorems were crue.8 That did not quite work out. Contrary co Aristotle's hope, in the natural and human sciences observation and measurement are still essential to establishing the truth of theories. Only in mathematics, and in some closely related fields like computer science and just possibly ethics, is proof in the full sense feasible. 9 Thus mathematics remains tl,e ideal training ground for tl,e human faculty of understanding and proof. Mathematics at the Centre of Western Education \'i/estern education has not lost sight of the point of mamematics and has always made it central to education, despite the fact that it is quite hard to learn and subject to a certain degree of customer resistance. Medieval liberal tertiary education, the preparation for specialised studies such as theology, law and medicine, was divided into two levels, the llivi11111 and me q11nd1ivi11111. The 11iv1i1111 consisted of grammar, logic and rhetoric, chat is studies in words and how chev work. Then came the q11ndtivi11111: arithmetic, g~omecry, music (meory) and as~ronomy, all of which are matl,ematics in one form or anomer.'O Then the Italian merchant schools of the lacer MidcUe Ages made the remarkable discovery that calculation with indefinitely large numbers could be reduced co rules and taught co seven-year-olds.11 Basic numeracy became widespread along witl, basic literacy. The new technologies of early modern times found a population able to understand and manipulate them. 7 Briefly in James Franklin, "The mathematical world," Aeon 7 Apr 2014 https://aeon.co/essays/aristotle-was-right-about-mathematics-after-all; fully in James Franklin, An Aristotelian Realist Philosophy of Mathematics: Mathematics as the science of quantity and structure (Basingstoke: Macmillan, 2014). 8 Richard D. McKirahan, Principles and Proofs: Aristotle'; theory of demonstrative science (Princeton: Princeton University Press, 1992). • See Svetlana V. Drachova et al, "Teaching mathematical reasoning principles for software correctness and its assessment," ACM Transactions on Computing Education 15, no. 3 (2015): article 15; and James Franklin, "On the parallel between mathematics and morals," Philosophy 79 (2004): 97-119. 10 David L. Wagner, ed., The Seven Uberol Arts in the Middle Ages (Bloomington, Ind: Indiana University Press, 1983). 11 Richard W. Hadden, On the Shoulders of Merchants: Exchange and the mathematical conception of nature in early modern Europe (Albany NY: SUNY Press, 1994). 1"i2 Mathematics, Core of the Past and Hope of the Future There was some backslidi~g ~rom the humanises of the Renaissance, who preferred words, but the Scientific Revolution of the seventeenth century was a_nd was ~een to be based on mathematics and data instead of wordv disputaaon. GaliJeo explains: * If what we are _discuss_ing were a poinr of law or of the humanities, in which neither true ñr taJse exists, one might trust in subtlety of mind and readiness of tongue and_ in the greater experience of the writers, and expect him who exceUecl in those things to make his ~easoning more plausible, and one might judge it to be the bes~. But in narural sciences whose conclusions are true and necessary and have nothing to do with human will, one must take care not to place oneself in tlie clefcnse of error; for here a thousand Demostheneses and a rhousand J\ristotles would be left in the lurch by e\*ery* mediocre wit who happened to hit upon the truth for himself.12 \\:then the Jesuit missionary i\ latteo Ricci reached China in 1582, he soon found tha_t the _Chinese scholars were particularly impressed with Western mathemaacal ~c1ence. T~e first work translated into Chinese by him and his collaborator Xu Guang91 ,ms the first six books of Euclid's Ele111m(s. Ricci says in his diary: Nothing pleased the Chinese as much as• the volume on the Elements of Euclid. This p~rhaps was _due c~ the fact tbat no people esteem mathematics as highly as the Ch1_n_ese, desp1t~ rhe1r method of reaching, in which rhey propose all kinds of prop~sitions but without demonstrations. Tbe result of such a system is that anyone.'~ free to exercise his imagination relative to mathematics without offering a defi111t1ve proof of anything.13 Tha_c is quite right about the difference between Western and other mathemaãs. N on-? 1escern ñachemãcs was in many ways impressive, especially in Bab}lon, India and China, but 1r looks more like modern computer science than ?,odern mathemaacs: a series of recipes for calculating ramer than an orgarused body of proofs of tl1eorems.14 . ~n Victorian England, the study of Euclid was presumed suitable for murung boys of the upper classes in the intellectual casks that awaited chem u Galileo, Dialogue Concerning the Two Chief World Systems trans s Drake (2nd d B k I . u * sity of California Press, 1967), 53.54_ ' * * e , er e ey. mvera Chino in the Sixteenth Cent~ry_: The Journals of Matthew Ricci, trans. L.J. Gallagher (New York: Random House, 1953),_ 4,76; details in Peter M. Engelfriet, Euclid in Chino: The Genesis of First Chinese Tr~nslo'.10~ of Euc//~ s Elements Books 1-_VI (Jihe yuanben; Beijing, 1607) and its Reception up to 1723 (Leide~. _Brill, 1998), ty~lcal postmõern1st complaints about the resulting "loss of indigenous mathematics in Sara N. Hottinger, Inventing the Mathematician: Gender, Race and Our Cultural Underf:anding of Mathematics (Albany NY: SUNY Press, 2016), 121-2. ?eorge Gh~verghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (3'd ed Pnnceton: Pnnceton University Press, 2011). ' 153 James Franklin such as governing India. Animated debate proceeded merely as to whether Euclid was best swallowed whole o r whether modern re-hashings were easier but not oversimplified; Charles Lurwidge Dodgson (Lewis Carroll)'s book E11clid and His Modem B.ivals defended teaching straight Euclid. 15 The upshot of this long process was that mathematics-and mathematics at a substantially high level-has become a compulsory pare of education across tbe board. That has applied no t just in Western countries. The Soviet Union, for all its Marxist ideology in the hwn anities and Lysenkoisc delusions in biology, left pure mathematics alone and maintained a very high standard of mathematics education in schools.16 East Asian countries have surpassed Western ones in school mathematics education, having successfully grafted Western mathematics onto their cultural traditions.17 T hird World countries are doing their best co caccl, up. The culnire of research mathematics is the same worldwide, with the same symbols used everywhere and vi rtually everything written in English. Mathematics is a universal culture-as international and standardised as air traffic control but impacting vascJy more people. Enemies of Mathematics Naturally, the enemies of Western civilisation and of rationality have not taken the achievements of macJ,ematics lying down. The Greeks in Athens in the fifth cenrnry BC, multi-talented as they were, invented not o nly Western civilisation but how to complain about it. The Sophist Gorgias of Leontini, in the course of making a lot of money corrup ting the youth of Athens around 420, defended the propositions: Nothing exists If anything existed, it could not be known If anything were known, ir could not be communicated.18 That just about covers everything, and cJ,e poscmodernists of the lace twentieth century did not have much to add to it (except prolixity, obviously). They found plenty of life left in Gorgias's insights, and recycled them in such forms as: is Charles Lutwidge Dodgson, Euclid and His Modern Rivals (London: Macmillan, 1879); Rafael Montoito and Antonio Vicente Marafioti Garnica, "Lewis Carroll, Education and the Teaching of Geometry in Victorian England," Hist6ria da Educa,iio 19, *no. 45 (2015): 9-27. 16 Alexander Karp and Bruce R. Vogeli, Russian Mathematics Education: History and World Significance (New Jersey: World Scientific, 2010). 17 Frederick K.S. Leung, Klaus-D. Graf and Francis J. Lopez-Real, Mathematics Education in Different Cultural Traditions: A Camporotive Study of East Asia and the West: The 13th ICM/ study {New York: Springer, 2010). 11 Bruce McComiskey, "Gorgias, " On Non-Existence": Sextus Empiricus, "Against the Logicians" 1.6587, translated from the Greek text in Hermann Diels's "Die Fragmente der Vorsokratiker"," Philosophy & Rhetoric 30 (1997): 45-49. 1'i4 Mathematics, Core of the Past and Hope o f the Future l. D oubt whether there is anv solid realitv out the , fi . least label "na.. ,, 1 • . , re or science to know (or at 2 . . , . • ive ~ ie assumpnon thar there is such a rcalin•) * i\lmnrnin that even 1f there is some sort f 1* I •1 I . . b * o rea 1ty our tiere we ea 1 w 1at 1t 1s ecause we are trappc I . . , , nnot ,now b . • t 111 our own evolut1onaril1"-deter * I rams/cultural understandings/specific 11* . . . / '. mince tions19 . istonc1ues reacrronary educa3. Allege that language is incapable of co . . rive reality, as it cannot refer directly to ;~~gsuñc~~nhg ~ny tmrh _about objec- . • t e1r properties The evils of posrmodernism are a well s d. d fi hearsed again.20 Here we confine . I -.tu ie teld and need not be reI ourse ves to tbe case of mat! * N ra Jy, the most shameless and foolhard . in .. ' . . ' 1emac1cs. acukeen to make a name for themsel ~ - . the uracionalist camp have been defence. Sokal and Bricmont's ; . ve~b} f trying to kñck over this last line of l111poslm-e.r, easily collected a whoxlposhe o posftmode~nist absurdities, lntellec/11al h ~ e c apter o garblmgs of h * f t e best F rench theorises. Gilles D I , mat emaacs rom of "theorv" had *a1 .r e ~uze, one of the most worshipped gurus J , a spec1 taste ror pieces of h . ematics), such as: mat ematJcs (or apparent matbThe respective independence of ,*ariablcs a . . them is at a higher power than the first Tf p~ear~ Ill mathemaucs when one of in the function is not con fin "d 1 * 1 iat is w 1Y I legel shows that variabilitv l'. to va ues t iat can be cl d ('/3 * left undetermined (a = 2b~ b . iange and 4/ 6) or are , 1 ur re9t11res o ne of the va * bJ b . power (f/x = P). For iris then that a ., . • ~la es to eat a higher ferential relation r!,)'/ dx, in which tl~e ~~ta;on eã be directly determined as di fables is that of disappearing or b . b ) erermmaaon o_f the value of the \"arinite speeds .. .2' emg Orn, even though It is wrested from infiThe writings of tbe irrationalists abollt m I . no impact on mathematicians i\l I , . . at iemaacs have bad absolutely complainers. le has been all . r at ,emffaocdians ~,ave nor read one word of any . water o a uck s back to th b have remamed unaware of and unaffected b cl 11 bem ecause they Th d * ' Y ,e se -out v the hu * * e pro ucoon of theorems and th Ii . , manmes. modelling and airline scheduling ha,~e app: caododn of naJJ1athematics co climate cee e , tot, y untroubled by any 19 James Franklin "Stove's Di 24. , scovery of the Worst Argument in the World," Philosophy 77 (2002), 61520 One of the earliest and best refutations is Ra mon . . . Saussureon Literary Theory (Basingstok . M Y. d Talhs, Not Soussure: A Cnt1que of Postwas brilliant. e. acmillan, 1997)Needless to say it was as ineffective as it 21 Gilles Deleuze and Felix Guattari, What is Philosa h , . . . 122, in Alan Sokal and Jean Bricmont Intel/ t I P y. (New York. Columbia University Press 1994) Australian imitations in James Fran'*11' C ec u? Impostures (London: Profile Books, 1998) 1i o-1- , ~ n, orruptmg the Youth* A H" t ., * ' ' (Sydney: Macleay Press, 2003), 368_ * 15 ory a, Philosophy in Australia 1 c:c James Franklin li * f f ding of doubts as to the fallibili t)* of proof (but subject co the po a cs o un ' course). ' the ood news. The bad news is that it is oche~v_ise in mat~ematics T~at s Th g k like Paul Ernest's Social Constr11ctw1s111 as a Pbdosopl[)' of educaaon ere woe s l 'T ard : , . ' .* 1 ' indeed.22 It may be that t ,e paper ow, M.athemtrftcs are taken vet) senous) . G d Levier's Hioher Superstition c * * 1 b ' that was analysed m ross an o a renurust age ra h . . al of the field 23 but it is not hard to find hat beyond w at is typ1c, ' l . was s~me,~ d d lik I . e *tract from Educational Studies in Mat~e111altcs, text sail being pro uce e c 11s x 2004: . . e of mathematics is an inscicutional frame that funcThe supposed apolmcal natur *1tl11*n schools This paper disrupts the . * . cific power strncturcs w * . uons co sustain spe . ( b d . of knowledge constructed in * n tbat machernaucs as a o ) . common ~ssui:npoo '. free from entrenched ideological motives. Using siruac~d h1_scor'.cal ml omentes~ l~xamines the wavs in which novice mathematics namtUve 109u1ry, t 1c pap * * . . . Ii • 24 teachers negotiate the intersection of curriculum and msuruuonal po ucs . .. nds ma ' be observed by following up in Google Scholar the _41 Recent ere . _) . h . 'C .. ising with Foucault: cowards a gu1dworks that cite this article, sue as ñc . cl . d tion' 2016. i.ng framework for socio-political scud1e~ '.n ma ,emaocs e uca ' The research field remains active. J\nd acav1sc. * **Now Mathemaucs as it is . . h e i.n such distant regions as the Those o utside mathematics, espec1:i"y t os d . he lase century or so. . . ' have missed some important tren s in t humanmes, ma) d . kn some verv old pure mathemacP I with a general school e ucaaon ow ' . . eop e . . ,d algebrn have a rough idea of scausacs, ics such as anthmeoc, geomecry _at f , d . need sore is used i.n physics, d h ' . n mac mathemat1cs o some a ' a . . an a, ~ a noduofimance That is noc exactly wrong but is a very partial view. compurmg an * . . The truth is more complex and 1nteresdt1ng. b f disciplines at the edge of I the mid-century, there appeare a num er o . " . . . n . .* 1, 11 d the " formal" o r "mathemaacal sciences o r sc1-mathemat1cs, vaaous ) ea e . h ' ms enuinecring control theof lexin.*-operat1ons researc , S) see o* ' ences o comp . *; . d h 25 To take just o ne example o f the ry, theoretical computer science ã ~t ers. . . eeds \en h of stops, kind of thing they do, consider tram umecabling. T ram sp , gt -------------- . ., . h fcs (Albany NY: SUNY Press, 1998). 12 Paul Ernest, Social Constructivism as a Philosophy o, Mat emo , (800+ citations on Google Scholar) . . . . The Academic Left and its Quarrels Wi th Sci• n Paul R. Gross and Norman Levitt, Higher Supers11t1on. . J h H k*ns University Press, 1994), 113-5. . • ence (Baltimore: a ns op 1 . 1 th Portical Axis* The Interior Voice of D1ssent1ng " Elizabeth de Freitas, "Plotting lntersect1~ns_A ongh e t.' SS 12oo'4)* 259-274. h " Ed cotiono/ Stud1es in Mat emo ,cs * Mathematics Teac ers, u . . h Ph.losophers' Stone," Studies in History and i.s James Franklin, "The Formal Sciences Discover t e I Philosophy of Science 25 (1994): 513-33. 156 Mathematics, Core of the Past and Hope of the Future and crew rosters create (<.1uantitative) conscraincs that train journeys must satisfy, on top of which there are delays and breakdowns of various degrees of predictability. Those constraints must be described mathematicaUy and software must be programmed to ti.mecable the trains, ideally so as to minimise delays. Though developed originally for use in various such applications, these bodies of knowledge are in tl1emselves purely mathematical disciplines (though often now housed in faculties of engineering). *Mo re recently, the siwation has changed through the abili ty of computers to deal with large amounts of data. Where once the strength of computers was in calcL1lation, the focus has moved, as the rise of the phrase "information technology" suggests, co the processing of the flood of data. There are unstoppable streams of daca arriving from satellites, from telescopes, from weather buoys, from medical imaging machines, from supermarket scanners, from wire services sen*ing up tick data of all the world's financial transactions. Huge databases await mining and matching. J\s the science journalise Mitchell \".1ald rop says, "drink from the firehose of data" .26 D ara interpretation tasks that the brain finds easy, like opening the eyes and seei.ng what's in front of chem, remain very challenging for compmers. Computer vision is still ynreliable on the problem of listing the objects in a natural scene. While there is plenty of activity in these fields, it is recognjsed chat the capabilities of che hardware and software in storing and accessing the data have raced well ahead of the mathematical algorithms needed co make sense of the data. There is soph.isticared technology for recording and displaying Pap smears, say, but for recogrusing whether they are cancerous, the method of choice is still to have a trained human looking at them-or at lease, supervising any results generated aucomaticaUy. That is the t)'Pical situation. Data mining, interpretation of pathology results, target recognition and the like are all held up because che mathematics of pattern recognition, and statistical methods appropriate to large data sees in general, remain in a grossly primitive state. Surely one of the main directions for future mathematics is to sore that out. The problems co be solved include fraL1d detectio n and finding missing planes.27 The future o f mathematics is hard to predict, but perhaps less so than me future of just about everything else. Mathematics has been going a very long time, and its changes have been expansions rather than revolutions. 1,Iathe26 M. Mitchell Waldrop, "Learning to Drink From A Firehose," Sdence 248 (1990): 674*5. 27 Bart Baesens, Veronique Van Vlasselaer and Wouter Verbeke, Fraud Anolytics Using Descriptive, Predictive and Social Network Techniques: A guide to data science for fraud detection (Hoboken: Wiley, 2015).; "How Statisticians Found Air France Flight 447 Two Years After It Crashed into the Al* lantic." MIT Technology Review, May 27, 2014, https://www.technologyreview.com/s/527506/howstatisticians-found-air-f ra nce-flight -44 7 -two-years-after-it-crashed-into-at lantic/; Sam Davey, Nell Gordon, Ian Holland, Mark Rutten and Jason Williams, Bayesian Methods in the Search for MH370 (Singapore: Springer, 2016). 1 ~ ., James Franklin matics does not go back to square one, and there is some sense of what will come next from the pattern of the unsolved problems of today. Undoubtedly many advances in mathematics in the near term will be simply cracking the problems that seem today to be in the course of yielding. Not long ago speech recognition by computer looked intractable, but the algorithms advanced and it is now, if not perfect, _usable e~ough to answer phones for large organisations. Just now at the cuwng edge 1s the recent s~1dden improvement in Google Translate arising from a powerful stat1st1cal learning algorithm applied to huge corpora of translated texts.28 Some problems arc a litde farther "out there". A big one that is not so _far going very well but surely ought to be reasonably sol:rable 1s that of ext:acang causalitv from data. Among the many amazmg thmgs that human 111fants learn v~ry fast is how to infer what causes what from looking at how things behave. They can manipulate a few close things, but most things can only be observed from a distance, and causality is not something that can be directly observed. Yet infants can work out what causes what and hence predict what will happen next. It is a difficult problem because "correlation does not imply causalicy".29 If we could learn the algorithms behind that, we could trawl through medical data to learn the causes of disease. \V/e would be able to determine automatically whether, for example, lower cholesterol is a cause or symprom of lower heart disease. The causes of global war~ing \~ould be established beyond doubt and we would know what 111tervent1ons will work. A different kind of problem lies at, so to speak, d1e opposite end of the spectrum from big data. It concerns the estimation of "extreme risks"risks of major disasters that are not exactly like anything that has happened y~t, such as terrorist attacks or major quarantine incursions. 1t is "data-free stanstics", in chat there are no data, or hardly any, direcdy relevant to the event to be predicted. In that case one must combine the little data of some rel~vance (such as the occurrence of somewhat similar events) ,vith expert opm1on _on possible scenarios. How to use data to keep opinion honest is a challenging problem.30 Plenty of problems remain, too, in mad1ematical finance, the area most popular as an employment destination for maths graduates. It would be good to understand how to apply mathematics to enhance instead of undermme 28 Gideon Lewis-Kraus, "The Great A.I. Awakening," New York Times Magazine, Dec 14, 2016, h ttps://www.nytimes.com/2016/12/14 / ma ga zi n e/t he-great -a i-awa keni ng .html 29 Divorce rate in Maine correlates with per capita consumption of margarine, correlation= 99.26% (http://www.tylervigen.com/spurious-correlations) 30 James Franklin, Scott A. Sisson, Mark A. Burgman and Jennifer K. Martin, "Evaluating Extreme Risks in Invasion Ecology: Learning from Banking Compliance," Diversity and Distributions 14 (2008): 58191. Mathematics, Core of the Past and Hope of the Future the stability of the global financial system. Medicine too presents a huge number of problems needmg mathematics, in areas like gene expression.31 Directions for Mathematics Education We are used to a stream of reports of falling standards in schools, falling enrolme~!s m advanc~d mad,_s, an_d the better performance of our Asian neighbours. The first ili111!5, plamly, 1s to make sure all students study an appropriate level of matbemancs, and do not, for example, ditch mathematics because easier sub_jects gain higher HSC marks. 33 Efforts to explain the need for mathematics to students and their parents need to continue. It is true however, d1at enrolments by the brighter smdems in university mathematics have held up well in recent years, especially in those courses related to jobs in finance. . Having sai~ that, the mathematics taught needs a fev,; adjustments to be suitable for an mtelligent person in the present centurv.34 Svllabuses at school and_ univers_ity arc dominated b?' pure mathematics, about ~umbers and algebraic techmques that are promised co be useful in the "real world" in an indefinite future .. lt is indeed necessary co study those things, bur an exclusive focus on them 1s both unmotivating and untrue to real mathematics. * The first thing that needs adding is some serious mad1ernatical modellina. ~fodelling is _the process of describing smne real-world structure or proble~1 111 rnathcmãcal. term_s, so_ iliac mathematical techniques can be brought to bear on solvmg it. It IS quite a different skill from pure mathematics. Take a problem like: is. it feas ible to tow an iceberg from Antarctica ro provide fresh water for Adelaide? That does not require solving a set mathematical problem like "Simpljfy (x 3)(x3 + 4x 7)" and the skills involved are not the same. It needs a team to model what would need to happen ro cow an iceberg and work out what quantitative information is needed-such as the size of icebergs, rates of melting, feasible speeds of towing, Adelaide's water demands- and _finding d1at information. The outcome should be a clearly written report adv1smg what the result 1s and setting out the reasons clearly.35 " E.g. Ahmet Ay and David N. Arnosti, "Mathematical Modelling of Gene Expression: A Guide for the Perplexed Biologist," Critical Reviews in Biochemistry and Molecular Biology 46, no. 2 (2011): 137151. 32 Most recently Samantha Hutchinson, "Australia looks to China for Maths, Science Lessons," The Australian 11/7/2017; figures in Frank Barrington and Michael Evans, "Participation in Year 12 Mathematics 2004-2014," Australian Mathematical Sciences Institute report, http://amsi.org.au/pu blications/participation-in-year-12-mathematics-2004-2014/ . 33 Eryk Bagshaw, "HSC Maths: Students Studying Advanced Maths Stung with Lower Marks in ATAR" Sydney Morning Herald, 19/5/2015. ' 30 James Franklin, "Mathematics for the Intelligent," Gazette of the Austrolion Mathematicol Society 24 (1997): 2-3. "~obert B. Banks, Towing Icebergs, Foiling Dominoes ond Other Adventures in Applied Mathematics (Prmceton: Pnnceton University Press, 1998); UNSW's course 'Mathematical Modelling For Real 159 James Franklin Statistics too needs a higher profile in mathematics education, as dealing with data and reaching conclusions from it are now the main uses of mathematics. In fact mathematics syllabuses have been moving in that direction, at both tertiary and secondary levels.36 However syllabuses have been slower to adapt to the statistical needs of the "big data" thã is ~ooding iño data wa_rehouses from the sophisticated hardware collect1ng it. Extractmg meamng from huge datasets in real time is not quite the same task as the one for which classical statistical methods were developed, of extracting the most from smaU and expensive datasets such as in medicine. The next thing that needs adding is a course o n proof. Although mathematics advertises itself as "teaching you how co think"37 and, as we have seen, proof is central ro the Western mathematical tradition, m~thematics degrees rarely include an explicit course on how to prove mathematical results. It 1s an easy enough skill to pick up but it is not inborn. Here is a simple example: Prove that the square of every even number is even. Proof: Let x be an even number (so that the result has generality by applying to eve!]' even number) Sox= 2 x k for some whole number k (that is the meaning of "even") So _,._:?. = (2 x k)2 = 2 x 2 X k2 (basic algebra) which is even (again from the meaning of "even": it is twice some whole number) Therefore the result is proved: for every even number, its square is even. Understanding that does not require any stroke of genius: it is a perfectly mechanical application of the definition of "even" and simple algebraic manipulations. Such proof techniques can be and should be taught directly.38 . . Mathematical communication also needs to be taught. Commumcatmg via graphs and diagrams is not the same as communicating via text. Ma_chematical professionals face tl1e task of communicating their recommendations to people who cannot really understand them, and the challenges of cõmunicating a simplified but sufficiently honest account of the mathematics are considerable. Those employers who habitually complain about the poor communication skills of graduates may wane to consider the fact that they are World Systems' is described at https://www.maths.unsw.edu.au/courses/math3041-mathematicalmodelling-real-world-systems 36 NSW Education Standards Authority, Mathematics K-10 syllabus, Statistics and Probability, https:// syllabus. n e sa. nsw. ed u. au/mathematics/mathematics-kl 0/ statistics-and* probab iii ty / "Research on this tricky question in Matthew Inglis and Nina Attridge, Does Mothemotical Study Develop Logicol Thinking? Testing the Theory of Formal Discipline (New Jersey: World Scientific, 2017). 38 James Franklin and Albert Daoud, Proof in Mothemotics: An Introduction (Sydney: Kew Books, 2010). 160 Mathematics, Core of the Past and Hope of the Future acquiring their supply of graduates absolutely free and draw the obvious conclusion. ~fathematics has enormous effects in che "real world", as shown by such headlines as 'The Formula That Killed Wall Street'39 and the fact that the military l:as always been an enthusiastic investor in mathematically-based technologtes.40 It fol lows that education in mathematics should include attention :o ethical questions, in the same way as is normal i11 medical education. The 1ssu~s for professional mathematicians are much the same as in other information _management professions-duty of care, conilict of interest, honesty 111 drawmg and commurucaang conclusions, confidentialitv and whistleblowing, and the likebut the effects of malfeasance can be orders of magnitudes larger than 111 most professions.41 Conclusion For tl1ose who wished to retain their sanity amid the stress of twentiethcentury cultural chaos, where was there to escape to? In the humanities world, there was always the past, and many a cultural refugee from various Modernisms recuperated through communion with ivfonteverdi, or Vermeer, or Jane Austen. But fõ tl1ose who preferred their culture still living and bre~thing, the most extensive vandal-free space was science and mathematics. Two regions of science stayed particularly free of any modern nervousness about themselves. One was _engine~rir:g, for the obvious reason that bridge constrnction on cultural relativist prmc1ples is forbidden by the laws of nature as s trtctly as by those of man. The other was mathematics. . Mathemati_cs has several advantages as a cultural counterweight to relativ1sms ãd sce~t1:1sms. Everyone knows something about it-in fact quite a lot ~b?ut Jt-:-:so it 1s not necessary to ~ake the word of experts about everything 10 Jt, as Jt is for, say, quantum physics. Secondly, the truths in it are subject to proof, and what is proved does not become unproved (though it can be proved bet~er).4~ Fõ the~e reas~ns math_ematics has always been an unfail.ing support for rat1onalist views, views which exalt the capacity of the human 39 Felix Salmon, _"Recipe For Disaster: The Formula That Killed Wall Street," Wired 23/2/2009, https://www. w I red .com/2009/02/wp-quant/ ' 0 Examples in Nicholas Daras, ed, Applications of Mathematics ond Informatics in Military Science (New York: Springer, 2012). "James Fr~nklin, "A 'Professional issues and ethics in mathematics' course," Gazette of the Australian Mat_hemat,cal_Soc,ety 32 (2005): 98-100; an Australian case study of the misuse of actuarial information in David Jackson QC, Report of the Special Commission of Inquiry into the Medical Research and Compensation Foundation, 2004, eh. 23 ~2ttpJ/www._;r.james_hardie.c0m.au/jh/asbestos_compensation/special_commission_of_inquiry.jsp. It 1s sometimes claimed that non-Euclidean geometries are a counterexample where Euc/'d' f " d" . , 1 s proo s we_re overturne . That 1s not correct; the case is examined in James Franklin, An Aristotelian Realist Philosophy of Mathematics: Mathematics as the science of quontity and structure (Basingstoke: Macmillan, 2014), 160-2. 1 (.1 James Franklin mind co find out the cruch. Conversely, mathematics has been ~ p_er_eñal thorn in the side of opinions chat abase human knowledge, and claim 1_c 1s liñ ited by sense experience, cultural horizons or one's persoñ educauo n an? perspective. Any culture or person chat can count co 4 has discovered chat - + 2 = 4 and should any fear arise of losing a grasp of that truth, resort co countiñ stones will quickly relieve any anxiety. The truths of mathematics, unfortunately, cannot defeñ themselves, a_s they are not physical things with a caus~ action o n the pl?s1cal w?rld._ Neither ethical nor mathematical truths and ideals c~n fig~t tanks, or blizzar~s ~f alJe ations about history or politics (though again, neither can they be ltquidac! d by chose enemies). They depend on human minds co attune co chem to act on their behalf-to implement those ideals and teach chem co the next generation. . . r . cl t Bolstered by the continuing success of machemaucs 10 app 1cauons, ,~ makes an education in mathematics the surest way forward to a grasp of eternal verities. 162 Art Teaching as Part of a Liberal Education Christopher Allen Most art education curricL1la, whether in primary schools, secondary schools or vocational colleges, seem to have been written by people who have no real understanding of art, either as practitioners, hiscorians o r critics. Art classes, especially in primary school, arc often treated as a kind of playtime, a break from serious academic subjects, and when art is taken a little more seriously, it is either thought of as a form of self-expression o r as a way of articulating some kind of social critique. Bm pupils are hardly ever taught any discipline of drawing, painting or any other art form that might make expression or articulation possible. Nor are they caught arr history, which would allow them to reflect critically on their own practice; instead they are provided with a pre-cligested set of ideological messages which they are encouraged to use both as inspiration for their own work and in tl1e interpretation of any works by otl1er artists tl1ey are shown. This essay will begin by asking what role the teaching of arr plays in a liberal education, and wiU then go on co suggest what approaches co the teaching of both art practice and art history, theory and criticism might best serve those ends. It will propose tl1at art practice develops the intellect and the imagination in ways different from but complementary to other disciplines such as mathematics and languages, and that arr history expands our understanding of human experience in a way comparable to the study of literature and music. Art, in its many manifestations, means cliffcrent things to different countries in the modern world: in Japan, for example, the aestl1etic sensibility remains central to culrural life; in Italy, the art and architecrure of past centuries is a source of pride even co uncultivated people, and opera remains a kind of popular music; the French cake great pride in their literature and cinema, and in lran such poets as Hafez and Sa'adi arc loved by learned and unlearned alike. ln the British traclition, literature has always been richer and more highly regarded than either music or pain ting and sculpture, and perhaps that is why the English-speaking countries have comparatively little appreciation of the visual ans even today. \Xie have biennales and art fairs, but these contrived