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CHAPTER FOUR

The Value of Mathematics

Unfortunately, the reigning mediocrity of our days demands a heavy toll of anyscientist who dares stray from routine publication of technicalities, over into theblue yonder of speculation.

Gian-Carlo Rota

In one of his last writings, Gian-Carlo Rota sounds a dramatic alarm for the survival of mathematics. It was the nal occasion for continuing his re ection on the value of this discipline, in keeping with a tradition that has vaunted some of theleading mathematicians of the 20th century. This nal chapter, inspired by the titleof a celebrated text by Poincar, is an attempt to nd in the very perils that threatenmathematics occasions to rethink its value for scienti c and philosophic research.In this way we shall have an occasion to return to and investigate more thoroughlysome aspects of the phenomenological sense of intentionality that were examinedin Section 2.2.

Rota maintains that the queen of sciences is threatened by ignorance of its

results and widespread hostility towards its practitioners.1

Among the numerousand heterogeneous causes that foster such animosity he singles out the convic-tion that mathematics must possess many applications. 2 This contention has been

wrongly taken by a heterogeneous throng of major social gures (in politics andin the military, from entrepreneurs to educators) to mean immediate and direct ap -plicability. But such applicability is the exception rather than the rulea fact thatgenerates frustration and a misguided sense of uselessness that (directly or indi-rectly) leads to a reduction of research funding, a fall in the number of universitymathematics students 3 and, by a sort of chain reaction, fosters sterile ethical andepistemological debates on pure science and technology.

1 Rota (2000), p. VII.2 Ibid., pp. VIIIIX.3 See the essay Ten Lessons for the Survival of a Mathematics Department in Rota (1997a), pp. 204208.

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Chapter Four 88

This desire for ensured and instantaneous (we could even say preventive) ap -plicability is the result of numerous factors rooted in the current cultural situation,

perhaps the most pernicious of them being a widespread economism, hypnotizedby the present and the immediate future, which, aiming at short-term gain, ignoresthe history of mathematics and the history of science in general. 4

Such a viewpoint is so narrow that company-mindedness would be a betterterm for it. Not by chance, the global and long-term vision of political economyis now being increasingly morti ed by that business economics whose successcan be measured by the spread of its aggressive terminology and of its criteria ofevaluation, which judge other disciplines on the basis of the short-sighted schemaof the double entry.

These criticisms do not mean rejecting any sort of planning dimension, or-ganization of resources, and hierarchy of priorities, on the basis of an anarchisttheory of knowledge taken to the extreme. 5 They intend, rather, to denounce theincapacity to range over a broader horizon that couples the short-term with thelong; the incapacity to view the great technological conquests as a composition ofmany small heterogeneous results, apparently devoid of applicability.

For those who judge such considerations to be obvious, I recall Rotas warn-ing that mathematics risks becoming a curiosity that we will soon be showing our

children at the zoo of endangered intellectual species (Rota, 2000, p. VII). It is myintention, then, to develop his sorrowful exhortations into re ections on the rela -tion between the teleological and nonteleological aspects, and the theoretical resultsand technological applications, of mathematical research. My re ection avails itselfof Rotas fundamental philosophical reference, represented by phenomenology,and begins with his radical critique of the myth of progress.

4.1 The Myth of Progress

A look at Rotas views on the concept of progress will help us bring the gen-eral framework for his critique of the company-minded interpretation ofmathematics into focus. Rota, in fact, considered progress to be incompatible

with contemporary mathematics devoid of unitariness, characterized by his-torical discontinuity and looking, at times, towards the past (see Rota, 1990a;see 1997a, p. 117).

This was a recurrent theme in Rotas conversations with friends, students, andcolleagues, even if it was barely mentioned in his writings; it sheds considerable

light on the axiological correlates of his philosophical re ections, centered on thetheory of knowledge. In this regard a short section of the text of his course of

4 We recall that Rota attenuates the distinction between mathematical and natural sciences; seeRota (1993a), p. 28.5 See Feyerabend (1975), and Palombi (2004), pp. 3435.


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The Value of Mathematics 89

lectures on phenomenology at MIT is particularly signi cant, in which the problemis dealt with in speci cally economic terms. 6

Progress, here, is de ned as the myth that the human condition is characterizedby a constant tension towards material and spiritual improvement. Rota denouncesthe irrationality of this conviction on the basis of the precise limits of the food,energy, and environmental resources of the Earth. Furthermore, he stigmatizes theassessment of development based exclusively on the increase of such factors asthe statistical average of life expectancy, caloric intake, average income, or GrossDomestic Product. This, for Rota, is a way of thinking that considers just one partof the planet and completely neglects other fundamental aspects of human life. 7

The myth of progress would be untenable if different factors were considered,such as the increase in mental illnesses, 8 or free time, or the time parents in the Westdedicate to their children. This untenability would be no less evident if we were toconsider the same quantitative factors, but referred to the countries of the thirdand fourth world. 9

These problems have been staring us in the face for decades, and yet one con-tinues to regard them as momentary dif culties that will be resolved by the veryprogress that has created them, without structurally modifying our mode of pro-ducing, consuming, living, and thinking. Rota attributes this conviction (which in

many respects is irrational) to reductionism, in the sense of an intense desire to beable to understand by oversimplifying and reducing the world to one fundamentallevel (Rota, 1991a, p. 30).

The analysis of the concept of progress and of the philosophies of historyconnected with it is a major theoretical and axiological question that represented apoint of reference for Rota that I cannot go into here. 10 Rather, I intend to show thatRota analyzed the semantic area of the word progress in order to underline theteleological components. In this perspective the concept of progress becomes aphilosohpical interpretation of human life that attens every aspect of existence ona vector drawn towards an end.

4.2 Intentionality Is Not a TeleologyRota opposes this cumbersome primacy of teleology in the interpretation of thegrowth of scienti c knowledge not only on the terrain of history, economics, and

6 Rota (1991a), pp. 2930.7 Ibid., p. 30.8 The question was already raised by Freud; see (19291930).9 For chronological reasons, and for reasons of academic and cultural contiguity, it is probablethat Rotas re ections were in uenced by a famous report by a group of MIT scholars, emphasiz -ing the limits of the current model of industrial development; see Meadows, Meadows, Randers,and Behrens (1972).10 We note that Rota interprets it as a sort of legacy of Jewish monotheism; see Rota (1991a), p. 30.


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tive in perceptive space. 14 In fact the shifting of the viewpoint makes it possible toaccede to parts of an object that were not visible before, on the condition that the

previously focalized parts vanish. Analogously, consciousness can be assimilated toa sort of intentional horizon that delimits and divides a eld of vision composedof backgrounds and foregrounds, of visible, shadow, and blind areas. This pro-duces a distortion that, making it possible to generalize shading to all intentionalphenomenon, allows us to clarify some aspects of Rotas thinking (Rota, 1973a; see1986a, pp. 167173).

Let us focus on the very concepts we have been engaged with here. At pres-ent, the foreground of the readers attention is occupied by a phenomenologicalproblem that represents the object that saturates the intentional structure. In themargins of this eld further themes progressively shade off, while others are out -side the horizon of present consciousness altogether.

Just as the movement of the eyeballs and the bending of the head allow thereader to see the objects located around or behind the book, so the shifting ofhis or her attention makes it possible to make other things explicit; for example,the concept of progress we have previously examined. In this way the content ofthese pages leaves the focus (in the optical sense) of the intentional eld to entermore peripheral areas, until it is concealed in shadow areas that at the moment are

invisible. This route (analogously to the shifting of the glance) is not reducible to anend, nor is it totally active, since other themes can interfere with the argument weattempt to focus on, as a lazy pupilor a tired, or overly curious scholarknowsall too well.

One of the central elements of the generalization that permits us to pass froma perceived to an ideal object is represented precisely by the focus of the eld ofconsciousness that constitutes a sort of foreground of attention. Going back toour parallel with the search for the house keys, we emphasize the fact that this focus(analogously to the glance) can encounter other objects that may be unimportant,or that may be signi cant. In this second case they capture our attention becausethey are intermediate stages of our original search, or because they are useful forother projects. They assert themselves in the intentional horizon even without hav-ing been deliberately sought after, because they have a force of their own.

As universal, the intentional structure also applies to mathematical entitiesthat, like all objects, are grasped through shadings that, from one time to the next,bring out their particular aspects. This accounts for the importance of the differ-ent axiomatizations, proofs, and interpretations of the same mathematical object

that permit the researcher to grasp its unsuspected aspects and properties. Rotasuggests that phenomenology can be considered as the explication of three ideas:as, already and beyond (Rota, 1991a, p. 81). These unexpected aspects and prop -erties express in the domain of mathematics the function of the beyond that14 See Husserl (1913), pp. 4142.


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Chapter Four 92

Rota uses programmatically to characterize thealways necessary and never de-nitivephenomenological overcoming of every perspective limit. In fact, doing

research means becoming aware of prejudices that stop us from seeing what isin front of us. 15

4.3 Teleological Aspects of Mathematical Research The aim of my analysis of nonteleological aspects has been to diminish the ele-ments of nality and design in the horizon of science, in order to understand theirfunction better. In this regard we recall that Rota, in keeping with his antireduction-ist convictions, always speaks of ends and never end and describes the motorof mathematics in terms of a multiplicity of equally legitimate goals. 16 Rota in thisrespect is the heir of a liberal and pluralist tradition, fueled by the texts of Croce,

which he articulates both in the epistemological and the axiological sense, justifyingthe reciprocal in uence between mathematics and philosophy that characterizes hisintellectual biography.

It is not fortuitous that Rotas research, centered on the nite and on the dis -crete, goes against the grain of most of 20th century mathematics, dedicated to thein nite and the continuous. 17 If we interpret multiplicity as the corollary of a dis-crete vision of the real 18 then Rotas antireductionism can be interpreted as a sort

of philosophical parallel of his mathematical research. In fact, it can be maintainedthat in the multiple [...] there is a sort of original polymorphism of reality [...], anoriginal plurality of phenomena. 19

Based on what we have said up to now, it is necessary to articulate this mul-tiplicity further. In the same way in which the end is only a particular case ofthe universal phenomenon of intentional relation, so technological applicability isonly one of the possible senses of end. Consider that the values represented bysimplicity and beauty on the one hand, and speci c problems on the other, can beequally important nalities for mathematical research. 20 Hence it is possible to un-derstand that an articulated axiology sustainsalbeit implicitlyRotas re ectionson mathematical research.

In particular, Rota judges especially signi cant the con dential (or, more pre -cisely, secret )21 list of problems that represents the motivation of every mathemati-

15 Rota (1985a), p. 101.16 Ibid., p. 102.17 Lanciani (2005), p. 13.18 Ibid., p. 14.19 Loc. cit.20 See Rota (1997a), pp. 121133, and Palombi (2004).21 In an article published towards the end of his life Rota lists ten unsolved problems he particu-larly has at heart, even though they are apparently far from his elds of research. He justi es thiscontradiction by saying that the closer to our heart a mathematical theory lies, the harder it is totalk about it (Rota, 1998c, p. 45).


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cians work, and on which any new concept or sector is systematically put to the test.Rota describes this way of working as a full and proper heuristic method, which, in

primis , leads the mathematician to sift the fruits of this constant effort22

through arigorous process of selection based on logic and proof, in order to eliminate all thecrackpot ideas that inevitably arise. 23 Among the results of this complex process(sustained both by the will and by reason) there are many theorems that were laterjudged to be brilliant. 24

This is a phenomenon that has at times been synthesized under the exotic termserendipity, as sagacious observation [...] of one who [...] observes without makingconclusions on the truth content of what one sees , with a suspension of judgment ( Garella,2006, p. 29, emphasis in original). It is a dynamic that involves both a sort of phe -nomenological epoch and a kind of combinatorics of the possible cases. Both fac-tors are particularly important in Rotas investigation.

This esoteric aspect of the work of mathematicians could constitute a partialjusti cation for that sort of heterogenesis of ends thanks to which many interme -diate stages necessary to prove a theorem or to conceive a new sector of mathemat-ics are the resultnot of a comprehensive and conscious effort but, rather, areassembled a posteriori.

To describe this complex historical dynamic we can nd no better com -

parison than the one with the discovery of America, the fruit of a completelydifferent project, which intended to nd a new and shorter route to the Indies.It cannot be denied that the undertaking was pursued with systematicity andrationality; but it must also be said that the new continent was found in anentirely unexpected way. These considerations show why it is misleading tobelieve that

mathematics, and scienti c problems more generally, are formulated and solved inresponse to practical necessities [...]. Apart from science ction novels, events of

this type occur very rarely [...]. What does occur is very different. When a problemof application arises, the technologists throw themselves upon the scienti c litera -ture and go hunting for something that can be of help [...] usually [...] developedelsewhere for totally different reasons, or for no reason at all. (Rota, 1985e, p. 136)

Rota, in support of his thesis, draws up an interesting list of historic results,achieved with no regard for immediate technological objectivessuch things asthe discovery of the fast Fourier transform, of cluster analysis, of formal gram-mars, and of complex numbers (Rota, 1985e, p.137). But, in my opinion, the story

of Professor Smith of the University of Terranovaa pseudonym behind whichRota conceals some fundamental detailsis even more interesting (Rota, 1985e,

22 Rota (1985a), p. 97.23 Ibid., p. 96.24 Ibid., p. 97.


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Chapter Four 94

pp. 135136). 25 Professor Smith is looking for algorithms to subdivide rectanglesinto smaller ones, and publishes the results of thisapparently marginalsec-

tor of research in serious but second-rate journals. The importance of his workchanges dramatically with the advent of the microelectronic revolution, with chipsproduced on silicon wafers thanks to rectangular jigs. At that moment the algo-rithms for constructing rectangles are transformed from a sort of mathematicalgame of little importance, justi ed solely by the value of freedom of research, intoan industrial tool of such great economic value that IBM technicians immediatelyset out on a pilgrimage to Terranova. 26 For Rota the moral of this story is that

the glory of mathematics [...] lies [...] in the exciting and wonderful fact that a

theory developed to tackle a certain type of question often turns out to be theonly way of solving entirely different problems, far from the ones for which thetheory was conceived. These coincidences happen so frequently that they mustnecessarily belong to the deep essence of mathematics, and no philosophy ofmathematics can avoid explaining them. 27

4.4 The Indirect

I am convinced that Rotas challenge represents an important contribution to theepistemology of mathematics. For this reason, in my brief, programmatic conclu-sions, I shall seek to valorize it.

We have seen how mathematics imposes lateral thinking 28 on the researchersince, as Rotafollowing George Polyaaf rms, no mathematical problem isever solved directly (Rota, 1985a, p. 97; see Polya, 1962, p. 132). Our question,now, is the following: what is the general philosophical meaning of this indirect ?First, it presupposes a unitary conception of mathematical reality of a Husserlianstamp (Husserl, 1959, Appendix VI, The Origin of Geometry ). This is indispensableif the research of every scholar, developed from different starting points and dif-ferent directions, is not to be transformed into solipsistic activity. A philosophicinterpretation of this sort is endowed with indubitable heuristic value as is shown,for example, by Langlands program that, proposing a series of conjectures onthe possible connections between disparate areas of mathematics, represented25 Rota insisted that this little story [...] is substantially authentic (ibid., p. 136). For this reasonit seems interesting to me that, according to the opinion of some friends and colleagues ofRota, this pseudonym would hide William Thomas Tutte (19172002), an Anglo-Canadian

mathematician who was one of the greatest specialists in cryptography and graph theory of thelast century.26 Rota (1985e) pp. 13513627 Ibid., p. 22.28 I have extrapolated this expression from Du Sautoy (2003) to render an attitude that I previ -ously referred to as a sidelong glance.


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the hinterland of the proof of Fermats last theorem executed by Wiles (Odi-freddi, 2000a, pp. 2728).

Second, emphasis on the indirect nature of mathematical proof subtends adistinctive context of discovery that it would be interesting to propose in the lightof Bachelards psychoanalysis of objective knowledge. 29 In this respect one mightconsider the proof of a theorem as a need that is primarily intellectual in nature.

The satisfaction of this need stems from a logic of discovery described by Rota asthe determination of the researcher, who un aggingly tackles the problem fromevery possible direction. This self-discipline, on the one hand, possesses an autono-mous value of its own; on the other, it must be understood as the epiphenomenonof deep and continuous work, not controllable by the mathematician who slowlyleads the discovery towards consciousness. 30

This interpretation appears to repropose some aspects of the general neuroti-cizing character of human culture theorized by Freud. 31 In fact we are confronted

with the impossibility of directly satisfying a need whose satisfaction is conditionedby the capacity to wait and to carry out complex and articulated strategies. Thecapacity to defer the satisfaction of a need for a long time, to satisfy it indirectly orsubstitutively, represents a reserve of sense and energy that is speci cally charac -teristic of human culture.

At this point the perniciousness of the strictly economistic interpretation ofscience we noted at the beginning of this chapter has been made clear. The claim tobe able to force scienti c research into an increasingly narrow bureaucratic struc -ture in order to reduce it completely to a planning constituted by objectives, dead-lines, and hierarchies is counterproductive.

Rotas professional experience was that of a mathematician involved in the aca-demic, governmental, and military management of research in the United States;hence he can hardly be accused of humanistic ingenuousness. His defense of theindirect and his criticisms of a widespread interpretation of mathematics bear inmind not only the freedom of research but also the development of technology,the world of value, and that of practice.

29 As a matter of fact, Bachelard limited himself to expressing the need for this undertakingand, rhapsodically, to drawing up an interesting list of historical cases, rather than developinga detailed analysis of a scienti c theory, in reference to a speci c psychoanalytic doctrine. SeeBachelard (1938).30 This perspective is different from the one we have pursued here, following Rota. While I cannot

go into the intricate problem of the relations between phenomenology and psychoanalysis onthis occasion, I do wish to point out a distinction of great importance for our purposes: namely,that the former takes the form of an investigation of consciousness, of which the unconsciousrepresents the edge, while the latter explains consciousness against the background of the un-conscious. I refer the reader interested in a psychoanalytic reading of science to Palombi (2003).31 See Freud (19291930).

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