Francesco BOLDIZZONI - Arnaldo CANZIANI - Portale di Ateneo … · Boccardo, da Edgeworth in poi,...

Università degli Studi Dipartimento didi Brescia Economia Aziendale

Dicembre 2005

Paper numero 50

Francesco BOLDIZZONI - Arnaldo CANZIANI

MATHEMATICS AND ECONOMICS:USE, MISUSE, OR ABUSE?From Walrasian Deductivism

to Demaria, Brambilla and De Finetti

Università degli Studi di BresciaDipartimento di Economia AziendaleContrada Santa Chiara, 50 - 25122 Bresciatel. 030.2988.551-552-553-554 - fax 030.295814e-mail: [email protected]

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MATHEMATICS AND ECONOMICS:

USE, MISUSE, OR ABUSE?

From Walrasian Deductivism to Demaria, Brambilla and De Finetti

by

Francesco BOLDIZZONI Bocconi University

Arnaldo CANZIANI University of Brescia

Paper presented to the annual meeting of ESHET - European Society for the History of Economic Thought,

Stirling (UK), June 9-12, 2005

Summary

Presentazione ............................................................................................ 1

1. On the use of mathematics in economics, or elegant paths to utopia..11

2. The economics of mathematical deductivism ..................................... 14

3. The economics of mathematical descriptivism ................................... 18

4. Pioneering mathematical economics: the algorithms of reality .......... 22

4.1. Demaria: capturing the exogenous world, or indetermination............................................................................ 22

4.2. Brambilla: non-linear fluctuations and inferences ..................... 26

4.3. De Finetti: subjective probabilities against rational choice ....... 30

References ............................................................................................... 35

Mathematics and Economics: Use, Misuse, or Abuse?

Presentazione

1. Il call for papers 2005 (Stirling) della European Society for the

History of Economic Thought intitolava tematicamente <Economics and other Disciplines>.

Così, riflettendo sull’evoluzione di quel pensiero dal periodo della sua formalizzazione – dunque ormai da centocinquant’anni –, immediato si affacciò nella mente a chi scrive il binomio <economia e matematiche>.

Si trattava infatti di un tema così noto sin dai tempi di Cournot (e prima) da essersi ormai tramutato da storico in storiografico. Inoltre, di un tema che aveva attraversato la speculazione economicista degli ultimi due secoli sia con varietà di accenti e di esiti, sia con interrogativi – logici prima che metodologici, inutile qui ricordare v. Mises e altri – relativi alle applicazioni, anzi alla medesima fondatezza di quel binomio. Infine, di un tema dalle prorompenti applicazioni contemporanee, omonime ma eterotassiche, che spaziano dal Modellierung alla teoria dei giochi ai problemi di ottimizzazione (finanziaria, creditizia, assicurativa), dai Kantorovich Memorials all’Istituto ‘Th. Stieltjes’ (Olanda) al CEMI dell’Accademia Russa delle Scienze.

Curiosamente, i papers riconducibili a siffatto soggetto risultarono – al Convegno – di minoranza assoluta se non, sul totale, sporadici: per <altre discipline> s’intendevano talora, infatti, l’antropologia o la psicologia cognitiva o altre ancora, ma non le matematiche. Così, per riprendere una ormai antica osservazione crociana, ci si poteva interrogare sul punto: era quello un problema inavvertito? pejus, problema inesistente? anzi, irrimediabilmente peggio, era forse un problema da tempo superato e risolto, dunque ormai soltanto reliquato e vivente nella falsa immaginazione degli autori?

2. L’ultima ipotesi – che è tuttavia anche risposta e storiografica e

teoretica – potrebbe sembrare la più immediata, e nel contempo diffusa. Da un lato, infatti, il lettore non specialista ha presente la

modellizzazione algoritmica ormai pervasiva di tutte le branche dell’Economia politica, e di alcune in via forse definitiva: trattare di <economia e matematica> potrebbe dunque sembrargli superfluo, se non forse per dibattere dell’opportunità d’uso di uno o altro algoritmo-tipo in

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Francesco Boldizzoni - Arnaldo Canziani

funzione dei differenti sotto-insiemi disciplinarî (ad esempio la teoria dei campi gravitazionali nell’economia spaziale da Adolf Lösch in poi1).

D’altro lato, viceversa, il lettore specialista conosce – dell’Economia politica – la storia degli approcci via via naturalistici, poi formalizzati, infine storicisti, quindi pragmatici o eclettici fino al neo-formalismo da Arrow-Debreu in poi. E probabilmente li legge – quegli sviluppi – in parallelo all’avvenuta evoluzione delle gnoseologie anglosassoni, dal positivismo al pragmatismo agli strascichi del Wiener Kreis al neo-positivismo, dunque quale espressione (anticipazione?) delle stesse.

In sostanza: allo studioso non-specialista il plesso <economia e matematiche> può sembrare problema speciale di campi speciali; mentre allo studioso specialista quel binomio pare antitesi oggi così antica da rassomigliare a reliquia: le matematiche sono ormai consustanziali agli sviluppi moderni dell’Economia politica, e ne esprimono lo statuto disciplinare in modo plasticamente aderente all’epistemologia contemporanea. Una scienza economica, dunque, la quale ha ormai incorporato organicamente in sé la riflessione quantitativa di tipo analitico2.

3. In realtà, le interpretazioni (percezioni?) precedenti paiono vere e false

nel contempo giacché – per continuare con Croce – mal chiarite dal punto di vista speculativo. Il problema infatti – a un’analisi attenta – sembra essere più sottile e complesso, e involgere la natura stessa di un’economia che si voglia scienza. Non per combinazione, del resto, esso è in trattazione da sempre: ab antiquo, quindi da Walras, da Pareto, da v. Bortkiewicz, da Boccardo, da Edgeworth in poi, fino ai contributi contemporanei di L.V. Kantorovich e altri3. E in fondo deriva – lo sappiamo dal tempo di Hume –

1 E.F. Arcangeli, Un riesame dell’Ordine spaziale dell’economia di August

Lösch, <Giornale degli Economisti e annali di Economia>, luglio-agosto 1983, pp. 483-506.

2 Per l’economia quale <econofisica> cfr. ora I. Kondor, J. Krtesz (Eds), Econophysics: An Emerging Science, Dordrecht, Kluwer, 1999.

3 L.V. Kantorovich, Mathematics in Economics: Achievements, Difficulties, Perspectives, Nobel Lecture 1975.

Cfr. inoltre S.C. Dow, The Use of Mathematics in Economics, <ESRC Birmingham Seminar 1999>; Understanding the Relationship between Mathematics and Economics, <The Journal of Post Keynesian Economics>, 2003, issue 4, pp. 547-560; F. Facchini, Usage des mathématiques et scientificité de la science économique, Parigi, <Colloque Gide>, 1999; K. Vela Velupillai, The Unreasonable Ineffectiveness of Mathematics in Economics, Università di Trento, Dipartimento di Economia, Discussion Paper 6-2004; E.R. Weintraub, We Need a Sociology of Mathematical Economics, Editorial, <Economics and Sociology Mail

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dalla presunzione di molti in tema di valore esaustivamente conoscitivo della ragion matematica.

A chi scrive, la situazione oggettiva sembrava dunque – lontano da

unilateralismi metodologici – ben sintetizzabile nel titolo che, forse un po’ provocatoriamente, implicitamente stigmatizzava misuses e abuses quali utilizzazioni ritenute fuori campo – o comunque improprie-incoerenti-inopportune-inesatte – degli strumenti matematici.

Lasciando quindi da banda le trattazioni – interessanti, e talora doverose,

del tipo mathematics for economics –, per le altre si sostiene qui che, prima di svolgere trattazioni monistiche più o meno matematizzate (comunque denominandole <Economia>), una serie di Autori dovrebbe forse interrogarsi sulle fondazioni (nel senso di Grundlagen).

A riguardo del tema, infatti, possa essere o meno l’Economia regina scientiarum, pare doveroso chiarire in via previa:

a. la natura dei fatti economici quale <campo husserliano>4; b. l’individuazione dei problemi economici quale interpretazione

soggettiva di quel campo; c. la formazione delle teoresi economiciste quali proposte – nel

contempo realistiche e nomotetiche – da formularsi con metodi rispondenti alla natura del campo stesso.

Sul punto, per anticipare in forme brevi, ritengono gli scriventi (del resto

aderendo in larga misura a teorie sempiterne) (§ 1): - i fatti economici essere fatti inscindibilmente sociali; - dunque l’Economia essere scienza sociale, da trattarsi con i metodi

sintetici proprî di queste; - ancora, essa partecipare non solo delle Sozialwissenschaften, ma anche

dello status singolare di Naturwissenschaften quali l’astronomia, la meteorologia, la fisica dei quanti, impossibilitate a sperimentare e tuttavia pronte a modellizzare in forme anche statistico-inferenziali.

Ciò impone – per quanto concerne il binomio <matematica ed economia> – di evitare misuses and abuses, e implica dunque la critica sia del descrittivismo matematico in senso latamente keynesiano (§ 3), sia prima

List>, February 2001; How Economics Became a Mathematical Science, Durham, Duke University Press, 2002.

4 E. Husserl, Logische Untersuchungen, Halle a.d.S., Max Niemeyer, vol. I, 19284, passim, ma in particolare le pp. 4-7, nonché 105-109 (nella trad. it. del Saggiatore, per i tipi di Arnoldo Mondadori, Milano 19882, vol. I, pp. 24-26 e 121-124).

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ancora del deduttivismo matematico di tipo Walras-Pareto, e di quest’ultimo il rifiuto (§ 2).

Quelle premesse comportano viceversa l’adozione di uses, cioè degli ‘algoritmi del realismo’, ricondotti in questo paper al pensiero di tre famosi studiosi italiani, non importa quanto noti nel mondo anglosassone (§ 4): Giovanni Demaria (per il trattamento dell’indeterminazione dinamica, 4.1.), Francesco Brambilla (per le inferenze non-lineari, 4.2.), Bruno De Finetti (per le probabilità soggettive 4.3.).

4. Il trattamento del descrittivismo e del deduttivismo matematici è –

come si diceva – differenziato, e più severo per il secondo. Mentre infatti a livello introduttivo il primo comunque individua rilevanti connessioni pragmatiche – salvo poi trattarle in modo altamente discutibile, e in alcuni casi erroneo –, il deduttivismo applica viceversa il logo univocamente astratto delle scienze fisico-matematiche ad un campo sociale.

E poiché quest’ultimo approccio, forse per la sua facilità argomentativa, non cessa di richiamare adepti (alcuni dei quali insigniti del Nobel), pare opportuno chiarire brevemente nel seguito alcune fra le ragioni di quel rifiuto.

Non dunque impuntature di (ex) giovinastri insolenti, ma causali logiche e metodologiche, radicate nella riflessione economicista classica, e nondimeno nel pensiero filosofico antico come ripreso poi – dal punto di vista gnoseologico – dall’idealismo italiano. Ragioni la cui presentazione contribuisce dunque a chiarire che chi scrive è anzi in favore della formalizzazione matematico-statistica – gli uses citati –, ma contrario all’uso povero e dogmatico della stessa, in forme non importa se andanti o sofisticate5.

Negli impianti walras-paretiani tradizionali, infatti – sintetizzando assai –

l’economia sociale viene intesa a-temporalmente, e rinserrata in limitati sistemi lineari, quando non in un’unica y = f (x) comunque differenziabile.

Così, il discorso economico che ne deriva perde la propria natura multiforme e i propri nessi prammatici. Quella teoresi tratta infatti tutte le variabili indipendenti come una sola, e impone loro gli andamenti intrinseci all’algoritmo prescelto, obbliganti nello stesso: si dice <Economia>, ma è solamente un’applicazione – talora altamente irrazionale – di matematiche non di rado elementari.

5 Cfr. e.g. M.A. Khan and N.C. Yannelis (Eds), Equilibrium Theory in Infinite

Dimensional Spaces (‘Studies in Economic Theory’, vol. I), Berlin, Springer-Verlag, 1991.

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A questo punto non importa censurare l’anti-realismo degli esiti: è sufficiente prima ancora riconoscere le soluzioni quali pseudo-soluzioni, rinchiuse all’interno del puro piano algoritmico loro proprio.

Così intesa, quell’Economia politica si dissolve nella forma fantastica d’un matematismo con parvenza economica. Esatto, indiscutibile e indubbiamente elegante nei propri confini, in quanto riferito a una scienza sociale e di applicazione esso risulta – nelle premesse e conclusioni – perennemente implausibile.

Dal punto di vista scientifico, infatti, la scuola neoclassica è scuola solo fintamente speculativa: per quante domande si ponga, essa già ne conosce – o almeno ne antivede – le risposte, dogmaticamente rinchiuse ex ante nelle soluzioni degli algoritmi che si utilizzeranno.

Si tratta appunto di un <sapere concluso> sin dal tempo della sua nascita, cioè ormai dal tempo di Cournot; si tratta insomma di un’<economia-conclusione> che, come la filosofia di Cartesio:

teorizza solo all’interno di un sistema già tutto dispiegato, ed esprime la domanda soltanto perché crede di essere in possesso della risposta. La sua filosofia è, dunque, nel sapere compiuto, dove non vive più il problema, né il dubbio, né il provvisorio. Anzi lo svolgimento del pensiero di Cartesio fa legittimamente ritenere che il dubbio sia stato teorizzato a posteriori, in funzione della rinnovata certezza del cogito 6.

In realtà, l’economia neoclassica non vuole conoscere la realtà empirico-sociale, né i problemi fenomenici: li riforma ambedue. Pseudo-realtà e pseudo-problemi dunque, immaginati solo per poter applicare gli algoritmi ex ante prescelti. Si ottiene così una conoscenza esatta, ferrea, definitiva, rifugio dello scetticismo conoscitivo che si abbranca ad a priori: ne derivano convincimenti di impronta profondamente mistica, esiti di tipo inevitabilmente dogmatico.

5. Certo, a questo punto nasce, fra gli altri, un problema ulteriore. Si

sostiene infatti che nelle matematiche moderne – grosso modo dopo Hilbert, dunque con l’assiomatizzazione –, rilevino semmai i) la coerenza deduttiva, ii) la potenza generalizzatrice, e pertanto iii) le relazioni e non le quantità. Altrettanto dunque nelle economie (matematiche) da Neumann-Morgenstern

6 U. Spirito, Il problema della filosofia, oggi: la filosofia come teoreticità pura,

in AA.VV., Atti del XVI Congresso Nazionale di Filosofia promosso dalla Società Filosofica Italiana – Il problema della filosofia oggi, Roma-Milano, Fratelli Bocca Editori, 1953, pp. 86-92, alle pp. 86-87 (con abbreviazioni).

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a Debreu7, al di là del potere esplicativo e del problema meramente eventuale del realismo.

Ahinoi, nihil sub sole novi: si tratta soltanto della forma ulteriore – più sofisticata? – dell’ipostatizzazione precedente. L’annotazione di cui sopra sarebbe infatti salvifica se reggesse all’obiezione che dice trattarsi – nel campo husserliano in discorso – non dell’interna coerenza degli a priori astratti e delle loro conseguenze altrettanto astratte, quanto piuttosto dell’applicazione di quelli a una scienza empirica e sociale, dunque di conseguenze che rimangono astratte nonché anti-realistiche proprio quando dovrebbero risultare empiriche, e se del caso nomotetiche. (Gli assiomatici del resto, mentre tacciano quest’ultimo problema di inesistente, nel contempo tuttavia non si peritano – quelle deduzioni anti-realistiche – di trasformarle in pratiche, sic).

Quand’anche infine quell’ipotesi si volesse accettare, rimarrebbe comunque il problema logico del rinserramento nel logo astratto: torna, <più grande e più bella che pria>, l’ipostasi citata più sopra, che <cancella l’alterità del reale> (ancora Spirito) e si rinchiude nel monismo spirituale delle proprie forme, questa volta assiomatiche.

6. Riguardo ai citati problemi, e peraltro sulla base di teoresi note, si è

dunque desiderato esporre posizioni sia critiche sia propositive – certo discutibili, ma ben ferme – in sede d’un Convegno specialistico che sembrava appunto adatto ad accoglierle, giacché intitolava L’economia e le altre discipline. Posizioni le quali riprendono alcune fondate critiche a impostazioni diffuse per collegarsi – riproponendoli – a interi sistemi teoretici originali, non per combinazione italiani. E nel largo disinteresse della speculazione contemporanea per tali approcci è in parte consolante che la medesima prolusione al Convegno in parola si intitolasse The Social Science of Economics8, come pure che si sia or ora autorevolmente sostenuta le necessità di riformare ab imis molta dell’Economia politica contemporanea9.

7 Del quale, sul tema, cfr. da ultimo The Mathematization of Economic Theory,

<American Economic Review>, 1991, issue 1, pp. 1-7. 8 B.J. Loasby, The social Science of Economics, 9th ESHET Conference,

University of Stirling, June 9th, 2005. Del medesimo Autore cfr. Cognition and Markets, Manchester, <CRIC Workshop>, 2000; Time, Knowledge, and evolutionary Dynamics: Why Connections Matter, <Journal of Evolutionary Economics>, 2001, issue 4, pp. 393-412

9 K.W. Rothschild, New Worlds – New Approaches: A Note on Future Research Strategies, <Kyklos>, 2005, issue 3, pp. 439-447.

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Quelle ipotesi speculative vengono qui presentate come tali, in forma semplicemente ripulita e prefata: non per rigidità intellettuale o per amor sui, ma appunto quale work in progess, nonché per proporne originalità e fondatezza al Lettore nell’attesa, tempo permettendo, della successiva puntata.

Milano, Università Commerciale “L. Bocconi” Brescia, Università degli Studi gli autori*

* Pur frutto d’un comune pensiero, le parti del paper che segue possono venire

così attribuite: i paragrafi 1., 4.2., 4.3. ad Arnaldo Canziani; i paragrafi 2., 3., 4.1. a Francesco Boldizzoni.

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The human understanding is of its nature prone to suppose the existence

of more order and regularity in the world than it finds.

(Francis Bacon, Novum Organum, 1620, t. I)

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1. On the use of mathematics in economics, or elegant paths to utopia

Were we Kuhnians or not, and due to this maybe doubtful followers of the evolution-revolution model, nevertheless the history of economics during the nineteenth century is normally described according to juxtaposed phases, i.e. classical v. marginal (or neo-classical) economics.

To follow in scholars’ tracks, thus accepting descriptions even if stylized ones, the core of the marginalist revolution is currently identified by:

1. the transition from an ‘objective’ theory of value to its ‘subjective’

determination through marginal utility; 2. the instrumental adoption of mathematical language within

economics.

Now, to be true nor particularly parochial, both these approaches were already traced back to Italian thinkers of the 16th, 17th and 18th centuries, respectively Davanzati and Galiani as far as subjective values are concerned (Bianchini 2002, Boldizzoni 2005, not to mention Schumpeter 1954), Ceva and others for the use of mathematics (Bianchini 1977).

Far from philologism, those authors are quoted here as they represent the very first phase of a peculiar attitude, i.e. the use of mathematics to express (to substantiate?) the economic reasoning: a well-known way of deducting largely utilized before, within, after, and independently of utilitarianism.

The mathematical approach to economics – better, to modelling of economic facts – is anyway something deeper and more involving: i) it enters the world of economics in the early eighteenth century, ii) it reaches its first top in the period 1871-1911 (from Jevons’ Theory of Political Economy to Pareto’s Économie mathématique – see sect. 2); iii) it develops afterwards in very different ways with the leaders of the economic profession from Wicksell to Cassel to Frisch to our days.

After the collapse of marginalism in fact – maybe when the same

Vilfredo Pareto turns to sociology having got the consciousness (1906-07) of the <pure> economics being an absurd goal –, a different use of mathematics enters the field: still a deductive one, but under less stylized (fancy?) apriori assumptions. A more descriptive use therefore appears as i) the intent was to move from (more) realistic approaches, ii) the ultimate goal was to reach a factual, workable description of the functioning of economic world instead of the abstract ofelimity of the single agent.

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This way, after the First World War – that is to say after the age of Second Industrial Revolution – the need for a larger realism increased, not by a chance giving origin to the crumbling away of the neo-classical orthodoxy after the critiques and the new outcomes of P. Sraffa (1925-26), E. Chamberlin, J. Robinson, J.M. Keynes, M. Kalecki, R. Harrod. No wonder that this phase is normally recalled under Shackle’s well-known label of <years of high theory> (sect. 3).

Evolutions and revolutions follow anyway each other: after World War II

economics opened to a new era, the renewed axiomatic, anti-Keynesian approach of Arrow and Debreu (1954).

After some fifty years, this approach still represents the mainstream as far as mathematical economics is concerned, while its apriori have been also recently resumed by the same Gérard Debreu as follows:

Being denied a sufficiently secure experimental base, economic theory has to adhere to the rules of logical discourse and must renounce the facility of internal inconsistency. A deductive structure that tolerates a contradiction does so under the penalty of being useless since any statement can be derived flawlessly and immediately from that contradiction. In its mathematical form, economic theory is open to an efficient scrutiny for logical errors (1991, p. 3).

The aforementioned quotation lays itself open to the following –

objective nor subjective – critiques, relevant here as largerly referable to the whole vulgata of this stream:

1. the rules of logical discourse – of a Wittgenstein-type or not – should be obeyed in any case, building over <sufficient> or <insufficient> experimental bases as a scientific pre-requisite; in the absence of which we could land on false conclusions – inconsistent ones – also in case of large, iper-sufficient experimental bases, the former being the so-called <false inferences> or, that is the same, <the theories of the false experience>; in a word, consistency has to be considered a tool and not an aim in itself, i.e. a <necessary-but-not-sufficient> condition;

2. in addition, to speak of <insufficient> experimental base confesses the unconscious apriori of the Naturwissenschaften, which are based on the <experiment> (this anyway only according to common people’s vulgata, as nor astronomy nor meteorology nor in a way quantum physics are able to <experiment>);

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3. in the field of economics – application science – the core problem does not consist in <experiments> but, on the contrary, in the fair adherence to facts, to be studied through the special methods of the Sozialwissenschaften, variously rooted in such epistemic logical systems as Aristotle’s, Kant’s, Husserl’s or others (only one-sided scholars can imagine that solely mathematical deduction can assure us about consistency, only inexperienced or unpractised ones can think themselves capable to ignore the methods proper of social sciences).

And to end with a not so lateral addition, a possible psychanalitical

interpretation of the exclusively mathematical approach to problems in the field of economics could be easily proposed.

It refers to the scholars we spoke about, scholars not by a chance of technical an imprint and mathematical a culture. Due to education in fact, and training as well – Walras was an astronomer, Pareto an engineer, Jevons and Edgeworth chemists, and so on – they had got the habit of defining empirical goals – a combination of elements – reaching them deductively step by step.

The same they did reducing the field of the economy to a natural one, shutting out the true nature of the world while reducing it to one (two) variables, squeezing the relations between them in merely linear (or second order) equations in the absence of time, and at the end of the story resolving these equations.

This way they were able – in short – to walk down the paths they were formed in: step by step deductivism, logical validity, sure conclusions, solutions to be imposed inasmuch they were forcely <true> ones. They were ending where they had (un)consciously projected to end, i.e. where they started. From their own mathematical point of view they became able to be true, sure, logical, consistent and so on, spite of the fact that – what a shame – the world was quite different from both their own theories and the image they had got of it.

According to the proposals of this paper, and spite of the aforementioned

critiques, the use of mathematics in the field of economics is anyway of overwhealming an importance (sect. 4), provided we do not speak of Walras’ algebra, nor of Pareto’s mechanics nor of (pure) axiomatism in its old or renewed forms.

Provided we speak, on the contrary, of a social science with its own logic, methods, practice and consistencies. Within economics, therefore, special branches of mathematics are fundamental additive (in some cases

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explorative) tools, inasmuch they only are in a position to propel us to such further frontiers as no qualitative imagination can make us to achieve with equivalent accurateness.

Provided we speak, in addition, of such special tools – among the so many possible from Georgescu-Roegen to Palomba – able to permit us to treat such peculiar problems as subjective probabilities (De Finetti, sect. 4.1), dynamic indeterminacy (Demaria, sect. 4.2), non-linear fluctuations (Brambilla, sect. 4.3), and so on.

2. The economics of mathematical deductivism

In the first decades of the nineteenth century, economics records the inklings of the break in continuity with the Smith-Ricardo orthodoxy. A Smithian topos is resumed (the exchange), anyway changing its intrinsic nature: from the objective value to a subjective approach.

The exchange, prior to society according to Adam Smith, is now vice versa defined in its equilibria by two joint utilities (Auguste Walras), as it will be defined later on by the junction of marginal utilities up to their equivalence.

The plenty of researches, and methods as well, by which this problem was tackled, was such as to make it understood – in those very years – as the very problem of economic science.

Anyway, to be true, two crucial problems (at least) can be individuated within it:

1. do the (reputed) equilibria-prices effectively represent the core

problem of economic science? 2. more specifically, can the individuation of the optimum – as regards

the determination of all prices for every commodity in every exchange-process of all the markets of the entire world – be truly defined a <scientific> problem?

1) The progress of sciences certainly depends on the type of core

problems the leading scholars individuated in the climate of their ages, the problems i.e. they choosed as their own Lebenswerk. Anyway, once fronted with the crucial problems of economic science from Aristotle onwards – production, distribution, cycle, and later on growth – we can ask ourselves a twofold question: has the formation of (hypothetical) prices of exchange-equilibrium between (or among) individuals the same relevance? Have its solutions a heuristic value?

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Very likely, both its choice and its over-evaluation arose from a specific reason, a simultaneously historical, social and ideological one: in 1867 the first volume of Das Kapital had appeared, where the <value> stood for the pattern of social injustice.

Now, Léon Walras and Vilfredo Pareto were rooted in a bourgeois environment, the first of them in addition having been influenced by his father, and by Cournot as well. It is so easy to understand why, from their own point of view, the term <value> – and its concept too – was to be removed substituting it, better, transforming it from the scientific standpoint.

From the epistemic point of view in general, as well as from the point of view of the Weltanschauungen 1870-1900, which better tool than mathematics in order to make the concept a scientific one? Thanks to it, a new <theory of value> could be proposed once established on <utility>.

In addition, these renewed foundations seemed to be in accordance with the Second Industrial Revolution running in those very years. Thanks to this, a relative plenty of both capitals and commodities were now at disposal, which made relevant also from the material point of view the problem of choice among alternatives.

2) No doubt the scientific-making process is always – partly – a

conventional one: no scientific problem in fact was born as such. However, we could ask ourselves if the marginal solution to the problem of equilibrium-prices was not a too-much conventional one. One one side it lacked real uniformity, on the other it had too much of it (of course a merely conventional one).

Neo-classical conventionalism imposes in fact from inside too much stylized uniformity to the model: due to the so many restrictive conditions, scholars come to exact but untrue theories inasmuch as they were established on anti-realistic bases: they were, in short, purely irrational ones.

Scholars fully know that, moving from the restrictive conditions of Walras, the continuation of the story recorded anyway continuous revivals or changes. If on one side Pareto interrogates himself on the point, choosing after well-known trials to abandon the field of economics, others (Arrow and Debreu 1954 e.g.) in resuming axiomatism gave it a larger perimeter, the proof of the fundamental properties of equilibium included.

Pareto turning to sociology means a gallant failure, but also Debreu’s trial (1959) seems to be undermined by highly conditioning, anti-realistic premisses. His solution consists in fact not only in multiple restrictive conditions, but also in <particularly irksome> ones (Screpanti and Zamagni

15


1989, p. 343). Among them, with special reference to existence, the following ones:

As contracts are stipulated in the present, for all commodities produced and delivered not only in the present but also in the future, futures markets must exist for the commodities available in all future periods – a hypothesis of which it is not even worth raising problems of realism […]. Besides this, it is necessary to assume that economic agents are endowed with perfect foresight, as they know with precision, as consumers, the future evolution of their own preferences and, as producers, the future evolution of technology (ibidem).

Thus, he tried to avoid this peculiarity by introducing the notion of uncertainty, but he did so in a way which was no less odd, i.e. by attributing a further specification to goods, one relating to the ‘state of the world’. Thus, for example, a sack of corn available here and now would be a different commodity, not only from that available in another city now, or from that available here tomorrow, but also from that available here tomorrow in the case that there is an earthquake or any other act of God tonight (ibidem).

The further problem of uniqueness (1970, 1976) was so-to-say solved by Debreu stating that most economies are <regular> ones, i.e. characterized by a discrete and finite set of equilibria. Provided that the suggested frequence is correct, to fulfil that hypothesis it should be anyhow necessary the following conditions to exist:

1. individual demand-functions are differentiable; 2. every commodity is desirable.

Now the hypothesis of differentiability seems rather brave. Economists are used to treating it as if it were normal, but this does not mean it is sensible; it only means that the economist’s education is generally successful in developing special gifts in the suspension of his critical faculties. First, the differentiability hypothesis presupposes that individuals are able to formulate a precise demand, for example with regard to the variation in the price of cars, not only for any number of cars but for any fraction of them; then, even worse, it implies that it is possible to determine the rate of variation of the demand for cars corresponding to every infinitesimal variation of price. And even more ridicolous, if

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possible, is the hypothesis of desirability; which implies that, for example, when the price of water approaches zero, individuals will tend to voluntarily drown themselves or to try to hoard the seas (Screpanti and Zamagni 1989, p. 347).

These remarks are so well-known ones as to be presented since decades in textbooks. To these critiques, peculiar to mathematical analysis, other have been anyway added (see e.g. sect. 4.1), presented in one or other form in various moments of the history of economic thought, and particularly referred to the use of equations within the economic field:

1. the searchers of economic optima are perfect agents inasmuch as

they are one-sided ones, i.e. fictious; 2. maxima (and minima) are optima from the economic point of view

only within steady-state and further limitative conditions; 3. individual optima differ from social ones.

Given these (and others) well-known flaws, the aforementioned suspect

(see sect. 1) gains legitimacy: the introduction of economic variables and concepts into the properties of adopted algorithms is a device – in some cases a dodge – to impose the Weltanschauungen characteristic of engineers, and their toolbox as well, by the sub-system of them converted to economics. As a result, the equilibria they propose – or impose? – are deceptive ones from the descriptive point of view as well as the prescriptive one. Due to individual behaviours, to collective actions, to the passing of time and changing of external conditions, those equilibria do not have in fact the static nature of an arch, a vault or a bridge, nor the properties neither the exact definableness of their equilibrium of forces.

This predicament in knowledge has an unquestionable source. Their logic in fact, which comes from their breeding ground, is largely alien to economic reasoning, as it hinges phenomenologically upon demonstration and its steps. Due to this i) problems are solved, ii) generalizations are suggested, iii) results are predicted, sad to say by mathematical tools not at the level of mastering reality.

The result is a logic (an algebra, an analysis) void of raison d’etre bringing forward wrong empirical results. No wonder anyway: due to these premisses those economical engineers usually behave in fact as abstract <social constructivists> as they are – after all or first of all? – non-Galilean practical prescribers. Different solutions could be anyway impossible ones, once that mentality is used out of its own field into others, where it continues to apply the equilibria equations of mechanics, the system of equations of rational mechanics.

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On the contrary, <pure> mathematicians in the sense of Cournot and Marshall use the mathematical instruments as an argumentative technique, neutral with respect to goals but at a level of assuring deductive rigour. They are clear-eyed critical about the intrinsic utility of the instruments, and not so carried away with the Newtonian approach of the application and equilibrium of forces.

Other social scientists in addition – that is a confirmation – elaborated an organic theory of distribution skipping the theory of equilibrium, an essential step vice versa according to its proposers (see sect. 3).

3. The economics of mathematical descriptivism

Differently from Edgeworth and others, the trial is accomplished in Marshall – for some decades at least – to merge economic theory and facts especially in the form of history. The result was a set of <stylized facts> – among them the representative firm, the inter-temporal industrial equilibria, the Marshallian cross –, which allowed Marshall to be not as deductive as the marginalists had been, nor entirely empirical: the result was a <conventional descriptivism>. It was maybe due to these reasons that, in a famous letter we all know, he assigned to mathematics the pure role of assuring consistency as a pre-requisite. Within his system, he does not need a foundative mathematics, this not being the centre of his universe: for example he builds the demand curve on empirical foundations, without recurring to any utility concept whatsoever.

Reactions and critiques to different pillars of the Marshallian edifice tend to emerge as early as in the early twenties, Marshall still alive.

In his article of 1922 on Empty Economic Boxes, J. Clapham put under spot-light the irrelevance of Marshall’s increasing-constant-decreasing costs for applied economics. Just three years later, P. Sraffa targeted on the other hand the logical consistency of Marshall’s theories, particularly underlining that competitive equilibrium cannot escape the dilemma of being either uncoherent or irrelevant.

Following these critiques, also Marshall’s former followers started to shell the Marshallian edifice, better, to build differently in the field of both micro- and macro-economics.

From the microeconomics standpoint (Robinson 1933) they criticized in particular:

1. the consistency of the concept of representative firm, 2. the modern inappropriateness of product homogeneity,

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3. the truthfulness – especially after the First World War – of the perfect competition.

Building on juxtaposed concepts to Marshallian ones – i.e. recognizing product differentiation, accepting firms’ asymmetries, cancelling any <representative> firm whatsoever –, the revolutionary concept of market power slowly caught on within economics: it can with no doubt being interpreted in an endogenous way (or traced back to some form of endogeneity, e.g. collusion), but it can on the other side give life to a whole new economic approach as in Pouvoir et économie by F. Perroux (1973).

Spite of the different conclusions they reached – or proposed –, Marshall’s disciples marched in the master’s steps from the methodological point of view, especially as far as mathematical economics is concerned. They accepted in fact Marshall’s invitation to burn out the mathematical framework once satisfied the basic requirement of consistency, going rarely beyond the use of analytical algebra and geometry.

Joan Robinson, J.M. Keynes and others – while criticizing Marshall – followed him also in a further methodological approach: the need for realism. Sraffa’s and Sraffian critiques to partial competitive equilibrium, largely based on its unique validity in the case of constant costs (in which anyway conclusions are similar to the classical ones) did not carry the Cambridge microeconomic theory back to Walras and the general equilibrium theory: those critiques drove the first Marshallians farther.

From the macroeconomics standpoint (J.M. Keynes, R. Harrod, and the Keynesian revolution in general) they crowded out the very foundations of the spontaneous equilibrium between aggregate supply and demand, a blueprint keeping its own place since J.B. Say (and maybe A. Smith) onwards. An innovative macroeconomic approach led over the Second World War to our days by the same J. Robinson, by N. Kaldor, by M. Kalecki, and in general by the whole galaxy of post-Keynesian economists (Canziani 1977).

In both cases, mathematics proved anyway to be a purely descriptive tool.

This attitude can be expressed by a plenty of well-known examples, hereby focussed on the three main topics of economics during the <years of high theory>:

a) multiplier, b) accelerator, c) the search for a <golden> rule.

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A. Multiplier

Given ΔC = cΔY 0 < c < 1 [1] and ΔY = ΔA + ΔC [2] substituting [1] into [2] we have: ΔY = (1 / 1 – c) ΔA [3]

The mathematical bulk of the multiplier-principle due to Lord Kahn is

therefore a non-linear but simple finite difference equation, the elegant way to express in its own nature the geometric progression of Ancient Greek mathematicians. Spite of its simplicity, or maybe due to this, this tool is anyway a powerful one: thanks to its descriptive strenght, it makes an appeal to some of our most hidden and ancestral intuitions.

B. Accelerator

I = a ΔY* [4] where a = Kt / Yt , 0 < a < 1 [5]

Once more, the mathematical bulk of Aftalion’s principle is a finite

difference equation: it keeps its own powerfulness, spite of the rather fancy representation of firms’ behaviour. Entrepreneurs, in fact, can satisfy (expected) increases in demand ΔY* first increasing to 100% the rate of machinery-exploitment, second by recalling to use wholly depreciated machineries, third recurring to agreements with other firms (and so on) before investing. In addition, they generally decide to invest only in case the future demand-trend is defined, in this process being forerunners, or on due time, or followers from case to case (Canziani 2005).

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C. Towards the golden rule

Given: St = sYt where s = 1 – c [6] It = aΔYt* , [7] once supposed St = It [8] and the equivalence between expected and effective demand, we have: sYt = aΔYt* ≡ aΔYt [9]

from which follows:

G = ΔYt/Yt = s / a . [10]

In this case too, the underpinning is given by a finite difference equation applied to complex economic phenomena.

We could here once-for-all ask ourselves if the economy is expressed in

true analytical terms. Or do in case the adopted mathematical tools superimpose a rigid structure, determined by a fixed set of constraints? Under (C) e.g., in fact, we suppose:

• the pure endogeneity of the model, tautologically determined (Y is

conditioned by s which is a function of Y); • a and s to be independent variables, a in addition giving room for the

aforementioned critiques; • the stableness of both s and a, the former due to the exixtence of two

only social classes (defined by fixed behaviours), the latter due to its instantaneous determination for every entrepreneur of every firm of every size in every industry.

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4. Pioneering mathematical economics: the algorithms of reality

4.1. Demaria: capturing the exogenous world, or indetermination Strongly influenced – and motivated – by the scientific revolutions of the

early twentieth century as well as by the German heritage of the historical school, Giovanni Demaria (1899-1999) graduated in Turin and specialized himself later on in Venice University under Gustavo Del Vecchio.

Particularly, curiously gifted in both mathemaics and history, he was deeply interested in the search for economic (and social) uniformities, expressed by advanced mathematical tools.

It was maybe due to the influence of Del Vecchio his interpretation of the economy as the dynamics of economic systems, that is to say its never-ending change within different historical periods. He tried in fact for all his life-long to capture through his approaches the structural constants as well as the novelties of change, joining them both in his models. As a result he placed indetermination as the very premise of our studies in the field of economics, that one being a three-faced one: logical, dynamic, permanent.

1. Logic indetermination is defined as the cognitive limit stemming

from the human impossibility/incapacity to observe facts with such a precision as to be able to establish objective links among them;

2. dynamic indetermination means the impossibility of describing the (economic) future as the joint result of both past and present, due to the emerging of sponaneous novelties or <creative originalities> (the best known among them being the <creative acts> proposed by Schumpeter);

3. permanent indetermination, endemically pervading economic systems in the long period, due to the fact that <Economics is not an autonomous science. Economic factors and variables alone are not enough to determine the properties of the system and the equilibrium solutions of the problem. Other largely non-economic variables must be introduced into the model> (Demaria 1996, p. 19).

This way, indetermination exists due to the fact that the economic world

is pervaded and dominated by exogeneity, that is to say the whole sphere of natural, moral, social, cultural, political variables influencing the economic behaviour of agents. Economists must take them into account in one way or another, in order not to falsify the reality: principles as <consumers maximize utility> or <growth is established on the accumulation of capital> – logical and fitting the pure economic behaviour according to Pareto – are,

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to be true, neither <pure> nor <neutral>, as they undermean on the contrary specific a priori assumptions as regards the social structure or the personal psychological nature. This could be defined the illusion of economics. Either we make explicit such reductive hypotheses, or we land in the sea of arbitrarity.

Similar critiques to the so-called orthodox theories (neo-classical ones or not) were in those very years largely widespread, before the new shutting of most economics in the cage of axioms and formulae. Gunnar Myrdal said among others:

Economic theorists […] have long been disposed to arrive at general propositions and then postulate them as valid for every time, place, and culture. There is a tendency in contemporary economic theory to follow this path to the extreme. For such confidence in the constructs of economic reasoning, there is no empirical justification […]. As long as their use is restricted to our part of the world this pretense of generality may do little harm. But when theories and concepts designed to fit the special conditions of the Western world […] are used in the study of underdeveloped countries in South Asia, where they do not fit, the consequences are serious (1968, vol. I, pp. 16-17).

Economics – in order not to be false or purely fancy as the neo-classical

theories are – has one only alternative: a) the description from time to time of portions of economic reality, each time clarifying the limitative hypotheses they are based on (this choice has been adopted by both Keynesians on one side, by German historical school and American institutionalism on the other); b) the transformation to science through the search for meta-economic categories:

establish our scientific reflections on universalia means looking for an economic science more complete and less unoperational than the today-one, rich in temporarily descriptions, artificial taxonomies, useless methods, sillogisms which cover, to be true, mere identities. Economics, if merely an endogenous one, tends to be secluded within special, arbitrary premises, and crowded by purely mental nor realistic constructs: its end is only contingency, or phantasy. Its sub-scientific status depends on its lack of universalia (Demaria 1974, p. 15).

Such a project needed for a substantial and massive approach, for such a

conclusion as the huge building of Demaria’s Trattato di logica economica (1962-1974).

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To include extra-economic variables gives us anyway a substantial problem: those being of qualitative a nature, they are not quantifiable ones. We can anyway quantify their effects on the endogenous world (increases, decreases, change of sign), that is to say the variables economists are interested in. To study power-structures in fact, or cultural latencies, or political systems belongs in fact to sociology, anthropology, political science: economists are merely interested in the effects of them – and their changes – on economic systems as such.

This way, Demaria chooses a mathematical system (later called DGS, Demaria General System) able to both a) describe the endogenous quantities of the economy, b) be opened to exogenous influences through <degrees of freedom> in the technical meaning of the word. It combines some 30 endogenous variables and a double set of exogenous ones, permanent and erratic (propagatori and entelechiani respectively): the former presenting the structure of the economic system, the latter the influences and changes exogenously produced.

Just to give a rough idea of the functioning of the system, we resume here a parallel model related to development economics (Boldizzoni 2000).

The degree of <mechanization> of economies (the so-called capital/labour ratio) can be expressed by the following equation, function of ρ, parametric in a and b:

)1(/ ρaebLKk −−== ),1( +∞∈ρ , , 0>a 0>b

where ρ represents the coefficient of <aim-oriented rationality> (Max Weber) of a given society at a given time, that is to say a synthetic index of its modernization. It is normally modified by the factor a, while b represents on the contrary the basic propension to substitute labour with capital (it is inversely proportional, within modern capitalistic economies, to the level of union protection 1/b).

With modernization – and its parallel rationalization (the governance over techniques included) – societies tend to substitute capital to work due to well-known advantages, or necessities or hopes; it will be anyway their cultures to determine the quantum of such a process. Just to give a case, b can be high in today United States of America due to the acceptance of technological unemployment (or to its <frictionality>), due at the same time to the non-acceptance of social protection costs payed through productive inefficiency (and vice-versa for many countries of Europe, were unions have a substantial role in slowing the rate of automatization, or had).

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Fig. 1 – k-values for a=1 and several values of b

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0,01 1 2 3 4 5 6 7 8 9 10

ρ

k=K

/L

b=1b=2b=4

Generally speaking, on the other hand, Demaria’s system represents – as

a proposal at least – a global framework, ex ante solving the whole economy once dipped within history. As we mentioned before, Demaria’s mathematical system (DGS) includes some 30 endogenous variables and a double set of exogenous ones, both permanent and erratic:

• the former present the skeleton of the economic system, the emerging

and visible structure of economic processes of production-distribution-consumption of wealth (i.e. quantities of productive factors; their costs; prices; monetary values and so on);

• the latter express a) the structural – or long-period – dynamics of societies (demography, psychology, technology, politics); b) the sudden and apparently casual shocks hitting the economies: natural disasters, wars, pandemies, financial crises, and so on.

This system, in Demaria’s words, is characterized – differently with respect to others, in particular with common deductive-deterministic and econometric modelling – by three theoretical requirements:

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1) recognition of the secondary role of endogenous economic variables in changes in the dimension and causality of economic life; 2) indisputable superiority of the exogenous variables of propagation in determining the dimension and orientation of endogenous economic variables; 3) inclusion of exogenous shock variables to explain the violent positive and negative growth processes and the originality of economic life as well as the non-homogeneity of its monetary structure over time (Montesano 1978, p. 709, summarized).

The DGS is given by Demaria in a first [...] approximation in the form of linear algebraic systems: a11x1 + a12x2 + … + a1nxn = b1a21x1 + a22x2 + … + a2nxn = b2……………………………… an1x1 + an2x2 + … + annxn = bn or else in the matrix form: AX = B where A is the matrix of the coefficients, that is of the aij’s expressing the effect of the propagatori on the respective xj’s, X is the column vector of the unknown xj’s expressing the endogenous macro variables, and B is the column vector <known terms> expressing the unforseeable movement, but ascertainable a posteriori, of the exogenous shock variables. […] This system is not stochastic since it does not take into consideration random effects. Neither does it take into consideration the lags or leads which characterize the actions of exogenous variables, sometimes in ways which cannot be ignored (ibid., p. 710).

4.2. Brambilla: non-linear fluctuations and inferences According to Bayesians, in the world of statistics frequentism-type

analyses represent in so many cases a second (third?) best, as if they were merely documentary, or mechanical ones.

As a matter of fact, frequentism – and near every scientific tool as well – can be sublimated in <comprehending categories> of higher and higher heuristic value. Spite of the condescension by which it is sometimes treated,

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frequentism in fact can be processed at different levels, each of them of importance:

a) positive ascertainment of the frequencies related to the core-variables of

populations as an integer; b) transformation of such an ascertainment into <distributive functions>

expressing by a law the set of values assumed by the core-variables i) within the population, ii) within its temporal transforms;

c) sample-inferential analyses, intended to clarify the distributive functions through the analysis of a sample taken from the population: from it we derive both the laws of the sampling analysis and the characteristics the sample needs in order to act as a basis for inference (consistency, non-distortion and so on);

d) transformation of the sample-inference into the larger-one related to the distributive function of the population which the sample has been taken from.

This final inference is maybe the most challenging and hard task for

statistics, the same which made Corrado Gini say: Applied statistics is still largely an art, as always it will be to a certain degree. In its success, intuition – or, if you prefer, genius – plays and will play a relevant role. This characteristics will appear as an advantage to some of us, as a disvantage to others with respect to other fields. Anyway, it explains so many of the divergence and contradictions among its scholars (Gini 1939).

In particular, objectivity and subjectivity prove to be especially fused within the analyses of the d)-type: the objectivity of data as well and the subjectivity which chooses and interpretates them, and as an end casts them into one or another – ex ante equi-probable – distribution function.

This problem specially relates to a further, well-known problem, an unsolved one from ancient times: the dis-order proper to the world. Depending in fact from the variables choosen by scholars as well as from their i) separableness, ii) time-space extension, iii) aggregation, the results of the analyses can justify two opposite pictures of the world: a) an endless and stopless variability versus b) its catalysis into usually hidden <structures of order>, no matter if it deals with a distributive order or an evolutionary one (trend).

Both the (unconscious) drives and the transferences of the scholars largely determine the research paths, and the Spirit of the Time (Zeitgeist) as well – during the nineteenth century, e.g., the world is still a unity of sense

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and meaning. As a matter of fact, a large contribution is also given by the nature of reality and its characteristics; they belong anyhow to dynamics, where they join the evolution of present structures (the aforementioned propagatori) as well as exogenous shocks (entelechiani).

Francesco Brambilla had performed his own upper and university studies

between 1920 and 1940, during that fast, roaring but irregular years generally marked by Versailles, by the crash of 1929, by the new authoritative regimes of Italy (1922) and Germany (1933), by the three-wars-period (China-Japan, Ethiopia, Spain), and at its end by the Second World War. During years – in addition – scientifically marked on one side by the coming in power of idealism in its different forms (Croce-Gentile in Italy, Husserl-Jaspers-Heidegger in the German world), on the other by the breakthrough of the Wiener Kreis and Wittgenstein.

After his university studies, Brambilla was for some moments voluntary assistant to Zappa (the founder of the Italian Economia aziendale, a type of Betriebswirtschaftslehre) and Demaria as well, anyway being probably too much original and too much a character to remain in their train.

Later on, when he individuated statistics as his own field, he entered into a cluster in those very times characterized by:

1. its revival as an autonomous science by The Grammar of Science of

K. Pearson (1892), which in addition presented it as a further dimension of knowledge;

2. the innovative contributions by Yule, Student, R.A. Fisher, and especially by the Kuhnian revolution by the same Fisher (1922);

3. the evolution of this revolution through and after the Wiener Kreis up to 1940, with particular reference to the calculus of probabilities, better, to the same notion of probability, so replaying some thirty years later the crisis that mathematical foundations had lived with Cantor, Russell, Frege and Peano;

4. the criticizing the concept of <chance> to substitute it with axiomatization (Kolmogorov; De Finetti) or frequentism (von Mises and Reichenbach as influenced also by R.A. Fisher, F.P. Ramsey, J.M. Keynes).

In his theoretical approach, Brambilla was probably influenced by the

essay Das harmonische Gesetz der Statistik (1923), which identified the Gauss-distribution as the hidden but widespread and never-ending representation of phenomena, so proposing its characteristics as the substantial, common – and in addition harmonious – structure of the world.

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It was maybe due to this proposal – or hypothesis – that he orientated himself towards the study of variability in general: in those very years the logical nature of the variability of phenomena was still to be clarified, but the field had just then recorded the undoubted methodological advancement of the Neyman-Pearson theory of first and second level biases (1928).

In the same 1933, in addition, had Harold Hotelling proposed a powerful transformation system to sort out the information stemming from world-phenomena: the theory of main components, which Brambilla described in the following way some sixty years later:

The world appears us with so many facets: it is observable through such a plenty of variables that they seem to multiplicate its apparent dimensions beyond measure. Anyway, when each statistical variable is represented through a vector as long as its mean square error, and the correlation (i.e. the distance between two variables) is measured by the cosine of the angle between vectors, all of a sudden, plastically, a polyhedron springs out from the mass of our information: at every vertex we find we find a variable, that is to say one of the so many facets of the reality we observed (Brambilla 1986, p. 7).

The laurea thesis of Brambilla (1936) had been dedied to The Theory of Correlation and the Confluential Analysis of Ragnar Frisch, as he had been <fascinated by the power of mathematical tools in transformating the statistical variables to make the hidden structure emerge from the chaos of the variability of phenomena>. So he choosed the variability (1959) as its own Lebenswerk, unifying both the <analysis of the hidden structure> and the <main components theory> in order:

i) to explain the logical nature of variability, ii) to individuate its laws of distribution in space-time.

After having softwarized the Hotelling method, his research program went along the following stages:

a. <to sift the tangle of the data expressing phenomena in order to

positionate the so many apparent variables within the new space of latent factors, normally only two, this way identifying a plane>;

b. to interpretate the position of points by a general interpretative tool: the oscillations of systems around their equilibria according to Bendixson, Krylov, Poincaré (Brambilla 1949-1959),

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c. trying to read the position of points along an ellipsis, i.e. according to a cyclical structure of phenomena;

d. enriching later on this system through Thom’s theorem of classification of equilibrium structures, and the related catastrophes theory.

Anyhow, a basic methodological problem lay behind these studies:

<were the found uniformities intrinsic properties of the mathematical operators and their algebrae, or were they intrinsic to the nature of phenomena?> (e.g., <under some very general conditions, the sum of n standardized variables tends, as n grows, to stable distributions: is this stability a property of the sum-operator by which we read the variability, or is it a property of an inner nature?>).

As an answer, Brambilla concluded saying that we have three only possible interpretations referred to the dynamics of every system whatsoever:

1. the wave motion, 2. the theory of potential, 3. the equation of diffusive processes which, in its steady-state, is

expressed by Pearson equation (this due to the <high number of distributions of frequency which we can describe through the solutions of Pearsons’ systems>).

4.3. De Finetti: subjective probabilities against rational choice Bruno De Finetti deserves no introduction for the English reader, at least

as far as the theory of subjective probabilities is concerned (De Finetti 1974-75; see also e.g.: Lindley 1986; Cifarelli and Regazzini 1996; Jackson 1999).

Along years, a number of his contributions were anyway dedied to mathematical economics, better, to the use of mathematical operators in different fields of economics. Now, De Finetti being first of all a first-order (applied) mathematician, he immediately perceived – and censored – so many of the misuses-abuses of mathematics within the economic field, concluding as such at the top of his career:

Within axiomatism, one can demonstrate that a special property – e.g. the existence of an equilibrium – exists in a given problem under given hypotheses (or ‘axioms’) so and so. From the mathematical point of view, every result of this kind, provided it is correct, is a correct result, and nothing more. But

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what really matters is the contribution to economics, this depending not on the thruthfulness of the result, but to its relevance. Were the axioms chosen in a trivial way in order to facilitate the demonstration, the same result is normally a trivial one. Were the terms presented as if they always apply (or with few exceptions), the result is a deceptive one. The method is very proper, and an instructive one, if the analysis is professedly carried out to clarify the vlidity of conclusions (De Finetti 1969, p. 182).

In addition, quoting R. Frisch, De Finetti says:

Both too many modern contributions, and whole theories enjoying today great favour, are void of any effective interest for economics as well as far from any occasion of empirical application. They are exercises in which one toys with abstruse statements expressing unreal or trifling problems: they don’t belong to econometrics but, vice versa, to play-o-metrics (ibid., p. 176).

Far from skepticism, these considerations were on one side induced by

the widespread diffusion of an economic a priori thought to be treated in mathematical forms; on the other, they were motivated by the very passion he always had – since 1943, and before – for realistic, fruitful treatments of true economic problems through the use of sophisticately simple mathematical tools (e.g. the geometry of hyper-planes).

The re-writing of the whole problem begins with the acknowledgement of the merely dialectic notion of mathematics, so refusing the old-fashioned (and fallacious) predicative one. According to the former, no absolute basis whatsoever, nor unique and everlasting foundations, neither the perfect certitude it was supposed to introduce in every scientific field transforming them into the <kingdom of absoluteness>. After in fact the crises a) of causality due to probabilism, b) of space-time representations due to relativity, c) of the standard conceptualizations of the physical world due to intedermination, mathematics is merely connoted by limited, conventional bases, able anyway to give life to useful Baedeckers for still unknown lands.

Mathematics in fact is always at a level of finding new instruments, and ways: the best example is maybe Einstein’s critique of the Michelson-Morley problem: he criticized the concept of (space-time) simultaneity through pseudo-Euclidean 4-dimensions geometry as well as the absolute differential calculus of Levi-Civita and Ricci-Curbastro, opening the gate to the invention of relativity.

This peculiar role could be defined as the <vivifying function> of mathematics, which acts like Ophelia in Shakespeare’s Hamlet: she is

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seductive, essential, but a little bit crazy. This way, <vivifying function> means its capability:

• to criticize not only theories but their a priori as well, • to transform old theories into radically new ones, which keep intact

only the true core of the former ones, • to develop unknown analytical fields favourable for the (applied)

sciences of tomorrow. Now, as far as the empirical side is concerned, one major reversal arises:

according to Bridgman’s tatsächlich Zielpunkt intended as a rule of scientific reasoning, the processes of measuring and verifying represent the very nature of the research: concepts make sense inasmuch they express the results of a) experiments, b) measurements.

In particular, as far as economics is concerned, its crisis can be referred to its i) apriori, ii) concepts, iii) goals. The first step to be taken in order to solve these problem consists in the radical reconsideration of its principles and concepts, even though acquired and/or of a reputed evidence: the solution will stem from the subversion of the core bulks of the vulgata, anyway keeping its previous analytical forms.

A first general critique regards the fields of political economy, as the analyses have to be set free from any principle and/or data peculiar to specific economic systems: its goal seem to consist <in the analysis of the possibilities of satisfying both individual and collective needs and desires subjected to reciprocal constraints>.

Under this light, classical economics sounds as a pure fetishism, but at the end of the story also its critics obey to the same principles, and use concepts of value only within old frameworks. From this point of view, Pareto’s system is not only an empty and unconclusive one as now (1948) largely admitted: its unconclusiveness derives from the fact that he polluted his theory by the hurried acceptance of both premisses and prejudices taken from classical economics.

Within economic science, the use of both terminologies and concepts characteristic of classical economics means the self-exclusion from the analyses of effective economic problems, as well as from systematic theories as well.

To study problems under restrictions implies in fact i) prejudice the solutions, b) to prevent the study of the effects of a single restriction whatsoever.

The solution consists on the contrary in the reversal of the whole procedures: the search for general theories, not subjected to any a priori

32


restriction. This enables us a) to identify the dependence from specific economic systems, b) to define the features to be impressed to economic organizations in order to reach well-defined solutions.

So, as a matter of fact, first of all it is necessary to abandon the idea of economic freedom as a preliminary question; second, to abandon the analyses of given, a priori systems to dictate their economic theory. It is necessary on the contrary to study the general problems of optima from a general point of view:

b. considering economic organization (and freedom) from a purely

functional standpoint, c. after this, evaluating how many <degrees of freedom> it is

convenient to give to individuals in order to attain <optimal> behaviours.

From the economic point of view, vice versa, we should: 1. define every <situation Р given>, 2. define the nature of functions φ (P) = φ (x, y).

In this case, answers are nor obvious nor univocal ones, as 3. the <degree of precision> necessary to make (P) determinate

depends on the nature of the problem as well as on the levels of analysis;

4. preference-criteria φ differ according to different economic systems.

The indetermination which follows is peculiar to practical sciences like mechanics (and others), and the mathematical methods applied are necessarily unable to treat problems without isolating partial sets from the organic complex of phenomena. According to these methods and procedures, simplifications are anyway useful: the optima strictly defined will be truer under less severe conditions, and this helps to take steps to solutions by degrees.

33


34


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DIPARTIMENTO DI ECONOMIA AZIENDALE PAPERS PUBBLICATI∗:

1. Arnaldo CANZIANI, La ricerca nelle scienze sociali: note metodologiche e pre-metodologiche, novembre 1998.

2. Daniela M. SALVIONI, Controllo di gestione e comunicazione nell’azienda pubblica, aprile 1999.

3. Arnaldo CANZIANI, Giovanni Demaria nei ricordi di un allievo, luglio 1999. 4. Rino FERRATA, Tecnologia e mercato: i criteri di scelta dei metodi di valutazione,

luglio 1999. 5. Giuseppe BERTOLI, Salvatore VICARI, L'impresa diversificata come organizzazione

che apprende, dicembre 1999. 6. Virna FREDDI, Attività economica e impresa nella concezione economicista, febbraio

2000. 7. Virna FREDDI, L'approccio Resource-based alla teoria dell'impresa: fattori interni e

competitività aziendale, febbraio 2000. 8. Maria MARTELLINI, Sviluppo, imprese e società, maggio 2000. 9. Arnaldo CANZIANI, Per la critica della teoresi zappiana, e delle sue forme di

conoscenza, dicembre 2000. 10. Giuseppe BERTOLI, Gabriele TROILO, L'evoluzione degli studi di marketing in

Italia. Dalle origini agli anni settanta, dicembre 2000. 11. Giuseppe BERTOLI, Profili di efficienza delle procedure concorsuali. Il concordato

preventivo nell’esperienza del tribunale di Brescia, dicembre 2000. 12. Daniele RONER, Domanda e offerta di beni economici. Rassegna critica

dall’irrealismo neoclassico alla differenziazione dei prodotti, marzo 2001. 13. Elisabetta CORVI, Le valenze comunicative del bilancio annuale. I risultati di

un'indagine empirica, luglio 2001. 14. Ignazio BASILE, Nicola DONINELLI, Roberto SAVONA, Management Styles of

Italian Equity Mutual Funds, agosto 2001. 15. Arnaldo CANZIANI, I processi competitivi fra economia e diritto, settembre 2001. 16. André Carlo PICHLER, L'Economic Value Added quale metodo di valutazione del

capitale economico e strumento di gestione aziendale, dicembre 2001. 17. Monica VENEZIANI, Economicità aziendale e capacità informativa del bilancio nelle

aziende cooperative agricole, dicembre 2001. 18. Pierpaolo FERRARI, La gestione del capitale nelle principali banche internazionali,

febbraio 2002. 19. Giuseppe BERTOLI, Bruno BUSACCA, Il valore della marca. Modello evolutivo e

metodi di misurazione, marzo 2002. 20. Paolo Francesco BERTUZZI, La gestione del rischio di credito nei rapporti

commerciali, aprile 2002. 21. Vincenzo CIOFFO, La riforma dei servizi a rete e l'impresa multiutility, maggio 2002. 22. Giuseppe MARZO, La relazione tra rischio e rendimento: proposte teoriche e ricerche

empiriche, giugno 2002. 23. Sergio ALBERTINI, Francesca VISINTIN, Corporate Governance e performance

innovativa nel settore delle macchine utensili italiano, luglio 2002. 24. Francesco AVALLONE, Monica VENEZIANI, Models of financial disclosure on the

Internet: a survey of italian companies, gennaio 2003.

∗ Serie depositata a norma di legge

39

25. Anna CODINI, Strutture organizzative e assetti di governance del non profit, ottobre 2003.

26. Annalisa BALDISSERA, L’origine del capitale nella dottrina marxiana, ottobre 2003. 27. Annalisa BALDISSERA, Valore e plusvalore nella speculazione marxiana, ottobre

2003. 28. Sergio ALBERTINI, Enrico MARELLI, Esportazione di posti di lavoro ed

importazione di lavoratori:implicazioni per il mercato locale del lavoro e ricadute sul cambiamento organizzativo e sulla gestione delle risorse umane, dicembre 2003.

29. Federico MANFRIN, Sulla natura del controllo legale dei conti e la responsabilità dei revisori esterni, dicembre 2003.

30. Rino FERRATA, Le variabili critiche nella misurazione del valore di una tecnologia, aprile 2004.

31. Giuseppe BERTOLI, Bruno BUSACCA, Co-branding e valore della marca, aprile 2004. 32. Arnaldo CANZIANI, La natura economica dell’impresa, giugno 2004. 33. Angelo MINAFRA, Verso un nuovo paradigma per le Banche Centrali agli inizi del

XXI secolo?, luglio 2004. 34. Yuri BIONDI, Equilibrio e dinamica economica nell’impresa di Maffeo Pantaleoni,

agosto 2004. 35. Yuri BIONDI, Gino Zappa lettore degli Erotemi di Maffeo Pantaleoni, agosto 2004. 36. Mario MAZZOLENI, Co-operatives in the Digital Era, settembre 2004. 37. Claudio TEODORI, La comunicazione via WEB delle imprese italiane quotate: un

quadro d’insieme, dicembre 2004. 38. Elisabetta CORVI, Michelle BONERA, La comunicazione on line nel settore della

distribuzione dell’energia elettrica, dicembre 2004. 39. Yuri BIONDI, Zappa, Veblen, Commons: azienda e istituzioni nel formarsi

dell’Economia Aziendale, dicembre 2004. 40. Federico MANFRIN, La revisione del bilancio di esercizio e l’uso erroneo degli

strumenti statistici, dicembre 2004. 41. Monica VENEZIANI, Effects of the IFRS on Financial Communication in Italy:

Impact on the Consolidated Financial Statement, gennaio 2005. 42. Anna Maria TARANTOLA RONCHI, Domenico CERVADORO, L’industria

vitivinicola di Franciacorta: un caso di successo, marzo 2005. 43. Paolo BOGARELLI, Strumenti economico aziendali per il governo delle aziende

familiari, marzo 2005. 44. Anna CODINI, I codici etici nelle cooperative sociali, luglio 2005. 45. Francesca GENNARI, Corporate Governance e controllo della Brand Equity

nell’attuale scenario competitivo, luglio 2005. 46. Yuri BIONDI, The Firm as an Entity: Management, Organisation, Accounting, agosto

2005. 47. Giuseppe BERTOLI, Bruno BUSACCA, Luca MOLTENI, Consumatore, marca ed

“effetto made in”: evidenze dall’Italia e dagli Stati Uniti, novembre 2005. 48. Pier-Luca BUBBI, I metodi basati sui flussi: condizioni e limiti di applicazione ai fini

della valutazione delle imprese aeroportuali, novembre 2005. 49. Simona FRANZONI, Le relazioni con gli stakeholder e la responsabilità d’impresa,

dicembre 2005.

40

Università degli Studi Dipartimento didi Brescia Economia Aziendale

Dicembre 2005

Paper numero 50

Francesco BOLDIZZONI - Arnaldo CANZIANI

MATHEMATICS AND ECONOMICS:USE, MISUSE, OR ABUSE?From Walrasian Deductivism

to Demaria, Brambilla and De Finetti

Università degli Studi di BresciaDipartimento di Economia AziendaleContrada Santa Chiara, 50 - 25122 Bresciatel. 030.2988.551-552-553-554 - fax 030.295814e-mail: [email protected]

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