ARTI GRAFICHE APOLLONIO
Università degli Studi
di Brescia
Dipartimento di
Economia Aziendale
Francesco BOLDIZZONI - Arnaldo CANZIANI
MATHEMATICS AND ECONOMICS:
USE, MISUSE, OR ABUSE?
From Walrasian Deductivism
to Demaria, Brambilla and De Finetti
Paper numero 50
Università degli Studi di Brescia
Dipartimento di Economia Aziendale
Contrada Santa Chiara, 50 - 25122 Brescia
tel. 030.2988.551-552-553-554 - fax 030.295814
e-mail: [email protected]
Dicembre 2005
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 poi 1 ).
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 neopositivismo, 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 analitico 2 .
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 altri 3 . 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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Mathematics and Economics: Use, Misuse, or Abuse?
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-incoerentiinopportune-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 121124).
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Francesco Boldizzoni - Arnaldo Canziani
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
sofisticate 5 .
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, SpringerVerlag, 1991.
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Mathematics and Economics: Use, Misuse, or Abuse?
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’<economiaconclusione> 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 empiricosociale, 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).
5
Francesco Boldizzoni - Arnaldo Canziani
a Debreu 7 , 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 Economics 8 , come pure che si sia or ora autorevolmente
sostenuta le necessità di riformare ab imis molta dell’Economia politica
contemporanea 9 .
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.
6
Mathematics and Economics: Use, Misuse, or Abuse?
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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Francesco Boldizzoni - Arnaldo Canziani
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Mathematics and Economics: Use, Misuse, or Abuse?
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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Francesco Boldizzoni - Arnaldo Canziani
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Mathematics and Economics: Use, Misuse, or Abuse?
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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Francesco Boldizzoni - Arnaldo Canziani
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-butnot-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>);
12
Mathematics and Economics: Use, Misuse, or Abuse?
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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Francesco Boldizzoni - Arnaldo Canziani
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 exchangeequilibrium between (or among) individuals the same relevance? Have its
solutions a heuristic value?
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Mathematics and Economics: Use, Misuse, or Abuse?
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
Francesco Boldizzoni - Arnaldo Canziani
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
16
Mathematics and Economics: Use, Misuse, or Abuse?
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.
17
Francesco Boldizzoni - Arnaldo Canziani
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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Mathematics and Economics: Use, Misuse, or Abuse?
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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Francesco Boldizzoni - Arnaldo Canziani
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).
20
Mathematics and Economics: Use, Misuse, or Abuse?
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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Francesco Boldizzoni - Arnaldo Canziani
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 neverending 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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Mathematics and Economics: Use, Misuse, or Abuse?
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).
23
Francesco Boldizzoni - Arnaldo Canziani
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:
k = K / L = b(1 − e − aρ )
ρ ∈ (1,+∞ ) , a > 0 , b > 0
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).
24
Mathematics and Economics: Use, Misuse, or Abuse?
Fig. 1 – k-values for a=1 and several values of b
4,5
4
k=K/L
3,5
3
b=1
2,5
b=2
2
b=4
1,5
1
0,5
0
0,01
1
2
3
4
5
6
7
8
9
10
ρ
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 productiondistribution-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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Francesco Boldizzoni - Arnaldo Canziani
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 = b1
a21x1 + 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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Mathematics and Economics: Use, Misuse, or Abuse?
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
27
Francesco Boldizzoni - Arnaldo Canziani
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 threewars-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.
28
Mathematics and Economics: Use, Misuse, or Abuse?
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 worldphenomena: 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),
29
Francesco Boldizzoni - Arnaldo Canziani
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 197475; 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
30
Mathematics and Economics: Use, Misuse, or Abuse?
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 MichelsonMorley 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
31
Francesco Boldizzoni - Arnaldo Canziani
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
Mathematics and Economics: Use, Misuse, or Abuse?
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
Francesco Boldizzoni - Arnaldo Canziani
34
Mathematics and Economics: Use, Misuse, or Abuse?
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DIPARTIMENTO DI ECONOMIA AZIENDALE
PAPERS PUBBLICATI ∗ :
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Arnaldo CANZIANI, La ricerca nelle scienze sociali: note metodologiche e premetodologiche, novembre 1998.
Daniela M. SALVIONI, Controllo di gestione e comunicazione nell’azienda pubblica,
aprile 1999.
Arnaldo CANZIANI, Giovanni Demaria nei ricordi di un allievo, luglio 1999.
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luglio 1999.
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che apprende, dicembre 1999.
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2000.
Virna FREDDI, L'approccio Resource-based alla teoria dell'impresa: fattori interni e
competitività aziendale, febbraio 2000.
Maria MARTELLINI, Sviluppo, imprese e società, maggio 2000.
Arnaldo CANZIANI, Per la critica della teoresi zappiana, e delle sue forme di
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∗ Serie depositata a norma di legge
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quadro d’insieme, dicembre 2004.
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distribuzione dell’energia elettrica, dicembre 2004.
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dell’Economia Aziendale, dicembre 2004.
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dicembre 2005.
40
ARTI GRAFICHE APOLLONIO
Università degli Studi
di Brescia
Dipartimento di
Economia Aziendale
Francesco BOLDIZZONI - Arnaldo CANZIANI
MATHEMATICS AND ECONOMICS:
USE, MISUSE, OR ABUSE?
From Walrasian Deductivism
to Demaria, Brambilla and De Finetti
Paper numero 50
Università degli Studi di Brescia
Dipartimento di Economia Aziendale
Contrada Santa Chiara, 50 - 25122 Brescia
tel. 030.2988.551-552-553-554 - fax 030.295814
e-mail: [email protected]
Dicembre 2005