The aristotelian paradigm of distributive justice: the golden triangle

International Association for Greek Philosophy - IAGP

23rd International Conference of Philosophy 1

23232323RDRDRDRD INTERNATIONAL CONFERENCE OF PHILOSOPHYINTERNATIONAL CONFERENCE OF PHILOSOPHYINTERNATIONAL CONFERENCE OF PHILOSOPHYINTERNATIONAL CONFERENCE OF PHILOSOPHYVOULIAGMENI VOULIAGMENI VOULIAGMENI VOULIAGMENI ---- ATHENS: 17ATHENS: 17ATHENS: 17ATHENS: 17THTHTHTH –––– 22222222NDNDNDND OF JULY 2011OF JULY 2011OF JULY 2011OF JULY 2011

Dr. Maria Antonietta Salamone, Profesor of Philosophy, University Complutense of Madrid

Ναός τηςΝαός τηςΝαός τηςΝαός της SegestaSegestaSegestaSegesta ((((ΣικελίαΣικελίαΣικελίαΣικελία) ) ) )



THE ARISTOTELIAN PARADIGM THE ARISTOTELIAN PARADIGM THE ARISTOTELIAN PARADIGM THE ARISTOTELIAN PARADIGM OF DISTRIBUTIVE JUSTICE: OF DISTRIBUTIVE JUSTICE: OF DISTRIBUTIVE JUSTICE: OF DISTRIBUTIVE JUSTICE:

THE GOLDEN TRIANGLETHE GOLDEN TRIANGLETHE GOLDEN TRIANGLETHE GOLDEN TRIANGLE



The objective of this paper is to offer the mathematical andphilosophical solution to the geometrical theorem that has beenput forward by Aristotle in the Nicomachean Ethics in relation withthe search for the medium term of Distributive Justice.According to this interpretation, in fact, the just distribution of theeconomical and political rights and duties among citizens must bea mean proportional (golden section) between the threedistributive criteria: virtue (aristocracy), liberty (democracy),and wealth (oligarchy). As a matter of fact, in the application ofthe solution of the theorem in the Politics, it becomes scientificallyapparent that for Aristotle the best political constitution is theAristocratic Republic. Finally, in this paper it is underlined not onlythe metaphysical foundation of the Aristotelian Ethics and Politicsbut also the mediating function of the mathematical sciences inthe Aristotelian epistemology and ontology.



In the Fifth Book of the Nicomachean Ethics Aristotleestablishes a link, even etymological, between justice, law,equality -so that it will be said that a law is politically just orright when establishes a certain form of equality amongcitizens-, and distinguishes between two kind of justices inview of the kind of equality that is to be carried out: on the oneside, the distributive justice, on the other side, the correctivejustice, in turn divided into commutative and restorative justice.Aristotle, in fact, recognizes two meanings in the word justice:

1. the just is the lawful or the political justice (the whole justice);2. the just is the equality or the distributive and corrective justices

(the parts of the justice). Thus, the unjust is the unlawful andthe inequality.



The political justice is identified with the constitution of eachRepublic, while the distributive justice refers principally to thedistribution of the political and economical rights and dutiesamong the citizens. The distributive justice is differentiated,furthermore, from the corrective justice which regulates in anormative way the private relations among the citizens punishing,consequently, those who violate the laws. Aristotle tries to provethat, in each one of these parts of justice, the best consists in theestablishment of a certain medium term, either geometrical orarithmetical, according to what is said about the distributive or thecorrective justice.

Political justice(The whole Justice: lawful)

Distributive and Corrective Justice(The parts of Justice: equality)



Moreover, as regards the distributive justice, Aristotle manifeststhat, as no one comes to an agreement about the “criterion” thathas to be adopted at the time of distributing the economical andpolitical rights and duties among citizens – but the democrats putit in liberty, the oligarchs in wealth and the aristocrats in virtue–and as it is impossible to construct a proportion on the basis ofthe one single term, the medium term of the distributive justicehas to be a mean proportional among the three types of criterion;and the unjust what violates the proportion; in fact, this meanproportional is the just proportion. The man who is unjust,actually, is the man who takes for himself more benefits andfewer costs than what corresponds to him; inequality, in fact, is aparticular vice that has to be distinct from the others «being itsmotive the pleasure that arises from gain», in other words fromthe profit.



In fact, in an oligarchic constitution the criterion of a distributive justice is the wealth: in this case, the distribution of the political and economical rights and duties is calculated in accordance with the level of income and of property of the citizens, in other words proportionally according to census.

In an aristocratic constitution the criterion of distribution of thepublic goods is the virtue and, consequently, the political andeconomical responsibilities are distributed proportionally in thelevel of goodness and education of their citizens.



While in a democratic constitution the criterion on which thedistributive justice is founded is the condition of the freecitizen, in other words, liberty and, as a consequence, thewealth and the honors must be distributed in a proportionaland equitable way among all the free citizens.

In addition to this, in order to discover the mean proportionalof the distributive justice, we must find a figure that complieswith the conditions which were indicated by Aristotle, as thegreat Philosopher asks us to solve the following geometricaltheorem:



GEOMETRICAL THEOREM GEOMETRICAL THEOREM GEOMETRICAL THEOREM GEOMETRICAL THEOREM ((((Nicomachean EthicsNicomachean EthicsNicomachean EthicsNicomachean Ethics, Book V, 3), Book V, 3), Book V, 3), Book V, 3)

1) The medium term of the distributive justice is a geometricalproportion which has been formed by four terms:

A : B = C : D

2) The four terms of this geometrical proportion have acontinuous relation:

A : B = B : C

3) Nonetheless this continuous proportion is not divisible inonly one dimension, but in two or three; therefore it isnot applied to the Linear Geometry (longitudes), but to thePlan Geometry (polygons) or to the Solid Geometry(polyhedrons);



GEOMETRICAL THEOREM GEOMETRICAL THEOREM GEOMETRICAL THEOREM GEOMETRICAL THEOREM ((((Nicomachean EthicsNicomachean EthicsNicomachean EthicsNicomachean Ethics, Book V, 3), Book V, 3), Book V, 3), Book V, 3)

4) «The conjunction between the straight lines AC andBD is what is just in distribution, and this speciesof the just is a mean proportional, and the unjust what violatesthe proportion; in fact, this mean proportional is the justproportion»;

5) The λόγος that expresses the essence or the property of thegeometrical proportion is the following: «Mathematicians callthis kind of proportion geometrical; for it is in geometricalproportion that it follows that the whole is to the whole aseither part is to the corresponding part».



J. Tricot, as well as the majority of the moderns interpreters, hastried to identify the Aristotelian definition of the geometricalproportion with the Proposition 18 of the Fifth Book of theElements: «If magnitudes, taken jointly, be proportionals theyshall also be proportionals when taken separately». This propertyis known as componendo, and states that in a proportion«the sum of the antecedents is to the sum of the consequents, aseach antecedent is to its own consequent». If A : B = C : D, then

(A + B) : (C + D) = C : D

In my opinion, this interpretation is explicitly erroneous given that,as I am going to demonstrate mathematically as well asphilosophically, the Aristotelian λόγος refers very clearly to theproportion most characteristic or continuous par excellence:

(A+B) : A = A : B



This proportion, applied to the linear Geometry, corresponds towhat Euclid called “division in mean and extreme ratio”, or goldenproportion, that is the most logic and important asymmetricdivision because of its arithmetical, geometrical, architectonical,esthetical and ethical consequences: «A straight line is said to becut in extreme and mean ratio, when the whole is to the greatersegment, as the greater is to the less» (Elements, Book VI,Proposition 30). Sectioning a line in mean and extreme ratiomeans, actually, dividing it into two asymmetric segments so thatthe greater is the mean proportional between the less and thewhole. In fact, the golden proportion applied to the line AB willgive the result that the whole segment (AC + CB) is to the greatersegment (AC) as the greater segment (AC) is to the less (CB) =number Φ 1,618 or golden ratio. This also means to make an“additive partition”.



That the geometrical concept of “the whole” doesn’t coincide withthe arithmetical concept of “the sum”, or that the whole is morethan the sum of its parts, is something that Aristotle alreadysucceeded in clearing up effectively when he states the generalprinciple of holism: «In the case of all things which have severalparts and in which totality is not, as it were, a mere heap, but thewhole is something besides the parts, there is a cause»(Metaphysica, Book VII, 6). See also the definition of Euclid:«The whole is greater than its part» (Elements, Book 1).Furthermore, taking into consideration that we have to find themean proportional between the three criteria of distribution (virtue,wealth and liberty), the geometrical figure will have to be,in an obligatory way, a triangle which is, actually, divisible andcontinuous in two dimensions (longitude and altitude = polygon),and not in only one (longitude = line). Therefore, the geometricaltheorem could be reformulated in the following manner: To dividea given triangle in extreme and mean ratio or golden proportion.



In effect, the geometrical theorem which has been set by Aristotlecan be resolved, in my opinion, exclusively if we take intoconsideration the Elements of Euclid: «To describe an isoscelestriangle, having each of the angles at the base double of thethird angle» (Proposition 10 of the Book IV). Actually, there is onlyone polygon with the unique and sublime ratio which is explainedby Aristotle: it is about the “Golden Triangle” in which not only thearea of the whole triangle is to the area of the greater triangle asthe area of the greater triangle is to the area of the smallertriangle [«the whole is to the whole (area)» = (B+Γ) : B = B : Γ],but also the length of the whole segment is to the length of thelonger segment as the length of the longer segment is to thelength of the shorter segment [«as either part is to thecorresponding part (longitude)» = (AD+DC) : AD = AD : DC].



THE ARISTOTELIAN PARADIGM OF DISTRIBUTIVE THE ARISTOTELIAN PARADIGM OF DISTRIBUTIVE THE ARISTOTELIAN PARADIGM OF DISTRIBUTIVE THE ARISTOTELIAN PARADIGM OF DISTRIBUTIVE JUSTICE: THE GOLDEN TRIANGLEJUSTICE: THE GOLDEN TRIANGLEJUSTICE: THE GOLDEN TRIANGLEJUSTICE: THE GOLDEN TRIANGLE



In effect, in the Golden Triangle «the conjunction between thestraight lines AC and BD is what is just in distribution, and thisspecies of the just is a mean proportional, and the unjustwhat violates the proportion; in fact, this mean proportional is thejust proportion». Actually, the diagonal BD divides the segmentAC in golden proportion, with AD being the mean proportional(golden section) between the whole segment and the less.In such a way it is finally demonstrated that the golden section isthe mean proportional among the three different criteria of thedistributive justice: virtue, wealth and liberty.Moreover, in the Politics, when the time comes to define whichone is the best political constitution, Aristotle takes fordemonstrated in the Ethics that what is just in distribution has tobe a mean proportional among the three different criteria, whichfinally has a mathematical name and surname: the GoldenSection.



What must be done now is to give it also a political nameand surname, in other words: Aristocratic Republic, providedthat the aim of the political community (“the final cause”) is notonly «the prevention of mutual crime» (restorative justice) or«the sake of exchange» (commutative justice), but above all«the sake of a perfect and self-sufficing life» (distributive justice) and,as a consequence, the first criterion which must be taken inconsideration has to be necessarily the political virtue. That's why Iput the virtue on the top of the triangle as well as the wealth and theliberty at the bottom.Actually, if we apply the solution of the geometrical theorem(“the formal cause”) to the correspondent political constitutions -Democracy, Oligarchy and Aristocracy-, we should have the followingGolden Triangle in which the diagonal BD cut the segment AC ingolden proportion; in a way that the Aristotelian best constitution willbe identified with the golden section AD, in other words with a meanproportional among the three different constitutions: the AristocraticRepublic (the mixed constitution).



THE ARISTOCRATIC REPUBLIC THE ARISTOCRATIC REPUBLIC THE ARISTOCRATIC REPUBLIC THE ARISTOCRATIC REPUBLIC



Finally, what is politically just or right for Aristotle is arrangedaccording to the united interest of the whole political community -including the richest (oligarchy), the best (aristocracy) and thepoorest (democracy). It is not arranged just for the benefit of asingular part or faction: «Now in most states the form calledrepublic exists, for the fusion goes no further than the attempt tounite the freedom of the poor and the wealth of the rich (…). Butas there are three grounds on which men claim an equal share inthe government, freedom, wealth, and virtue (…), it is clear thatthe admixture of the two elements, that is to say, of the rich andpoor, is to be called a republic or constitutional government; andthe union of the three is to be called aristocratic republic or thegovernment of the best, and more than any other form ofgovernment, except the true and ideal, has a right to this name»(Politica, Book IV).



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The aristotelian paradigm of distributive justice: the golden triangle

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