23rd World Congress of Philosophy (WCP 2013) Athens, 4 – 10 August 2013

School of Philosophy, National & Kapodistrian University of Athens, Greece

Abstract & Paper Submission Form (All fields marked with * should be completed and written in English to avoid technical problems)

1) I HAVE SUBMITTED PARTICIPATION FORM No 1: □ Yes

Country:

* First (Given) Name (Mr., Mrs., Ms.): Constantinos

* Last (Family) Name: Tsiantis

* Title: □ Prof. □ Dr. □ MPhil. □ MA □ BA: Prof.

* Affiliation—Institution (Teaching or Research): TEI of Athens

* E-mail Address: cotsiant@live.com

2) PAPER TITLE (Please note that the duration of speaking time is 20΄ minutes)

The mathematical law of the Athenian participatory Democracy

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4) ABSTRACT (a maximum of 200 words-keywords are not included)

The mathematical law of the Athenian participatory Democracy

Constantinos N. Tsiantis

Ph.D. University of Maryland, Emeritus Professor

Summary:

The representation of a population is not a conventional subjective choice, neither does itdepend upon

ideal and non-ascertained statistical conditions. The mathematic law that we present solves this

problem and can become a foundation for the political-social sciences: If a population of size N is

constituted by m groups of citizens and each group possesses percentage wi (i=1, 2,… m) of the entire

population (w1+ w2+… + wm =1), then the minimum number of citizens n (sample) that have to be

selected from all the population groups (n = n1+ n2+… + nm, where ni>0) for the representation of

the entire population is given by the relation:

This relationship explains the size of the Parliament in the Athenian Democracy, hence its name. It has

been successfully tested in statistical sampling and may be tested in other areas as well.

Key words: Athenian democracy, mathematical law of participation, representation

5) PAPER (a maximum of 1800 words)

The mathematical law of the Athenian participatory Democracy

Constantinos N. Tsiantis

Ph.D. University of Maryland, Emeritus Professor

1. The founders of democracy

Democracy was the political system developed in Athens through the legislative and political work of

Solon, Kleishenes, Ephialtes and Pericles. Athens was the first city-state of Greece that was

transformed into polis and was ruled directly by its body of citizens. Citizenship emerged in Athens as

a historical necessity and not as a utopia1.

Solon, one of the seven sages of ancient Greece (lawmaker, philosopher and poet), was granted

the consensus by the conflicting social parties in Athens to resolve conflict and proceed to the

necessary political and social reforms (594/3 B.C.). Solon split the Athenians into four classes on the

basis of their income and ability to perform military service. He legislated the seisachtheia: the

deletion of debts. He established the Boule (Parliament) of the four hundred and gave great strength to

the Ecclesia of Demos (the assembly of citizens), which had as its purpose to select the archons and

decide upon the bills prepared by the Boule and upon all other major issues. Solon also established the

people's court, the Heliaia, and recognized the right of citizens to appeal against judgmental decisions

made by the archons and the Supreme Court (Areopagus). Those where the first steps towards

democracy2.

Kleisthens (508 B.C.) split the city-state of Athens into a large number of equal municipalities

(demes) to form the 10 tribes. In each tribe existed citizens of different income from various areas of

Attica.

Kleistenes also established the Boule of the five hundred which consisted of citizens, thirty

years old and above, selected by draw annually from each tribe in equal number3. The main task of

the parliament was to prepare the legislative drafts (probouleymata) for the Ecclesia of Demos, which

required a minimum number of 6.000 citizens to form a quorum4. From that assembly were elected

the 10 generals who governed the military and the state itself. The Ecclesia also served in the role of

the supreme court (Heliaia ) ruling over all issues with the exception of murder cases and religious

matters, which remained the responsibility of the Areopagus.

Ephialtes (462 B.C.) gave independent status to Heliaia5. In the new Heliaia, the 6000

Athenians were bound by oath to the proper exercise of their duties.

Ephialtes was assassinated shortly after his reforms and his leadership in the democratic party of

Athens was taken by Pericles6. Under the reign of Pericles (461- 429 B.C.) Athens experienced its

Golden Age.

2. The mathematical aspect of the Athenean Parliament

It is reasonable to think that the size of the Athenian Boule was not arbitrary. However, during the

twenty five centuries that have passed since that glorious epoch, the problem of justifying the size of

the Athenian Boule has never been posed as a mathematical problem. The development of statistical

science in the first half of the 20th century offered the context for exploring this problem through the

concept of sample size and Newman’s concept of representation7.

We met the problem some years ago while we were conducting the national statistical study

Phronisis. However, for epistemological reasons, we made the decision to avoid the available

statistical methods in computing the sample size. Instead we decided, trusting our philosophical

curiosity, to seek out the mathematical formula of sample size computation starting from scratch.

3. The problem and its solution.

3.1 Stating the problem:

A population of size N consists of m classes of subjects, with N1 subjects in class-1, N2 subjects in

class-2, ..., and Nm subjects in class-m, where N1+N2+...+Nm=N. What is the optimum sample

standing for the population? Specifically: (i) what is the sample size n (n<N), and (ii) what is its

composition, i.e. the size of its classes n1, n2, ..., nm, where n1+n2+...+nm=n and ni>0?

Definitions: wi ≡ Ni/N and λi ≡ ni/n. Σwi=1 and Σλi=1 ( i=1,2,...,m).

3.2 The Athenian solution

We consider first a sample n taken randomly from the population N. The probability to find one

subject of class-1 in the population is

1

1

1w

N

Np (1)

The probability that one subject (of whatever class) from the population N is included in the sample n

is

N

np (2)

The probability 11p that one subject from the population class-1 is included in the sample n is then

ppp111

(3)

We consider now the method posed by the problem, the so called stratified sampling: From a

population N we receive a sample n by taking at random n1 subjects from the population class-1 (call

it event E1), n2 subjects from the population class-2 (event E2),...,nm subjects from the population

class-m (event Em). Since we have selected at random n1 subjects from the population class-1, the

probability p11 (Eq.3) becomes here equal to the product n1*p11, which we signify by p1n. Thus, the

probability that one subject from the population class-1 is included in the sample n becomes (after

replacing)

N

nwnp

n 111 (4)

This inclusion probability p1n must be equal to the probability that one subject of class-1 is found in

the sample: λ1=n1/n . Thus we have:

1111 N

nwnp

n (5)

We repeat the aforementioned process for all the events Ei (i=1, 2, ..., m). Thus respectively we have:

mmmmn

n

n

N

nwnp

N

nwnp

N

nwnp

.................

2222

1111

(6)

The events Ei can be considered independent. Therefore, the probability of their intersection (under

stratified sampling) is the product of their probabilities:

mnnnm

pppEEEP ...)...(2121

or

N

nwn

N

nwn

N

nwn

mmm.........

221121 (7)

From Eq.7, after replacing ni=λi*n (i=1, 2,…, m), we get

m

m

m Nwwwn ...21

2 (8)

Whereupon the law of population representation is derived:

m

m

athwww

Nnn

...21

(9)

Since Eq.9 justifies the size of the Boule in ancient Athens (§ 4.1), it will be from now on referred to

as the Athenian law of representation8 or personal participation. The terms are relative: (i) When

the population is completely homogeneous: m=1, then nath=N1/2

. (ii) When it is completely

inhomogeneous: m=N, then nath=N. This is the (theoretical) case of direct participation of all citizens.

The composition of the sample (the number of subjects per class) is then (by definition):

minnathii

,...,2,1 (10)

The minimum sample size n is achieved when all the population classes are of equal size: wi’s are

equal to 1/m. Then nmin=(N*m)1/2

. This means that the number of classes m plays a decisive role in the

size of parliament (or sample size).

3.3. The hypergeometric solution

The Athenian solution coincides with the solution n=x*N/(1+x), where x is the solution of the

equation log(x)+a/x-a*x=k (where x=n/(N-n), a=1/(2N) and k constant). This equation is derived from

the hypergeometric probability distribution, after applying Stirling’s approximation and Lagrange’s

multiplier method for optimum probability. The maximum probability is achieved when wi = λi

(i=1,2,…,m). The logarithmic equation does not have a closed solution. The solution we propose leads

to the equation

exp(k)))*1/Nlambertw(-*N(-2*)1/N/exp(k)lambertw(-*N1

))1/N/exp(k)lambertw(-*exp(k))*1/Nlambertw(-*N(1*N 2

hypn (11)

where lambertw(.) is the Lambert W function. If, by using the Athenian solution, we set

)(2

1

2

1log

athathath

ath

nNnnN

nk

, (12)

then we find that, with high accuracy, nhyp equals to nath.

4. Applications

4.1.Determining the size of the Kleisthenean Boule.

Sinclair (1988) argues that around the end of the 6th century the number of citizens eligible for the

Boule could not have exceeded thirty thousands but “could exist at least 20000 (and rather 25000)

citizens”9 . He suggests also taking into account the number of extra deputies.

By taking N=25000 and ten tribes (m=10) of equal size (w1=w2=...=w10=w=1/m), we

receive

50025000010*25000*mNnn

ath

If N=30000, as Hansen (2006) argues

10, then nath=548. By adopting Sinclair’s suggestion, this

number is considered reasonable.

4.2 The size of participatory Democracies demanded today

(i) Implementing the mathematical law to Greece and using the data of the June 2012 elections

(N=7.800.000, m=8, w=(0.30, 0.27, 0.12, 0.075, 0.069, 0.062, 0.045, 0.059)), we get nath=6715

deputies. This requires a Boule of five hundred in each of the thirteen administrative regions.

(ii) Implementing the law to the European Union (N=350.000.000, m=27, equal percentages), we

get nath=97211 deputies.

(iii) Implementing the law to the world community (N=4.5 billion citizens, m=180, equal

percentages) we get nath= 900000 deputies.

Participatory democracy sounds strange here. However it is feasible. A number of practical

scenarios can be proposed for its implementation. Science and technology have transformed the globe

to a village. Two decades ago economic markets succeeded in their global electronic interconnection

with millions of transactions per day11

. It would seem ironic for politics to resist democratization by

employing the arguments of realism and utopia.

In Conclusion

The mathematical law of representation or direct participation provides a valid solution to a basic

political-social problem which has remained forgotten for centuries. The law can be extended into

different fields12

. However, it takes particular importance against a world that is abolishing freedom

and justice. This law calls for us to use numbers in a wise way. To trust citizens and science in making

decisions. To remove oligarchies dominating our destinies. To study and establish the contemporary

scheme of Athenian democracy. “Let it be”13

.

References

1 Clotz, Gustave. Ancient Greek at Work, London: Routledge and Kegan, 1926. He argues that

‘‘the vast majority of slaves came from war’’, p.192. 2 For the high public positions they elected people exclusively from the two higher social classes: the

nobeles (pentakosiomedimnoi) and the knights (horsemen). The farmers (zeygites) were elected for

inferior positions. The thetai (metics, outliders), became for the first time eligible (after a certain

period of residence) to vote in the Ecclesia (Assembly) but not to be themselves elected.

http://el.wikipedia.org/wiki, Solon 3 Manitakis, Antonios. “The Athenian Democracy as example of peoples’ self-determination through

the self-government of Demos”, in the Honorary volume for Ioannis Manoledakis, vol. III,

Sakkoulas Publ., Thessalonica, 2007, p. 43-64.

4 http://en.wikipedia.org/wiki/Cleisthenes

5Sakellariou, M. B. The Athenian Democracy, Academic publications of Crete, Heraklion 2000.

6 Rigas, Christos G. The Democracy of Ephialtis: The power of citizen, Eleusis Publications, Athens

2008. 7 Neyman, Jerzy (1934). On the two different aspects of representative method: The method of

stratified sampling and the method of purposive selection, Journal of Royal Statistical Society, 97

(4), 558-625. 8 Tsiantis, C. N. “Computing the sample size of multivariate populations: the Athenian law of

representation and sampling”, in Long, C. et al (Eds) Recent Advances on Applied Mathematics,

Harvard USA, March 24-26, 2008, WSEAS Press, pp.301-306. 9 Sinclair, R. K, Democracy and Participation in Athens, Cambridge University Press, 1988. Tr.in

Greek ( E. Tamvaki), Karthamitsas Publ.,, Athens 1997, pp.125-126. 10

Mogens Herman Hansen, Studies in the Population of Aigina, Athens and Eretria. Historisk-

filosofiske Meddelelser 94. Copenhagen: The Royal Danish Academy of Sciences and Letters,

2006. 11

Tsiantis, George J. Technology and contemporary Democracy, Ionia Publications, Athens 2012. 12

Tsiantis, C. N. “Sample size computing for factorial designs: An extension of the Athenian

representative method”, in Jegdic K, Simeonov P, Zafiris V (Eds) Recent Advances in Applied

Mathematics, Vol.1, UHD, Huston,2009, WSEAS Press, pp.171-176. 13

Tsiantis, C. N. The refoundation of Democracy, 25th International Conference of Philosophy: The

Concept of Form and Way of Life, Αthens, 2013.