In Which Order Did Platon Write Critias, Philebus, Politicus, Sophist, and Timaeus?

In which order did Platon write Critias, Philebus, Politicus, Sophist, and Timaeus?

Nils Lid Hjort

Department of Mathematics, University of Oslo

NordStat, Tromsø, June 2021

1 of XV Platon: from Republic to Laws

Perhaps you, likeΠ λατωνως, for stylistic or rhetorical reasons, are very careful with your sentence endings. Then your clausula is SLSLS You can work through your last 1000 phrases or sentences, sort your endings into the2 5 = 32 diﬀerent clausulae, and count them. Perhaps your clausulae patterns carry your stylistic ﬁngerprint. Scholars agree that A: Republic( Politeia, Staten) comes several years before B: Laws( Nomoi, Lovene). We can estimate the probability vectors

pA =( pA,1,..., pA,32),

pB =( pB,1,..., pB,32), with good precision. For ﬁve other dialogues we can attempt to assess 32-vectors p1,..., p5 and place them on a probability bridge fromA toB. 2 of XV From Politeia (c. 375 b.C.) to Nomoi (c. 350 b.C.) – his style has changed.

3 of XV From Republic to Laws – with ﬁve other dialogues in between: 8 6 4 from Republic to Laws from Republic 2

0 5 10 15 20 25 30 clausulae

4 of XV The evolution from Republic to Laws is signiﬁcant: 4 2 0 −2 Republic minus Laws minus Republic −4 −6

0 5 10 15 20 25 30 clausulae

5 of XV Some rhythms have changed more than others

Cases where pA much bigger than pB : clausula pA pB diff sd wald 9 1 0 0 1 0 2.8 0.6 2.2 0.296 7.424 13 0 1 0 0 1 4.6 1.1 3.5 0.381 9.194 22 1 0 0 1 1 4.2 0.6 3.6 0.350 10.296

Cases where pA much smaller than pB : clausula pA pB diff sd wald 10 1 0 0 0 1 4.6 8.8 -4.2 0.573 -7.330 16 0 0 0 1 1 2.5 5.7 -3.2 0.455 -7.040 30 1 1 1 0 1 4.1 8.8 -4.7 0.562 -8.358

6 of XV Measuring the distance toA and toB

First we read A: Republic and B: Laws carefully, and estimate pA and pB well (large sample sizes: 3778 and 3783 sentences). We then read the ﬁve intermediate dialogues, counting clausulae of the 32 types: Critias (Kritias, 150 phrases), Philebus (Filebos, 958 phrases), Politicus (Politikos, 770 phrases), Sophist (Soﬁstis, 919 phrases), Timaeus (Timaios, 762 phrases).

Idea: for each of these, with p = (p1,..., p32), put up a good distance to A and distance to B, then hope that

H(p) = d(p, pA) − d(p, pB )

can line up the ﬁve from negative H(pA) to positive H(pB ). 7 of XV There are many versions of distance to A and distance to B, in the 32-dimensional space of p = (p1,..., p32). I go for

k X n pj o dw (p, pA)= wj pj log − (pj − pA,j ) , pA,j j=1 k X n pj o dw (p, pB )= wj pj log − (pj − pB,j ) , pB,j j=1 yielding

H(p) = dw (p, pA) − dw (p, pB ) k X n pB,j o = wj pj log − (pB,j − pAj ) . pA,j j=1

Weights wj can be selected to emphasise some clausulae more than others. With all wj = 1, we have plain Kullback–Leibler distances (and some terms disappear).

8 of XV Estimating H(p), with measure of precision

For the given dialogue, with N = (N1,..., N32) multinomial (n, p1,..., p32), we do bpj = Nj /n, with familiar variances and covariances. Hence

k k k 2 X nX 2 X o Var cj bpj = (1/n) cj pj − cj pj , j=1 j=1 j=1 and we can read off standard errors, and find confidence curves, for

H(bp) = dw (bp, pA) − dw (bp, pB ) k X n pB,j o = w p log − (p − p ) . j bj B,j Aj pA,j j=1 For the ﬁve dialogues, I compute & display H(p) − H(p) b cc(H(p)) = 1 − 2 Φ . τb

9 of XV Conﬁdence curves for the ﬁve H(p)

From A: Rep to B: Laws, with w = 1 for the 10 most important of the 32 clausulae: Tim, Soph, Crit, Pol, Phil. 1.0 0.8 0.6 0.4 confidence 0.2

0.0 Ti SoCr PoPh

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 dw(p,pA) − dw(p,pB) 10 of XV My approach, above: a certain but low correlation with that of Cox and Brandwood (JRSS B, 1959). They considered p1−λpλ p = A,j B,j for j = 1,..., k, j R(λ) with λ ∈ [0, 1], which for given corpus leads to log-likelihood

k X `n(λ) = Nj {(1 − λ) log pA,j + λ log pB,j } − n log R(λ). j=1 and suﬃcient statistic k X Nj pB,j Sn,cox = log . n pA,j j=1 With my diﬀerence between weighted KL distances I use

k X nNj pB,j o Hn = wj log − (pB,j − pA,j ) . n pA,j j=1

11 of XV Markov models for the clausulae

Instead of nonparametric estimation of p = (p1,..., p32), may consider models of lower dimension. 2-step memory Markov model:

Pr(0, 1, 1, 0, 1) = π01 p01,1 p11,0 p10,1

etc. This means 3 para for the πi,j and 4 para for the pi,j,1, so reduction from 31 para to 7 para.

3-step memory Markov model:

Pr(0, 1, 1, 0, 1) = π011 p011,0 p110,1

etc. Here 7 para for the πi,j,k and 8 para for the pi,j,k,1, so reduction from 31 para to 15 para.

AIC and FIC say: 3-step memory, with 15 para, does well (but 2-step memory, with 7 para, is too coarse for Platon).

12 of XV Raw estimates Nj /n, black, via 2-step Markov, red and via 3-step Markov, blue, for Timaeus: 6 5 4 3 direct, with 2−step, 3−step Markov direct, with 2−step, 2

0 5 10 15 20 25 30 clausulae 13 of XV Estimators for 3-step memory Markov model

Estimating all Pr(i, j, k, l, m): No additional structure: dim = 31 (used above). With 3-step Markov model

Pr(i, j, k, l, m) = πi,j,k pi,j,k,l pj,k,l,m has dim = 7 + 8 = 15: somewhat better precision, using

Mi,j,k Ci,j,k,l πbi,j,k = , bpi,j,k,l = . n Ci,j,k,0 + Ci,j,k,1 With sustained eﬀorts: zero-mean normal limits for all √ √ n(πbi,j,k − πi,j,k ), n(bpi,j,k,1 − pi,j,k,1), ∗ can then reach better precision, using pj instead of bpj : k ∗ X n ∗ pB,j o H(p ) = wj pj log − (pB,j − pA,j ) . pA,j j=1 Results: somewhat slimmer conﬁdence curves, similar to those above. 14 of XV Remarks

A: I’ve used the sentence-endings data from Cox and Brandwood (JRSS B, 1959), but done more: they constructed a simple ad hoc score, whereas I have deﬁned distances dw (p, pA) − dw (p, pB ), estimated these via Markov models, leading to higher precision, etc. B: This is ﬁne, given these data alone: Tim, Soph, Crit, Pol, Phil, with Soph and Crit being close. Can use the II-CC-FF paradigm of Cunen and Hjort (SJS, 2021) to combine my analysis summary with those of others, using other aspects of the dialogues, expert opinions among Platon scholars, etc.

C: Which models for p = (p1,..., p32) are the best? Can throw it to the categorial FIC machinery of Jullum and Hjort (Statistica Sinica, 2017). D: There are other problems with similar characteristics. In which order did Marcus Aurelius and Fronto write all their letters? Were BWV 4, 106, 150 really composed already in M¨uhlhausen?

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