*Figure: Graphical representation of the quantification of dialectics.*

A sequence S of P philosophers along a given period of time would incorporate the P most prominent and visible philosophers in that interval. The use of such a criterion to build the time-sequence for the philosophers implies in not necessarily uniform time-intervals between each pair of subsequent entries.

The set of C measurements used to characterize the philosophers define a C−dimensional feature space which will be henceforth referred to as the philosophical space. The characteristic vector v⃗_{i} of each philosopher i defines a respective philosophical state in the philosophical space. Given a set of P philosophers, the average state at time i, i ≤ P, is defined as

a⃗_{i} = 1/i ∑_{k=1}^{i} v⃗_{k}

The opposite state of a given philosophical state v⃗_{i} is defined as:

r⃗_{i} = v⃗_{i} +2(a⃗_{i} −v⃗_{i}) = 2a⃗_{i} − v⃗_{i}

The opposition vector of philosophical state v⃗_{i} is given by D⃗_{i} = r⃗_{i} − v⃗_{i}. The opposition amplitude of that same state is defined as ||D⃗_{i}||.

An emphasis move taking place from the philosophical state v⃗_{i} is any displacement from v⃗_{i} along the direction −r⃗_{i}. A contrary move from the philosophical state v⃗_{i} is any displacement from v⃗_{i} along the direction r⃗_{i}.

Given a time-sequence S of P philosophers, the philosophical move implied by two successive philosophers i and j corresponds to the M⃗_{i,j} vector extending from v⃗_{i }to v⃗_{j} , i.e.

M⃗_{i,j} = v⃗_{j} – v⃗_{i}

In principle, an innovative or differentiated philosophical move would be such that it departs substantially from the current philosophical state v⃗_{i}. Decomposing innovation moves into two main subtypes: opposition and skewness.

The opposition index W_{i,j} of a given philosophical move M⃗_{i,j} is defined as

W_{i,j} = 〈M⃗_{i,j}, D⃗_{i}〉/ ||D⃗_{i}||^{2}

This index quantifies the intensity of opposition of that respective philosophical move, in the sense of having a large projection along the vector D⃗_{i}. It should also be noticed that the repetition of opposition moves lead to little innovation, as it would imply in an oscillation around the average state. The skewness index s_{i,j} of that same philosophical move is the distance between v⃗_{j} and the line L defined by the vector D⃗_{i}, and therefore quantifies how much the new philosophical state departs from the respective opposition move. Actually, a sequence of moves with zero skewness would represent more trivial oscillations within the opposition line L_{i}.

We also suggest an index to quantify the dialectics between a triple of successive philosophers i, j and k. More specifically, the philosophical state v⃗_{i} is understood as the thesis, the state j is taken as the antithesis, with the synthesis being associated to the state v⃗_{k}. The hypothesis that k is the consequence, among other forces, of a dialectics between the views v⃗_{i} and v⃗_{j} can be expressed by the fact that the philosophical state v⃗_{k} be located near the middle line ML_{i,j} defined by the thesis and antithesis (i.e. the points which are at an equal distance to both v⃗_{i} and v⃗_{j}) relatively to the opposition amplitude ||D⃗_{i}||.

Therefore, the counter-dialectic index is defined as

ρ_{i→k} = d_{i→k} /||M⃗_{i,j}||

where d_{i→k} is the distance between the philosophical state v⃗_{k} and the middle-line M_{Li,j} between v⃗_{i} and v⃗_{j}. Note that 0 ≤ d_{i→k} ≤ 1. The choice of counter-dialectics instead of dialectics is justified to maintain compatibility with the use of a distance from point to line as adopted for the definition of skewness….