Hegel and Topos Theory. Thought of the Day 46.0

The intellectual feat of Lawvere is as important as Gödel’s formal undecidability theorem, perhaps even more. But there is a difference between both results: whereas Gödel led to a blind alley, Lawvere has displayed a new and fascinating panorama to be explored by mathematicians and philosophers. Referring to the positive results of topos theory, Lawvere says:

A science student naively enrolling in a course styled “Foundations of Mathematics” is more likely to receive sermons about unknowability… than to receive the needed philosophical guide to a systematic understanding of the concrete richness of pure and applied mathematics as it has been and will be developed. (Categories of space and quantity)

One of the major philosophical results of elementary topos theory, is that the way Hegel looked at logic was, after all, in the good track. According to Hegel, formal mathematical logic was but a superficial tautologous script. True logic was dialectical, and this logic ruled the gigantic process of the development of the Idea. Inasmuch as the Idea was autorealizing itself through the opposition of theses and antitheses, logic was changing but not in an arbitrary change of inferential rules. Briefly, in the dialectical system of Hegel logic was content-dependent.

Now, the fact that every topos has a corresponding internal logic shows that logic is, in quite a precise way, content-dependent; it depends on the structure of the topos. Every topos has its own internal logic, and this logic is materially dependent on the characterization of the topos. This correspondence throws new light on the relation of logic to ontology. Classically, logic was considered as ontologically aseptic. There could be a multitude of different ontologies, but there was only one logic: the classical. Of course, there were some mathematicians that proposed a different logic: the intuitionists. But this proposal was due to not very clear speculative epistemic reasons: they said they could not understand the meaning of the attributive expression “actual infinite”. These mathematicians integrated a minority within the professional mathematical community. They were seen as outsiders that had queer ideas about the exact sciences. However, as soon as intuitionistic logic was recognized as the universal internal logic of topoi, its importance became astronomical. Because it provided, for the first time, a new vision of the interplay of logic with mathematics. Something had definitively changed in the philosophical panorama.

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Dialectics: Mathematico-Philosophical Sequential Quantification. Drunken Risibility.

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=1ⁱ 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….