# Badiou’s Materiality as Incorporeal Ontology. Note Quote.

Badiou criticises the proper form of intuition associated with multiplicities such as space and time. However, his own ’intuitions’ are constrained by set theory. His intuition is therefore as ‘transitory’ as is the ontology in terms of which it is expressed. Following this constrained line of reasoning, however, let me now discuss how Badiou encounters the question of ‘atoms’ and materiality: in terms of the so called ‘atomic’ T-sets.

If topos theory designates the subobject-classifier Ω relationally, the external, set-theoretic T-form reduces the classificatory question again into the incorporeal framework. There is a set-theoretical, explicit order-structure (T,<) contra the more abstract relation 1 → Ω pertinent to categorical topos theory. Atoms then appear in terms of this operator <: the ‘transcendental grading’ that provides the ‘unity through which all the manifold given in an intuition is united in a concept of the object’.

Formally, in terms of an external Heyting algebra this comes down to an entity (A,Id) where A is a set and Id : A → T is a function satisfying specific conditions.

Equaliser: First, there is an ‘equaliser’ to which Badiou refers as the ‘identity’ Id : A × A → T satisfies two conditions:

1) symmetry: Id(x, y) = Id(y, x) and
2) transitivity: Id(x, y) ∧ Id(y, z) ≤ Id(x, z).

They guarantee that the resulting ‘quasi-object’ is objective in the sense of being distinguished from the gaze of the ‘subject’: ‘the differences in degree of appearance are not prescribed by the exteriority of the gaze’.

This analogous ‘identity’-function actually relates to the structural equalization-procedure as appears in category theory. Identities can be structurally understood as equivalence-relations. Given two arrows X ⇒ Y , an equaliser (which always exists in a topos, given the existence of the subobject classifier Ω) is an object Z → X such that both induced maps Z → Y are the same. Given a topos-theoretic object X and U, pairs of elements of X over U can be compared or ‘equivalized’ by a morphism XU × XUeq ΩU structurally ‘internalising’ the synthetic notion of ‘equality’ between two U-elements. Now it is possible to formulate the cumbersome notion of the ‘atom of appearing’.

An atom is a function a : A → T defined on a T -set (A, Id) so that
(A1) a(x) ∧ Id(x, y) ≤ a(y) and
(A2) a(x) ∧ a(y) ≤ Id(x, y).
As expressed in Badiou’s own vocabulary, an atom can be defined as an ‘object-component which, intuitively, has at most one element in the following sense: if there is an element of A about which it can be said that it belongs absolutely to the component, then there is only one. This means that every other element that belongs to the component absolutely is identical, within appearing, to the first’. These two properties in the definition of an atom is highly motivated by the theory of T-sets (or Ω-sets in the standard terminology of topological logic). A map A → T satisfying the first inequality is usually thought as a ‘subobject’ of A, or formally a T-subset of A. The idea is that, given a T-subset B ⊂ A, we can consider the function
IdB(x) := a(x) = Σ{Id(x,y) | y ∈ B}
and it is easy to verify that the first condition is satisfied. In the opposite direction, for a map a satisfying the first condition, the subset
B = {x | a(x) = Ex := Id(x, x)}
is clearly a T-subset of A.
The second condition states that the subobject a : A → T is a singleton. This concept stems from the topos-theoretic internalization of the singleton-function {·} : a → {a} which determines a particular class of T-subsets of A that correspond to the atomic T-subsets. For example, in the case of an ordinary set S and an element s ∈ S the singleton {s} ⊂ S is a particular, atomic type of subset of S.
The question of ‘elements’ incorporated by an object can thus be expressed externally in Badiou’s local theory but ‘internally’ in any elementary topos. For the same reason, there are two ways for an element to be ‘atomic’: in the first sense an ‘element depends solely on the pure (mathematical) thinking of the multiple’, whereas the second sense relates it ‘to its transcendental indexing’. In topos theory, the distinction is slightly more cumbersome. Badiou still requires a further definition in order to state the ‘postulate of materialism’.
An atom a : A → T is real if if there exists an element x ∈ T so that a(y) = Id(x,y) ∀ y ∈ A.
This definition gives rise to the postulate inherent to Badiou’s understanding of ‘democratic materialism’.
Postulate of Materialism: In a T-set (A,Id), every atom of appearance is real.
What the postulate designates is that there really needs to exist s ∈ A for every suitable subset that structurally (read categorically) appears to serve same relations as the singleton {s}. In other words, what ‘appears’ materially, according to the postulate, has to ‘be’ in the set-theoretic, incorporeal sense of ‘ontology’. Topos theoretically this formulation relates to the so called axiom of support generators (SG), which states that the terminal object 1 of the underlying topos is a generator. This means that the so called global elements, elements of the form 1 → X, are enough to determine any particular object X.
Thus, it is this specific condition (support generators) that is assumed by Badiou’s notion of the ‘unity’ or ‘constitution’ of ‘objects’. In particular this makes him cross the line – the one that Kant drew when he asked Quid juris? or ’Haven’t you crossed the limit?’ as Badiou translates.
But even without assuming the postulate itself, that is, when considering a weaker category of T-sets not required to fulfill the postulate of atomism, the category of quasi-T -sets has a functor taking any quasi-T-set A into the corresponding quasi-T-set of singletons SA by x → {x}, where SA ⊂ PA and PA is the quasi-T-set of all quasi-T-subsets, that is, all maps T → A satisfying the first one of the two conditions of an atom designated by Badiou. It can then be shown that, in fact, SA itself is a sheaf whose all atoms are ‘real’ and which then is a proper T-set satisfying the ‘postulate of materialism’. In fact, the category of T-Sets is equivalent to the category of T-sheaves Shvs(T, J). In the language of T-sets, the ‘postulate of materialism’ thus comes down to designating an equality between A and its completed set of singletons SA.

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# Priest’s Razor: Metaphysics. Note Quote.

The very idea that some mathematical piece employed to develop an empirical theory may furnish us information about unobservable reality requires some care and philosophical reflection. The greatest difficulty for the scientifically minded metaphysician consists in furnishing the means for a “reading off” of ontology from science. What can come in, and what can be left out? Different strategies may provide for different results, and, as we know, science does not wear its metaphysics on its sleeves. The first worry may be making the metaphysical piece compatible with the evidence furnished by the theory.

The strategy adopted by da Costa and de Ronde may be called top-down: investigating higher science and, by judging from the features of the objects described by the theory, one can look for the appropriate logic to endow it with just those features. In this case (quantum mechanics), there is the theory, apparently attributing contradictory properties to entities, so that a logic that does cope with such feature of objects is called forth. Now, even though we believe that this is in great measure the right methodology to pursue metaphysics within scientific theories, there are some further methodological principles that also play an important role in these kind of investigation, principles that seem to lessen the preferability of the paraconsistent approach over alternatives.

To begin with, let us focus on the paraconsistent property attribution principle. According to this principle, the properties corresponding to the vectors in a superposition are all attributable to the system, they are all real. The first problem with this rendering of properties (whether they are taken to be actual or just potential) is that such a superabundance of properties may not be justified: not every bit of a mathematical formulation of a theory needs to be reified. Some of the parts of the theory are just that: mathematics required to make things work, others may correspond to genuine features of reality. The greatest difficulty is to distinguish them, but we should not assume that every bit of it corresponds to an entity in reality. So, on the absence of any justified reason to assume superpositions as a further entity on the realms of properties for quantum systems, we may keep them as not representing actual properties (even if merely possible or potential ones).

That is, when one takes into account other virtues of a metaphysical theory, such as economy and simplicity, the paraconsistent approach seems to inflate too much the population of our world. In the presence of more economical candidates doing the same job and absence of other grounds on which to choose the competing proposals, the more economical approaches take advantage. Furthermore, considering economy and the existence of theories not postulating contradictions in quantum mechanics, it seems reasonable to employ Priest’s razor – the principle according to which one should not assume contradictions beyond necessity – and stick with the consistent approaches. Once again, a useful methodological principle seems to deem the interpretation of superposition as contradiction as unnecessary.

The paraconsistent approach could take advantage over its competitors, even in the face of its disadvantage in order to accommodate such theoretical virtues, if it could endow quantum mechanics with a better understanding of quantum phenomena, or even if it could add some explanatory power to the theory. In the face of some such kind of gain, we could allow for some ontological extravagances: in most cases explanatory power rules over matters of economy. However, it does not seem that the approach is indeed going to achieve some such result.

Besides that lack of additional explanatory power or enlightenment on the theory, there are some additional difficulties here. There is a complete lack of symmetry with the standard case of property attribution in quantum mechanics. As it is usually understood, by adopting the minimal property attribution principle, it is not contentious that when a system is in one eigenstate of an observable, then we may reasonably infer that the system has the property represented by the associated observable, so that the probability of obtaining the eigenvalue associated is 1. In the case of superpositions, if they represented properties of their own, there is a complete disanalogy with that situation: probabilities play a different role, a system has a contradictory property attributed by a superposition irrespective of probability attribution and the role of probabilities in determining measurement outcomes. In a superposition, according to the proposal we are analyzing, probabilities play no role, the system simply has a given contradictory property by the simple fact of being in a (certain) superposition.

For another disanalogy with the usual case, one does not expect to observe a sys- tem in such a contradictory state: every measurement gives us a system in particular state, never in a superposition. If that is a property in the same foot as any other, why can’t we measure it? Obviously, this does not mean that we put measurement as a sign of the real, but when doubt strikes, it may be a good advice not to assume too much on the unobservable side. As we have observed before, a new problem is created by this interpretation, because besides explaining what is it that makes a measurement give a specific result when the system measured is in a superposition (a problem usually addressed by the collapse postulate, which seems to be out of fashion now), one must also explain why and how the contradictory properties that do not get actualized vanish. That is, besides explaining how one particular property gets actual, one must explain how the properties posed by the system that did not get actual vanish.

Furthermore, even if states like 1/√2 (| ↑x ⟩ + | ↓x ⟩) may provide for an example of a  candidate of a contradictory property, because the system seems to have both spin up and down in a given direction, there are some doubts when the distribution of probabilities is different, in cases such as 2/√7 | ↑x ⟩ + √(3/7) | ↓x ⟩. What are we to think about that? Perhaps there is still a contradiction, but it is a little more inclined to | ↓x⟩ than to | ↑x⟩? That is, it is difficult to see how a contradiction arises in such cases. Or should we just ignore the probabilities and take the states composing the superposition as somehow opposed to form a contradiction anyway? That would put metaphysics way too much ahead of science, by leaving the role of probabilities unexplained in quantum mechanics in order to allow a metaphysical view of properties in.

# Metempsychosis of the Ancients’ Veritability

The impenetrable veil of arcane secrecy was thrown over the sciences taught in the sanctuary. This is the cause of the modern depreciation of the ancient philosophies. Much of Plato’s public teachings and writings had therefore to consist of blinds or half-truths or allegories, and just as Jesus spoke in parables, so the Mysteries were ever reserved for special groups of neophytes – and, needless to say, they did not reach the Church of the days of Constantine, which never held the keys of the Mysteries, and hence can hardly be said to have lost them.

The ancient philosophers seem to be generally held, even by the least prejudiced of modern critics, to have lacked that profundity and thorough knowledge in the exact sciences of which a couple of last centuries and present are so boastful. It is even questioned whether they understood that basic scientific principle: ex nihilo nihil fit. If they suspected the indestructibility of matter at all – say these commentators – it was not in consequence of a firmly established formula, but only through intuitional reasoning and by analogy. The philosophers themselves had to be initiated into perceptive mysteries, before they could grasp the correct idea of the ancients in relation to this most metaphysical subject. Otherwise – outside such initiation – for every thinker there will be a “Thus far shalt thou go and no further,” mapped out by his intellectual capacity, as clearly and unmistakably as there is for the progress of any nation or race in its cycle by the law of karma. Much of current agnostic speculation on the existence of the “First Cause” is little better than veiled materialism — the terminology alone being different. Even so a thinker as Herbert Spencer speaks of the “Unknowable” occasionally in terms that demonstrate the lethal influence of materialistic thought which, like the deadly sirocco, has withered and blighted most of current ontological speculation. For instance, when he terms the “First Cause” — the Unknowable — a “power manifesting through phenomena,” and an “infinite eternal Energy,” (?) it is clear that he has grasped solely the physical aspect of the mystery of Being — the energies of cosmic substance only. The co-eternal aspect of the ONE REALITY — cosmic ideation — (as to its noumenon, it seems nonexistent in his mind) — is absolutely omitted from consideration.

The doctrine of metempsychosis has been abundantly ridiculed by scientists and rejected by theologians, yet if it had been properly understood in its application to the indestructibility of matter and the immortality of spirit, it would have been perceived that it is a sublime conception. Should we not first regard the subject from the standpoint of the ancients before venturing to disparage its teachers? The solution of the great problem of eternity belongs neither to religious superstition nor to gross materialism. The harmony and mathematical equiformity of the double evolution – spiritual and physical – are elucidated only in the universal numerals of Pythagoras, who built his system entirely upon the so-called metrical speech of the Hindu Vedas.

# Symmetry, Cohomology and Homotopy. Note Quote Didactic.

Given a compact Kähler manifold Y, we know that the cohomology groups H(Y,C) have a Hodge decomposition Hp,q. Now because we have Poincaré duality, and the comparisons between singular and De Rham cohomologie, we know that any other space Z that has the same homotopy type as Y will have the same cohomology groups. Consequently they will share the same Hodge diamond, thus its symmetries.

This means that the symmetry of the Hodge diamond is mostly attached to the homotopy type of Y . This is not surprising anymore because it can already be seen from the equivalence of the De Rham cohomology which is analytic, and the Betti cohomology which is something purely simplicial. In fact, it can also be seen from the (smooth) homotopy invariance of De Rham cohomology.

This symmetry can be understood using the Quillen-Segal formalism as follows. Given Y , let’s consider Ytop ∈ Top. We have a Quillen equivalence U : Top → sSetQ, where U = Sing is the singular functor whose left adjoint is the geometric realization. When we consider the comma category sSetQU[Top] = sSet ↓ U, we are literally creating in French a “trait d’union”, between the two categories. And when we consider the subcategory of Quillen-Segal objects then we have a triangle that descends to a triangle of equivalences between the homotopy categories. In fact there is a much better statement.

It turns out that if we choose the Joyal model structure sSetJ, we get the Homotopy hypothesis.

A fibrant replacement of Y in the model category sSetQU[Top], is a trivial fibration F → U(Y), where F is fibrant in sSetQ, that is a Kan complex. But a Kan complex is exactly an ∞-groupoid. ∞-Groupoids generalize groupoids, and still are category-like. In particular we can take their opposite (or dual), just like we consider the opposite category Cop of a usual category.

Given Y as above, we can think of the mirror of Y as the opposite ∞-groupoid Fop. A good approximation of Fop can be obtained by the schematization functor à la Toën applied to the simplicial set (quasicategory) underlying Fop. We can take as model for F the fundamental ∞-groupoid Π(Y). And depending on the dimension it’s enough to stop at the corresponding n-groupoid. Toën schematization functor can also be obtained from the Quillen-Segal formalism applied to the embedding

U : Sh(Var(C)) ֒→ sPresh(Var(C),

where on the right hand side we consider the model category of simplicial presheaves à la Jardine-Joyal. The representability of the π0 of the schematization has to be determined by descent along the equivalence type.