Badiou, Heyting Algebras cross the Grothendieck Topoi. Note Quote.

Let us commence by introducing the local formalism that constitutes the basis of Badiou’s own, ‘calculated phenomenology’. Badiou is unwilling to give up his thesis that the history of thinking of being (ontology) is the history of mathematics and, as he reads it, that of set theory. It is then no accident that set theory is the regulatory framework under which topos theory is being expressed. He does not refer to topoi explicitly but rather to the so called complete Heyting algebras which are their procedural equivalents. However, he fails to mention that there are both ‘internal’ and ‘external’ Heyting algebras, the latter group of which refers to local topos theory, while it appears that he only discusses the latter—a reduction that guarantees that indeed that the categorical insight may give nothing new.

Indeed, the external complete Heyting algebras T then form a category of the so called T-sets, which are the basic objects in the ‘world’ of the Logics of Worlds. They local topoi or the so called ‘locales’ that are also ‘sets’ in the traditional sense of set theory. This ‘constitution’ of his worlds thus relies only upon Badiou’s own decision to work on this particular regime of objects, even if that regime then becomes pivotal to his argument which seeks to denounce the relevance of category theory.

This problematic is particularly visible in the designation of the world m (mathematically a topos) as a ‘complete’ (presentative) situation of being of ‘universe [which is] the (empty) concept of a being of the Whole’ He recognises the ’impostrous’ nature of such a ‘whole’ in terms of Russell’s paradox, but in actual mathematical practice the ’whole’ m becomes to signify the category of Sets – or any similar topos that localizable in terms of set theory. The vocabulary is somewhat confusing, however, because sometimes T is called the ‘transcendental of the world’, as if m were defined only as a particular locale, while elsewhere m refers to the category of all locales (Loc).

An external Heyting algebra is a set T with a partial order relation <, a minimal element μ ∈ T , a maximal element M ∈ T . It further has a ‘conjunction’ operator ∧ : T × T → T so that p ∧ q ≤ p and p ∧ q = q ∧ p. Furthermore, there is a proposition entailing the equivalence p ≤ q iff p ∧ q = p. Furthermore p ∧ M = p and μ ∧ p = μ for any p ∈ T .

In the ‘diagrammatic’ language that pertains to categorical topoi, by contrast, the minimal and maximal elements of the lattice Ω can only be presented as diagrams, not as sets. The internal order relation ≤ Ω can then be defined as the so called equaliser of the conjunction ∧ and projection-map

≤Ω →^e Ω x Ω →_π1^∧ L

The symmetry can be expressed diagrammatically by saying that

is a pull-back and commutes. The minimal and maximal elements, in categorical language, refer to the elements evoked by the so-called initial and terminal objects 0 and 1.

In the case of local Grothendieck-topoi – Grothendieck-topoi that support generators – the external Heyting algebra T emerges as a push-forward of the internal algebra Ω, the logic of the external algebra T := γ ∗ (Ω) is an analogous push-forward of the internal logic of Ω but this is not the case in general.

What Badiou further requires of this ‘transcendental algebra’ T is that it is complete as a Heyting algebra.

A complete external Heyting algebra T is an external Heyting algebra together with a function Σ : PT → T (the least upper boundary) which is distributive with respect to ∧. Formally this means that ΣA ∧ b = Σ{a ∧ b | a ∈ A}.

In terms of the subobject classifier Ω, the envelope can be defined as the map Ω^t : Ω^Ω → Ω¹ ≅ Ω, which is internally left adjoint to the map ↓ seg : Ω → Ω^Ω that takes p ∈ Ω to the characteristic map of ↓ (p) = {q ∈ Ω | q ≤ p}27.

The importance the external complete Heyting algebra plays in the intuitionist logic relates to the fact that one may now define precisely such an intuitionist logic on the basis of the operations defined above.

The dependence relation ⇒ is an operator satisfying

p ⇒ q = Σ{t | p ∩ t ≤ q}.

(Negation). A negation ¬ : T → T is a function so that

¬p =∑ {q | p ∩ q = μ},

and it then satisfies p ∧ ¬p = μ.

Unlike in what Badiou calls a ‘classical world’ (usually called a Boolean topos, where ¬¬ = 1_Ω), the negation ¬ does not have to be reversible in general. In the domain of local topoi, this is only the case when the so called internal axiom of choice is valid, that is, when epimorphisms split – for example in the case of set theory. However, one always has p ≤ ¬¬p. On the other hand, all Grothendieck-topoi – topoi still materially presentable over Sets – are possible to represent as parts of a Boolean topos.

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Dark Matter as an Ode to Ma Kali. Note Quote.

Arcane knowledge provides some answers assuming we ask the questions. If Isis is “Infinite Stars, Infinite Space”, then what is Nepthys? Being the opposite side of Isis we have to assume she plays a part in Universe. And, if Kali’s re-creation of Universe is possible, then can we see it in the process? The answer to both of these lies in the Dark Matter. This is very intriguing but resolved in the connection in Isis’s dark twin, Nepthys. She is dark (like Kali) because she is hidden, manifested but unseen. It is speculated that she became dominant when Isis was shedding lunar blood (sacred to Kali), this is when the unfertile seeds are being discarded. For the aspirant this is a time of great power, and danger. Nepthys is the goddess of the night magicks, the red magick of Vamamarg sometimes referred to as the “left hand path”. Hers is the force of re-creation which is so vital to the growth of the aspirant. IAO, Isis-creator, Apophis (Set, husband of Nepthys)-destroyer, and Osiris-re-creator. In Tantra, Kali is all three. She gives birth to Universe, devours it when all life has expended its energy, and re-creates it from the seeds of the old Universe. It’s uncertain whether there is enough Dark Matter to cause the collapse of Universe, but clearly if there is a chance, it is in this manifestation of the Dark Goddess. Her body is the body of matter that lies “between” known spaces and stars, her power is felt in the pull of matter itself, “Love is the law, love under will” is the axion of gravity where all particles seek to unite with all others. Her books are written in the night sky, her rites are the rites of ancient humans awed by the power of the Great Sleep, and equally awed by it’s power of re-creation. If Kali/Nepthys manifests at the end of time, it will be as the mouths of numerous black holes, each larger one devouring the smaller, uniting in one undifferentiated monad of space-time, not only matter sucked in but the net of creation on which it resides as well. In the Dark Matter is the new creation. Dark matter is maddeningly shy. More like a de-terrestrial-centric potency for sure with none of the considerations for earthlings.

Alpha

Alpha is known as the difference between a fund’s expected returns and its actual returns. Alpha has a very close relationship with another financial term known as beta. Beta is a measure used to determine a fund’s expected returns. Along with being the term used for expected returns, beta is also associated with the level of risk.

Alpha is commonly considered the active return on an investment, working as a gauge to determine how a fund is performing against the average. In some cases, the alpha can be construed as the value a portfolio manager can bring to a fund. A smart manager will be capable of exceeding the expected returns, bringing a positive alpha. A manager who is not as successful and does not perform as expected will yield a negative alpha. However, a positive alpha can also be due to luck with the markets. There is no way to determine which is the case.

Calculating the alpha for a fund can be tricky and involves a number of factors. The formula for alpha is:

Alpha = r – R_f – beta * (R_m – R_f)

r = the security’s or portfolio’s return

R_f = the risk-free rate of return

beta = systemic risk of a portfolio

R_m = the market return

The final result is a number, either negative or positive, depending on the performance of the fund. The higher your beta, the more difficult it is for your alpha to be a positive number.

With everything else being equal, the market likely would be more efficient if all companies followed the unwritten rules – or if they were required to reward their shareholders by systematically increasing the stock price via, say, buybacks with a formulaic relation between the required buybacks and earnings (among other details). As it stands, the supply and demand is driven by what appears to be a rather random perception/interpretation of the earnings announcements (among other information) by market participants. This leads to volatility and mispricings at various time horizons. These mispricings are then arbitraged away by what can be generically termed as “mean-reversion” (or “contrarian”) strategies. For a given “mean-reversion” time horizon there might also exist opportunities to profit via what can be generically termed as “momentum” strategies on accordingly shorter (and, in some cases, longer) time horizons.

One important ingredient that is implicitly assumed in the above discussion is market impact and executions. Even if every company under the Sun followed the unwritten rules, due to a large number of market players and virtual impossibility to predict supply and demand imbalances or their precise timings, mispricings and inefficiencies are inevitable. Longer horizon strategies thereby create arbitrage opportunities on somewhat shorter horizons; strategies on such scales create arbitrage opportunities on yet shorter time scales; and so on – all the way down to HFT (high frequency trading) strategies. While on the longest horizon time scales the strategies are mostly long (mutual and pension funds, holding companies, etc.), on shorter horizons strategies can be dollar neutral, hence seemingly creating profit “out of thin air” – which does not take into account substantial monetary and human capital involved.

Carnap’s Topological Properties and Choice of Metric. Note Quote.

Husserl’s system is ontologically, a traditional double hierarchy. There are regions or spheres of being, and perfectly traditional ones, except that (due to Kant’s “Copernican revolution”) the traditional order is reversed: after the new Urregion of pure consciousness come the region of nature, the psychological region, and finally a region (or perhaps many regions) of Geist. Each such region is based upon a single highest genus of concrete objects (“individua”), corresponding to the traditional highest genera of substances: in pure consciousness, for example, Erlebnisse; in nature, “things” (Dinge). But each region also contains a hierarchy of abstract genera – genera of singular abstracta and of what Husserl calls “categorial” or “syntactic” objects (classes and relations). This structure of “logical modifications,” found analogously in each region, is the concern of logic. In addition, however, to the “formal essence” which each object has by virtue of its position in the logical hierarchy, there are also truths of “material” (sachliche) essence, which apply to objects as members of some species or genus – ultimately, some region of being. Thus the special sciences, which are individuated (as in Aristotle) by the regions they study, are each broadly divided into two parts: a science of essence and a science of “matters of fact.” Finally, there are what might be called matters of metaphysical essence: necessary truths about objects which apply in virtue of their dependence on objects in prior regions, and ultimately within the Urregion of pure consciousness.

This ontological structure translates directly into an epistemological one, because all being in the posterior regions rests on positing Erlebnisse in the realm of pure consciousness, and in particular on originary (immediate) rational theoretical positings, i.e. “intuitions.” The various sciences are therefore based on various types of intuition. Sciences of matters of fact, on the one hand, correspond to the kinds of ordinary intuition, analogous to perception. Sciences of essence, on the other hand, and formal logic, correspond to (formal or material) “essential insight” (Wesensschau). Husserl equates formal- and material-essential insight, respectively, as sources of knowledge, to Kant’s analytic and synthetic a priori, whereas ordinary perceptual intuition, the source of knowledge about matters of fact, corresponds to the Kantian synthetic a posteriori. Phenomenology, finally, as the science of essence in the region of pure consciousness, has knowledge of the way beings in one region are dependent on those in another.

In Carnap’s doctoral thesis, Der Raum, he applies the above Husserlian apparatus to the problem of determining our sources of knowledge about space. Is our knowledge of space analytic, synthetic a priori, or empirical? Carnap answers, in effect: it depends on what you mean by “space.” His answer foreshadows much of his future thought, but is also based directly on Husserl’s remark about this question in Ideen I: that, whereas Euclidean manifold is a formal category (logical modification), our knowledge of geometry as it applies to physical objects is a knowledge of material essence within the region of nature. Der Raum is largely an expansion and explication of that one remark. Our knowledge of “formal space,” Carnap says, is analytic, i.e. derives from “formal ontology in Husserl’s sense,” but our knowledge of the “intuitive space” in which sensible objects are necessarily found is synthetic a priori, i.e. material-essential. There is one important innovation: Carnap claims that essential (a priori) knowledge of intuitive space extends only to its topological properties, whereas the full structure of physical space requires also a choice of metric. This latter choice is informed by the actual behavior of objects (e.g. measuring rods), and knowledge of physical space is thus in part a posteriori – as Carnap also says, a knowledge of “matters of fact.” But such considerations never force the choice of one metric or another: our knowledge of physical space also depends on “free positing”. This last point, which has no equivalent in Husserl, is important. Still more telling is that Carnap compares the choice involved here to a choice of language, although at this stage he sees this as a mere analogy. On the whole, however, the treatment of Der Raum is more or less orthodoxly Husserlian.

Conjuncted: Hobbes’ Authoritarianism (2)

Hobbes built up a theory of most thorough going collectivism but a rationale of such a collectivism was the peace and the security of the person and property of the individual, which gives a tinge of individualism to the theory of Hobbes. He even allowed his individual the right to resist his Sovereign if the latter attacked the individual’s life for whose preservation the contract was entered into. In certain contingencies the individual could withdraw the allegiance to the Sovereign who was not supporting the individual’s life. “The obligation of the subjects to the Sovereign is understood to last as long, and no longer, than the power lasteth, by which he is able to protect them.” A man has a right to disobey his Sovereign if the latter commands to “kill, wound or maim himself; or not to resist those that assault him; or to abstain from the use of food, medicine or any other thing without which he cannot live.” An individual has the right to refuse allegiance to a deposed Sovereign. Hobbes is an individualist, since his entire system is based on the individual psychology of fear and self-defence. In so far as Hobbes equates the right of resistance of an individual to his capacity to resist, the right equally vanishes against the strength of the Sovereign. The right of the individual to resist the Sovereign if the individual’s life is endangered implies that the individual is the judge as to when his life is endangered. On the Hobbesian view of human nature, the individual will misuse this right and resist the Sovereign as often as he can. This will destroy what Hobbes wanted to create i.e. unlimited absolutism of the Sovereign.

Religion:

Hobbes always considered hereditary absolute monarchy as the best form of the State. He does consider the possibility of elected monarchy, under which the ‘people is sovereign in property’, but not ‘in use’. Hobbes always maintained the distinction between the natural and the artificial State. He distinguished between the ‘commonwealth by acquisition’ and the ‘commonwealth by institution’; the former being based on natural force and the latter on voluntary subjection to an elected Government. Hobbes stresses on the fact that the law of nature is obligatory not only on the basis of Sovereign command but also as delivered in the ‘word of God’. Hobbes mentions solicitude for the eternal salvation of the subjects. With dual intentions, Hobbes becomes an interpreter; of the Bible in the first place to make use of the authority of the scriptures themselves. He does criticize religion in his most important three discourses. We have seen how he answers the question, “on what authority does one believe that scriptures are the word of God?”

Historicity:

Hobbes turning to history is filled with philosophic intentions. Hobbes reiterated the fact that it is history and not philosophy that gives man, prudence. The study is concerned with the historicity of its material; that the clear knowledge of application of the norms which obtain for human actions is the knowledge of actions, which have taken place in the past. Philosophy seeks general precepts, while the study seeks the application and realization of the precepts, the conditions and results of those precepts.

Through history a reader is to be taught which kinds of aims are salutary or destructive. History is often taken up to remedy man’s disobedience. It can be stated that the development, at least in the 16^th century, justifies the assertion that the reason why philosophy turned to history is the repression of the morality of obedience.

Contribution to political philosophy:

The cardinal contribution made by Hobbes in the field of political philosophy was his doctrine of sovereignty. The concept of sovereignty had begun to develop in the hands of Machiavelli, Bodin and Grotius, but Hobbes was facile princeps to give it the shape and content, which sovereignty holds today. He was one of the first to see that the idea of sovereignty lay at the root of any State. His Leviathan aroused the indignation of almost all-important interests in England. His Erastianism (adherence of the supposed doctrine of Erastus. Subordinating ecclesiastical to secular power), was distasteful to the Church. Devoted churchmen found it absolutely intolerable that the Church was a mere department of the State. The monarchists, who adhered to the belief in the Divine Right of Kings, did not appreciate his secular theory based on a social contract. The Royalists on the other hand did not like the Hobbesian view of sovereignty because it justified the de facto Government of a successful dictator as much as that of a legitimate monarchy, and justified the absolutism of a Parliament as much as that of a King. Hobbes thought of the Divine Right of the State as compared to the Divine Right of the King. The parliamentarians viewed with scorn the opposition of Hobbes to mixed Government and constitutional checks. Although his political philosophy was little noticed in England, it created a stir on the continent. While Machiavelli had separated politics from religion and morals, Hobbes not only kept them separate but subordinated religion and morals to politics. Hobbes scored over Machiavelli in his exaltation of the State; for Machiavelli was never so absolutist as to declare that the laws of nature and the laws of God were to find their expression only through the interpretation and the will of the sovereign. The sovereignty of Hobbes was indivisible and unlimited. Hobbes knew that the basis of moral and legal right was reason, but to Hobbes, this reason was the reason of the sovereign expressed through his will only. Hobbes was an individualist in so far as he believed in the natural equality of men. His cardinality as a political philosopher lies in his deriving logically from a mass of free and equal individuals the concept of an omnipotent State. The brilliance of Hobbes was that he turned the theory of early liberalism to the defence of unlimited absolutism at a time when absolutism, born of Divine Right of Kings, was quickly losing its theoretical applications and practical implications.

CRITICISM OF HOBBES AS A POLITICAL PHILOSOPHER

The Hobbesian theory of social contract implies that man brings with him to the social contract ‘rights’ of the State of nature, which are devoid of social function, social recognition and hence they could only be powers. Hobbes proceeds to evolve his civil society on the basis of social contract, which suddenly transforms the chaos of the State of nature into the ordered civil society. Social contract is itself made in the State of nature. The Hobbesian sovereign is the representative of the people. But what guarantee is there that this ‘representative’ of the people will ‘represent’ the people by following public opinion and looking after the public welfare. John Locke attacked Hobbesian social contract according to which, when men quitting the State of nature entered into society, they agreed that all of them but one should be under the restraint of laws; but that he should still retain all the liberty of the State of nature, increased with power and made licentious by impunity. This is to think that men are so foolish that they take care to avoid what mischief polecats and foxes, may do them, but are content, nay, think it safety, to be devoured by lions. Hobbes builds up his theory on the basis of pleasure-pain theory and evolves a master slave relationship. Hobbes regards matter and motion alone as real. His dogmatic materialism lives him little scope for freedom of human will. He is neither fully utilitarian, nor fully an idealist.

The unsoundness of Hobbes’ State of nature was a State of war of all against all in which the cardinal virtues are force and fraud. So, its clear that Hobbes’ man is anti-social. How could such a man go against his own  nature and suddenly enter a State not of war, but of peace, a State in which force and fraud are deliberately set aside, a State which is founded upon the ideas of right and justice, and in which acts of wrong and injustice are put under the double ban of public disapproval and of positive prohibition? With Hobbes, the self-interest of an individual before the contract is suddenly changed into his duties towards the sovereign after the contract. If men are not all force and fraud, they do not need an absolute sovereign; if they are; they cannot render passive obedience to him for all time after the contract. With Hobbes, it was Absolutism or Anarchy. The only remedy for the good behaviour of men was the coercive power of the sovereign. Hobbes failed to realize that there were other characteristics besides the fear of law and punishment, which kept men from relapsing into anarchy, viz., reason, religion and public opinion based on the faculty of reason. The Leviathan of Hobbes essentially states, ‘This State is a necessary evil, an instrument to defend men against their savage instincts, not to achieve a free and a progressive civilization’. Hobbesian position is that in the case of sovereign, might is right. 

Hobbes was a materialist and a rationalist. His philosophy vindicated the absolute sovereignty of whatever Government happened to be in power. Hobbes believed that human nature was bad but he held that it could be rendered moral by the State, which must be preserved at all costs. It must be realized that Hobbesian State is authoritarian and not totalitarian.

Unlike the totalitarian system, it is based on contractual obligations. In his State, there is equality of all before law and there is no privileged ruling class. Unlike the totalitarian State, it insists on the outward conformity of the subjects to law and not inner conformity to opinions and beliefs. His State, unlike the totalitarian, does not swallow the individual.