Black Hole Entropy in terms of Mass. Note Quote.

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If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

GD = G11/Ld = l911/Ld —– (1)

A Schwartzschild black hole of mass M has a radius

Rs ~ M(1/(D-3)) GD(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4GD —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ RsD-2 —– (4)

Thus

S ~ 1/GD (MGD)(D-2)/(D-3) ~ M(D-2)/(D-3) GD1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

Nmin(S) = MRs = M(MGD)1/D-3 = S —– (6)

We see that the value of Nmin in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = Nmin = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑3 of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T33 —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N2.

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N2T43 —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N23((E/N23))3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M2/N R —– (11)

Plugging (11) into (10)

S = N23(M2R/N23)3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

The Biological Kant. Note Quote.

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The biological treatise takes as its object the realm of physics left out of Kant’s critical demarcations of scientific, that is, mathematical and mechanistic, physics. Here, the main idea was that scientifically understandable Nature was defined by lawfulness. In his Metaphysical Foundations of Natural Science, this idea was taken further in the following claim:

I claim, however, that there is only as much proper science to be found in any special doctrine on nature as there is mathematics therein, and further ‘a pure doctrine on nature about certain things in nature (doctrine on bodies and doctrine on minds) is only possible by means of mathematics’.

The basic idea is thus to identify Nature’s lawfulness with its ability to be studied by means of mathematical schemata uniting understanding and intuition. The central schema, to Kant, was numbers, so apt to be used in the understanding of mechanically caused movement. But already here, Kant is very well aware of a whole series of aspects of spontaneuosly experienced Nature is left out of sight by the concentration on matter in movement, and he calls for these further realms of Nature to be studied by a continuation of the Copernican turn, by the mind’s further study of the utmost limits of itself. Why do we spontaneously see natural purposes, in Nature? Purposiveness is wholly different from necessity, crucial to Kant’s definition of Nature. There is no reason in the general concept of Nature (as lawful) to assume that nature’s objects may serve each other as purposes. Nevertheless, we do not stop assuming just that. But what we do when we ascribe purposes to Nature is using the faculties of mind in another way than in science, much closer to the way we use them in the appreciation of beauty and art, the object of the first part of the book immediately before the treatment of teleological judgment. This judgment is characterized by a central distinction, already widely argued in this first part of the book: the difference between determinative and reflective judgments, respectively. While the judgment used scientifically to decide whether a specific case follows a certain rule in explanation by means of a derivation from a principle, and thus constitutes the objectivity of the object in question – the judgment which is reflective lacks all these features. It does not proceed by means of explanation, but by mere analogy; it is not constitutive, but merely regulative; it does not prove anything but merely judges, and it has no principle of reason to rest its head upon but the very act of judging itself. These ideas are now elaborated throughout the critic of teleological judgment.

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In the section Analytik der teleologischen Urteilskraft, Kant gradually approaches the question: first is treated the merely formal expediency: We may ascribe purposes to geometry in so far as it is useful to us, just like rivers carrying fertile soils with them for trees to grow in may be ascribed purposes; these are, however, merely contingent purposes, dependent on an external telos. The crucial point is the existence of objects which are only possible as such in so far as defined by purposes:

That its form is not possible after mere natural laws, that is, such things which may not be known by us through understanding applied to objects of the senses; on the contrary that even the empirical knowledge about them, regarding their cause and effect, presupposes concepts of reason.

The idea here is that in order to conceive of objects which may not be explained with reference to understanding and its (in this case, mechanical) concepts only, these must be grasped by the non-empirical ideas of reason itself. If causes are perceived as being interlinked in chains, then such contingencies are to be thought of only as small causal circles on the chain, that is, as things being their own cause. Hence Kant’s definition of the Idea of a natural purpose:

an object exists as natural purpose, when it is cause and effect of itself.

This can be thought as an idea without contradiction, Kant maintains, but not conceived. This circularity (the small causal circles) is a very important feature in Kant’s tentative schematization of purposiveness. Another way of coining this Idea is – things as natural purposes are organized beings. This entails that naturally purposeful objects must possess a certain spatio-temporal construction: the parts of such a thing must be possible only through their relation to the whole – and, conversely, the parts must actively connect themselves to this whole. Thus, the corresponding idea can be summed up as the Idea of the Whole which is necessary to pass judgment on any empirical organism, and it is very interesting to note that Kant sums up the determination of any part of a Whole by all other parts in the phrase that a natural purpose is possible only as an organized and self-organizing being. This is probably the very birth certificate of the metaphysics of self-organization. It is important to keep in mind that Kant does not feel any vitalist temptation at supposing any organizing power or any autonomy on the part of the whole which may come into being only by this process of self-organization between its parts. When Kant talks about the forming power in the formation of the Whole, it is thus nothing outside of this self-organization of its parts.

This leads to Kant’s final definition: an organized being is that in which all that is alternating is ends and means. This idea is extremely important as a formalization of the idea of teleology: the natural purposes do not imply that there exists given, stable ends for nature to pursue, on the contrary, they are locally defined by causal cycles, in which every part interchangeably assumes the role of ends and means. Thus, there is no absolute end in this construal of nature’s teleology; it analyzes teleology formally at the same time as it relativizes it with respect to substance. Kant takes care to note that this maxim needs not be restricted to the beings – animals – which we spontaneously tend to judge as purposeful. The idea of natural purposes thus entails that there might exist a plan in nature rendering processes which we have all reasons to disgust purposeful for us. In this vision, teleology might embrace causality – and even aesthetics:

Also natural beauty, that is, its harmony with the free play of our epistemological faculties in the experience and judgment of its appearance can be seen in the way of objective purposivity of nature in its totality as system, in which man is a member.

An important consequence of Kant’s doctrine is that their teleology is so to speak secularized in two ways: (1) it is formal, and (2) it is local. It is formal because self-organization does not ascribe any special, substantial goal for organisms to pursue – other than the sustainment of self-organization. Thus teleology is merely a formal property in certain types of systems. This is why teleology is also local – it is to be found in certain systems when the causal chain form loops, as Kant metaphorically describes the cycles involved in self-organization – it is no overarching goal governing organisms from the outside. Teleology is a local, bottom-up, process only.

Kant does not in any way doubt the existence of organized beings, what is at stake is the possibility of dealing with them scientifically in terms of mechanics. Even if they exist as a given thing in experience, natural purposes can not receive any concept. This implies that biology is evident in so far as the existence of organisms cannot be doubted. Biology will never rise to the heights of science, its attempts at doing so are beforehand delimited, all scientific explanations of organisms being bound to be mechanical. Following this line of argument, it corresponds very well to present-day reductionism in biology, trying to take all problems of phenotypical characters, organization, morphogenesis, behavior, ecology, etc. back to the biochemistry of genetics. But the other side of the argument is that no matter how successful this reduction may prove, it will never be able to reduce or replace the teleological point of view necessary in order to understand the organism as such in the first place.

Evidently, there is something deeply unsatisfactory in this conclusion which is why most biologists have hesitated at adopting it and cling to either full-blown reductionism or to some brand of vitalism, subjecting themselves to the dangers of ‘transcendental illusion’ and allowing for some Goethe-like intuitive idea without any schematization. Kant tries to soften up the question by philosophical means by establishing an crossing over from metaphysics to physics, or, from the metaphysical constraints on mechanical physics and to physics in its empirical totality, including the organized beings of biology. Pure mechanics leaves physics as a whole unorganized, and this organization is sought to be established by means of mediating concepts’. Among them is the formative power, which is not conceived of in a vitalist substantialist manner, but rather a notion referring to the means by which matter manages to self-organize. It thus comprehends not only biological organization, but macrophysic solid matter physics as well. Here, he adds an important argument to the critic of judgment:

Because man is conscious of himself as a self-moving machine, without being able to further understand such a possibility, he can, and is entitled to, introduce a priori organic-moving forces of bodies into the classification of bodies in general and thus to distinguish mere mechanical bodies from self-propelled organic bodies.

Metaphysical Continuity in Peirce. Thought of the Day 122.0

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Continuity has wide implications in the different parts of Peirce’s architectonics of theories. Time and time again, Peirce refers to his ‘principle of continuity’ which has not immediately anything to do with Poncelet’s famous such principle in geometry, but, is rather, a metaphysical implication taken to follow from fallibilism: if all more or less distinct phenomena swim in a vague sea of continuity then it is no wonder that fallibilism must be accepted. And if the world is basically continuous, we should not expect conceptual borders to be definitive but rather conceive of terminological distinctions as relative to an underlying, monist continuity. In this system, mathematics is first science. Thereafter follows philosophy which is distinguished form purely hypothetical mathematics by having an empirical basis. Philosophy, in turn, has three parts, phenomenology, the normative sciences, and metaphysics. The first investigates solely ‘the Phaneron’ which is all what could be imagined to appear as an object for experience: ‘ by the word phaneron I mean the collective total of all that is in any way or in any sense present to the mind, quite regardless whether it corresponds to any real thing or not.’ (Charles Sanders Peirce – Collected Papers of Charles Sanders Peirce) As is evident, this definition of Peirce’s ‘phenomenology’ is parallel to Husserl’s phenomenological reduction in bracketing the issue of the existence of the phenomenon in question. Even if it thus is built on introspection and general experience, it is – analogous to Husserl and other Brentano disciples at the same time – conceived in a completely antipsychological manner: ‘It religiously abstains from all speculation as to any relations between its categories and physiological facts, cerebral or other.’ and ‘ I abstain from psychology which has nothing to do with ideoscopy.’ (Letter to Lady Welby). The normative sciences fall in three: aesthetics, ethics, logic, in that order (and hence decreasing generality), among which Peirce does not spend very much time on the former two. Aesthetics is the investigation of which possible goals it is possible to aim at (Good, Truth, Beauty, etc.), and ethics how they may be reached. Logic is concerned with the grasping and conservation of Truth and takes up the larger part of Peirce’s interest among the normative sciences. As it deals with how truth can be obtained by means of signs, it is also called semiotics (‘logic is formal semiotics’) which is thus coextensive with theory of science – logic in this broad sense contains all parts of philosophy of science, including contexts of discovery as well as contexts of justification. Semiotics has, in turn, three branches: grammatica speculativa (or stekheiotics), critical logic, and methodeutic (inspired by mediaeval trivium: grammar, logic, and rhetoric). The middle one of these three lies closest to our days’ conception of logic; it is concerned with the formal conditions for truth in symbols – that is, propositions, arguments, their validity and how to calculate them, including Peirce’s many developments of the logic of his time: quantifiers, logic of relations, ab-, de-, and induction, logic notation systems, etc. All of these, however, presuppose the existence of simple signs which are investigated by what is often seen as semiotics proper, the grammatica speculativa; it may also be called formal grammar. It investigates the formal condition for symbols having meaning, and it is here we find Peirce’s definition of signs and his trichotomies of different types of sign aspects. Methodeutic or formal rhetorics, on the other hand, concerns the pragmatical use of the former two branches, that is, the study of how to use logic in a fertile way in research, the formal conditions for the ‘power’ of symbols, that is, their reference to their interpretants; here can be found, e.g., Peirce’s famous definitions of pragmati(ci)sm and his directions for scientific investigation. To phenomenology – again in analogy to Husserl – logic adds the interest in signs and their truth. After logic, metaphysics follows in Peirce’s system, concerning the inventarium of existing objects, conceived in general – and strongly influenced by logic in the Kantian tradition for seeing metaphysics mirroring logic. Also here, Peirce has several proposals for subtypologies, even if none of them seem stable, and under this headline classical metaphysical issues mix freely with generalizations of scientific results and cosmological speculations.

Peirce himself saw this classification in an almost sociological manner, so that the criteria of distinction do not stem directly from the implied objects’ natural kinds, but after which groups of persons study which objects: ‘the only natural lines of demarcation between nearly related sciences are the divisions between the social groups of devotees of those sciences’. Science collects scientists into bundles, because they are defined by their causa finalis, a teleologial intention demanding of them to solve a central problem.

Measured on this definition, one has to say that Peirce himself was not modest, not only does he continuously transgress such boundaries in his production, he frequently does so even within the scope of single papers. There is always, in his writings, a brief distance only from mathematics to metaphysics – or between any other two issues in mathematics and philosophy, and this implies, first, that the investigation of continuity and generality in Peirce’s system is more systematic than any actually existing exposition of these issues in Peirce’s texts, second, that the discussion must constantly rely on cross-references. This has the structural motivation that as soon as you are below the level of mathematics in Peirce’s system, inspired by the Comtean system, the single science receives determinations from three different directions, each science consisting of material and formal aspects alike. First, it receives formal directives ‘from above’, from those more general sciences which stand above it, providing the general frameworks in which it must unfold. Second, it receives material determinations from its own object, requiring it to make certain choices in its use of formal insights from the higher sciences. The cosmological issue of the character of empirical space, for instance, can take from mathematics the different (non-)Euclidean geometries and investigate which of these are fit to describe spatial aspects of our universe, but it does not, in itself, provide the formal tools. Finally, the single sciences receive in practice determinations ‘from below’, from more specific sciences, when their results by means of abstraction, prescission, induction, and other procedures provide insights on its more general, material level. Even if cosmology is, for instance, part of metaphysics, it receives influences from the empirical results of physics (or biology, from where Peirce takes the generalized principle of evolution). The distinction between formal and material is thus level specific: what is material on one level is a formal bundle of possibilities for the level below; what is formal on one level is material on the level above.

For these reasons, the single step on the ladder of sciences is only partially independent in Peirce, hence also the tendency of his own investigations to zigzag between the levels. His architecture of theories thus forms a sort of phenomenological theory of aspects: the hierarchy of sciences is an architecture of more and less general aspects of the phenomena, not completely independent domains. Finally, Peirce’s realism has as a result a somewhat disturbing style of thinking: many of his central concepts receive many, often highly different determinations which has often led interpreters to assume inconsistencies or theoretical developments in Peirce where none necessarily exist. When Peirce, for instance, determines the icon as the sign possessing a similarity to its object, and elsewhere determines it as the sign by the contemplation of which it is possible to learn more about its object, then they are not conflicting definitions. Peirce’s determinations of concepts are rarely definitions at all in the sense that they provide necessary and sufficient conditions exhausting the phenomenon in question. His determinations should rather be seen as descriptions from different perspectives of a real (and maybe ideal) object – without these descriptions necessarily conflicting. This style of thinking can, however, be seen as motivated by metaphysical continuity. When continuous grading between concepts is the rule, definitions in terms of necessary and sufficient conditions should not be expected to be exhaustive.

Geometry and Localization: An Unholy Alliance? Thought of the Day 95.0

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There are many misleading metaphors obtained from naively identifying geometry with localization. One which is very close to that of String Theory is the idea that one can embed a lower dimensional Quantum Field Theory (QFT) into a higher dimensional one. This is not possible, but what one can do is restrict a QFT on a spacetime manifold to a submanifold. However if the submanifold contains the time axis (a ”brane”), the restricted theory has too many degrees of freedom in order to merit the name ”physical”, namely it contains as many as the unrestricted; the naive idea that by using a subspace one only gets a fraction of phase space degrees of freedom is a delusion, this can only happen if the subspace does not contain a timelike line as for a null-surface (holographic projection onto a horizon).

The geometric picture of a string in terms of a multi-component conformal field theory is that of an embedding of an n-component chiral theory into its n-dimensional component space (referred to as a target space), which is certainly a string. But this is not what modular localization reveals, rather those oscillatory degrees of freedom of the multicomponent chiral current go into an infinite dimensional Hilbert space over one localization point and do not arrange themselves according according to the geometric source-target idea. A theory of this kind is of course consistent but String Theory is certainly a very misleading terminology for this state of affairs. Any attempt to imitate Feynman rules by replacing word lines by word sheets (of strings) may produce prescriptions for cooking up some mathematically interesting functions, but those results can not be brought into the only form which counts in a quantum theory, namely a perturbative approach in terms of operators and states.

String Theory is by no means the only area in particle theory where geometry and modular localization are at loggerheads. Closely related is the interpretation of the Riemann surfaces, which result from the analytic continuation of chiral theories on the lightray/circle, as the ”living space” in the sense of localization. The mathematical theory of Riemann surfaces does not specify how it should be realized; if its refers to surfaces in an ambient space, a distinguished subgroup of Fuchsian group or any other of the many possible realizations is of no concern for a mathematician. But in the context of chiral models it is important not to confuse the living space of a QFT with its analytic continuation.

Whereas geometry as a mathematical discipline does not care about how it is concretely realized the geometrical aspects of modular localization in spacetime has a very specific geometric content namely that which can be encoded in subspaces (Reeh-Schlieder spaces) generated by operator subalgebras acting onto the vacuum reference state. In other words the physically relevant spacetime geometry and the symmetry group of the vacuum is contained in the abstract positioning of certain subalgebras in a common Hilbert space and not that which comes with classical theories.

The Mystery of Modality. Thought of the Day 78.0

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The ‘metaphysical’ notion of what would have been no matter what (the necessary) was conflated with the epistemological notion of what independently of sense-experience can be known to be (the a priori), which in turn was identified with the semantical notion of what is true by virtue of meaning (the analytic), which in turn was reduced to a mere product of human convention. And what motivated these reductions?

The mystery of modality, for early modern philosophers, was how we can have any knowledge of it. Here is how the question arises. We think that when things are some way, in some cases they could have been otherwise, and in other cases they couldn’t. That is the modal distinction between the contingent and the necessary.

How do we know that the examples are examples of that of which they are supposed to be examples? And why should this question be considered a difficult problem, a kind of mystery? Well, that is because, on the one hand, when we ask about most other items of purported knowledge how it is we can know them, sense-experience seems to be the source, or anyhow the chief source of our knowledge, but, on the other hand, sense-experience seems able only to provide knowledge about what is or isn’t, not what could have been or couldn’t have been. How do we bridge the gap between ‘is’ and ‘could’? The classic statement of the problem was given by Immanuel Kant, in the introduction to the second or B edition of his first critique, The Critique of Pure Reason: ‘Experience teaches us that a thing is so, but not that it cannot be otherwise.’

Note that this formulation allows that experience can teach us that a necessary truth is true; what it is not supposed to be able to teach is that it is necessary. The problem becomes more vivid if one adopts the language that was once used by Leibniz, and much later re-popularized by Saul Kripke in his famous work on model theory for formal modal systems, the usage according to which the necessary is that which is ‘true in all possible worlds’. In these terms the problem is that the senses only show us this world, the world we live in, the actual world as it is called, whereas when we claim to know about what could or couldn’t have been, we are claiming knowledge of what is going on in some or all other worlds. For that kind of knowledge, it seems, we would need a kind of sixth sense, or extrasensory perception, or nonperceptual mode of apprehension, to see beyond the world in which we live to these various other worlds.

Kant concludes, that our knowledge of necessity must be what he calls a priori knowledge or knowledge that is ‘prior to’ or before or independent of experience, rather than what he calls a posteriori knowledge or knowledge that is ‘posterior to’ or after or dependant on experience. And so the problem of the origin of our knowledge of necessity becomes for Kant the problem of the origin of our a priori knowledge.

Well, that is not quite the right way to describe Kant’s position, since there is one special class of cases where Kant thinks it isn’t really so hard to understand how we can have a priori knowledge. He doesn’t think all of our a priori knowledge is mysterious, but only most of it. He distinguishes what he calls analytic from what he calls synthetic judgments, and holds that a priori knowledge of the former is unproblematic, since it is not really knowledge of external objects, but only knowledge of the content of our own concepts, a form of self-knowledge.

We can generate any number of examples of analytic truths by the following three-step process. First, take a simple logical truth of the form ‘Anything that is both an A and a B is a B’, for instance, ‘Anyone who is both a man and unmarried is unmarried’. Second, find a synonym C for the phrase ‘thing that is both an A and a B’, for instance, ‘bachelor’ for ‘one who is both a man and unmarried’. Third, substitute the shorter synonym for the longer phrase in the original logical truth to get the truth ‘Any C is a B’, or in our example, the truth ‘Any bachelor is unmarried’. Our knowledge of such a truth seems unproblematic because it seems to reduce to our knowledge of the meanings of our own words.

So the problem for Kant is not exactly how knowledge a priori is possible, but more precisely how synthetic knowledge a priori is possible. Kant thought we do have examples of such knowledge. Arithmetic, according to Kant, was supposed to be synthetic a priori, and geometry, too – all of pure mathematics. In his Prolegomena to Any Future Metaphysics, Kant listed ‘How is pure mathematics possible?’ as the first question for metaphysics, for the branch of philosophy concerned with space, time, substance, cause, and other grand general concepts – including modality.

Kant offered an elaborate explanation of how synthetic a priori knowledge is supposed to be possible, an explanation reducing it to a form of self-knowledge, but later philosophers questioned whether there really were any examples of the synthetic a priori. Geometry, so far as it is about the physical space in which we live and move – and that was the original conception, and the one still prevailing in Kant’s day – came to be seen as, not synthetic a priori, but rather a posteriori. The mathematician Carl Friedrich Gauß had already come to suspect that geometry is a posteriori, like the rest of physics. Since the time of Einstein in the early twentieth century the a posteriori character of physical geometry has been the received view (whence the need for border-crossing from mathematics into physics if one is to pursue the original aim of geometry).

As for arithmetic, the logician Gottlob Frege in the late nineteenth century claimed that it was not synthetic a priori, but analytic – of the same status as ‘Any bachelor is unmarried’, except that to obtain something like ‘29 is a prime number’ one needs to substitute synonyms in a logical truth of a form much more complicated than ‘Anything that is both an A and a B is a B’. This view was subsequently adopted by many philosophers in the analytic tradition of which Frege was a forerunner, whether or not they immersed themselves in the details of Frege’s program for the reduction of arithmetic to logic.

Once Kant’s synthetic a priori has been rejected, the question of how we have knowledge of necessity reduces to the question of how we have knowledge of analyticity, which in turn resolves into a pair of questions: On the one hand, how do we have knowledge of synonymy, which is to say, how do we have knowledge of meaning? On the other hand how do we have knowledge of logical truths? As to the first question, presumably we acquire knowledge, explicit or implicit, conscious or unconscious, of meaning as we learn to speak, by the time we are able to ask the question whether this is a synonym of that, we have the answer. But what about knowledge of logic? That question didn’t loom large in Kant’s day, when only a very rudimentary logic existed, but after Frege vastly expanded the realm of logic – only by doing so could he find any prospect of reducing arithmetic to logic – the question loomed larger.

Many philosophers, however, convinced themselves that knowledge of logic also reduces to knowledge of meaning, namely, of the meanings of logical particles, words like ‘not’ and ‘and’ and ‘or’ and ‘all’ and ‘some’. To be sure, there are infinitely many logical truths, in Frege’s expanded logic. But they all follow from or are generated by a finite list of logical rules, and philosophers were tempted to identify knowledge of the meanings of logical particles with knowledge of rules for using them: Knowing the meaning of ‘or’, for instance, would be knowing that ‘A or B’ follows from A and follows from B, and that anything that follows both from A and from B follows from ‘A or B’. So in the end, knowledge of necessity reduces to conscious or unconscious knowledge of explicit or implicit semantical rules or linguistics conventions or whatever.

Such is the sort of picture that had become the received wisdom in philosophy departments in the English speaking world by the middle decades of the last century. For instance, A. J. Ayer, the notorious logical positivist, and P. F. Strawson, the notorious ordinary-language philosopher, disagreed with each other across a whole range of issues, and for many mid-century analytic philosophers such disagreements were considered the main issues in philosophy (though some observers would speak of the ‘narcissism of small differences’ here). And people like Ayer and Strawson in the 1920s through 1960s would sometimes go on to speak as if linguistic convention were the source not only of our knowledge of modality, but of modality itself, and go on further to speak of the source of language lying in ourselves. Individually, as children growing up in a linguistic community, or foreigners seeking to enter one, we must consciously or unconsciously learn the explicit or implicit rules of the communal language as something with a source outside us to which we must conform. But by contrast, collectively, as a speech community, we do not so much learn as create the language with its rules. And so if the origin of modality, of necessity and its distinction from contingency, lies in language, it therefore lies in a creation of ours, and so in us. ‘We, the makers and users of language’ are the ground and source and origin of necessity. Well, this is not a literal quotation from any one philosophical writer of the last century, but a pastiche of paraphrases of several.

Discontinuous Reality. Thought of the Day 61.0

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Convention is an invention that plays a distinctive role in Poincaré’s philosophy of science. In terms of how they contribute to the framework of science, conventions are not empirical. They are presupposed in certain empirical tests, so they are (relatively) isolated from doubt. Yet they are not pure stipulations, or analytic, since conventional choices are guided by, and modified in the light of, experience. Finally they have a different character from genuine mathematical intuitions, which provide a fixed, a priori synthetic foundation for mathematics. Conventions are thus distinct from the synthetic a posteriori (empirical), the synthetic a priori and the analytic a priori.

The importance of Poincaré’s invention lies in the recognition of a new category of proposition and its centrality in scientific judgment. This is more important than the special place Poincaré gives Euclidean geometry. Nevertheless, it’s possible to accommodate some of what he says about the priority of Euclidean geometry with the use of non-Euclidean geometry in science, including the inapplicability of any geometry of constant curvature in physical theories of global space. Poincaré’s insistence on Euclidean geometry is based on criteria of simplicity and convenience. But these criteria surely entail that if giving up Euclidean geometry somehow results in an overall gain in simplicity then that would be condoned by conventionalism.

The a priori conditions on geometry – in particular the group concept, and the hypothesis of rigid body motion it encourages – might seem a lingering obstacle to a more flexible attitude towards applied geometry, or an empirical approach to physical space. However, just as the apriority of the intuitive continuum does not restrict physical theories to the continuous; so the apriority of the group concept does not mean that all possible theories of space must allow free mobility. This, too, can be “corrected”, or overruled, by new theories and new data, just as, Poincaré comes to admit, the new quantum theory might overrule our intuitive assumption that nature is continuous. That is, he acknowledges that reality might actually be discontinuous – despite the apriority of the intuitive continuum.

Poincaré and Geometry of Curvature. Thought of the Day 60.0

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It is not clear that Poincaré regarded Riemannian, variably curved, “geometry” as a bona fide geometry. On the one hand, his insistence on generality and the iterability of mathematical operations leads him to dismiss geometries of variable curvature as merely “analytic”. Distinctive of mathematics, he argues, is generality and the fact that induction applies to its processes. For geometry to be genuinely mathematical, its constructions must be everywhere iterable, so everywhere possible. If geometry is in some sense about rigid motion, then a manifold of variable curvature, especially where the degree of curvature depends on something contingent like the distribution of matter, would not allow a thoroughly mathematical, idealized treatment. Yet Poincaré also writes favorably about Riemannian geometries, defending them as mathematically coherent. Furthermore, he admits that geometries of constant curvature rest on a hypothesis – that of rigid body motion – that “is not a self evident truth”. In short, he seems ambivalent. Whether his conception of geometry includes or rules out variable curvature is unclear. We can surmise that he recognized Riemannian geometry as mathematical, and interesting, but as very different and more abstract than geometries of constant curvature, which are based on the further limitations discussed above (those motivated by a world satisfying certain empirical preconditions). These limitations enable key idealizations, which in turn allow constructions and synthetic proofs that we recognize as “geometric”.

Holonomies: Philosophies of Conjugacy. Part 1.

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Suppose that N is an irreducible 2n-dimensional Riemannian symmetric space. We may realise N as a coset space N = G/K with Gτ ⊂ K ⊂ (Gτ)0 for some involution τ of G. Now K is (a covering of) the holonomy group of N and similarly the coset fibration G → G/K covers the holonomy bundle P → N. In this setting, J(N) is associated to G:

J(N) ≅ G ×K J (R2n)

and if K/H is a K-orbit in J(R2n) then the corresponding subbundle is G ×K K/H = G/H and the projection is just the coset fibration. Thus, the subbundles of J(N) are just the orbits of G in J(N).

Let j ∈ J (N). Then G · j is an almost complex submanifold of J (N) on which J is integrable iff j lies in the zero-set of the Nijenhuis tensor NJ.

This focusses our attention on the zero-set of NJ which we denote by Z. In favourable circumstances, the structure of this set can be completely described. We begin by assuming that N is of compact type so that G is compact and semi-simple. We also assume that N is inner i.e. that τ is an inner involution of G or, equivalently, that rankG = rankK. The class of inner symmetric spaces include the even-dimensional spheres, the Hermitian symmetric spaces, the quaternionic Kähler symmetric spaces and indeed all symmetric G-spaces for G = SO(2n+1), Sp(n), E7, E8, F4 and G2. Moreover, all inner symmetric spaces are necessarily even-dimensional and so fit into our framework.

Let N = G/K be a simply-connected inner Riemannian symmetric space of compact type. Then Z consists of a finite number of connected components on each of which G acts transitively. Moreover, any G-flag manifold is realised as such an orbit for some N.

The proof for the above requires a detour into the geometry of flag manifolds and reveals an interesting interaction between the complex geometry of flag manifolds and the real geometry of inner symmetric spaces. For this, we begin by noting that a coset space of the form G/C(T) admits several invariant Kählerian complex structures in general. Using a complex realisation of G/C(T) as follows: having fixed a complex structure, the complexified group GC acts transitively on G/C(T) by biholomorphisms with parabolic subgroups as stabilisers. Conversely, if P ⊂ GC is a parabolic subgroup then the action of G on GC/P is transitive and G ∩ P is the centraliser of a torus in G. For the infinitesimal situation: let F = G/C(T) be a flag manifold and let o ∈ F. We have a splitting of the Lie algebra of G

gC = h ⊕ m

with m ≅ ToF and h the Lie algebra of the stabiliser of o in G. An invariant complex structure on F induces an ad h-invariant splitting of mC into (1, 0) and (0, 1) spaces mC = m+ ⊕ m− with [m+, m+] ⊂ m+ by integrability. One can show that m+ and m are nilpotent subalgebras of gC and in fact hC ⊕ m is a parabolic subalgebra of gC with nilradical m. If P is the corresponding parabolic subgroup of GC then P is the stabiliser of o and we obtain a biholomorphism between the complex coset space GC/P and the flag manifold F.

Conversely, let P ⊂ GC be a parabolic subgroup with Lie algebra p and let n be the conjugate of the nilradical of p (with respect to the real form g). Then H = G ∩ P is the centraliser of a torus and we have orthogonal decompositions (with respect to the Killing inner product)

p = hC ⊕ n, gC = hC ⊕ n ⊕ n

which define an invariant complex structure on G/H realising the biholomorphism with GC/P.

The relationship between a flag manifold F = GC/P and an inner symmetric space comes from an examination of the central descending series of n. This is a filtration 0 = nk+1 ⊂ nk ⊂…⊂ n1 = n of n defined by ni = [n, ni−1].

We orthogonalise this filtration using the Killing inner product by setting

gi = ni+1 ∩ ni

for i ≥ 1 and extend this to a decomposition of gC by setting g0 = hC = (g ∩ p)C and g−i = gfor i ≥ 1. Then

gC = ∑gi

is an orthogonal decomposition with

p = ∑i≤0 gi, n = ∑i>0 g

The crucial property of this decomposition is that

[gi, gj] ⊂ gi+j

which can be proved by demonstrating the existence of an element ξ ∈ h with the property that, for each i, adξ has eigenvalue √−1i on gi. This element ξ (necessarily unique since g is semi-simple) is the canonical element of p. Since ad ξ has eigenvalues in √−1Z, ad exp πξ is an involution of g which we exponentiate to obtain an inner involution τξ of G and thus an inner symmetric space G/K where K = (Gτξ)0. Clearly, K has Lie algebra given by

k = g ∩ ∑i g2i

Harmonies of the Orphic Mystery: Emanation of Music

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As the Buddhist sage Nagarjuna states in his Seventy Verses on Sunyata, “Being does not arise, since it exists . . .” In similar fashion it can be said that mind exists, and if we human beings manifest its qualities, then the essence and characteristics of mind must be a component of our cosmic source. David Bohm’s theory of the “implicate order” within the operations of nature suggests that observed phenomena do not operate only when they become objective to our senses. Rather, they emerge out of a subjective state or condition that contains the potentials in a latent yet really existent state that is just awaiting the necessary conditions to manifest. Thus within the explicate order of things and beings in our familiar world there is the implicate order out of which all of these emerge in their own time.

Clearly, sun and its family of planets function in accordance with natural laws. The precision of the orbital and other electromagnetic processes is awesome, drawing into one operation the functions of the smallest subparticles and the largest families of sun-stars in their galaxies, and beyond even them. These individual entities are bonded together in an evident unity that we may compare with the oceans of our planet: uncountable numbers of water molecules appear to us as a single mass of substance. In seeking the ultimate particle, the building block of the cosmos, some researchers have found themselves confronted with the mystery of what it is that holds units together in an organism — any organism!

As in music where a harmony consists of many tones bearing an inherent relationship, so must there be harmony embracing all the children of cosmos. Longing for the Harmonies: Themes and Variations from Modern Physics is a book by Frank Wilczek, an eminent physicist, and his wife Betsy Devine, an engineering scientist and freelance writer. The theme of their book is set out in their first paragraph:

From Pythagoras measuring harmonies on a lyre string to R. P. Feynman beating out salsa on his bongos, many a scientist has fallen in love with music. This love is not always rewarded with perfect mastery. Albert Einstein, an ardent amateur of the violin, provoked a more competent player to bellow at him, “Einstein, can’t you count?”

Both music and scientific research, Einstein wrote, “are nourished by the same source of longing, and they complement one another in the release they offer.” It seems to us, too, that the mysterious longing behind a scientist’s search for meaning is the same that inspires creativity in music, art, or any other enterprise of the restless human spirit. And the release they offer is to inhabit, if only for a moment, some point of union between the lonely world of subjectivity and the shared universe of external reality.

In a very lucid text, Wilczek and Devine show us that the laws of nature, and the structure of the universe and all its contributing parts, can be presented in such a way that the whole compares with a musical composition comprising themes that are fused together. One of the early chapters begins with the famous lines of the great astronomer Johannes Kepler, who in 1619 referred to the music of the spheres:

The heavenly motions are nothing but a continuous song for several voices (perceived by the intellect, not by the ear); a music which, through discordant tensions, through sincopes [sic] and cadenzas, as it were (as men employ them in imitation of those natural discords) progresses towards certain pre-designed quasi six-voiced clausuras, and thereby sets landmarks in the immeasurable flow of time. — The Harmony of the World (Harmonice mundi)

Discarding the then current superstitions and misinformed speculation, through the cloud of which Kepler had to work for his insights, Wilczek and Devine note that Kepler’s obsession with the idea of the harmony of the world is actually rooted in Pythagoras’s theory that the universe is built upon number, a concept of the Orphic mystery-religions of Greece. The idea is that “the workings of the world are governed by relations of harmony and, in particular, that music is associated with the motion of the planets — the music of the spheres” (Wilczek and Devine). Arthur Koestler, in writing of Kepler and his work, claimed that the astronomer attempted

to bare the ultimate secret of the universe in an all-embracing synthesis of geometry, music, astrology, astronomy and epistemology. The Sleepwalkers

In Longing for the Harmonies the authors refer to the “music of the spheres” as a notion that in time past was “vague, mystical, and elastic.” As the foundations of music are rhythm and harmony, they remind us that Kepler saw the planets moving around the sun “to a single cosmic rhythm.” There is some evidence that he had association with a “neo-Pythagorean” movement and that, owing to the religious-fomented opposition to unorthodox beliefs, he kept his ideas hidden under allegory and metaphor.

Shakespeare, too, phrases the thought of tonal vibrations emitted by the planets and stars as the “music of the spheres,” the notes likened to those of the “heavenly choir” of cherubim. This calls to mind that Plato’s Cratylus terms the planets theoi, from theein meaning “to run, to move.” Motion does suggest animation, or beings imbued with life, and indeed the planets are living entities so much grander than human beings that the Greeks and other peoples called them “gods.” Not the physical bodies were meant, but the essence within them, in the same way that a human being is known by the inner qualities expressed through the personality.

When classical writers spoke of planets and starry entities as “animals” they did not refer to animals such as we know on Earth, but to the fact that the celestial bodies are “animated,” embodying energies received from the sun and cosmos and transmitted with their own inherent qualities added.

Many avenues open up for our reflection upon the nature of the cosmos and ourselves, and our interrelationship, as we consider the structure of natural laws as Wilczek and Devine present them. For example, the study of particles, their interactions, their harmonizing with those laws, is illuminating intrinsically and, additionally, because of their universal application. The processes involved occur here on earth, and evidently also within the solar system and beyond, explaining certain phenomena that had been awaiting clarification.

The study of atoms here on earth and their many particles and subparticles has enabled researchers to deduce how stars are born, how and why they shine, and how they die. Now some researchers are looking at what it is, whether a process or an energy, that unites the immeasurably small with the very large cosmic bodies we now know. If nature is infinite, it must be so in a qualitative sense, not merely a quantitative.

One of the questions occupying the minds of cosmologists is whether the universal energy is running down like the mechanism of an unwinding Swiss watch, or whether there is enough mass to slow the outward thrust caused by the big bang that has been assumed to have started our cosmos going. In other words, is our universe experiencing entropy — dying as its energy is being used up — or will there be a “brake” put upon the expansion that could, conceivably, result in a return to the source of the initial explosion billions of years ago? Cosmologists have been looking for enough “dark mass” to serve as such a brake.

Among the topics treated by Wilczek and Devine in threading their way through many themes and variations in modern physics, is what is known as the mass-generating Higgs field. This is a proposition formulated by Peter Higgs, a Scottish physicist, who suggests there is an electromagnetic field that pervades the cosmos and universally provides the electron particles with mass.

The background Higgs field must have very accurately the same value throughout the universe. After all, we know — from the fact that the light from distant galaxies contains the same spectral lines we find on Earth — that electrons have the same mass throughout the universe. So if electrons are getting their mass from the Higgs field, this field had better have the same strength everywhere. What is the meaning of this all-pervasive field, which exists with no apparent source? Why is it there? (Wilczek and Devine).

What is the meaning? Why is it there? These are among the most important questions that can be asked. Though physicists may provide profound mathematical equations, they will thereby offer only more precise detail as to what is happening. We shall not receive an answer to the “What” and the “Why” without recourse to meta-physics, beyond the realm of brain-devised definitions.

The human mind is limited in its present stage of evolution. It may see the logical necessity of infinity referent to space and time; for if not infinity, what then is on the other side of the “fence” that is our outermost limit? But, being able to perceive the logical necessity of infinity, the finite mind still cannot span the limitless ranges of space, time, and substance.

If we human beings are manifold in our composition, and since we draw our very existence and sustenance from the universe at large, our conjoint nature must be drawn from the sources of life, substance, and energy, in which our and all other cosmic lives are immersed.

As the authors conclude their fascinating work:

“The worlds opened to our view are graced with wonderful symmetry and uniformity. Learning to know them, to appreciate their many harmonies, is like deepening an acquaintance with some great and meaningful piece of music — surely one of the best things life has to offer.”

Oncontology

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In conventional oncological terms, the process of metastasis is the wild overgrowth of cells to the detriment of the body, resulting in either growths that are benign or malignant. Apoptosis (PCD) is the process by which the cell receives a signal to stop production at a previously prescribed genetic point. The process is twofold: to retain proper cell function integral to the organism, and to remove potentially harmful or lethal elements in the cell which could endanger the organism as a whole. There are only two ways by which cells perish: either by some external agent (e.g. toxic chemicals, fire, removal) or by being induced to perish (i.e. apoptosis). Firstly, apoptosis is necessary in the organism; for instance, the uterine wall sheds during menstruation; the surplus “webbed” tissue between the fingers and toes on the fetus; the fusing of bone plates when the growth period is at an end; the resorption of the tadpole tail in the development of a frog; and so on. Secondly, apoptosis is necessary for the destruction of cells injurious to the organism such as virally infected cells, cells with corrupt DNA, or damaged or cancerous cells. Apoptosis occurs in two ways: removing or blocking all positive stimulus to the cell necessary for the cell’s continuance (one can envision that apoptosis is a kind of siege-craft, cutting all supply lines to the cellular castle), and the inducement of negative signals such as increased oxidation in the cell, aberrant absorption of proteins, the release of particular molecules that bind to the receptors of the cell’s surface which activate the apoptotic process.

Metastasis and apoptosis do not exhaust one another in some sort of dialectical exchange toward finality. They are not a coupling unit, but processes by which we may name desire or ontology. To assert that they cancel one another out in equilibrium is to “gorgonify” the “cacophysical” reality of Being. In the realm of biological science, there is a moment of equilibrium in the body: a certain quantity of cells will match the creation and destruction ratio to achieve a brief period of “plateau” called homeostasis, but this is hardly measurable or significant, since it may last a matter of seconds in the life of any body, the duration of this perhaps inconsequential or even impossible. This is an abstract idealization issued from the laboratories of biological science that may be able to measure such equal ratios in the simplest of organisms and assume that more complex bodies will also follow the same rule, or to simplify the results according to approximations of equilibrium. But our notion of bodies is much more extensive and intensive – we include more than just the life of an “organism”; we include everything that can be said has being. This includes books, plants, rocks, radios, and  even cities. Metastasis and apoptosis are derelict forces, two faces of desire. It is not a measure of zombifying ontology with a series of empty concepts. Immobility is effaced by perpetual be-comings, announced by the manifest process of unlimited production and unlimited expiration, both what Spinoza would call “potentia” and Nietzsche would call “will to power” as the constant mobilization of differences. Thought crudely apopticizes bodies, whereas bodies succumb to a biological apoptosis. Thought thinks it hypostasizes being, but the true process underlying being is metastasis. These processes strafe through being and it is our thought that attempts to transcendentally retrofit being through clumsy and ashen installations meant to prolong the tradition of thinking through as many ages and bodies as “humanly” possible. We know all too well the DeleuzoGuattarian de/re-territorializations, and how Pynchon’s Pirate could do as such to the cuisine attached to the banana. We know the real rhapsodic geometries (a rhapsoid?) that inhere within phylo- and ontogenetics. But, in the end, as it functions for Derrida in the domain of language-meaning to which we are all condemned to pursue like the ever-reticent horizon, the law of necessarily probable failure inheres in ontology as well, and what remains is to commit considerable study to the mechanics of this “failure”. Even the functions of symbiosis (not to be confused with equilibrium) where bacteria provides a benefit to the body does not endanger what we say here about metastasis since we are considering the metastasis-apoptosis phenomenon without demonstrating a prejudice in favour of the sustainable functions of the body, but rather isolating the principle of metastasis as descriptive of the troubling philosophical concept of becoming.

Oncontology