The unicellular organism has thin filaments protruding from its cell membrane, and in the absence of any stimuli, it simply wanders randomly around by changing between two characteristical movement patterns. One is performed by rotating the flagella counterclockwise. In that case, they form a bundle which pushes the cell forward along a curved path, a ‘run’ of random duration with these runs interchanging with ‘tumbles’ where the flagella shifts to clockwise rotation, making them work independently and hence moving the cell erratically around with small net displacement. The biased random walk now consists in the fact than in the presence of a chemical attractant, the runs happening to carry the cell closer to the attractant are extended, while runs in other directions are not. The sensation of the chemical attractant is performed temporally rather than spatially, because the cell moves too rapidly for concentration comparisons between its two ends to be possible. A chemical repellant in the environment gives rise to an analogous behavioral structure – now the biased random walk takes the cell away from the repellant. The bias saturates very quickly – which is what prevents the cell from continuing in a ‘false’ direction, because a higher concentration of attractant will now be needed to repeat the bias. The reception system has three parts, one detecting repellants such as leucin, the other detecting sugars, the third oxygen and oxygen-like substances.

The cell’s behavior forms a primitive, if full-fledged example of von Uexküll’s functional circle connecting specific perception signs and action signs. Functional circle behavior is thus no privilege for animals equipped with central nervous systems (CNS). Both types of signs involve categorization. First, the sensory receptors of the bacterium evidently are organized after categorization of certain biologically significant chemicals, while most chemicals that remain insignificant for the cell’s metabolism and survival are ignored. The self-preservation of metabolism and cell structure is hence the ultimate regulator which is supported by the perception-action cycles described. The categorization inherent in the very structure of the sensors is mirrored in the categorization of act types. Three act types are outlined: a null-action, composed of random running and tumbling, and two mirroring biased variants triggered by attractants and repellants, respectively. Moreover, a negative feed-back loop governed by quick satiation grants that the window of concentration shifts to which the cell is able to react appropriately is large – it so to speak calibrates the sensory system so that it does not remain blinded by one perception and does not keep moving the cell forward on in one selected direction. This adaptation of the system grants that it works in a large scale of different attractor/repellor concentrations. These simple signals at stake in the cell’s functional circle display an important property: at simple biological levels, the distinction between signs and perception vanish – that distinction is supposedly only relevant for higher CNS-based animals. Here, the signals are based on categorical perception – a perception which immediately categorizes the entity perceived and thus remains blind to internal differences within the category.

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.

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.

Let |M| be a topological space. A presheaf of commutative algebras F on X is an assignment

U ↦ F(U), U open in |M|, F(U) is a commutative algebra, such that the following holds,

(1) If U ⊂ V are two open sets in |M|, ∃ a morphism r_{V, U}: F(V) → F(U), called the restriction morphism and often denoted by r_{V, U}(ƒ) = ƒ|_{U}, such that

(i) r_{U, U} = id,

(ii) r_{W, U} = r_{V, U} ○ r_{W, V}

A presheaf ƒ is called a sheaf if the following holds:

(2) Given an open covering {U_{i}}_{i∈I} of U and a family {ƒ_{i}}_{i∈I}, ƒ_{i} ∈ F(U_{i}) such that ƒ_{i}|_{Ui ∩ Uj} = ƒ_{j}|_{Ui ∩ Uj} ∀ i, j ∈ I, ∃ a unique ƒ ∈ F(U) with ƒ|_{Ui} = ƒ_{i}

The elements in F(U) are called sections over U, and with U = |M|, these are termed global sections.

The assignments U ↦ C^{∞}(U), U open in the differentiable manifold M and U ↦ O_{X}(U), U open in algebraic variety X are examples of sheaves of functions on the topological spaces |M| and |X| underlying the differentiable manifold M and the algebraic variety X respectively.

In the language of categories, the above definition says that we have defined a functor, F, from top(M) to (alg), where top(M) is the category of the open sets in the topological space |M|, the arrows given by the inclusions of open sets while (alg) is the category of commutative algebras. In fact, the assignment U ↦ F(U) defines F on the objects while the assignment

U ⊂ V ↦ r_{V, U}: F(V) → F(U)

defines F on the arrows.

Let |M| be a topological space. We define a presheaf of algebras on |M| to be a functor

F: top(M)^{op} → (alg)

The suffix “op” denotes as usual the opposite category; in other words, F is a contravariant functor from top(M) to (alg). A presheaf is a sheaf if it satisfies the property (2) of the above definition.

If F is a (pre)sheaf on |M| and U is open in |M|, we define F|_{U}, the (pre)sheaf F restricted to U, as the functor F restricted to the category of open sets in U (viewed as a topological space itself).

Let F be a presheaf on the topological space |M| and let x be a point in |M|. We define the stalk F_{x} of F, at the point x, as the direct limit

lim_{→} F(U)

where the direct limit is taken ∀ the U open neighbourhoods of x in |M|. F_{x} consists of the disjoint union of all pairs (U, s) with U open in |M|, x ∈ U, and s ∈ F(U), modulo the equivalence relation: (U, s) ≅ (V, t) iff ∃ a neighbourhood W of x, W ⊂ U ∩ V, such that s|_{W} = t|_{W}.

The elements in F_{x} are called germs of sections.

Let F and G be presheaves on |M|. A morphism of presheaves φ: F → G, for each open set U in |M|, such that ∀ V ⊂ U, the following diagram commutes

Equivalently and more elegantly, one can also say that a morphism of presheaves is a natural transformation between the two presheaves F and G viewed as functors.

A morphism of sheaves is just a morphism of the underlying presheaves.

Clearly any morphism of presheaves induces a morphism on the stalks: φ_{x}: F_{x} → G_{x}. The sheaf property, i.e., property (2) in the above definition, ensures that if we have two morphisms of sheaves φ and ψ, such that φ_{x} = ψ_{x} ∀ x, then φ = ψ.

We say that the morphism of sheaves is injective (resp. surjective) if x is injective (resp. surjective).

On the notion of surjectivity, however, one should exert some care, since we can have a surjective sheaf morphism φ: F → G such that φ_{U}: F(U) → G(U) is not surjective for some open sets U. This strange phenomenon is a consequence of the following fact. While the assignment U ↦ ker(φ(U)) always defines a sheaf, the assignment

U ↦ im( φ(U)) = F(U)/G(U)

defines in general only a presheaf and not all the presheaves are sheaves. A simple example is given by the assignment associating to an open set U in R, the algebra of constant real functions on U. Clearly this is a presheaf, but not a sheaf.

We can always associate, in a natural way, to any presheaf a sheaf called its sheafification. Intuitively, one may think of the sheafification as the sheaf that best “approximates” the given presheaf. For example, the sheafification of the presheaf of constant functions on open sets in R is the sheaf of locally constant functions on open sets in R. We construct the sheafification of a presheaf using the * étalé space*, which we also need in the sequel, since it gives an equivalent approach to sheaf theory.

Let F be a presheaf on |M|. We define the *étalé space *of F to be the disjoint union ⊔_{x∈|M|} F_{x}. Let each open U ∈ |M| and each s ∈ F(U) define the map š_{U}: U ⊔_{x∈|U|} F_{x}, š_{U}(x) = s_{x}. We give to the étalé space the finest topology that makes the maps š continuous, ∀ open U ⊂ |M| and all sections s ∈ F(U). We define F_{et} to be the presheaf on |M|:

U ↦ F_{et}(U) = {š_{U}: U → ⊔_{x∈|U|} F_{x}, š_{U}(x) = s_{x} ∈ F_{x}}

Let F be a presheaf on |M|. A sheafification of F is a sheaf F^{~}, together with a presheaf morphism α: F → F^{~ }such that

(1) any presheaf morphism ψ: F → G, G a sheaf factors via α, i.e. ψ: F →^{α} F^{~} → G,

(2) F and F^{~ }are locally isomorphic, i.e., ∃ an open cover {U_{i}}_{i∈I} of |M| such that F(U_{i}) ≅ F^{~}(U_{i}) via α.

Let F and G be sheaves of rings on some topological space |M|. Assume that we have an injective morphism of sheaves G → F such that G(U) ⊂ F(U) ∀ U open in |M|. We define the quotient F/G to be the sheafification of the image presheaf: U ↦ F(U)/G(U). In general F/G (U) ≠ F(U)/G(U), however they are locally isomorphic.

Ringed space is a pair M = (|M|, F) consisting of a topological space |M| and a sheaf of commutative rings F on |M|. This is a locally ringed space, if the stalk F_{x} is a local ring ∀ x ∈ |M|. A morphism of ringed spaces φ: M = (|M|, F) → N = (|N|, G) consists of a morphism |φ|: |M| → |N| of the topological spaces and a sheaf morphism φ^{*}: O_{N} → φ_{*}O_{M}, where φ_{*}O_{M} is a sheaf on |N| and defined as follows:

(φ_{*}O_{M})(U) = O_{M}(φ^{-1}(U)) ∀ U open in |N|

Morphism of ringed spaces induces a morphism on the stalks for each

x ∈ |M|: φ_{x}: O_{N,|φ|(x)} → O_{M,x}

If M and N are locally ringed spaces, we say that the morphism of ringed spaces φ is a morphism of locally ringed spaces if φ_{x} is local, i.e. φ^{-1}_{x}(m_{M,x}) = m_{N,|φ|(x)}, where m_{N,|φ|(x)} and m_{M,x} are the maximal ideals in the local rings O_{N,|φ|(x)} and O_{M,x} respectively.

Within the Austrian economics (* here*,

Even if the Austrian economists’ conception of the *a priori* is decidedly objectivist and anti-subjectivist, it is important to remark their insistence on subjectivity within their ontological domain. The Austrian economics tradition is famous exactly for their emphasis on the role of subjectivity in economy. From Carl Menger onwards, they protest against the mainstream economical assumption that the economic agent in the market is fully rational, knows his own preferences in detail, has constant preferences over time, has access to all prices for a given commodity at a given moment, etc. Thus, von Mises’ famous criticism of socialist planned economy is built on this idea: the system of ever-changing prices in the market constitutes a dispersed knowledge about the conditions of resource allocation which is *a priori* impossible for any single agent – let alone, any central planner’s office – to possess. Thus, their conception of the objective *a priori* laws of the economic domain perhaps surprisingly had the implication that they warned against a too objectivist conception of economy not taking into account the limits of economic rationality stemming from the general limitations of the capacities of real subjects. Their ensuing liberalism is thus built on *a priori* conclusions about the relative unpredictability of economics founded on the role played by subjective intentionality. For the same reason, Hayek ended up with a distinction between simple and complex processes, respectively, cutting across all empirical disciplines, where only the former permit precise, predictive, quantitative calculi based on mathemathical modeling while the latter permit only recognition of patterns (which may also be mathematically modeled, to be sure, but without quantitative predictability). It is of paramount importance, though, to distinguish this emphasis on the ineradicable role of subjectivity in certain regional domains from Kantian-like ideas about the foundational role of subjectivity in the construction of knowledge as such. The Austrians are as much subjectivists in the former respect as they are objectivists in the latter. In the history of economics, the Austrians occupy a middle position, being against historicism on the one hand as well as against positivism on the other. Against the former, they insist that *a priori* structures of economy transgress history which does not possess the power to form institutions at random but only as constrained by *a priori* structures. And against the latter, they insist that the mere accumulation of empirical data subject to induction will never in itself give rise to the formation of theoretical insights. Structures of intelligible concepts are in all cases necessary for any understanding of empirical regularities – in so far, the Austrian *a priori* approach is tantamount to a non-skepticist version of the doctrine of ‘theory-ladenness’ of observations.

A late descendant of the Austrian tradition after its emigration to the Anglo-Saxon world (von Mises, Hayek, and Schumpeter were such emigrés) was the anarcho-liberal economist Murray Rothbard, and it is the inspiration from him which allows * Barry Smith* to articulate the principles underlying the Austrians as ‘fallibilistic apriorism’. Rothbard characterizes in a brief

there are two basic differences between the positivists’ model science of physics on the one hand, and sciences dealing with human actions on the other: the former permits experimental verification of consequences of hypotheses, which the latter do not (or, only to a limited degree, we may add); the former admits of no possibility of testing the premisses of hypotheses (like: what is gravity?), while the latter permits a rational investigation of the premisses of hypotheses (like: what is human action?). This state of affairs makes it possible for economics to derive its basic laws with absolute – a priori – certainty: in addition to the fundamental axiom – the existence of human action – only two empirical postulates are needed: ‘(1) the most fundamental variety of resources, both natural and human. From this follows directly the division of labor, the market, etc.; (2) less important, that leisure is a consumer good’. On this basis, it may e.g. be inferred, ‘that every firm aims always at maximizing its psychic profit’.

Rothbard draws forth this example so as to counterargue traditional economists who will claim that the following proposition could be added as a corollary: ‘that every firm aims always at maximizing its money profit’. This cannot be inferred and is, according to Rothbard, an economical prejudice – the manager may, e.g. prefer for nepotistic reasons to employ his stupid brother even if that decreases the firm’s financial profit possibilities. This is an example of how the Austrians refute the basic premiss of absolute rationality in terms of maximal profit seeking. Given this basis, other immediate implications are:

the means-ends relationship, the time-structure of production, time-preference, the law of diminishing marginal utility, the law of optimum returns, etc.

Rothbard quotes Mises for seeing the fundamental Axiom as a ‘Law of Thought’ – while he himself sees this as a much too Kantian way of expressing it, he prefers instead the simple Aristotelian/Thomist idea of a ‘Law of Reality’. Rothbard furthermore insists that this doctrine is not inherently political – in order to attain the Austrians’ average liberalist political orientation, the preference for certain types of ends must be added to the *a priori* theory (such as the preference for life over death, abundance over poverty, etc.). This also displays the radicality of the Austrian approach: nothing is assumed about the content of human ends – this is why they will never subscribe to theories about Man as economically rational agent or Man as necessarily economical egotist. All different ends meet and compete on the market – including both desire for profit in one end and idealist, utopian, or altruist goals in the other. The principal interest, in these features of economical theory is the high degree of awareness of the difference between the – extreme – synthetic *a priori* theory developed, on the one hand, and its incarnation in concrete empirical cases and their limiting conditions on the other.

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The most explicit influence from mathematics in semiotics is probably René Thom’s controversial * theory of catastrophes (here and here)*, with philosophical and semiotic support from Jean Petitot. Catastrophe theory is but one of several formalisms in the broad field of qualitative dynamics (comprising also chaos theory, complexity theory, self-organized criticality, etc.). In all these cases, the theories in question are in a certain sense phenomenological because the focus is different types of qualitative behavior of dynamic systems grasped on a purely formal level bracketing their causal determination on the deeper level. A widespread tool in these disciplines is phase space – a space defined by the variables governing the development of the system so that this development may be mapped as a trajectory through phase space, each point on the trajectory mapping one global state of the system. This space may be inhabited by different types of attractors (attracting trajectories), repellors (repelling them), attractor basins around attractors, and borders between such basins characterized by different types of topological saddles which may have a complicated topology.

Catastrophe theory has its basis in differential topology, that is, the branch of topology keeping various differential properties in a function invariant under transformation. It is, more specifically, the so-called * Whitney topology* whose invariants are points where the n

*One of the simpler catastrophes is the so-called cusp (a). It constitutes a meta-diagram, namely a diagram of the possible type-shifts of a simpler diagram (b), that of the equation ax ^{4} + bx^{2} + cx = 0. The upper part of (a) shows the so-called fold, charting the manifold of solutions to the equation in the three dimensions a, b and c. By the projection of the fold on the a, b-plane, the pointed figure of the cusp (lower a) is obtained. The cusp now charts the type-shift of the function: Inside the cusp, the function has two minima, outside it only one minimum. Different paths through the cusp thus corresponds to different variations of the equation by the variation of the external variables a and b. One such typical path is the path indicated by the left-right arrow on all four diagrams which crosses the cusp from inside out, giving rise to a diagram of the further level (c) – depending on the interpretation of the minima as simultaneous states. Here, thus, we find diagram transformations on three different, nested levels.*

The concept of transformation plays several roles in this formalism. The most spectacular one refers, of course, to the change in external control variables, determining a trajectory through phase space where the function controlled changes type. This transformation thus searches the possibility for a change of the subtypes of the function in question, that is, it plays the role of eidetic variation mapping how the function is ‘unfolded’ (the basic theorem of catastrophe theory refers to such unfolding of simple functions). Another transformation finds stable classes of such local trajectory pieces including such shifts – making possible the recognition of such types of shifts in different empirical phenomena. On the most empirical level, finally, one running of such a trajectory piece provides, in itself, a transformation of one state into another, whereby the two states are rationally interconnected. Generally, it is possible to make a given transformation the object of a higher order transformation which by abstraction may investigate aspects of the lower one’s type and conditions. Thus, the central unfolding of a function germ in Catastrophe Theory constitutes a transformation having the character of an eidetic variation making clear which possibilities lie in the function germ in question. As an abstract formalism, the higher of these transformations may determine the lower one as invariant in a series of empirical cases.

Complexity theory is a broader and more inclusive term covering the general study of the macro-behavior of composite systems, also using phase space representation. The theoretical biologist Stuart Kauffman (* intro*) argues that in a phase space of all possible genotypes, biological evolution must unfold in a rather small and specifically qualified sub-space characterized by many, closely located and stable states (corresponding to the possibility of a species to ‘jump’ to another and better genotype in the face of environmental change) – as opposed to phase space areas with few, very stable states (which will only be optimal in certain, very stable environments and thus fragile when exposed to change), and also opposed, on the other hand, to sub-spaces with a high plurality of only metastable states (here, the species will tend to merge into neighboring species and hence never stabilize). On the base of this argument, only a small subset of the set of virtual genotypes possesses ‘evolvability’ as this special combination between plasticity and stability. The overall argument thus goes that order in biology is not a pure product of evolution; the possibility of order must be present in certain types of organized matter before selection begins – conversely, selection requires already organized material on which to work. The identification of a species with a co-localized group of stable states in genome space thus provides a (local) invariance for the transformation taking a trajectory through space, and larger groups of neighboring stabilities – lineages – again provide invariants defined by various more or less general transformations. Species, in this view, are in a certain limited sense ‘natural kinds’ and thus naturally signifying entities. Kauffman’s speculations over genotypical phase space have a crucial bearing on a transformation concept central to biology, namely mutation. On this basis far from all virtual mutations are really possible – even apart from their degree of environmental relevance. A mutation into a stable but remotely placed species in phase space will be impossible (evolution cannot cross the distance in phase space), just like a mutation in an area with many, unstable proto-species will not allow for any stabilization of species at all and will thus fall prey to arbitrary small environment variations. Kauffman takes a spontaneous and non-formalized transformation concept (mutation) and attempts a formalization by investigating its condition of possibility as movement between stable genomes in genotype phase space. A series of constraints turn out to determine type formation on a higher level (the three different types of local geography in phase space). If the trajectory of mutations must obey the possibility of walking between stable species, then the space of possibility of trajectories is highly limited. Self-organized criticality as developed by Per Bak (

A ringed space is a broad concept in which we can fit most of the interesting geometrical objects. It consists of a topological space together with a sheaf of functions on it.

Let M be a differentiable manifold, whose topological space is Hausdorff and second countable. For each open set U ⊂ M , let C^{∞}(U) be the R-algebra of smooth functions on U .

The assignment

U ↦ C^{∞}(U)

satisfies the following two properties:

(1) If U ⊂ V are two open sets in M, we can define the restriction map, which is an algebra morphism:

r_{V, U} : C^{∞}(V) → C^{∞}(U), ƒ ↦ ƒ|_{U}

which is such that

i) r_{U, U} = id

ii) r_{W, U} = r_{V, U} ○ r_{W, V}

(2) Let {U_{i}}_{i∈I} be an open covering of U and let {ƒ_{i}}_{i∈I}, ƒ_{i} ∈ C^{∞}(U_{i}) be a family such that ƒ_{i}|_{Ui ∩ Uj} = ƒ_{j}| _{Ui ∩ Uj} ∀ i, j ∈ I. In other words the elements of the family {ƒ_{i}}_{i∈I} agree on the intersection of any two open sets U_{i} ∩ U_{j}. Then there exists a unique ƒ ∈ C^{∞}(U) such that ƒ|_{Ui} = ƒ_{i}.

Such an assignment is called a sheaf. The pair (M, C^{∞}), consisting of the topological space M, underlying the differentiable manifold, and the sheaf of the C^{∞} functions on M is an example of locally ringed space (the word “locally” refers to a local property of the sheaf of C^{∞} functions.

Given two manifolds M and N, and the respective sheaves of smooth functions C_{M}^{∞} and C_{N}^{∞}, a morphism ƒ from M to N, viewed as ringed spaces, is a morphism |ƒ|: M → N of the underlying topological spaces together with a morphism of algebras,

ƒ^{*}: C_{N}^{∞}(V) → C_{M}^{∞}(ƒ^{-1}(V)), ƒ^{*}(φ)(x) = φ(|ƒ|(x))

compatible with the restriction morphisms.

Notice that, as soon as we give the continuous map |ƒ| between the topological spaces, the morphism ƒ^{*} is automatically assigned. This is a peculiarity of the sheaf of smooth functions on a manifold. Such a property is no longer true for a generic ringed space and, in particular, it is not true for supermanifolds.

A morphism of differentiable manifolds gives rise to a unique (locally) ringed space morphism and vice versa.

Moreover, given two manifolds, they are isomorphic as manifolds iff they are isomorphic as (locally) ringed spaces. In the language of categories, we say we have a fully faithful functor from the category of manifolds to the category of locally ringed spaces.

The generalization of algebraic geometry to the super-setting comes somehow more naturally than the similar generalization of differentiable geometry. This is because the machinery of algebraic geometry was developed to take already into account the presence of (even) nilpotents and consequently, the language is more suitable to supergeometry.

Let X be an affine algebraic variety in the affine space A^{n} over an algebraically closed field k and let O(X) = k[x_{1},…., x_{n}]/I be its coordinate ring, where the ideal I is prime. This corresponds topologically to the irreducibility of the variety X. We can think of the points of X as the zeros of the polynomials in the ideal I in A^{n}. X is a topological space with respect to the Zariski topology, whose closed sets are the zeros of the polynomials in the ideals of O(X). For each open U in X, consider the assignment

U ↦ O_{X}(U)

where O_{X}(U) is the k-algebra of regular functions on U. By definition, these are the functions ƒ X → k that can be expressed as a quotient of two polynomials at each point of U ⊂ X. The assignment U ↦ O_{X}(U) is another example of a sheaf is called the structure sheaf of the variety X or the sheaf of regular functions. (X, O_{X}) is another example of a (locally) ringed space.

Defaultable bonds are bonds that have a positive possibility of default. Most corporate bonds and some government bonds are defaultable. When a bond defaults, its coupon and principal payments will be altered. Most of the time, only a portion of the principal, and sometimes, also a portion of the coupon, will be paid. A defaultable (T, x) – bond with maturity T > 0 and credit rating x ∈ I ⊆ [0, 1], is a financial contract which pays to its holder 1 unit of currency at time T provided that the writer of the bond hasn’t bankrupted till time T. The set I stands for all possible credit ratings. The bankruptcy is modeled with the use of a so called loss process {L(t), t ≥ 0} which starts from zero, increases and takes values in the interval [0, 1]. The bond is worthless if the loss process exceeds its credit rating. Thus the payoff profile of the (T, x) – bond is of the form

1_{{LT ≤ x}}

The price P(t, T, x) of the (T, x) – bond is a stochastic process defined by

P(t, T, x) = 1_{{LT ≤ x}}e^{−∫tT f(t, u, x)du}, t ∈ [0, T] —– (1)

where f (·, ·, x) stands for an x-forward rate. The value x = 1 corresponds to the risk-free bond and f(t, T, 1) determines the short rate process via f(t, t, 1), t ≥ 0.

The (T, x) – bond market is thus fully determined by the family of x-forward rates and the loss process L. This is an extension of the classical non-defaultable bond market which can be identified with the case when I is a singleton, that is, when I = {1}.

The model of (T, x) – bonds does not correspond to defaultable bonds which are directly traded on a real market. For instance, in this setting the bankruptcy of the (T, x_{2}) – bond automatically implies the bankruptcy of the (T, x_{1}) – bond if x_{1} < x_{2}. In reality, a bond with a higher credit rating may, however, default earlier than that with a lower one. The (T, x) – bonds are basic instruments related to the pool of defaultable assets called Collateralized Debt Obligations (CDOs), which are actually widely traded on the market. In the CDO market model, the loss process L(t) describes the part of the pool which has defaulted up to time t > 0 and F(L_{T}), where F as some function, specifies the CDO payoff at time T > 0. In particular, (T, x) – bonds may be identified with the digital-type CDO payoffs with the function F of the form

F(z) = F_{x}(z) := 1_{[0,x]}(z), x ∈ I, z ∈ [0,1]

Then the price of that payoff p_{t}(F_{x}(L_{T})) at time t ≤ T equals P(t, T, x). Moreover, each regular CDO claim can be replicated, and thus also priced, with a portfolio consisting of a certain combination of (T, x) – bonds. Thus it follows that the model of (T, x) – bonds determines the structure of the CDO payoffs. The induced family of prices

P(t, T, x), T ≥ 0, x ∈ I

will be a CDO term structure. On real markets the price of a claim which pays more is always higher. This implies

P(t, T, x_{1}) = p_{t}(Fx_{1}(L_{T})) ≤ p_{t}(F_{x2}(L_{T})) = P(t, T, x_{2}), t ∈ [0, T], x_{1} < x_{2}, x_{1}, x_{2} ∈ I —– (2)

which means that the prices of (T, x) – bonds are increasing in x. Similarly, if the claim is paid earlier, then it has a higher value and hence

P(t, T_{1}, x) = p_{t}(F_{x}(L_{T1})) ≥ p_{t}(F_{x}(L_{T}2)) = P(t, T_{2}, x), t ∈ [0, T_{1}], T_{1} < T_{2}, x ∈ I —– (3)

which means that the (T, x) – bond prices are decreasing in T. The CDO term structure is monotone if both (2) and (3) are satisfied. Surprisingly, monotonicity of the (T, x) – bond prices is not always preserved in mathematical models even if they satisfy severe no-arbitrage conditions.

The purpose of quantum gravity phenomenology is to analyze the physical consequences arising from various models of quantum gravity. One hope for obtaining an experimental grasp on quantum gravity is the generic prediction arising in many (but not all) quantum gravity models that ultraviolet physics at or near the Planck scale, M_{Planck} = 1.2 × 10^{19} GeV/c^{2}, (or in some models the string scale), typically induces violations of Lorentz invariance at lower scales. Interestingly most investigations, even if they arise from quite different fundamental physics, seem to converge on the prediction that the breakdown of Lorentz invariance can generically become manifest in the form of modified dispersion relations

ω^{2} = ω_{0}^{2} + (1 + η_{2}) c^{2}k^{2} + η_{4}(ħ/M_{Lorentz violation})^{2} + k^{4} + ….

where the coefficients η_{n} are dimensionless (and possibly dependent on the particle species under consideration). The particular inertial frame for these dispersion relations is generally specified to be the frame set by cosmological microwave background, and M_{Lorentz violation} is the scale of Lorentz symmetry breaking which furthermore is generally assumed to be of the order of M_{Planck}.

Although several alternative scenarios have been considered to justify the modified kinematics,the most commonly explored avenue is an effective field theory (EFT) approach. Here, the focus is explicitly on the class of non-renormalizable EFTs with Lorentz violations associated to dispersion relations. Even if this framework as a “test theory” is successful, it is interesting to note that there are still significant open issues concerning its theoretical foundations. Perhaps the most pressing one is the so called naturalness problem which can be expressed in the following way: The lowest-order correction, proportional to η_{2}, is not explicitly Planck suppressed. This implies that such a term would always be dominant with respect to the higher-order ones and grossly incompatible with observations (given that we have very good constraints on the universality of the speed of light for different elementary particles). If one were to take cues from observational leads, it is assumed either that some symmetry (other than Lorentz invariance) enforces the η_{2} coefficients to be exactly zero, or that the presence of some other characteristic EFT mass scale μ ≪ M_{Planck} (e.g., some particle physics mass scale) associated with the Lorentz symmetry breaking might enter in the lowest order dimensionless coefficient η_{2}, which will be then generically suppressed by appropriate ratios of this characteristic mass to the Planck mass: η_{2} ∝ (μ/M_{Planck})^{σ} where σ ≥ 1 is some positive power (often taken as one or two). If this is the case then one has two distinct regimes: For low momenta p/(M_{Planck}c) ≪ (μ/M_{Planck})^{σ} the lower-order (quadratic in the momentum) deviations will dominate over the higher-order ones, while at high energies p/(M_{Planck}c) ≫ (μ/M_{Planck})^{σ} the higher order terms will be dominant.

The naturalness problem arises because such a scenario is not well justified within an EFT framework; in other words there is no natural suppression of the low-order modifications. EFT cannot justify why only the dimensionless coefficients of the n ≤ 2 terms should be suppressed by powers of the small ratio μ/M_{Planck}. Even worse, renormalization group arguments seem to imply that a similar mass ratio, μ/M_{Planck} would implicitly be present also in all the dimensionless n > 2 coefficients, hence suppressing them even further, to the point of complete undetectability. Furthermore, without some protecting symmetry, it is generic that radiative corrections due to particle interactions in an EFT with only Lorentz violations of order n > 2 for the free particles, will generate n = 2 Lorentz violating terms in the dispersion relation, which will then be dominant. Naturalness in EFT would then imply that the higher order terms are at least as suppressed as this, and hence beyond observational reach.

A second issue is that of universality, which is not so much a problem, as an issue of debate as to the best strategy to adopt. In dealing with situations with multiple particles one has to choose between the case of universal (particle-independent) Lorentz violating coefficients η_{n}, or instead go for a more general ansatz and allow for particle-dependent coefficients; hence allowing different magnitudes of Lorentz symmetry violation for different particles even when considering the same order terms (same n) in regards to momentum. Any violation of Lorentz invariance should be due to the microscopic structure of the effective space-time. This implies that one has to tune the system in order to cancel exactly all those violations of Lorentz invariance which are solely due to mode-mixing interactions in the hydrodynamic limit.

**Developing Countries**: The developing countries, marked in light blue, may prefer a fixed or managed exchange rate to a floating exchange rate. This is because sudden depreciation in their currency value poses a significant threat to the stability of their economies.

Here is * Von Mises* on Balance of Payments:

The surplus of the balance of payments that is not settled by the consignment of goods and services but by the transmission of money was long regarded as merely a consequence of the state of international trade. It is one of the great achievements of Classical political economy to have exposed the fundamental error in this view. It demonstrated that international movements of money are not consequences of the state of trade; that they constitute not the effect, but the cause, of a favorable or unfavorable trade balance. The precious metals are distributed among individuals and hence among nations according to the extent and intensity of their demand for money….the proposition is as true of money as of every other economic good, that its distribution among individual economic agents depends on its marginal utility … all economic goods, including of course money, tend to be distributed in such a way that a position of equilibrium among individuals is reached, when no further act of exchange that any individual could undertake would bring him any gain, any increase of subjective utility. In such a position of equilibrium, the total stock of money, just like the total stocks of commodities, is distributed among individuals according to the intensity with which they are able to express their demand for it in the market. Every displacement of the forces affecting the exchange ratio between money and other economic goods [i.e., the supply and demand for money] brings about a corresponding change in this distribution, until a new position of equilibrium is reached.

Balance of Payments – whether this is a monetary or a real phenomenon? The Monetary approach stresses that the balance of payments is essentially a monetary phenomenon. To be specific, it claims that money plays a vital role in determining the balance of payments. This approach does not deny the importance of non-monetary factors such as productivity changes, tariffs, government spending and taxation on the balance of payments. Indeed, it stresses the links between these factors and the money market. The Monetary approach does not assert that balance of payments problems are caused solely by monetary mismanagement or that monetary policy is the only possible cure. Rather, it emphasizes that the monetary process will bring about a cure of some kind (not necessarily very attractive) unless frustrated by deliberate monetary policy action. Policies that neglect or aggravate the monetary implications of deficits or surpluses will not be successful in their declared objectives.

The Monetary approach to the balance of payments applies specifically to a fixed exchange-rate regime. In this environment, it is the money supply that adjusts to money demand through international flows of money, which brings about equilibrium in the balance of payments. In this case, the money market determines the balance of payments. In a regime of flexible exchange rates, it is the money demand that adjusts to the money supply, which in turn is determined by the central bank. In a “managed float”, or “dirty float” scenario, in which exchange rates** **fluctuate from day to day, but central banks attempt to influence their countries’ exchange rates by buying and selling currencies, both international money flows and exchange rate changes are anticipated, given the nature of the central bank intervention.

The Keynesian approach focuses mainly on the merchandise trade account, with the capital account incorporated in the analysis at a later date. Disequilibrium in the balance of payments is caused by both the current account and the capital account. The current account, however, is considered to be more important than the capital account for the balance of payments disequilibrium. The relative prices and relative income levels of domestic and foreign countries are responsible for the balance of payments disequilibrium condition.

The relative prices and exchange rate of a country, vis-à-vis its trading partners, will determine the competitiveness of that country’s goods and services. An adverse movement in the relative price structure of a country, *ceteris paribus*, would decrease its competitiveness relative to its trading partners. Keynesians argue that a devaluation of a country’s currency will improve the trade account and enhance competitiveness. In time, equilibrium will be restored in the trade account. From the Monetary approach’s point of view, the money market is the principal vehicle, if not the only one, that is responsible for a balance of payments disequilibrium. For the Keynesians, it is differences in relative prices and domestic absorption rates that determine the balance of payments outcome. In short, real factors are more important than monetary factors in determining the balance of payments outcome.

The Monetary approach states that in the long run, monetary variables cannot affect real variables such as output, employment and, in the case of the balance of payments, the trade account. In the short run, * monetary variables will affect real variables*. This is true in the case of the money supply, money demand, exchange rates, and interest rates. The influence of interest rates on foreign direct investment is not made clear in the Monetary approach. This is because the approach does not distinguish between different types and durations of capital flows and the money account.

The Keynesian approach considers the balance of payments as a real phenomenon. Factors such as relative prices, devaluations, and aggregate demand affect the real variables in the short as well as the long run. In addition, capital flows are divided into short and long term. Protagonists of this approach argue that it is the short-term capital flows and long-term portfolio investments that respond to monetary factors.

For the Monetary approach, the demand for money is a stable function of a few variables. In a world of stable money demand functions, the assumptions of this approach are all valid. However, the Keynesian approach is not anchored on the premise of a stable demand function. In fact, for Keynesians, money demand functions are not considered stable since velocity is not stable. Hence, the Monetary approach’s conclusions are not valid for the Keynesians. The Monetary approach does not specifically identify whether it is the current account or the capital account that is responsible for a balance of payments deficit or surplus. However, it may be important to be able to clearly attribute the deficit or surplus to either the current account or the capital account. This would have an implication on the determination of the net worth of a country. For example, a change in the net worth of country will occur if the capital account is in deficit, *ceteris paribus*. However, net worth declines if the deficit is in the current account. The Keynesian approach specifically identifies which account is responsible for balance of payments deficits or surpluses. The reason for this identification is due to the importance of a country’s net worth over time.

The Monetary approach argues that the response of wages to a change in the money supply is not symmetric due to resistance from workers and unions to wage reductions following a decrease in the money supply. Conversely, with an increase in the money supply, wages rise following an increase in prices caused by the money supply expansion. Strong resistance to a drop in wages may be due to contractual agreements and institutional rigidities. According to the Monetary approach, a given change in the money supply is similar to a change in exchange rates in terms of percentage changes. For the Keynesians, however, this is not the case. It is their belief that in the real world, exchange rates are motivated in part by political reasons as well as economic and monetary factors. Keynesians argue that the reaction of wage earners and unions to a decrease in the money supply compared to their reaction to a devaluation is different. A decrease in the money supply will reduce nominal wages, and such a reduction is unacceptable to labor and labor unions. On the other hand, all other things remaining equal, a devaluation lowers the real wage rate by increasing domestic prices. However, a devaluation normally does not bring forth resistance from labor unions, despite the fact that an outcome similar to a decrease in the money supply is produced. It is possible that labor unions tend to focus on the immediate, direct effects on wages rather than on the delayed, indirect effects produced by a devaluation. In essence, the results of a decrease in the money supply or a devaluation are the same in the Keynesian view.

The Keynesian approach regards the exchange rate as a relative price of domestic and foreign goods. A change in relative prices will lead to a decrease in exchange rates. In the case of a devaluation, domestic prices of the devaluating country will decline in terms of foreign (goods) prices by the amount of the devaluation in percentage terms. In a regime of flexible exchange rates, the outcome of the current and capital accounts determines the exchange rate. According to the Keynesians, it is the capital account that exerts a more significant influence on exchange rates. This view is different from the Monetary approach, which argues that it is the money market outcome – money supply and money demand – that determines the exchange rate. The factors that influence the money supply and money demand will indirectly influence the exchange rate.

**Foreign Exchange Regimes**: The above map shows which countries have adopted which exchange rate regime. Dark green is for free float, neon green is for managed float, blue is for currency peg, and red is for countries that use another country’s currency.

Man, universe, gods and ritual are not considered separate entities but rather different manifestations of the same Śakti. Therefore, during a particular ritual every element of it is symbolic of something else. The flowers are representative of something else, the incense is representative of something else and so on. This viewpoint is based upon the crucial teaching that *“worldly and spiritual”* are the two faces of a same coin. One often thinks that “spirituality” is associated with something which is “within”, while “worldliness” is associated with something which is “without”. So, if you see a light “within”, that is a “spiritual” experience, while if you see a light “without”, that is a “worldly” experience. Besides, the worldliness is based on *“day-to-day experiences”*. It is approximately so. Tantricism *considers all to be the manifestation of Śakti*, the Divine Mother. So, an external light is as spiritual as an internal one and vice versa. In fact, there is neither spirituality nor worldliness because only one Supreme Consciousness is permeating everything and everyone.

Śakti or the Divine Mother is the core of all tantric practices. She is known as Kuṇḍalinī when residing in a living being. She is the bestower of the Supreme Bliss for all those followers that worship Her according to the sacred rituals and meditations contained in the Tantra-s. Her importance has been emphasized in Niruttaratantra:

बहूनां जन्मनामन्ते शक्तिज्ञानं प्रजायते।

शक्तिज्ञानं विना देवि निर्वाणं नैव जायते॥

Bahūnaṁ janmanāmante śaktijñānaṁ prajāyate|

Śaktijñānaṁ vinā devi nirvāṇaṁ naiva jāyate||

After (ante) many (bahūnām) births (janmanām), the knowledge (jñānam) of Śakti (śakti) is born *(in oneself)* (prajāyate). Oh goddess (devi)!, without (vinā) the knowledge (jñānam) of Śakti (śakti), Nirvāṇa *— final Liberation —* (nirvāṇam) does not (na eva) spring up (jāyate).

However, Tantricism should not be “strictly” equated to Shaktism, because there are groups of Śākta-s (followers of Śakti) which are not “tantric” at all. In turn, there are tantric groups that worship Śiva, Viṣṇu, etc. as well as Śakti.

Consequently, one may use a set of elements as representative of other realities. For example: a man represents Śiva and a woman represents Śakti. Then, their union *is representative* of that of Śiva and Śakti. Microcosm and macrocosm are closely allied to each other, because the two are the manifestation of only one Power. The following fragment extracted from the ancient Tantra-s clearly shows the aforesaid correlation between man, universe, gods and ritual. The sādhaka or practitioner is meditating on the Divine Mother (Śakti) in his heart lotus. He forms a mental image of Śakti there, and begins worshipping Her this way:

हृत्पद्मासनं दद्यात् सहस्रारच्युतामृतैः।

पाद्यं चरणयोर्दद्यान्मनसार्घ्यं निवेदयेत्॥

तेनामृतेनाचमनं स्नानीयमपि कल्पयेत्।

आकाशतत्त्वं वसनं गन्धं तु गन्धतत्त्वकम्॥

चित्तं प्रकल्पयेत् पुष्पं धूपं प्राणान् प्रकल्पयेत्।

तेजस्तत्त्वं च दीपार्थे नैवेद्यं च सुधाम्बुधिम्॥

अनाहतध्वनिं घण्टां वायुतत्त्वं च चामरम्।

नृत्यमिन्द्रियकर्माणि चाञ्चल्यं मनसस्तथा॥

पुष्पं नानाविधं दद्यादात्मनो भावसिद्धये।

अमायामनहङ्कारमरागममदं तथा॥

अमोहकमदम्भं च अद्वेषाक्षोभके तथा।

अमात्सर्यमलोभं च दशपुष्पं प्रकीर्तितम्॥

अहिंसा परमं पुष्पं पुष्पमिन्द्रियनिग्रहम्।

दयाक्षमाज्ञानपुष्पं पञ्चपुष्पं ततः परम्॥

इति पञ्चदशैर्पुष्पैर्भावपुष्पैः प्रपूजयेत्॥

Hṛtpadmāsanaṁ dadyāt sahasrāracyutāmṛtaiḥ|

Pādyaṁ caraṇayordadyānmanasārghyaṁ nivedayet||

Tenāmṛtenācamanaṁ snānīyamapi kalpayet|

Ākāśatattvaṁ vasanaṁ gandhaṁ tu gandhatattvakam||

Cittaṁ prakalpayet puṣpaṁ dhūpaṁ prāṇān prakalpayet|

Tejastattvaṁ ca dīpārthe naivedyaṁ ca sudhāmbudhim||

Anāhatadhvaniṁ ghaṇṭāṁ vāyutattvaṁ ca cāmaram|

Nṛtyamindriyakarmāṇi cāñcalyaṁ manasastathā||

Puṣpaṁ nānāvidhaṁ dadyādātmano bhāvasiddhaye|

Amāyāmanahaṅkāramarāgamamadaṁ tathā||

Amohakamadambham ca adveṣākṣobhake tathā|

Amātsaryamalobhaṁ ca daśapuṣpaṁ prakīrtitam||

Ahiṁsā paramaṁ puṣpamindriyanigraham|

Dayākṣamājñānapuṣpaṁ pañcapuṣpaṁ tataḥ param||

Iti pañcadaśairpuṣpairbhāvapuṣpaiḥ prapūjayet||

He gives (dadyāt… dadyāt) *(his)* heart (hṛt) lotus (padma) as the seat (āsanam), and the water for washing (pādyam) the feet (caraṇayoḥ) in the form of the nectars (amṛtaiḥ) flowing (cyuta) from Sahasrāra *— the supreme Cakra placed at the crown of the head–* (sahasrāra). He presents (nivedayet) the offering *— lit. water offered to a guest —* (arghyam) in the form of *(his)* mind (manasā).

He also (api) prepares (kalpayet) the water to be sipped from the palm of the hand *— a purificatory ceremony that is performed before any ritual or meal —* (ācamanam) *(as well as)* the water to be used in ablutions (snānīyam) by means of that very (tena) nectar (amṛtena). *(He gives)* the principle (tattvam) of Ākāśa *— ether or space–* (ākāśa) as the dress (vasanam), and the power of smelling (gandhatattvakam) as the odor (gandham).

He prepares (prakalpayet) *(his)* mind (manas) as the flower (vai) *(and)* arranges (prakalpayet) *(his)* vital energies (prāṇān) as incense (dhūpam). *(He)* also (ca) *(arranges)* the principle (tattvam) of Tejas *— fire —* (tejas) for it to act as (arthe) the lamp (dīpa), and (ca) the ocean (ambudhim) of nectar (sudhā) as the offering of food (naivedyam).

*(He prepares)* the Anāhata (anāhata) sound *— which keeps sounding constantly in the heart lotus —* (dhvanim) as the bell (ghaṇṭām), and (ca) the principle (tattvam) of Vāyu *–air–* (vāyu) as the fly-whisk made of tail of Yak (cāmaram). *(He offers)* the actions (karmāṇi) of the senses (indriya) as well as (tathā) the unsteadiness (cāñcalyam) of mind (manasaḥ) as dance (nṛtyam).

For realizing (siddhaye) the state (bhāva) of the Self (ātmanaḥ), he gives (dadyāt) flower(s) (puṣpam) of various sorts (nānāvidham): absence of delusion (amāyām), nonegotism (anahaṅkāram), dispassion and detachment (arāgam) as well as (tathā) absence of arrogance (amadam);…… absence of both bewilderment (amohakam) and (ca) deceit (adambham), as well as (tathā) nonmalevolence (adveṣa) and freedom from agitation (akṣobhake); absence of envy (amātsaryam) and (ca) liberty from covetousness (alobham)” — *(these virtues)* are named (prakīrtitam) the ten (daśa) flower(s) (puṣpam) –.

The supreme (paramam) flower(s) (puṣpam) *(known as)* Áhiṁsā *— nonviolence and harmlessness —* (ahiṁsā) and subjugation (nigraham) of the senses (indriya) *(along with)* the flower(s) (puṣpam) *(known as)* compassion (dayā), patience (kṣamā) and knowledge (jñāna), *(are)* therefore (tatas) the highest (param) five (pañca) flowers (puspam). Thus (iti), through *(these)* fifteen (pañcadaśaiḥ) flowers (puṣpaiḥ), *(which are actually fifteen)* flowers (puṣpaiḥ) formed from feelings (bhāva), he performs the worship (prapūjayet).

The sādhaka or practitioner uses every object in the ritual as representative of a virtue, state and so on. Therefore, one “must” be initiated in order to understand the Truth according to the Tantra-s, since only then the well-known vedic spirit of renunciation could be replaced for *“a reintegration of the worldly life to the purposes of Enlightenment”*. The “desire” and all values associated with it are then employed to achieve final Liberation. The tantric practitioner is both a master in spiritual matters and a master in worldly matters, because, in fact, there is no difference between “spiritual” and “worldly”. They are the two aspects in which the Divine Mother (Śakti) is manifested. So, a freed person is one who has transcended all pains and Saṁsāra (transmigration of the souls, that is, to be born and then to die, and to die and then to be born), and one who has acquired astonishing skills to lead a mundane life which is full of fulfillments.

मद्यपानेन मनुजो यदि सिद्धिं लभेत वै।

मद्यपानरताः सर्वे सिद्धिं गच्छन्तु पामराः॥११७॥

मांसभक्षणमात्रेण यदि पुण्यगतिर्भवेत्।

लोके मांसाशिनः सर्वे पुण्यभाजो भवन्त्विह॥११८॥

स्त्रीसम्भोगेन देवेशि यदि मोक्षं व्रजन्ति वै।

सर्वेऽपि जन्तवो लोके मुक्ताः स्युः स्त्रीनिषेवणात्॥११९॥

Madyapānena manujo yadi siddhiṁ labheta vai|

Madyapānaratāḥ sarve siddhiṁ gacchantu pāmarāḥ||117||

Māṁsabhakṣaṇamātreṇa yadi puṇyagatirbhavet|

Loke māṁsāśinaḥ sarve puṇyabhājo bhavantviha||118||

Strīsambhogena deveśi yadi mokṣaṁ vrajanti vai|

Sarve’pi jantavo loke muktāḥ syuḥ strīniṣevaṇāt||119||

If (yadi) a man (manujaḥ) really (vai) could attain (labheta) to Perfection (siddhim) by drinking (pānena) wine (madya), *(then)* may all (sarve) *(those)* vile (pāmarāḥ) people who are addicted to drinking (pānaratāḥ) wine (madya) achieve (gacchantu) Perfection (siddhim)!||117||

If (yadi) the achievement (gatiḥ) of Virtue (puṇya) would result (bhavet) from merely (mātreṇa) eating (bhakṣaṇa) meat (māṁsa), *(then)* may all (sarve) carnivorous beings (māṁsāśinaḥ) in this world (loke… iha) be (bhavantu) virtuous (puṇyabhājaḥ)!||118||

Oh goddess (deveśi)!, if (yadi) *(the beings)* indeed (vai) attain (vrajanti) to Liberation (mokṣam) through the enjoyment (sambhogena) of women (strī), *(then)* all (sarve) creatures (jantavaḥ) in this world (loke) would become (syuḥ) liberated (muktāḥ) by frequenting (niṣevaṇāt) women (strī)||119||