Ringed Spaces (2)

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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 rV, U: F(V) → F(U), called the restriction morphism and often denoted by rV, U(ƒ) = ƒ|U, such that

(i) rU, U = id,

(ii) rW, U = rV, U ○ rW, V

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

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

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 ↦ OX(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 ↦ rV, 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 Fx 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|. Fx 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 Fx 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

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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: φx: Fx → Gx. 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 φx = ψx ∀ 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 φU: 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| Fx. Let each open U ∈ |M| and each s ∈ F(U) define the map šU: U ⊔x∈|U| Fx, šU(x) = sx. 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 Fet to be the presheaf on |M|:

U ↦ Fet(U) = {šU: U → ⊔x∈|U| Fx, šU(x) = sx ∈ Fx}

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

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

(2) F and Fare locally isomorphic, i.e., ∃ an open cover {Ui}i∈I of |M| such that F(Ui) ≅ F~(Ui) 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 Fx 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*: ON → φ*OM, where φ*OM is a sheaf on |N| and defined as follows:

*OM)(U) = OM-1(U)) ∀ U open in |N|

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

x ∈ |M|: φx: ON,|φ|(x) → OM,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 φx is local, i.e. φ-1x(mM,x) = mN,|φ|(x), where mN,|φ|(x) and mM,x are the maximal ideals in the local rings ON,|φ|(x) and OM,x respectively.

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Revisiting Catastrophes. Thought of the Day 134.0

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 nth derivative of a function takes the value 0, graphically corresponding to minima, maxima, turning tangents, and, in higher dimensions, different complicated saddles. Catastrophe theory takes its point of departure in singularity theory whose object is the shift between types of such functions. It thus erects a distinction between an inner space – where the function varies – and an outer space of control variables charting the variation of that function including where it changes type – where, e.g. it goes from having one minimum to having two minima, via a singular case with turning tangent. The continuous variation of control parameters thus corresponds to a continuous variation within one subtype of the function, until it reaches a singular point where it discontinuously, ‘catastrophically’, changes subtype. The philosophy-of-science interpretation of this formalism now conceives the stable subtype of function as representing the stable state of a system, and the passage of the critical point as the sudden shift to a new stable state. The configuration of control parameters thus provides a sort of map of the shift between continuous development and discontinuous ‘jump’. Thom’s semiotic interpretation of this formalism entails that typical catastrophic trajectories of this kind may be interpreted as stable process types phenomenologically salient for perception and giving rise to basic verbal categories.

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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 ax4 + bx2 + 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 (How Nature Works the science of self-organized criticality) belongs to the same type of theories. Criticality is here defined as that state of a complicated system where sudden developments in all sizes spontaneously occur.

Ringed Spaces (1)

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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:

rV, U : C(V) → C(U), ƒ ↦ ƒ|U

which is such that

i) rU, U = id

ii) rW, U = rV, U ○ rW, V

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

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 CM and CN, 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*: CN(V) →  CM-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 An over an algebraically closed field k and let O(X) = k[x1,…., xn]/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 An. 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 ↦ OX(U)

where OX(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 ↦ OX(U) is another example of a sheaf is called the structure sheaf of the variety X or the sheaf of regular functions. (X, OX) is another example of a (locally) ringed space.

Defaultable Bonds. Thought of the Day 133.0

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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, x2) – bond automatically implies the bankruptcy of the (T, x1) – bond if x1 < x2. 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(LT), 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) = Fx(z) := 1[0,x](z), x ∈ I, z ∈ [0,1]

Then the price of that payoff pt(Fx(LT)) 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, x1) = pt(Fx1(LT)) ≤ pt(Fx2(LT)) = P(t, T, x2), t ∈ [0, T], x1 < x2, x1, x2 ∈ 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, T1, x) = pt(Fx(LT1)) ≥ pt(Fx(LT2)) = P(t, T2, x), t ∈ [0, T1], T1 < T2, 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.

Grand Unification Theory/(Anti-GUT): Emerging Symmetry, Topology in a Momentum Space. Thought of the Day 129.0

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Quantum phase transition between two ground states with the same symmetry but of different universality class – gapless at q < qc and fully gapped at q > qc – as isolated point (a) as the termination point of first order transition (b)

There are two schemes for the classification of states in condensed matter physics and relativistic quantum fields: classification by symmetry (GUT scheme) and by momentum space topology (anti-GUT scheme).

For the first classification method, a given state of the system is characterized by a symmetry group H which is a subgroup of the symmetry group G of the relevant physical laws. The thermodynamic phase transition between equilibrium states is usually marked by a change of the symmetry group H. This classification reflects the phenomenon of spontaneously broken symmetry. In relativistic quantum fields the chain of successive phase transitions, in which the large symmetry group existing at high energy is reduced at low energy, is in the basis of the Grand Unification models (GUT). In condensed matter the spontaneous symmetry breaking is a typical phenomenon, and the thermodynamic states are also classified in terms of the subgroup H of the relevant group G. The groups G and H are also responsible for topological defects, which are determined by the nontrivial elements of the homotopy groups πn(G/H).

The second classification method reflects the opposite tendency – the anti Grand Unification (anti-GUT) – when instead of the symmetry breaking the symmetry gradually emerges at low energy. This method deals with the ground states of the system at zero temperature (T = 0), i.e., it is the classification of quantum vacua. The universality classes of quantum vacua are determined by momentum-space topology, which is also responsible for the type of the effective theory, emergent physical laws and symmetries at low energy. Contrary to the GUT scheme, where the symmetry of the vacuum state is primary giving rise to topology, in the anti-GUT scheme the topology in the momentum space is primary while the vacuum symmetry is the emergent phenomenon in the low energy corner.

At the moment, we live in the ultra-cold Universe. All the characteristic temperatures in our Universe are extremely small compared to the Planck energy scale EP. That is why all the massive fermions, whose natural mass must be of order EP, are frozen out due to extremely small factor exp(−EP/T). There is no matter in our Universe unless there are massless fermions, whose masslessness is protected with extremely high accuracy. It is the topology in the momentum space, which provides such protection.

For systems living in 3D space, there are four basic universality classes of fermionic vacua provided by topology in momentum space:

(i)  Vacua with fully-gapped fermionic excitations, such as semiconductors and conventional superconductors.

(ii)  Vacua with fermionic excitations characterized by Fermi points – points in 3D momentum space at which the energy of fermionic quasiparticle vanishes. Examples are provided by the quantum vacuum of Standard Model above the electroweak transition, where all elementary particles are Weyl fermions with Fermi points in the spectrum. This universality class manifests the phenomenon of emergent relativistic quantum fields at low energy: close to the Fermi points the fermionic quasiparticles behave as massless Weyl fermions, while the collective modes of the vacuum interact with these fermions as gauge and gravitational fields.

(iii)  Vacua with fermionic excitations characterized by lines in 3D momentum space or points in 2D momentum space. We call them Fermi lines, though in general it is better to characterize zeroes by co-dimension, which is the dimension of p-space minus the dimension of the manifold of zeros. Lines in 3D momentum space and points in 2D momentum space have co-dimension 2: since 3−1 = 2−0 = 2. The Fermi lines are topologically stable only if some special symmetry is obeyed.

(iv) Vacua with fermionic excitations characterized by Fermi surfaces. This universality class also manifests the phenomenon of emergent physics, though non-relativistic: at low temperature all the metals behave in a similar way, and this behavior is determined by the Landau theory of Fermi liquid – the effective theory based on the existence of Fermi surface. Fermi surface has co-dimension 1: in 3D system it is the surface (co-dimension = 3 − 2 = 1), in 2D system it is the line (co- dimension = 2 − 1 = 1), and in 1D system it is the point (co-dimension = 1 − 0 = 1; in one dimensional system the Landau Fermi-liquid theory does not work, but the Fermi surface survives).

The possibility of the Fermi band class (v), where the energy vanishes in the finite region of the 3D momentum space and thus zeroes have co-dimension 0, and such topologically stable flat band may exist in the spectrum of fermion zero modes, i.e. for fermions localized in the core of the topological objects. The phase transitions which follow from this classification scheme are quantum phase transitions which occur at T = 0. It may happen that by changing some parameter q of the system we transfer the vacuum state from one universality class to another, or to the vacuum of the same universality class but different topological quantum number, without changing its symmetry group H. The point qc, where this zero-temperature transition occurs, marks the quantum phase transition. For T ≠ 0, the second order phase transition is absent, as the two states belong to the same symmetry class H, but the first order phase transition is not excluded. Hence, there is an isolated singular point (qc, 0) in the (q, T) plane, or the end point of the first order transition. The quantum phase transitions which occur in classes (iv) and (i) or be- tween these classes are well known. In the class (iv) the corresponding quantum phase transition is known as Lifshitz transition, at which the Fermi surface changes its topology or emerges from the fully gapped state of class (i). The transition between the fully gapped states characterized by different topological charges occurs in 2D systems exhibiting the quantum Hall and spin-Hall effect: this is the plateau-plateau transition between the states with different values of the Hall (or spin-Hall) conductance. The less known transitions involve nodes of co-dimension 3 and nodes of co-dimension 2.

The Fallacy of Deviant Analyticity. Thought of the Day 128.0

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Carnap’s thesis of pluralism in mathematics is quite radical. We are told that “any postulates and any rules of inference [may] be chosen arbitrarily”; for example, the question of whether the Principle of Selection (that is, the Axiom of Choice (AC)) should be admitted is “purely one of expedience” (Logical Syntax of Language); more generally,

The [logico-mathematical sentences] are, from the point of view of material interpretation, expedients for the purpose of operating with the [descriptive sentences]. Thus, in laying down [a logico-mathematical sentence] as a primitive sentence, only usefulness for this purpose is to be taken into consideration.

So the pluralism is quite broad – it extends to AC and even to ∏01-sentences. There are problems in maintaining ∏01-pluralism. One cannot, on pain of inconsistency, think that statements about consistency are not “mere matters of expedience” without thinking that ∏01-statements generally are not mere “matters of expedience”. The question of whether a given ∏01-sentence holds is not a mere matter of expedience; rather, such questions fall within the provenance of theoretical reason. One reason is that in adopting a ∏01-sentence one could always be struck by a counter-example. Other reasons have to do with the clarity of our conception of the natural numbers and with our experience to date with that structure. On this basis, for no sentence of first-order arithmetic is the question of whether it holds a mere matter of experience. Certainly this is the default view from which one must be moved.

What does Carnap have to say that will sway us from the default view, and lead us to embrace his radical form of pluralism? In approaching this question it is important to bear in mind that there are two general interpretations of Carnap. According to the first interpretation – the substantive – Carnap is really trying to argue for the pluralist conception. According to the second interpretation – the non-substantive – he is merely trying to persuade us of it, that is, to show that of all the options it is most “expedient”.

The most obvious approach to securing pluralism is to appeal to the work on analyticity and content. For if mathematical truths are without content and, moreover, this claim can be maintained with respect to an arbitrary mathematical system, then one could argue that even apparently incompatible systems have null content and hence are really compatible (since there is no contentual-conflict).

Now, in order for this to secure radical pluralism, Carnap would have to first secure his claim that mathematical truths are without content. But, he has not done so. Instead, he has merely provided us with a piece of technical machinery that can be used to articulate any one of a number of views concerning mathematical content and he has adjusted the parameters so as to articulate his particular view. So he has not secured the thesis of radical pluralism. Thus, on the substantive interpretation, Carnap has failed to achieve his end.

This leaves us with the non-substantive interpretation. There are a number of problems that arise for this version of Carnap. To begin with, Carnap’s technical machinery is not even suitable for articulating his thesis of radical pluralism since (using either the definition of analyticity for Language I or Language II) there is no metalanguage in which one can say that two apparently incompatible systems S1 and S2 both have null content and hence are really contentually compatible. To fix ideas, consider a paradigm case of an apparent conflict that we should like to dissolve by saying that there is no contentual-conflict: Let S1 = PA + φ and S2 = PA + ¬φ, where φ is any arithmetical sentence, and let the metatheory be MA = ZFC. The trouble is that on the approach to Language I, although in MT (metatheory) we can prove that each system is ω-complete (which is a start since we wish to say that each system has null content), we can also prove that one has null content while the other has total content (that is, in ω-logic, every sentence of arithmetic is a consequence). So, we cannot, within MT articulate the idea that there is no contentual-conflict. The approach to Language II involves a complementary problem. To see this note that while a strong logic like ω-logic is something that one can apply to a formal system, a truth definition is something that applies to a language (in our modern sense). Thus, on this approach, in MT the definition of analyticity given for S1 and S2 is the same (since the two systems are couched in the same language). So, although in MT we can say that S1 and S2 do not have a contentual-conflict this is only because we have given a deviant definition of analyticity, one that is blind to the fact that in a very straightforward sense φ is analytic in S1 while ¬φ is analytic in S2.

Now, although Carnap’s machinery is not adequate to articulate the thesis of radical pluralism in a given metatheory, under certain circumstances he can attempt to articulate the thesis by changing the metatheory. For example, let S1 = PA + Con(ZF + AD) and S2 = PA + ¬Con(ZF + AD) and suppose we wish to articulate both the idea that the two systems have null content and the idea that Con(ZF + AD) is analytic in S1 while ¬Con(ZF + AD) is analytic in S2. No single metatheory (on either of Carnap’s approaches) can do this. But it turns out that because of the kind of assessment sensitivity, there are two metatheories MT1 and MT2 such that in MT1 we can say both that S1 has null content and that Con(ZF + AD) is analytic in S1, while in MT2 we can say both that S2 has null content and that ¬Con(ZF + AD) is analytic in S2. But, of course, this is simply because (any such metatheory) MT1 proves Con(ZF+AD) and (any such metatheory) MT2 proves ¬Con(ZF+AD). So we have done no more than reflect the difference between the systems in the metatheories. Thus, although Carnap does not have a way of articulating his radical pluralism (in a given metalanguage), he certainly has a way of manifesting it (by making corresponding changes in his metatheories).

As a final retreat Carnap might say that he is not trying to persuade us of a thesis that (concerning a collection of systems) can be articulated in a given framework but rather is trying to persuade us to adopt a thorough radical pluralism as a “way of life”. He has certainly shown us how we can make the requisite adjustments in our metatheory so as to consistently manifest radical pluralism. But does this amount to more than an algorithm for begging the question? Has Carnap shown us that there is no question to beg? He has not said anything persuasive in favour of embracing a thorough radical pluralism as the “most expedient” of the options. The trouble with Carnap’s entire approach is that the question of pluralism has been detached from actual developments in mathematics.

Super-Poincaré Algebra: Is There a Case for Nontrivial Geometry in the Bosonic Sector?

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In one dimension there is no Lorentz group and therefore all bosonic and fermionic fields have no space-time indices. The simplest free action for one bosonic field φ and one fermionic field ψ reads

S = γ∫dt [φ.2 – i/2ψψ.] —– (1)

We treat the scalar field as dimensionless and assign dimension cm−1/2 to fermions. Therefore, all our actions will contain the parameter γ with the dimension [γ] = cm. (1) provides the first example of a supersymmetric invariant action and is invariant w.r.t. the following transformations:

δφ = −iεψ, δψ = −εφ ̇ —– (2)

The infinitesimal parameter ε anticommutes with fermionic fields and with itself. What is really important about transformations (2) is their commutator

δ2δ1φ = δ2(−ε1ψ) = iε1ε2φ ̇

δ1δ2φ = iε2ε1φ ̇ ⇒ [δ2, δ1] φ = 2iε1ε2φ ̇ —– (3)

Thus, from (3) we may see the main property of supersymmetry transformations: they commute on translations. In our simplest one-dimensional framework this is the time translation. This property has the followin form in terms of the supersymmetry generator Q:

{Q, Q} = −2P —– (4)

The anticommutator (4), together with

[Q, P] = 0 —– (5)

describe N = 1 super-Poincaré algebra in d = 1. The structure of N-extended super-Poincaré algebra includes N real super-charges QA , A = 1, . . . , N with the following anti commutators:

{QA, QB} = −2δABP, [QA, P] = 0 —– (6)

Let us stress that the reality of the supercharges is very important, as well as having the same sign in the r.h.s. of QA, QB ∀ QA.

From (2) we see that the minimal N = 1 supermultiplet includes one bosonic and one fermionic field. A natural question arises: how many components do we need, in order to realize the N-extended superalgebra (6)? In order to mimic the transformations (2) for all N supertranslations

δφi = −iεA(LA)i ψ, δψ = −εA (RA)iφ ̇i —– (7)

Here the indices i = 1,…,db and iˆ = 1,…,df count the numbers of bosonic and fermionic components, while (LA)i and (RA)i are N arbitrary, for the time being, matrices. The additional conditions one should impose on the transformations (7) are

  • they should form the N-extended superalgebra (6)
  • they should leave invariant the free action constructed from the involved fields.

When N > 8 the minimal dimension of the supermultiplets rapidly increases and the analysis of the corresponding theories becomes very complicated. For many reasons, the most interesting case seems to be the N = 8 supersymmetric mechanics. Being the highest N case of minimal N-extended supersymmetric mechanics admitting realization on N bosons (physical and auxiliary) and N fermions, the systems with eight supercharges are the highest N ones, among the extended supersymmetric systems, which still possess a nontrivial geometry in the bosonic sector. When the number of supercharges exceeds 8, the target spaces are restricted to be symmetric spaces.