Spinorial Algebra

Superspace is to supersymmetry as Minkowski space is to the Lorentz group. Superspace provides the most natural geometrical setting in which to describe supersymmetrical theories. Almost no physicist would utilize the component of Lorentz four-vectors or higher rank tensor to describe relativistic physics.

In a field theory, boson and fermions are to be regarded as diffeomorphisms generating two different vector spaces; the supersymmetry generators are nothing but sets of linear maps between these spaces. We can thus include a supersymmetric theory in a more general geometrical framework defining the collection of diffeomorphisms,

φ_i : R → R^d_L, i = 1,…, d_L —– (1)

ψ_α^ˆ : R → R^d_R, i = 1,…, d_R —– (2)

where the one-dimensional dependence reminds us that we restrict our attention to mechanics. The free vector spaces generated by {φ_i}_i=1^dL and {ψ_α^ˆ}_α^ˆdR are respectively V_L and V_R, isomorphic to R^d_L and R^d_R. For matrix representations in the following, the two integers are restricted to the case d_L = d_R = d. Four different linear mappings can act on V_L and V_R

M_L : V_L → V_R, M_R : V_R → V_L

U_L : V_L → V_L, U_R : V_R → V_R —– (3)

with linear map space dimensions

dimM_L = dimM_R = d_Rd_L = d²,

dimU_L = d_L² = d², dimU_R = d_R² = d² —– (4)

as a consequence of linearity. To relate this construction to a general real (≡ GR) algebraic structure of dimension d and rank N denoted by GR(d,N), two more requirements need to be added.

Defining the generators of GR(d,N) as the family of N + N linear maps

L_I ∈ {M_L}, I = 1,…, N

R_K ∈ {M_R}, K = 1,…, N —– (5)

such that ∀ I, K = 1,…, N, we have

L_I ◦ R_K + L_K ◦ R_I = −2δ_IKI_VR

R_I ◦ L_K + R_K ◦ L_I = −2δ_IKI_VL —– (6)

where I_VL and I_VR are identity maps on V_L and V_R. Equations (6) will later be embedded into a Clifford algebra but one point has to be emphasized, we are working with real objects.

After equipping V_L and V_R with euclidean inner products ⟨·,·⟩_VL and ⟨·,·⟩_VR, respectively, the generators satisfy the property

⟨φ, R_I(ψ)⟩V_L = −⟨L_I(φ), ψ⟩V_R, ∀ (φ, ψ) ∈ V_L ⊕ V_R —— (7)

This condition relates L_I to the hermitian conjugate of R_I, namely R_I^†, defined as usual by

⟨φ, R_I(ψ)⟩_VL = ⟨R_I^†(φ), ψ⟩_VR —– (8)

such that

R_I^† = R_I^t = −L_I —– (9)

The role of {U_L} and {U_R} maps is to connect different representations once a set of generators defined by conditions (6) and (7) has been chosen. Notice that (R_IL_J)_i^j ∈ U_L and (L_IR_J)_αˆ^βˆ ∈ U_R. Let us consider A ∈ {U_L} and B ∈ {U_R} such that

A : φ → φ′ = Aφ

B : ψ → ψ′ = Bψ —– (10)

with V_L as an example,

⟨φ, R_I(ψ)⟩_VL → ⟨Aφ, R_I B(ψ)⟩_VL

= ⟨φ,A^† R_I B(ψ)⟩_VL

= ⟨φ, R_I^′ (ψ)⟩_VL —– (11)

so a change of representation transforms the generators in the following manner:

L_I → L_I^‘ = B^†L_IA

R_I → R_I^′ = A^†R_IB —– (12)

In general (6) and (7) do not identify a unique set of generators. Thus, an equivalence relation has to be defined on the space of possible sets of generators, say {L_I, R_I} ∼ {L_I^‘, R_I^′} iff ∃ A ∈ {U_L} and B ∈ {U_R} such that L′ = B^†L_IA and R′ = A^†R_IB.

Moving on to how supersymmetry is born, we consider the manner in which algebraic derivations are defined by

δ_εφ_i = iε^I(R_I)_i^αˆψ^αˆ

δ_εψ_αˆ = −ε^I(L_I)_αˆⁱ∂_τφ_i —– (13)

where the real-valued fields {φ_i}_i=1^dL and {ψ_αˆ}_αˆ=1^dR can be interpreted as bosonic and fermionic respectively. The fermionic nature attributed to the V_R elements implies that M_L and M_R generators, together with supersymmetry transformation parameters ε^I, anticommute among themselves. Introducing the d_L + d_R dimensional space V_L ⊕ V_R with vectors

Ψ = (^ψ _φ) —– (14)

(13) reads

δ_ε(Ψ) = (^iεRψ _εL∂τφ) —– (15)

such that

[δ_ε1, δ_ε2]Ψ = iε₁^Iε₂^J (^{R_IL_J∂_τφ} _LIRJ∂τψ) – iε₂^Jε₁^I (^{R_JL_I∂_τφ} _LJRI∂τψ) = – 2iε₁^Iε₂^I∂_τΨ —– (16)

utilizing that we have classical anticommuting parameters and that (6) holds. From (16) it is clear that δ_ε acts as a supersymmetry generator, so that we can set

δ_QΨ := δ_εΨ = iε^IQ_IΨ —– (17)

which is equivalent to writing

δ_Qφ_i = i(ε^IQ_Iψ)_i

δ_Qψ_αˆ = i(ε^IQ_Iφ)_αˆ —– (18)

with

Q1 = (0LIH RI0) —– (19)

where H = i∂_τ. As a consequence of (16) a familiar anticommutation relation appears

{Q_I, Q_J} = − 2iδ_IJH —– (20)

confirming that we are about to recognize supersymmetry, and once this is achieved, we can associate to the algebraic derivations (13), the variations defining the scalar supermultiplets. However, the choice (13) is not unique, for this is where we could have a spinorial one,

δ_Qξ_αˆ = ε^I(L_I)_αˆⁱF_i

δ_QF_i = − iε^I(R_I)_i^αˆ∂_τξ_αˆ —– (21)

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Valencies of Predicates. Thought of the Day 125.0

Since icons are the means of representing qualities, they generally constitute the predicative side of more complicated signs:

The only way of directly communicating an idea is by means of an icon; and every indirect method of communicating an idea must depend for its establishment upon the use of an icon. Hence, every assertion must contain an icon or set of icons, or else must contain signs whose meaning is only explicable by icons. The idea which the set of icons (or the equivalent of a set of icons) contained in an assertion signifies may be termed the predicate of the assertion. (Collected Papers of Charles Sanders Peirce)

Thus, the predicate in logic as well as ordinary language is essentially iconic. It is important to remember here Peirce’s generalization of the predicate from the traditional subject-copula-predicate structure. Predicates exist with more than one subject slot; this is the basis for Peirce’s logic of relatives and permits at the same time enlarging the scope of logic considerably and approaching it to ordinary language where several-slot-predicates prevail, for instance in all verbs with a valency larger than one. In his definition of these predicates by means of valency, that is, number of empty slots in which subjects or more generally indices may be inserted, Peirce is actually the founder of valency grammar in the tradition of Tesnière. So, for instance, the structure ‘_ gives _ to _’ where the underlinings refer to slots, is a trivalent predicate. Thus, the word classes associated with predicates are not only adjectives, but verbs and common nouns; in short all descriptive features in language are predicative.

This entails the fact that the similarity charted in icons covers more complicated cases than does the ordinary use of the word. Thus,

where ordinary logic considers only a single, special kind of relation, that of similarity, – a relation, too, of a particularly featureless and insignificant kind, the logic of relatives imagines a relation in general to be placed. Consequently, in place of the class, which is composed of a number of individual objects or facts brought together by means of their relation of similarity, the logic of relatives considers the system, which is composed of objects brought together by any kind of relations whatsoever. (The New Elements of Mathematics)

This allows for abstract similarity because one phenomenon may be similar to another in so far as both of them partake in the same relation, or more generally, in the same system – relations and systems being complicated predicates.

But not only more abstract features may thus act as the qualities invoked in an icon; these qualities may be of widely varying generality:

But instead of a single icon, or sign by resemblance of a familiar image or ‘dream’, evocable at will, there may be a complexus of such icons, forming a composite image of which the whole is not familiar. But though the whole is not familiar, yet not only are the parts familiar images, but there will also be a familiar image in its mode of composition. ( ) The sort of idea which an icon embodies, if it be such that it can convey any positive information, being applicable to some things but not to others, is called a first intention. The idea embodied by an icon, which cannot of itself convey any information, being applicable to everything or nothing, but which may, nevertheless, be useful in modifying other icons, is called a second intention. 

What Peirce distinguishes in these scholastic standard notions borrowed from Aquinas via Scotus, is, in fact, the difference between Husserlian formal and material ontology. Formal qualities like genus, species, dependencies, quantities, spatial and temporal extension, and so on are of course attributable to any phenomenon and do not as such, in themselves, convey any information in so far as they are always instantiated in and thus, like other Second Intentions, in the Husserlian manner dependent upon First Intentions, but they are nevertheless indispensable in the composition of first intentional descriptions. The fact that a certain phenomenon is composed of parts, has a form, belongs to a species, has an extension, has been mentioned in a sentence etc. does not convey the slightest information of it until it by means of first intentional icons is specified which parts in which composition, which species, which form, etc. Thus, here Peirce makes a hierarchy of icons which we could call material and formal, respectively, in which the latter are dependent on the former. One may note in passing that the distinctions in Peirce’s semiotics are themselves built upon such Second Intentions; thus it is no wonder that every sign must possess some Iconic element. Furthermore, the very anatomy of the proposition becomes just like in Husserlian rational grammar a question of formal, synthetic a priori regularities.

Among Peirce’s forms of inference, similarity plays a certain role within abduction, his notion for a ‘qualified guess’ in which a particular fact gives rise to the formation of a hypothesis which would have the fact in question as a consequence. Many such different hypotheses are of course possible for a given fact, and this inference is not necessary, but merely possible, suggestive. Precisely for this reason, similarity plays a seminal role here: an

originary Argument, or Abduction, is an argument which presents facts in its Premiss which presents a similarity to the fact stated in the conclusion but which could perfectly be true without the latter being so.

The hypothesis proposed is abducted by some sort of iconic relation to the fact to be explained. Thus, similarity is the very source of new ideas – which must subsequently be controlled deductively and inductively, to be sure. But iconicity does not only play this role in the contents of abductive inference, it plays an even more important role in the very form of logical inference in general:

Given a conventional or other general sign of an object, to deduce any other truth than that which it explicitly signifies, it is necessary, in all cases, to replace that sign by an icon. This capacity of revealing unexpected truth is precisely that wherein the utility of algebraic formulae consists, so that the iconic character is the prevailing one.

The very form of inferences depends on it being an icon; thus for Peirce the syllogistic schema inherent in reasoning has an iconic character:

‘Whenever one thing suggests another, both are together in the mind for an instant. [ ] every proposition like the premiss, that is having an icon like it, would involve [ ] a proposition related to it as the conclusion [ ]’. Thus, first and foremost deduction is an icon: ‘I suppose it would be the general opinion of logicians, as it certainly was long mine, that the Syllogism is a Symbol, because of its Generality.’ …. The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation; namely deduction consists in constructing an icon or diagram the relation of whose parts shall present a complete analogy with those of the parts of the objects of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts. 

It then is no wonder that synthetic a priori truths exist – even if Peirce prefers notions like ‘observable, universal truths’ – the result of a deduction may contain more than what is immediately present in the premises, due to the iconic quality of the inference.

Tranche Declension.

With the CDO (collateralized debt obligation) market picking up, it is important to build a stronger understanding of pricing and risk management models. The role of the Gaussian copula model, has well-known deficiencies and has been criticized, but it continues to be fundamental as a starter. Here, we draw attention to the applicability of Gaussian inequalities in analyzing tranche loss sensitivity to correlation parameters for the Gaussian copula model.

We work with an R^N-valued Gaussian random variable X = (X₁, … , X_N), where each X_j is normalized to mean 0 and variance 1, and study the equity tranche loss

L_[0,a] = ∑_m=1^Nl_m1_[xm≤cm] – {∑_m=1^Nl_m1_[xm≤cm] – a}

where l₁ ,…, l_N > 0, a > 0, and c₁,…, c_N ∈ R are parameters. We thus establish an identity between the sensitivity of E[L_[0,a]] to the correlation r_jk = E[X_jX_k] and the parameters c_j and c_k, from where subsequently we come to the inequality

∂E[L_[0,a]]/∂r_jk ≤ 0

Applying this inequality to a CDO containing N names whose default behavior is governed by the Gaussian variables X_j shows that an increase in name-to-name correlation decreases expected loss in an equity tranche. This is a generalization of the well-known result for Gaussian copulas with uniform correlation.

Consider a CDO consisting of N names, with τ_j denoting the (random) default time of the j^th name. Let

X_j = φ_j^-1(F_j(τ_j))

where F_j is the distribution function of τ_j (relative to the market pricing measure), assumed to be continuous and strictly increasing, and φ_j is the standard Gaussian distribution function. Then for any x ∈ R we have

P[X_j ≤ x] = P[τ_j ≤ F_j^-1(φ_j(x))] = F_j(F_j^-1(φ_j(x))) = φ_j(x)

which means that X_j has standard Gaussian distribution. The Gaussian copula model posits that the joint distribution of the X_j is Gaussian; thus,

X = (X₁, …., X_n)

is an R^N-valued Gaussian variable whose marginals are all standard Gaussian. The correlation

τ_j = E[X_jX_k]

reflects the default correlation between the names j and k. Now let

p_j = E[τ_j ≤ T] = P[X_j ≤ c_j]

be the probability that the j^th name defaults within a time horizon T, which is held constant, and

c_j = φ_j⁻¹(F_j(T))

is the default threshold of the j^th name.

In schematics, when we explore the essential phenomenon, the default of name j, which happens if the default time τ_j is within the time horizon T, results in a loss of amount l_j > 0 in the CDO portfolio. Thus, the total loss during the time period [0, T] is

L = ∑_m=1^Nl_m1_[xm≤cm]

This is where we are essentially working with a one-period CDO, and ignoring discounting from the random time of actual default. A tranche is simply a range of loss for the portfolio; it is specified by a closed interval [a, b] with 0 ≤ a ≤ b. If the loss x is less than a, then this tranche is unaffected, whereas if x ≥ b then the entire tranche value b − a is eaten up by loss; in between, if a ≤ x ≤ b, the loss to the tranche is x − a. Thus, the tranche loss function t_{[a, b]} is given by

t_{[a, b]}(x) = 0 if x < a; = x – a, if x ∈ [a, b]; = b – a; if x > b

or compactly,

t_{[a, b]}(x) = (x – a)₊ – (x – b)₊

From this, it is clear that t_{[a, b]}(x) is continuous in (a, b, x), and we see that it is a non-decreasing function of x. Thus, the loss in an equity tranche [0, a] is given by

t_[0,a](L) = L − (L − a)₊

with a > 0.

CUSUM Deceleration. Drunken Risibility.

CUSUM, or cumulative sum is used for detecting and monitoring change detection. Let us introduce a measurable space (Ω, F), where Ω = R^∞, F = ∪_nF_n and F_n = σ{Y_i, i ∈ {0, 1, …, n}}. The law of the sequence  Y_i, i = 1, …., is defined by the family of probability measures {P_v}, v ∈ N*. In other words, the probability measure P_v for a given v > 0, playing the role of the change point, is the measure generated on Ω by the sequence Y_i, i = 1, … , when the distribution of the Y_i’s changes at time v. The probability measures P₀ and P_∞ are the measures generated on Ω by the random variables Y_i when they have an identical distribution. In other words, the system defined by the sequence Y_i undergoes a “regime change” from the distribution P₀ to the distribution P_∞ at the change point time v.

The CUSUM (cumulative sum control chart) statistic is defined as the maximum of the log-likelihood ratio of the measure P_v to the measure P_∞ on the σ-algebra F_n. That is,

C_n := max_0≤v≤n log dP_v/dP_∞|_Fn —– (1)

is the CUSUM statistic on the σ-algebra F_n. The CUSUM statistic process is then the collection of the CUSUM statistics {C_n} of (1) for n = 1, ….

The CUSUM stopping rule is then

T(h) := inf {n ≥ 0: max_0≤v≤n log dP_v/dP_∞|_Fn ≥ h} —– (2)

for some threshold h > 0. In the CUSUM stopping rule (2), the CUSUM statistic process of (1) is initialized at

C₀ = 0 —– (3)

The CUSUM statistic process was first introduced by E. S. Page in the form that it takes when the sequence of random variables Y_i is independent and Gaussian; that is, Y_i ∼ N(μ, 1), i = 1, 2,…, with μ = μ₀ for i < 𝜈 and μ = μ₁ for i ≥ 𝜈. Since its introduction by Page, the CUSUM statistic process of (1) and its associated CUSUM stopping time of (2) have been used in a plethora of applications where it is of interest to perform detection of abrupt changes in the statistical behavior of observations in real time. Examples of such applications are signal processing, monitoring the outbreak of an epidemic, financial surveillance, and computer vision. The popularity of the CUSUM stopping time (2) is mainly due to its low complexity and optimality properties in both discrete and continuous time models.

Let Y_i ∼ N(μ₀, σ₂) that change to Y_i ∼ N(μ₁, σ₂) at the change point time v. We now proceed to derive the form of the CUSUM statistic process (1) and its associated CUSUM stopping time (2). Let us denote by φ(x) = 1/√2π e^-x2/2 the Gaussian kernel. For the sequence of random variables Y_i given earlier,

C_n := max_0≤v≤n log dP_v/dP_∞|_Fn

= max_0≤v≤n log (∏_i=1^v-1φ(Y_i-μ₀)/σ ∏_i=vⁿφ(Y_i-μ₁)/σ)/∏_i=1ⁿφ(Y_i-μ₀)/σ

= 1/σ²max_0≤v≤n (μ₁ – μ₀)∑_i=vⁿ[Y_i – (μ₁ + μ₀)/2] —– (4)

In view of (3), let us initialize the sequence (4) at Y₀ = (μ₁ + μ₀)/2 and distinguish two cases.

a) μ₁ > μ₀: divide out (μ₁ – μ₀), multiply by the constant σ² in (4) and use (2) to obtain CUSUM stopping T⁺:

T⁺(h⁺) = inf {n ≥ 0: max_0≤v≤n ∑_i=vⁿ[Y_i – (μ₁ + μ₀)/2] ≥ h⁺} —– (5)

for an appropriately scaled threshold h⁺ > 0.

b) μ₁ < μ₀: divide out (μ₁ – μ₀), multiply by the constant σ² in (4) and use (2) to obtain CUSUM stopping T^–:

T^–(h^–) = inf {n ≥ 0: max_0≤v≤n ∑_i=vⁿ[(μ₁ + μ₀)/2 – Y_i] ≥ h^–} —– (6)

for an appropriately scaled threshold h^– > 0.

The sequences form a CUSUM according to the deviation of the monitored sequential observations from the average of their pre- and postchange means. Although the stopping times (5) and (6) can be derived by formal CUSUM regime change considerations, they may also be used as general nonparametric stopping rules directly applied to sequential observations.

Philosophizing Loops – Why Spin Foam Constraints to 3D Dynamics Evolution?

The philosophy of loops is canonical, i.e., an analysis of the evolution of variables defined classically through a foliation of spacetime by a family of space-like three- surfaces ∑_t. The standard choice is the three-dimensional metric g_ij, and its canonical conjugate, related to the extrinsic curvature. If the system is reparametrization invariant, the total hamiltonian vanishes, and this hamiltonian constraint is usually called the Wheeler-DeWitt equation. Choosing the canonical variables is fundamental, to say the least.

Abhay Ashtekar‘s insights stems from the definition of an original set of variables stemming from Einstein-Hilbert Lagrangian written in the form,

S = ∫e^a ∧ e^b ∧ R^cdε_abcd —– (1)

where, e^a are the one-forms associated to the tetrad,

e^a ≡ e^a_μdx^μ —– (2)

The associated SO(1, 3) connection one-form ϖ^a_b is called the spin connection. Its field strength is the curvature expressed as a two form:

R^a_b ≡ dϖ^a_b + ϖ^a_c ∧ ϖ^c_b —– (3)

Ashtekar’s variables are actually based on the SU(2) self-dual connection

A = ϖ − i ∗ ϖ —– (4)

Its field strength is

F ≡ dA + A ∧ A —– (5)

The dynamical variables are then (A_i, Eⁱ ≡ F⁰ⁱ). The main virtue of these variables is that constraints are then linearized. One of them is exactly analogous to Gauss’ law:

D_iEⁱ = 0 —– (6)

There is another one related to three-dimensional diffeomorphisms invariance,

trF_ijEⁱ = 0 —– (7)

and, finally, there is the Hamiltonian constraint,

trF_ijEⁱE^j = 0 —– (8)

On a purely mathematical basis, there is no doubt that Astekhar’s variables are of a great ingenuity. As a physical tool to describe the metric of space, they are not real in general. This forces a reality condition to be imposed, which is akward. For this reason it is usually prefered to use the Barbero-Immirzi formalism in which the connection depends on a free parameter, γ

Aⁱ_a + ϖⁱ_a + γKⁱ_a —– (9)

ϖ being the spin connection, and K the extrinsic curvature. When γ = i, Ashtekar’s formalism is recovered, for other values of γ, the explicit form of the constraints is more complicated. Even if there is a Hamiltonian constraint that seems promising, was isn’t particularly clear is if the quantum constraint algebra is isomorphic to the classical algebra.

Some states which satisfy the Astekhar constraints are given by the loop representation, which can be introduced from the construct (depending both on the gauge field A and on a parametrized loop γ)

W (γ, A) ≡ trPe^φ_γA —– (10)

and a functional transform mapping functionals of the gauge field ψ(A) into functionals of loops, ψ(γ):

ψ(γ) ≡ ∫DAW(γ, A) ψ(A) —– (11)

When one divides by diffeomorphisms, it is found that functions of knot classes (diffeomorphisms classes of smooth, non self-intersecting loops) satisfy all the constraints. Some particular states sought to reproduce smooth spaces at coarse graining are the Weaves. It is not clear to what extent they also approach the conjugate variables (that is, the extrinsic curvature) as well.

In the presence of a cosmological constant the hamiltonian constraint reads:

ε_ijkE^aiE^bj(F^k_ab + λ/3ε_abcE^ck) = 0 —– (12)

A particular class of solutions expounded by Lee Smolin of the constraint are self-dual solutions of the form

Fⁱ_ab = -λ/3ε_abcE^ci —– (13)

Loop states in general (suitable symmetrized) can be represented as spin network states: colored lines (carrying some SU(2) representation) meeting at nodes where intertwining SU(2) operators act. There is also a path integral representation, known as spin foam, a topological theory of colored surfaces representing the evolution of a spin network. Spin foams can also be considered as an independent approach to the quantization of the gravitational field. In addition to its specific problems, the hamiltonian constraint does not say in what sense (with respect to what) the three-dimensional dynamics evolve.

Complete Manifolds’ Pure Logical Necessity as the Totality of Possible Formations. Thought of the Day 124.0

In Logical Investigations, Husserl called his theory of complete manifolds the key to the only possible solution to how in the realm of numbers impossible, non-existent, meaningless concepts might be dealt with as real ones. In Ideas, he wrote that his chief purpose in developing his theory of manifolds had been to find a theoretical solution to the problem of imaginary quantities (Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy).

Husserl saw how questions regarding imaginary numbers come up in mathematical contexts in which formalization yields constructions which arithmetically speaking are nonsense, but can be used in calculations. When formal reasoning is carried out mechanically as if these symbols have meaning, if the ordinary rules are observed, and the results do not contain any imaginary components, these symbols might be legitimately used. And this could be empirically verified (Philosophy of Arithmetic_ Psychological and Logical Investigations with Supplementary Texts).

In a letter to Carl Stumpf in the early 1890s, Husserl explained how, in trying to understand how operating with contradictory concepts could lead to correct theorems, he had found that for imaginary numbers like √2 and √-1, it was not a matter of the possibility or impossibility of concepts. Through the calculation itself and its rules, as defined for those fictive numbers, the impossible fell away, and a genuine equation remained. One could calculate again with the same signs, but referring to valid concepts, and the result was again correct. Even if one mistakenly imagined that what was contradictory existed, or held the most absurd theories about the content of the corresponding concepts of number, the calculation remained correct if it followed the rules. He concluded that this must be a result of the signs and their rules (Early Writings in the Philosophy of Logic and Mathematics). The fact that one can generalize, produce variations of formal arithmetic that lead outside the quantitative domain without essentially altering formal arithmetic’s theoretical nature and calculational methods brought Husserl to realize that there was more to the mathematical or formal sciences, or the mathematical method of calculation than could be captured in purely quantitative analyses.

Understanding the nature of theory forms, shows how reference to impossible objects can be justified. According to his theory of manifolds, one could operate freely within a manifold with imaginary concepts and be sure that what one deduced was correct when the axiomatic system completely and unequivocally determined the body of all the configurations possible in a domain by a purely analytical procedure. It was the completeness of the axiomatic system that gave one the right to operate in that free way. A domain was complete when each grammatically constructed proposition exclusively using the language of the domain was determined from the outset to be true or false in virtue of the axioms, i.e., necessarily followed from the axioms or did not. In that case, calculating with expressions without reference could never lead to contradictions. Complete manifolds have the

distinctive feature that a finite number of concepts and propositions – to be drawn as occasion requires from the essential nature of the domain under consideration –  determines completely and unambiguously on the lines of pure logical necessity the totality of all possible formations in the domain, so that in principle, therefore, nothing further remains open within it.

In such complete manifolds, he stressed, “the concepts true and formal implication of the axioms are equivalent (Ideas).

Husserl pointed out that there may be two valid discipline forms that stand in relation to one another in such a way that the axiom system of one may be a formal limitation of that of the other. It is then clear that everything deducible in the narrower axiom system is included in what is deducible in the expanded system, he explained. In the arithmetic of cardinal numbers, Husserl explained, there are no negative numbers, for the meaning of the axioms is so restrictive as to make subtracting 4 from 3 nonsense. Fractions are meaningless there. So are irrational numbers, √–1, and so on. Yet in practice, all the calculations of the arithmetic of cardinal numbers can be carried out as if the rules governing the operations are unrestrictedly valid and meaningful. One can disregard the limitations imposed in a narrower domain of deduction and act as if the axiom system were a more extended one. We cannot arbitrarily expand the concept of cardinal number, Husserl reasoned. But we can abandon it and define a new, pure formal concept of positive whole number with the formal system of definitions and operations valid for cardinal numbers. And, as set out in our definition, this formal concept of positive numbers can be expanded by new definitions while remaining free of contradiction. Fractions do not acquire any genuine meaning through our holding onto the concept of cardinal number and assuming that units are divisible, he theorized, but rather through our abandonment of the concept of cardinal number and our reliance on a new concept, that of divisible quantities. That leads to a system that partially coincides with that of cardinal numbers, but part of which is larger, meaning that it includes additional basic elements and axioms. And so in this way, with each new quantity, one also changes arithmetics. The different arithmetics do not have parts in common. They have totally different domains, but an analogous structure. They have forms of operation that are in part alike, but different concepts of operation.

For Husserl, formal constraints banning meaningless expressions, meaningless imaginary concepts, reference to non-existent and impossible objects restrict us in our theoretical, deductive work, but that resorting to the infinity of pure forms and transformations of forms frees us from such conditions and explains why having used imaginaries, what is meaningless, must lead, not to meaningless, but to true results.

Metaphysical Would-Be(s). Drunken Risibility.

If one were to look at Quine’s commitment to similarity, natural kinds, dispositions, causal statements, etc., it is evident, that it takes him close to Peirce’s conception of Thirdness – even if Quine in an utopian vision imagines that all such concepts in a remote future will dissolve and vanish in favor of purely microstructural descriptions.

A crucial difference remains, however, which becomes evident when one looks at Quine’s brief formula for ontological commitment, the famous idea that ‘to be is to be the value of a bound variable’. For even if this motto is stated exactly to avoid commitment to several different types of being, it immediately prompts the question: the equation, in which the variable is presumably bound, which status does it have? Governing the behavior of existing variable values, is that not in some sense being real?

This will be Peirce’s realist idea – that regularities, tendencies, dispositions, patterns, may possess real existence, independent of any observer. In Peirce, this description of Thirdness is concentrated in the expression ‘real possibility’, and even it may sound exceedingly metaphysical at a first glance, it amounts, at a closer look, to regularities charted by science that are not mere shorthands for collections of single events but do possess reality status. In Peirce, the idea of real possibilities thus springs from his philosophy of science – he observes that science, contrary to philosophy, is spontaneously realist, and is right in being so. Real possibilities are thus counterposed to mere subjective possibilities due to lack of knowledge on the part of the subject speaking: the possibility of ‘not known not to be true’.

In a famous piece of self-critique from his late, realist period, Peirce attacks his earlier arguments (from ‘How to Make Our Ideas Clear’, in the late 1890s considered by himself the birth certificate of pragmatism after James’s reference to Peirce as pragmatism’s inventor). Then, he wrote

let us ask what we mean by calling a thing hard. Evidently that it will not be scratched by many other substances. The whole conception of this quality, as of every other, lies in its conceived effects. There is absolutely no difference between a hard thing and a soft thing so long as they are not brought to the test. Suppose, then, that a diamond could be crystallized in the midst of a cushion of soft cotton, and should remain there until it was finally burned up. Would it be false to say that that diamond was soft? […] Reflection will show that the reply is this: there would be no falsity in such modes of speech.

More than twenty-five years later, however, he attacks this argument as bearing witness to the nominalism of his youth. Now instead he supports the

scholastic doctrine of realism. This is usually defined as the opinion that there are real objects that are general, among the number being the modes of determination of existent singulars, if, indeed, these be not the only such objects. But the belief in this can hardly escape being accompanied by the acknowledgment that there are, besides, real vagues, and especially real possibilities. For possibility being the denial of a necessity, which is a kind of generality, is vague like any other contradiction of a general. Indeed, it is the reality of some possibilities that pragmaticism is most concerned to insist upon. The article of January 1878 endeavored to gloze over this point as unsuited to the exoteric public addressed; or perhaps the writer wavered in his own mind. He said that if a diamond were to be formed in a bed of cotton-wool, and were to be consumed there without ever having been pressed upon by any hard edge or point, it would be merely a question of nomenclature whether that diamond should be said to have been hard or not. No doubt this is true, except for the abominable falsehood in the word MERELY, implying that symbols are unreal. Nomenclature involves classification; and classification is true or false, and the generals to which it refers are either reals in the one case, or figments in the other. For if the reader will turn to the original maxim of pragmaticism at the beginning of this article, he will see that the question is, not what did happen, but whether it would have been well to engage in any line of conduct whose successful issue depended upon whether that diamond would resist an attempt to scratch it, or whether all other logical means of determining how it ought to be classed would lead to the conclusion which, to quote the very words of that article, would be ‘the belief which alone could be the result of investigation carried sufficiently far.’ Pragmaticism makes the ultimate intellectual purport of what you please to consist in conceived conditional resolutions, or their substance; and therefore, the conditional propositions, with their hypothetical antecedents, in which such resolutions consist, being of the ultimate nature of meaning, must be capable of being true, that is, of expressing whatever there be which is such as the proposition expresses, independently of being thought to be so in any judgment, or being represented to be so in any other symbol of any man or men. But that amounts to saying that possibility is sometimes of a real kind. (The Essential Peirce Selected Philosophical Writings, Volume 2)

In the same year, he states, in a letter to the Italian pragmatist Signor Calderoni:

I myself went too far in the direction of nominalism when I said that it was a mere question of the convenience of speech whether we say that a diamond is hard when it is not pressed upon, or whether we say that it is soft until it is pressed upon. I now say that experiment will prove that the diamond is hard, as a positive fact. That is, it is a real fact that it would resist pressure, which amounts to extreme scholastic realism. I deny that pragmaticism as originally defined by me made the intellectual purport of symbols to consist in our conduct. On the contrary, I was most careful to say that it consists in our concept of what our conduct would be upon conceivable occasions. For I had long before declared that absolute individuals were entia rationis, and not realities. A concept determinate in all respects is as fictitious as a concept definite in all respects. I do not think we can ever have a logical right to infer, even as probable, the existence of anything entirely contrary in its nature to all that we can experience or imagine. 

Here lies the core of Peirce’s metaphysical insistence on the reality of ‘would-be’s. Real possibilities, or would-bes, are vague to the extent that they describe certain tendential, conditional behaviors only, while they do not prescribe any other aspect of the single objects they subsume. They are, furthermore, represented in rationally interrelated clusters of concepts: the fact that the diamond is in fact hard, no matter if it scratches anything or not, lies in the fact that the diamond’s carbon structure displays a certain spatial arrangement – so it is an aspect of the very concept of diamond. And this is why the old pragmatic maxim may not work without real possibilities: it is they that the very maxim rests upon, because it is they that provide us with the ‘conceived consequences’ of accepting a concept. The maxim remains a test to weed out empty concepts with no conceived consequences – that is, empty a priori reasoning and superfluous metaphysical assumptions. But what remains after the maxim has been put to use, is real possibilities. Real possibilities thus connect epistemology, expressed in the pragmatic maxim, to ontology: real possibilities are what science may grasp in conditional hypotheses.

The question is whether Peirce’s revision of his old ‘nominalist’ beliefs form part of a more general development in Peirce from nominalism to realism. The locus classicus of this idea is Max Fisch (Peirce, Semeiotic and Pragmatism) where Fisch outlines a development from an initial nominalism (albeit of a strange kind, refusing, as always in Peirce, the existence of individuals determinate in all respects) via a series of steps towards realism, culminating after the turn of the century. Fisch’s first step is then Peirce’s theory of the real as that which reasoning would finally have as its result; the second step his Berkeley review with its anti-nominalism and the idea that the real is what is unaffected by what we may think of it; the third step is his pragmatist idea that beliefs are conceived habits of action, even if he here clings to the idea that the conditionals in which habits are expressed are material implications only – like the definition of ‘hard’; the fourth step his reading of Abbott’s realist Scientific Theism (which later influenced his conception of scientific universals) and his introduction of the index in his theory of signs; the fifth step his acceptance of the reality of continuity; the sixth the introduction of real possibilities, accompanied by the development of existential graphs, topology and Peirce’s changing view of Hegelianism; the seventh, the identification of pragmatism with realism; the eighth ‘his last stronghold, that of Philonian or material implication’. A further realist development exchanging Peirce’s early frequentist idea of probability for a dispositional theory of probability was, according to Fisch, never finished.

The issue of implication concerns the old discussion quoted by Cicero between the Hellenistic logicians Philo and Diodorus. The former formulated what we know today as material implication, while the latter objected on common-sense ground that material implication does not capture implication in everyday language and thought and another implication type should be sought. As is well known, material implication says that p ⇒ q is equivalent to the claim that either p is false or q is true – so that p ⇒ q is false only when p is true and q is false. The problems arise when p is false, for any false p makes the implication true, and this leads to strange possibilities of true inferences. The two parts of the implication have no connection with each other at all, such as would be the spontaneous idea in everyday thought. It is true that Peirce as a logician generally supports material (‘Philonian’) implication – but it is also true that he does express some second thoughts at around the same time as the afterthoughts on the diamond example.

Peirce is a forerunner of the attempts to construct alternatives such as strict implication, and the reason why is, of course, that real possibilities are not adequately depicted by material implication. Peirce is in need of an implication which may somehow picture the causal dependency of q on p. The basic reason for the mature Peirce’s problems with the representation of real possibilities is not primarily logical, however. It is scientific. Peirce realizes that the scientific charting of anything but singular, actual events necessitates the real existence of tendencies and relations connecting singular events. Now, what kinds are those tendencies and relations? The hard diamond example seems to emphasize causality, but this probably depends on the point of view chosen. The ‘conceived consequences’ of the pragmatic maxim may be causal indeed: if we accept gravity as a real concept, then masses will attract one another – but they may all the same be structural: if we accept horse riders as a real concept, then we should expect horses, persons, the taming of horses, etc. to exist, or they may be teleological. In any case, the interpretation of the pragmatic maxim in terms of real possibilities paves the way for a distinction between empty a priori suppositions and real a priori structures.

Fibrations of Elliptic Curves in F-Theory.

F-theory compactifications are by definition compactifications of the type IIB string with non-zero, and in general non-constant string coupling – they are thus intrinsically non-perturbative. F-theory may also seen as a construction to geometrize (and thereby making manifest) certain features pertaining to the S-duality of the type IIB string.

Let us first recapitulate the most important massless bosonic fields of the type IIB string. From the NS-NS sector, we have the graviton g_μν, the antisymmetric 2-form field B as well as the dilaton φ; the latter, when exponentiated, serves as the coupling constant of the theory. Moreover, from the R-R sector we have the p-form tensor fields C^(p) with p = 0,2,4. It is also convenient to include the magnetic duals of these fields, B⁽⁶⁾, C⁽⁶⁾ and C⁽⁸⁾ (C⁽⁴⁾ has self-dual field strength). It is useful to combine the dilaton with the axion into one complex field:

τ_IIB ≡ C⁽⁰⁾ + ie^-φ —– (1)

The S-duality then acts via projective SL(2, Z) transformations in the canonical manner:

τ_IIB → (aτ_IIB + b)/(cτ_IIB + d) with a, b, c, d ∈ Z and ad – bc = 1

Furthermore, it acts via simple matrix multiplication on the other fields if these are grouped into doublets (^B(2)_C⁽²⁾), (^B(6)_C⁽⁴⁾), while C⁽⁴⁾ stays invariant.

The simplest F-theory compactifications are the highest dimensional ones, and simplest of all is the compactification of the type IIB string on the 2-sphere, P¹. However, as the first Chern class does not vanish: C₁(P¹) = – 2, this by itself cannot be a good, supersymmetry preserving background. The remedy is to add extra 7-branes to the theory, which sit at arbitrary points z_i on the P¹, and otherwise fill the 7+1 non-compact space-time dimensions. If this is done in the right way, C₁(P¹) is cancelled, thereby providing a consistent background.

Encircling the location of a 7-brane in the z-plane leads to a jump of the perceived type IIB string coupling, τ_IIB →τ_IIB  +1.

To explain how this works, consider first a single D7-brane located at an arbitrary given point z₀ on the P¹. A D7-brane carries by definition one unit of D7-brane charge, since it is a unit source of C⁽⁸⁾. This means that is it magnetically charged with respect to the dual field C⁽⁰⁾, which enters in the complexified type IIB coupling in (1). As a consequence, encircling the plane location z₀ will induce a non-trivial monodromy, that is, a jump on the coupling. But this then implies that in the neighborhood of the D7-brane, we must have a non-constant string coupling of the form: τ_IIB(z) = 1/2πiIn[z – z₀]; we thus indeed have a truly non-perturbative situation.

In view of the SL(2, Z) action on the string coupling (1), it is natural to interpret it as a modular parameter of a two-torus, T², and this is what then gives a geometrical meaning to the S-duality group. This modular parameter τ_IIB = τ_IIB(Z) is not constant over the P¹ compactification manifold, the shape of the T² will accordingly vary along P¹. The relevant geometrical object will therefore not be the direct product manifold T² x P¹, but rather a fibration of T² over P¹

Fibration of an elliptic curve over P¹, which in total makes a K3 surface.

The logarithmic behavior of τ_IIB(z) in the vicinity of a 7-brane means that the T² fiber is singular at the brane location. It is known from mathematics that each of such singular fibers contributes 1/12 to the first Chern class. Therefore we need to put 24 of them in order to have a consistent type IIB background with C₁ = 0. The mathematical data: “T² fibered over P¹ with 24 singular fibers” is now exactly what characterizes the K3 surface; indeed it is the only complex two-dimensional manifold with vanishing first Chern class (apart from T⁴).

The K3 manifold that arises in this context is so far just a formal construct, introduced to encode of the behavior of the string coupling in the presence of 7-branes in an elegant and useful way. One may speculate about a possible more concrete physical significance, such as a compactification manifold of a yet unknown 12 dimensional “F-theory”. The existence of such a theory is still unclear, but all we need the K3 for is to use its intriguing geometric properties for computing physical quantities (the quartic gauge threshold couplings, ultimately).

In order to do explicit computations, we first of all need a concrete representation of the K3 surface. Since the families of K3’s in question are elliptically fibered, the natural starting point is the two-torus T². It can be represented in the well-known “Weierstraβ” form:

W_T² = y² + x³ + xf + g = 0 —– (2)

which in turn is invariantly characterized by the J-function:

J = 4(24f)³/(4f³ + 27g²) —– (3)

An elliptically fibered K3 surface can be made out of (2) by letting f → f₈(z) and g → g₁₂(z) become polynomials in the P¹ coordinate z, of the indicated orders. The locations z_i of the 7-branes, which correspond to the locations of the singular fibers where J(τ_IIB(z_i)) → ∞, are then precisely where the discriminant

∆(z) ≡ 4f₈³(z) + 27g₁₂²(z)

=: ∏_i=1²⁴(z –  z_i) vanishes.

The Coming Swarm DDoS Actions, Hacktivism, and Civil Disobedience on the Internet

On November 28, 2010, Wikileaks, along with the New York Times, Der Spiegel, El Pais, Le Monde, and The Guardian began releasing documents from a leaked cache of 2,51,287 unclassified and classified US diplomatic cables, copied from the closed Department of Defense network SIPrnet. The US government was furious. In the days that followed, different organizations and corporations began distancing themselves from Wikileaks. Amazon WebServices declined to continue hosting Wikileaks’ website, and on December 1, removed its content from its servers. The next day, the public could no longer reach the Wikileaks website at wikileaks.org; Wikileaks’ Domain Name System (DNS) provider, EveryDNS, had dropped the site from its entries on December 2, temporarily making the site inaccessible through its URL. Shortly thereafter, what would be known as the “Banking Blockade” began, with PayPal, PostFinance, Mastercard, Visa, and Bank of America refusing to process online donations to Wikileaks, essentially halting the flow of monetary donations to the organization.

Wikileaks’ troubles attracted the attention of anonymous, a loose group of internet denizens, and in particular, a small subgroup known as AnonOps, who had been engaged in a retaliatory distributed denial of service (DDoS) campaign called Operation Payback, targeting the Motion Picture Association of America and other pro-copyright, antipiracy groups since September 2010. A DDoS action is, simply, when a large number of computers attempt to access one website over and over again in a short amount of time, in the hopes of overwhelming the server, rendering it incapable of responding to legitimate requests. Anons, as members of the anonymous subculture are known, were happy to extend Operation Payback’s range of targets to include the forces arrayed against Wikileaks and its public face, Julian Assange. On December 6, they launched their first DDoS action against the website of the Swiss banking service, PostFinance. Over the course of the next 4 days, anonymous and AnonOps would launch DDoS actions against the websites of the Swedish Prosecution Authority, EveryDNS, Senator Joseph Lieberman, Mastercard, two Swedish politicians, Visa, PayPal, and amazon.com, and others, forcing many of the sites to experience at least some amount of downtime.

For many in the media and public at large, Anonymous’ December 2010 DDoS campaign was their first exposure to the use of this tactic by activists, and the exact nature of the action was unclear. Was it an activist action, a legitimate act of protest, an act of terrorism, or a criminal act? These DDoS actions – concerted efforts by many individuals to bring down websites by making repeated requests of the websites’ servers in a short amount of time – were covered extensively by the media. This coverage was inconsistent in its characterization but was open to the idea that these actions could be legitimately political in nature. In the eyes of the media and public, Operation Payback opened the door to the potential for civil disobedience and disruptive activism on the internet. But Operation Payback was far from the first use of DDoS as a tool of activism. Rather, DDoS actions have been in use for over two decades, in support of activist campaigns ranging from pro-Zapatistas actions to protests against German immigration policy and trademark enforcement disputes….

The Coming Swarm DDOS Actions, Hacktivism, and Civil Disobedience on the Internet

Knowledge Limited for Dummies….Didactics.

Bertrand Russell with Alfred North Whitehead, in the Principia Mathematica aimed to demonstrate that “all pure mathematics follows from purely logical premises and uses only concepts defined in logical terms.” Its goal was to provide a formalized logic for all mathematics, to develop the full structure of mathematics where every premise could be proved from a clear set of initial axioms.

Russell observed of the dense and demanding work, “I used to know of only six people who had read the later parts of the book. Three of those were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated.” The complex mathematical symbols of the manuscript required it to be written by hand, and its sheer size – when it was finally ready for the publisher, Russell had to hire a panel truck to send it off – made it impossible to copy. Russell recounted that “every time that I went out for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up.”

Momentous though it was, the greatest achievement of Principia Mathematica was realized two decades after its completion when it provided the fodder for the metamathematical enterprises of an Austrian, Kurt Gödel. Although Gödel did face the risk of being liquidated by Hitler (therefore fleeing to the Institute of Advanced Studies at Princeton), he was neither a Pole nor a Texan. In 1931, he wrote a treatise entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which demonstrated that the goal Russell and Whitehead had so single-mindedly pursued was unattainable.

The flavor of Gödel’s basic argument can be captured in the contradictions contained in a schoolboy’s brainteaser. A sheet of paper has the words “The statement on the other side of this paper is true” written on one side and “The statement on the other side of this paper is false” on the reverse. The conflict isn’t resolvable. Or, even more trivially, a statement like; “This statement is unprovable.” You cannot prove the statement is true, because doing so would contradict it. If you prove the statement is false, then that means its converse is true – it is provable – which again is a contradiction.

The key point of contradiction for these two examples is that they are self-referential. This same sort of self-referentiality is the keystone of Gödel’s proof, where he uses statements that imbed other statements within them. This problem did not totally escape Russell and Whitehead. By the end of 1901, Russell had completed the first round of writing Principia Mathematica and thought he was in the homestretch, but was increasingly beset by these sorts of apparently simple-minded contradictions falling in the path of his goal. He wrote that “it seemed unworthy of a grown man to spend his time on such trivialities, but . . . trivial or not, the matter was a challenge.” Attempts to address the challenge extended the development of Principia Mathematica by nearly a decade.

Yet Russell and Whitehead had, after all that effort, missed the central point. Like granite outcroppings piercing through a bed of moss, these apparently trivial contradictions were rooted in the core of mathematics and logic, and were only the most readily manifest examples of a limit to our ability to structure formal mathematical systems. Just four years before Gödel had defined the limits of our ability to conquer the intellectual world of mathematics and logic with the publication of his Undecidability Theorem, the German physicist Werner Heisenberg’s celebrated Uncertainty Principle had delineated the limits of inquiry into the physical world, thereby undoing the efforts of another celebrated intellect, the great mathematician Pierre-Simon Laplace. In the early 1800s Laplace had worked extensively to demonstrate the purely mechanical and predictable nature of planetary motion. He later extended this theory to the interaction of molecules. In the Laplacean view, molecules are just as subject to the laws of physical mechanics as the planets are. In theory, if we knew the position and velocity of each molecule, we could trace its path as it interacted with other molecules, and trace the course of the physical universe at the most fundamental level. Laplace envisioned a world of ever more precise prediction, where the laws of physical mechanics would be able to forecast nature in increasing detail and ever further into the future, a world where “the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule.”

What Gödel did to the work of Russell and Whitehead, Heisenberg did to Laplace’s concept of causality. The Uncertainty Principle, though broadly applied and draped in metaphysical context, is a well-defined and elegantly simple statement of physical reality – namely, the combined accuracy of a measurement of an electron’s location and its momentum cannot vary far from a fixed value. The reason for this, viewed from the standpoint of classical physics, is that accurately measuring the position of an electron requires illuminating the electron with light of a very short wavelength. But the shorter the wavelength the greater the amount of energy that hits the electron, and the greater the energy hitting the electron the greater the impact on its velocity.

What is true in the subatomic sphere ends up being true – though with rapidly diminishing significance – for the macroscopic. Nothing can be measured with complete precision as to both location and velocity because the act of measuring alters the physical properties. The idea that if we know the present we can calculate the future was proven invalid – not because of a shortcoming in our knowledge of mechanics, but because the premise that we can perfectly know the present was proven wrong. These limits to measurement imply limits to prediction. After all, if we cannot know even the present with complete certainty, we cannot unfailingly predict the future. It was with this in mind that Heisenberg, ecstatic about his yet-to-be-published paper, exclaimed, “I think I have refuted the law of causality.”

The epistemological extrapolation of Heisenberg’s work was that the root of the problem was man – or, more precisely, man’s examination of nature, which inevitably impacts the natural phenomena under examination so that the phenomena cannot be objectively understood. Heisenberg’s principle was not something that was inherent in nature; it came from man’s examination of nature, from man becoming part of the experiment. (So in a way the Uncertainty Principle, like Gödel’s Undecidability Proposition, rested on self-referentiality.) While it did not directly refute Einstein’s assertion against the statistical nature of the predictions of quantum mechanics that “God does not play dice with the universe,” it did show that if there were a law of causality in nature, no one but God would ever be able to apply it. The implications of Heisenberg’s Uncertainty Principle were recognized immediately, and it became a simple metaphor reaching beyond quantum mechanics to the broader world.

This metaphor extends neatly into the world of financial markets. In the purely mechanistic universe of classical physics, we could apply Newtonian laws to project the future course of nature, if only we knew the location and velocity of every particle. In the world of finance, the elementary particles are the financial assets. In a purely mechanistic financial world, if we knew the position each investor has in each asset and the ability and willingness of liquidity providers to take on those assets in the event of a forced liquidation, we would be able to understand the market’s vulnerability. We would have an early-warning system for crises. We would know which firms are subject to a liquidity cycle, and which events might trigger that cycle. We would know which markets are being overrun by speculative traders, and thereby anticipate tactical correlations and shifts in the financial habitat. The randomness of nature and economic cycles might remain beyond our grasp, but the primary cause of market crisis, and the part of market crisis that is of our own making, would be firmly in hand.

The first step toward the Laplacean goal of complete knowledge is the advocacy by certain financial market regulators to increase the transparency of positions. Politically, that would be a difficult sell – as would any kind of increase in regulatory control. Practically, it wouldn’t work. Just as the atomic world turned out to be more complex than Laplace conceived, the financial world may be similarly complex and not reducible to a simple causality. The problems with position disclosure are many. Some financial instruments are complex and difficult to price, so it is impossible to measure precisely the risk exposure. Similarly, in hedge positions a slight error in the transmission of one part, or asynchronous pricing of the various legs of the strategy, will grossly misstate the total exposure. Indeed, the problems and inaccuracies in using position information to assess risk are exemplified by the fact that major investment banking firms choose to use summary statistics rather than position-by-position analysis for their firmwide risk management despite having enormous resources and computational power at their disposal.

Perhaps more importantly, position transparency also has implications for the efficient functioning of the financial markets beyond the practical problems involved in its implementation. The problems in the examination of elementary particles in the financial world are the same as in the physical world: Beyond the inherent randomness and complexity of the systems, there are simply limits to what we can know. To say that we do not know something is as much a challenge as it is a statement of the state of our knowledge. If we do not know something, that presumes that either it is not worth knowing or it is something that will be studied and eventually revealed. It is the hubris of man that all things are discoverable. But for all the progress that has been made, perhaps even more exciting than the rolling back of the boundaries of our knowledge is the identification of realms that can never be explored. A sign in Einstein’s Princeton office read, “Not everything that counts can be counted, and not everything that can be counted counts.”

The behavioral analogue to the Uncertainty Principle is obvious. There are many psychological inhibitions that lead people to behave differently when they are observed than when they are not. For traders it is a simple matter of dollars and cents that will lead them to behave differently when their trades are open to scrutiny. Beneficial though it may be for the liquidity demander and the investor, for the liquidity supplier trans- parency is bad. The liquidity supplier does not intend to hold the position for a long time, like the typical liquidity demander might. Like a market maker, the liquidity supplier will come back to the market to sell off the position – ideally when there is another investor who needs liquidity on the other side of the market. If other traders know the liquidity supplier’s positions, they will logically infer that there is a good likelihood these positions shortly will be put into the market. The other traders will be loath to be the first ones on the other side of these trades, or will demand more of a price concession if they do trade, knowing the overhang that remains in the market.

This means that increased transparency will reduce the amount of liquidity provided for any given change in prices. This is by no means a hypothetical argument. Frequently, even in the most liquid markets, broker-dealer market makers (liquidity providers) use brokers to enter their market bids rather than entering the market directly in order to preserve their anonymity.

The more information we extract to divine the behavior of traders and the resulting implications for the markets, the more the traders will alter their behavior. The paradox is that to understand and anticipate market crises, we must know positions, but knowing and acting on positions will itself generate a feedback into the market. This feedback often will reduce liquidity, making our observations less valuable and possibly contributing to a market crisis. Or, in rare instances, the observer/feedback loop could be manipulated to amass fortunes.

One might argue that the physical limits of knowledge asserted by Heisenberg’s Uncertainty Principle are critical for subatomic physics, but perhaps they are really just a curiosity for those dwelling in the macroscopic realm of the financial markets. We cannot measure an electron precisely, but certainly we still can “kind of know” the present, and if so, then we should be able to “pretty much” predict the future. Causality might be approximate, but if we can get it right to within a few wavelengths of light, that still ought to do the trick. The mathematical system may be demonstrably incomplete, and the world might not be pinned down on the fringes, but for all practical purposes the world can be known.

Unfortunately, while “almost” might work for horseshoes and hand grenades, 30 years after Gödel and Heisenberg yet a third limitation of our knowledge was in the wings, a limitation that would close the door on any attempt to block out the implications of microscopic uncertainty on predictability in our macroscopic world. Based on observations made by Edward Lorenz in the early 1960s and popularized by the so-called butterfly effect – the fanciful notion that the beating wings of a butterfly could change the predictions of an otherwise perfect weather forecasting system – this limitation arises because in some important cases immeasurably small errors can compound over time to limit prediction in the larger scale. Half a century after the limits of measurement and thus of physical knowledge were demonstrated by Heisenberg in the world of quantum mechanics, Lorenz piled on a result that showed how microscopic errors could propagate to have a stultifying impact in nonlinear dynamic systems. This limitation could come into the forefront only with the dawning of the computer age, because it is manifested in the subtle errors of computational accuracy.

The essence of the butterfly effect is that small perturbations can have large repercussions in massive, random forces such as weather. Edward Lorenz was testing and tweaking a model of weather dynamics on a rudimentary vacuum-tube computer. The program was based on a small system of simultaneous equations, but seemed to provide an inkling into the variability of weather patterns. At one point in his work, Lorenz decided to examine in more detail one of the solutions he had generated. To save time, rather than starting the run over from the beginning, he picked some intermediate conditions that had been printed out by the computer and used those as the new starting point. The values he typed in were the same as the values held in the original simulation at that point, so the results the simulation generated from that point forward should have been the same as in the original; after all, the computer was doing exactly the same operations. What he found was that as the simulated weather pattern progressed, the results of the new run diverged, first very slightly and then more and more markedly, from those of the first run. After a point, the new path followed a course that appeared totally unrelated to the original one, even though they had started at the same place.

Lorenz at first thought there was a computer glitch, but as he investigated further, he discovered the basis of a limit to knowledge that rivaled that of Heisenberg and Gödel. The problem was that the numbers he had used to restart the simulation had been reentered based on his printout from the earlier run, and the printout rounded the values to three decimal places while the computer carried the values to six decimal places. This rounding, clearly insignificant at first, promulgated a slight error in the next-round results, and this error grew with each new iteration of the program as it moved the simulation of the weather forward in time. The error doubled every four simulated days, so that after a few months the solutions were going their own separate ways. The slightest of changes in the initial conditions had traced out a wholly different pattern of weather.

Intrigued by his chance observation, Lorenz wrote an article entitled “Deterministic Nonperiodic Flow,” which stated that “nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states.” Translation: Long-range weather forecasting is worthless. For his application in the narrow scientific discipline of weather prediction, this meant that no matter how precise the starting measurements of weather conditions, there was a limit after which the residual imprecision would lead to unpredictable results, so that “long-range forecasting of specific weather conditions would be impossible.” And since this occurred in a very simple laboratory model of weather dynamics, it could only be worse in the more complex equations that would be needed to properly reflect the weather. Lorenz discovered the principle that would emerge over time into the field of chaos theory, where a deterministic system generated with simple nonlinear dynamics unravels into an unrepeated and apparently random path.

The simplicity of the dynamic system Lorenz had used suggests a far-reaching result: Because we cannot measure without some error (harking back to Heisenberg), for many dynamic systems our forecast errors will grow to the point that even an approximation will be out of our hands. We can run a purely mechanistic system that is designed with well-defined and apparently well-behaved equations, and it will move over time in ways that cannot be predicted and, indeed, that appear to be random. This gets us to Santa Fe.

The principal conceptual thread running through the Santa Fe research asks how apparently simple systems, like that discovered by Lorenz, can produce rich and complex results. Its method of analysis in some respects runs in the opposite direction of the usual path of scientific inquiry. Rather than taking the complexity of the world and distilling simplifying truths from it, the Santa Fe Institute builds a virtual world governed by simple equations that when unleashed explode into results that generate unexpected levels of complexity.

In economics and finance, institute’s agenda was to create artificial markets with traders and investors who followed simple and reasonable rules of behavior and to see what would happen. Some of the traders built into the model were trend followers, others bought or sold based on the difference between the market price and perceived value, and yet others traded at random times in response to liquidity needs. The simulations then printed out the paths of prices for the various market instruments. Qualitatively, these paths displayed all the richness and variation we observe in actual markets, replete with occasional bubbles and crashes. The exercises did not produce positive results for predicting or explaining market behavior, but they did illustrate that it is not hard to create a market that looks on the surface an awful lot like a real one, and to do so with actors who are following very simple rules. The mantra is that simple systems can give rise to complex, even unpredictable dynamics, an interesting converse to the point that much of the complexity of our world can – with suitable assumptions – be made to appear simple, summarized with concise physical laws and equations.

The systems explored by Lorenz were deterministic. They were governed definitively and exclusively by a set of equations where the value in every period could be unambiguously and precisely determined based on the values of the previous period. And the systems were not very complex. By contrast, whatever the set of equations are that might be divined to govern the financial world, they are not simple and, furthermore, they are not deterministic. There are random shocks from political and economic events and from the shifting preferences and attitudes of the actors. If we cannot hope to know the course of the deterministic systems like fluid mechanics, then no level of detail will allow us to forecast the long-term course of the financial world, buffeted as it is by the vagaries of the economy and the whims of psychology.