# Malicious Machine Learnings? Privacy Preservation and Computational Correctness Across Parties. Note Quote/Didactics.

Multi-Party Computation deals with the following problem: There are n ≥ 2 parties P1, . . ., Pn where party Pi holds input ti, 1 ≤ i ≤ n, and they wish to compute together a functions = f (t1, . . . , tn) on their inputs. The goal is that each party will learn the output of the function, s, yet with the restriction that Pi will not learn any additional information about the input of the other parties aside from what can be deduced from the pair (ti, s). Clearly it is the secrecy restriction that adds complexity to the problem, as without it each party could announce its input to all other parties, and each party would locally compute the value of the function. Thus, the goal of Multi-Party Computation is to achieve the following two properties at the same time: correctness of the computation and privacy preservation of the inputs.

The following two generalizations are often useful:

(i) Probabilistic functions. Here the value of the function depends on some random string r chosen according to some distribution: s = f (t1, . . . , tn; r). An example of this is the coin-flipping functionality, which takes no inputs, and outputs an unbiased random bit. It is crucial that the value r is not controlled by any of the parties, but is somehow jointly generated during the computation.

(ii) Multioutput functions. It is not mandatory that there be a single output of the function. More generally there could be a unique output for each party, i.e., (s1, . . . , sn) = f(t1,…, tn). In this case, only party Pi learns the output si, and no other party learns any information about the other parties’ input and outputs aside from what can be derived from its own input and output.

One of the most interesting aspects of Multi-Party Computation is to reach the objective of computing the function value, but under the assumption that some of the parties may deviate from the protocol. In cryptography, the parties are usually divided into two types: honest and faulty. An honest party follows the protocol without any deviation. Otherwise, the party is considered to be faulty. The faulty behavior can exemplify itself in a wide range of possibilities. The most benign faulty behavior is where the parties follow the protocol, yet try to learn as much as possible about the inputs of the other parties. These parties are called honest-but-curious (or semihonest). At the other end of the spectrum, the parties may deviate from the prescribed protocol in any way that they desire, with the goal of either influencing the computed output value in some way, or of learning as much as possible about the inputs of the other parties. These parties are called malicious.

We envision an adversary A, who controls all the faulty parties and can coordinate their actions. Thus, in a sense we assume that the faulty parties are working together and can exert the most knowledge and influence over the computation out of this collusion. The adversary can corrupt any number of parties out of the n participating parties. Yet, in order to be able to achieve a solution to the problem, in many cases we would need to limit the number of corrupted parties. This limit is called the threshold k, indicating that the protocol remains secure as long as the number of corrupted parties is at most k.

Assume that there exists a trusted party who privately receives the inputs of all the participating parties, calculates the output value s, and then transmits this value to each one of the parties. This process clearly computes the correct output of f, and also does not enable the participating parties to learn any additional information about the inputs of others. We call this model the ideal model. The security of Multi-Party Computation then states that a protocol is secure if its execution satisfies the following: (1) the honest parties compute the same (correct) outputs as they would in the ideal model; and (2) the protocol does not expose more information than a comparable execution with the trusted party, in the ideal model.

Intuitively, the adversary’s interaction with the parties (on a vector of inputs) in the protocol generates a transcript. This transcript is a random variable that includes the outputs of all the honest parties, which is needed to ensure correctness, and the output of the adversary A. The latter output, without loss of generality, includes all the information that the adversary learned, including its inputs, private state, all the messages sent by the honest parties to A, and, depending on the model, maybe even include more information, such as public messages that the honest parties exchanged. If we show that exactly the same transcript distribution can be generated when interacting with the trusted party in the ideal model, then we are guaranteed that no information is leaked from the computation via the execution of the protocol, as we know that the ideal process does not expose any information about the inputs. More formally,

Let f be a function on n inputs and let π be a protocol that computes the function f. Given an adversary A, which controls some set of parties, we define REALA,π(t) to be the sequence of outputs of honest parties resulting from the execution of π on input vector t under the attack of A, in addition to the output of A. Similarly, given an adversary A′ which controls a set of parties, we define IDEALA′,f(t) to be the sequence of outputs of honest parties computed by the trusted party in the ideal model on input vector t, in addition to the output of A′. We say that π securely computes f if, for every adversary A as above, ∃ an adversary A′, which controls the same parties in the ideal model, such that, on any input vector t, we have that the distribution of REALA,π(t) is “indistinguishable” from the distribution of IDEALA′,f(t).

Intuitively, the task of the ideal adversary A′ is to generate (almost) the same output as A generates in the real execution or the real model. Thus, the attacker A′ is often called the simulator of A. The transcript value generated in the ideal model, IDEALA′,f(t), also includes the outputs of the honest parties (even though we do not give these outputs to A′), which we know were correctly computed by the trusted party. Thus, the real transcript REALA,π(t) should also include correct outputs of the honest parties in the real model.

We assumed that every party Pi has an input ti, which it enters into the computation. However, if Pi is faulty, nothing stops Pi from changing ti into some ti′. Thus, the notion of a “correct” input is defined only for honest parties. However, the “effective” input of a faulty party Pi could be defined as the value ti′ that the simulator A′ gives to the trusted party in the ideal model. Indeed, since the outputs of honest parties look the same in both models, for all effective purposes Pi must have “contributed” the same input ti′ in the real model.

Another possible misbehavior of Pi, even in the ideal model, might be a refusal to give any input at all to the trusted party. This can be handled in a variety of ways, ranging from aborting the entire computation to simply assigning ti some “default value.” For concreteness, we assume that the domain of f includes a special symbol ⊥ indicating this refusal to give the input, so that it is well defined how f should be computed on such missing inputs. What this requires is that in any real protocol we detect when a party does not enter its input and deal with it exactly in the same manner as if the party would input ⊥ in the ideal model.

As regards security, it is implicitly assumed that all honest parties receive the output of the computation. This is achieved by stating that IDEALA′,f(t) includes the outputs of all honest parties. We therefore say that our currency guarantees output delivery. A more relaxed property than output delivery is fairness. If fairness is achieved, then this means that if at least one (even faulty) party learns its outputs, then all (honest) parties eventually do too. A bit more formally, we allow the ideal model adversary A′ to instruct the trusted party not to compute any of the outputs. In this case, in the ideal model either all the parties learn the output, or none do. Since A’s transcript is indistinguishable from A′’s this guarantees that the same fairness guarantee must hold in the real model as well.

A further relaxation of the definition of security is to provide only correctness and privacy. This means that faulty parties can learn their outputs, and prevent the honest parties from learning theirs. Yet, at the same time the protocol will still guarantee that (1) if an honest party receives an output, then this is the correct value, and (2) the privacy of the inputs and outputs of the honest parties is preserved.

The basic security notions are universal and model-independent. However, specific implementations crucially depend on spelling out precisely the model where the computation will be carried out. In particular, the following issues must be specified:

1. The faulty parties could be honest-but-curious or malicious, and there is usually an upper bound k on the number of parties that the adversary can corrupt.
2. Distinguishing between the computational setting and the information theoretic setting, in the latter, the adversary is unlimited in its computing powers. Thus, the term “indistinguishable” is formalized by requiring the two transcript distributions to be either identical (so-called perfect security) or, at least, statistically close in their variation distance (so-called statistical security). On the other hand, in the computational, the power of the adversary (as well as that of the honest parties) is restricted. A bit more precisely, Multi-Party Computation problem is parameterized by the security parameter λ, in which case (a) all the computation and communication shall be done in time polynomial in λ; and (b) the misbehavior strategies of the faulty parties are also restricted to be run in time polynomial in λ. Furthermore, the term “indistinguishability” is formalized by computational indistinguishability: two distribution ensembles {Xλ}λ and {Yλ}λ are said to be computationally indistinguishable, if for any polynomial-time distinguisher D, the quantity ε, defined as |Pr[D(Xλ) = 1] − Pr[D(Yλ) = 1]|, is a “negligible” function of λ. This means that for any j > 0 and all sufficiently large λ, ε eventually becomes smaller than λ − j. This modeling helps us to build secure Multi-Party Computational protocols depending on plausible computational assumptions, such as the hardness of factoring large integers.
3. The two common communication assumptions are the existence of a secure channel and the existence of a broadcast channel. Secure channels assume that every pair of parties Pi and Pj are connected via an authenticated, private channel. A broadcast channel is a channel with the following properties: if a party Pi (honest or faulty) broadcasts a message m, then m is correctly received by all the parties (who are also sure the message came from Pi). In particular, if an honest party receives m, then it knows that every other honest party also received m. A different communication assumption is the existence of envelopes. An envelope guarantees the following properties: a value m can be stored inside the envelope, it will be held without exposure for a given period of time, and then the value m will be revealed without modification. A ballot box is an enhancement of the envelope setting that also provides a random shuffling mechanism of the envelopes. These are, of course, idealized assumptions that allow for a clean description of a protocol, as they separate the communication issues from the computational ones. These idealized assumptions may be realized by a physical mechanisms, but in some settings such mechanisms may not be available. Then it is important to address the question if and under what circumstances we can remove a given communication assumption. For example, we know that the assumption of a secure channel can be substituted with a protocol, but under the introduction of a computational assumption and a public key infrastructure.

# Econophysics: Financial White Noise Switch. Thought of the Day 115.0

What is the cause of large market fluctuation? Some economists blame irrationality behind the fat-tail distribution. Some economists observed that social psychology might create market fad and panic, which can be modeled by collective behavior in statistical mechanics. For example, the bi-modular distribution was discovered from empirical data in option prices. One possible mechanism of polarized behavior is collective action studied in physics and social psychology. Sudden regime switch or phase transition may occur between uni-modular and bi-modular distribution when field parameter changes across some threshold. The Ising model in equilibrium statistical mechanics was borrowed to study social psychology. Its phase transition from uni-modular to bi-modular distribution describes statistical features when a stable society turns into a divided society. The problem of the Ising model is that its key parameter, the social temperature, has no operational definition in social system. A better alternative parameter is the intensity of social interaction in collective action.

A difficult issue in business cycle theory is how to explain the recurrent feature of business cycles that is widely observed from macro and financial indexes. The problem is: business cycles are not strictly periodic and not truly random. Their correlations are not short like random walk and have multiple frequencies that changing over time. Therefore, all kinds of math models are tried in business cycle theory, including deterministic, stochastic, linear and nonlinear models. We outline economic models in terms of their base function, including white noise with short correlations, persistent cycles with long correlations, and color chaos model with erratic amplitude and narrow frequency band like biological clock.

The steady state of probability distribution function in the Ising Model of Collective Behavior with h = 0 (without central propaganda field). a. Uni-modular distribution with low social stress (k = 0). Moderate stable behavior with weak interaction and high social temperature. b. Marginal distribution at the phase transition with medium social stress (k = 2). Behavioral phase transition occurs between stable and unstable society induced by collective behavior. c. Bi-modular distribution with high social stress (k = 2.5). The society splits into two opposing groups under low social temperature and strong social interactions in unstable society.

Deterministic models are used by Keynesian economists for endogenous mechanism of business cycles, such as the case of the accelerator-multiplier model. The stochastic models are used by the Frisch model of noise-driven cycles that attributes external shocks as the driving force of business fluctuations. Since 1980s, the discovery of economic chaos and the application of statistical mechanics provide more advanced models for describing business cycles. Graphically,

The steady state of probability distribution function in socio-psychological model of collective choice. Here, “a” is the independent parameter; “b” is the interaction parameter. a Centered distribution with b < a (denoted by short dashed curve). It happens when independent decision rooted in individualistic orientation overcomes social pressure through mutual communication. b Horizontal flat distribution with b = a (denoted by long dashed line). Marginal case when individualistic orientation balances the social pressure. c Polarized distribution with b > a (denoted by solid line). It occurs when social pressure through mutual communication is stronger than independent judgment.

Numerical 1 autocorrelations from time series generated by random noise and harmonic wave. The solid line is white noise. The broken line is a sine wave with period P = 1.

Linear harmonic cycles with unique frequency are introduced in business cycle theory. The auto-correlations from harmonic cycle and white noise are shown in the above figure. Auto-correlation function from harmonic cycles is a cosine wave. The amplitude of cosine wave is slightly decayed because of limited data points in numerical experiment. Auto-correlations from a random series are an erratic series with rapid decade from one to residual fluctuations in numerical calculation. The auto-regressive (AR) model in discrete time is a combination of white noise term for simulating short-term auto-correlations from empirical data.

The deterministic model of chaos can be classified into white chaos and color chaos. White chaos is generated by nonlinear difference equation in discrete-time, such as one-dimensional logistic map and two-dimensional Henon map. Its autocorrelations and power spectra look like white noise. Its correlation dimension can be less than one. White noise model is simple in mathematical analysis but rarely used in empirical analysis, since it needs intrinsic time unit.

Color chaos is generated by nonlinear differential equations in continuous-time, such as three-dimensional Lorenz model and one-dimensional model with delay-differential model in biology and economics. Its autocorrelations looks like a decayed cosine wave, and its power spectra seem a combination of harmonic cycles and white noise. The correlation dimension is between one and two for 3D differential equations, and varying for delay-differential equation.

History shows the remarkable resilience of a market that experienced a series of wars and crises. The related issue is why the economy can recover from severe damage and out of equilibrium? Mathematically speaking, we may exam the regime stability under parameter change. One major weakness of the linear oscillator model is that the regime of periodic cycle is fragile or marginally stable under changing parameter. Only nonlinear oscillator model is capable of generating resilient cycles within a finite area under changing parameters. The typical example of linear models is the Samuelson model of multiplier-accelerator. Linear stochastic models have similar problem like linear deterministic models. For example, the so-called unit root solution occurs only at the borderline of the unit root. If a small parameter change leads to cross the unit circle, the stochastic solution will fall into damped (inside the unit circle) or explosive (outside the unit circle) solution.

# Stock Hedging Loss and Risk

A stock is supposed to be bought at time zero with price S0, and to be sold at time T with uncertain price ST. In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ, where τ is prior and close to T, and the n available put options are specified by their strike prices Ki (i = 1,2,··· ,n). As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget. The chosen put option is supposed to finish in-the-money at maturity, and the constraint of hedging expenditure is supposed to be binding.

Suppose the market price of the stock is S0 at time zero, the hedge ratio is h, the price of the put option is P0, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S0erT + hP0erT —– (1)

and the market price of the portfolio is

ST + h(K − Sτ)+ er(T − τ) —— (2)

therefore the loss of the portfolio is

L = S0erT + hP0erT − (ST +h(K − Sτ)+ er(T − τ)—– (3)

where x+ = max(x, 0), which is the payoff function of put option at maturity. For a given threshold v, the probability that the amount of loss exceeds v is denoted as

α = Prob{L ≥ v} —– (4)

in other words, v is the Value-at-Risk (VaR) at α percentage level. There are several alternative measures of risk, such as CVaR (Conditional Value-at-Risk), ESF (Expected Shortfall), CTE (Conditional Tail Expectation), and other coherent risk measures.

The mathematical model of stock price is chosen to be a geometric Brownian motion

dSt/St = μdt + σdBt —– (5)

where St is the stock price at time t (0 < t ≤ T), μ and σ are the drift and the volatility of stock price, and Bt is a standard Brownian motion. The solution of the stochastic differential equation is

St = S0 eσBt + (μ − 1/2σ2)t —– (6)

where B0 = 0, and St is lognormally distributed.

For a given threshold of loss v, the probability that the loss exceeds v is

Prob {L ≥ v} = E [I{X≤c1}FY(g(X) − X)] + E [I{X≥c1}FY (c2 − X)] —– (7)

where E[X] is the expectation of random variable X. I{X<c} is the index function of X such that I{X<c} = 1 when {X < c} is true, otherwise I{X<c} = 0. FY(y) is the cumulative distribution function of random variable Y, and

c1 = 1/σ [ln(k/S0) – (μ – 1/2σ2)τ]

g(X) = 1/σ [ln((S0 + hP0)erT − h(K − f(X))er(T − τ) − v)/S0 – (μ – 1/2σ2)T]

f(X) = S0 eσX + (μ−1σ2

c2 = 1/σ [ln((S0 + hP0)erT − v)/S0 – (μ – 1/2σ2)T]

X and Y are both normally distributed, where X ∼ N(0, √τ), Y ∼ N(0, √(T−τ)).

For a specified hedging strategy, Q(v) = Prob {L ≥ v} is a decreasing function of v. The VaR under α level can be obtained from equation

Q(v) = α —– (8)

The expectations can be calculated with Monte Carlo simulation methods, and the optimal hedging strategy which has the smallest VaR can be obtained from (8) by numerical searching methods.

# Potential Synapses. Thought of the Day 52.0

For a neuron to recognize a pattern of activity it requires a set of co-located synapses (typically fifteen to twenty) that connect to a subset of the cells that are active in the pattern to be recognized. Learning to recognize a new pattern is accomplished by the formation of a set of new synapses collocated on a dendritic segment.

Figure: Learning by growing new synapses. Learning in an HTM neuron is modeled by the growth of new synapses from a set of potential synapses. A “permanence” value is assigned to each potential synapse and represents the growth of the synapse. Learning occurs by incrementing or decrementing permanence values. The synapse weight is a binary value set to 1 if the permanence is above a threshold.

Figure shows how we model the formation of new synapses in a simulated Hierarchical Temporal Memory (HTM) neuron. For each dendritic segment we maintain a set of “potential” synapses between the dendritic segment and other cells in the network that could potentially form a synapse with the segment. The number of potential synapses is larger than the number of actual synapses. We assign each potential synapse a scalar value called “permanence” which represents stages of growth of the synapse. A permanence value close to zero represents an axon and dendrite with the potential to form a synapse but that have not commenced growing one. A 1.0 permanence value represents an axon and dendrite with a large fully formed synapse.

The permanence value is incremented and decremented using a Hebbian-like rule. If the permanence value exceeds a threshold, such as 0.3, then the weight of the synapse is 1, if the permanence value is at or below the threshold then the weight of the synapse is 0. The threshold represents the establishment of a synapse, albeit one that could easily disappear. A synapse with a permanence value of 1.0 has the same effect as a synapse with a permanence value at threshold but is not as easily forgotten. Using a scalar permanence value enables on-line learning in the presence of noise. A previously unseen input pattern could be noise or it could be the start of a new trend that will repeat in the future. By growing new synapses, the network can start to learn a new pattern when it is first encountered, but only act differently after several presentations of the new pattern. Increasing permanence beyond the threshold means that patterns experienced more than others will take longer to forget.

HTM neurons and HTM networks rely on distributed patterns of cell activity, thus the activation strength of any one neuron or synapse is not very important. Therefore, in HTM simulations we model neuron activations and synapse weights with binary states. Additionally, it is well known that biological synapses are stochastic, so a neocortical theory cannot require precision of synaptic efficacy. Although scalar states and weights might improve performance, they are not required from a theoretical point of view.

# Without Explosions, WE Would NOT Exist!

The scale factor S(t) obeys the Raychaudhuri equation

where κ is the gravitational constant and Λ the cosmological constant.

, the universe was radiation dominated (μrad ≫ μmat) at early times and matter dominated (μrad ≪ μmat) at late times; matter-radiation density equality occurred significantly before decoupling (the temperature Teq when this equality occurred was Teq ≃ 104K; at that time the scale factor was Seq ≃ 104S0, where S0 is the present-day value). The dynamics of both the background model and of perturbations about that model differ significantly before and after Seq.

Radiation was emitted by matter at the time of decoupling, thereafter travelling freely to us through the intervening space. When it was emitted, it had the form of blackbody radiation, because this is a consequence of matter and radiation being in thermodynamic equilibrium at earlier times. Thus the matter at z = zdec forms the Last Scattering Surface (LSS) in the early universe, emitting Cosmic Blackbody Background Radiation (‘CBR’) at 4000K, that since then has travelled freely with its temperature T scaling inversely with the scale function of the universe. As the radiation travelled towards us, the universe expanded by a factor of about 1100; consequently by the time it reaches us, it has cooled to 2.75 K (that is, about 3 degrees above absolute zero, with a spectrum peaking in the microwave region), and so is extremely hard to observe. It was however detected in 1965, and its spectrum has since been intensively investigated, its blackbody nature being confirmed to high accuracy (R. B. Partridge-3K_ The Cosmic Microwave Background Radiation). Its existence is now taken as solid proof both that the Universe has indeed expanded from a hot early phase, and that standard physics applied unchanged at that era in the early universe.

The thermal capacity of the radiation is hugely greater than that of the matter. At very early times before decoupling, the temperatures of the matter and radiation were the same (because they were in equilibrium with each other), scaling as 1/S(t) (Equation 1 above). The early universe exceeded any temperature that can ever be attained on Earth or even in the centre of the Sun; as it dropped towards its present value of 3 K, successive physical reactions took place that determined the nature of the matter we see around us today. At very early times and high temperatures, only elementary particles can survive and even neutrinos had a very small mean free path; as the universe cooled down, neutrinos decoupled from the matter and streamed freely through space. At these times the expansion of the universe was radiation dominated, and we can approximate the universe then by models with {k = 0, w = 1/3, Λ = 0}, the resulting simple solution of

3S ̇2/S2 = A/S3 + B/S4 + Λ/3 – 3k/S2 —– (3)

uniquely relating time to temperature:

S(t)=S0t1/2 , t=1.92sec [T/1010K]−2 —– (4)

(There are no free constants in the latter equation).

At very early times, even neutrinos were tightly coupled and in equilibrium with the radiation; they decoupled at about 1010K, resulting in a relic neutrino background density in the universe today of about Ων0 ≃ 10−5 if they are massless (but it could be higher depending on their masses). Key events in the early universe are associated with out of equilibrium phenomena. An important event was the era of nucleosynthesis, the time when the light elements were formed. Above about 109K, nuclei could not exist because the radiation was so energetic that as fast as they formed, they were disrupted into their constituent parts (protons and neutrons). However below this temperature, if particles collided with each other with sufficient energy for nuclear reactions to take place, the resultant nuclei remained intact (the radiation being less energetic than their binding energy and hence unable to disrupt them). Thus the nuclei of the light elements  – deuterium, tritium, helium, and lithium – were created by neutron capture. This process ceased when the temperature dropped below about 108K (the nuclear reaction threshold). In this way, the proportions of these light elements at the end of nucleosynthesis were determined; they have remained virtually unchanged since. The rate of reaction was extremely high; all this took place within the first three minutes of the expansion of the Universe. One of the major triumphs of Big Bang theory is that theory and observation are in excellent agreement provided the density of baryons is low: Ωbar0 ≃ 0.044. Then the predicted abundances of these elements (25% Helium by weight, 75% Hydrogen, the others being less than 1%) agrees very closely with the observed abundances. Thus the standard model explains the origin of the light elements in terms of known nuclear reactions taking place in the early Universe. However heavier elements cannot form in the time available (about 3 minutes).

In a similar way, physical processes in the very early Universe (before nucleosynthesis) can be invoked to explain the ratio of matter to anti-matter in the present-day Universe: a small excess of matter over anti-matter must be created then in the process of baryosynthesis, without which we could not exist today (if there were no such excess, matter and antimatter would have all annihilated to give just radiation). However other quantities (such as electric charge) are believed to have been conserved even in the extreme conditions of the early Universe, so their present values result from given initial conditions at the origin of the Universe, rather than from physical processes taking place as it evolved. In the case of electric charge, the total conserved quantity appears to be zero: after quarks form protons and neutrons at the time of baryosynthesis, there are equal numbers of positively charged protons and negatively charged electrons, so that at the time of decoupling there were just enough electrons to combine with the nuclei and form uncharged atoms (it seems there is no net electrical charge on astronomical bodies such as our galaxy; were this not true, electromagnetic forces would dominate cosmology, rather than gravity).

After decoupling, matter formed large scale structures through gravitational instability which eventually led to the formation of the first generation of stars and is probably associated with the re-ionization of matter. However at that time planets could not form for a very important reason: there were no heavy elements present in the Universe. The first stars aggregated matter together by gravitational attraction, the matter heating up as it became more and more concentrated, until its temperature exceeded the thermonuclear ignition point and nuclear reactions started burning hydrogen to form helium. Eventually more complex nuclear reactions started in concentric spheres around the centre, leading to a build-up of heavy elements (carbon, nitrogen, oxygen for example), up to iron. These elements can form in stars because there is a long time available (millions of years) for the reactions to take place. Massive stars burn relatively rapidly, and eventually run out of nuclear fuel. The star becomes unstable, and its core rapidly collapses because of gravitational attraction. The consequent rise in temperature blows it apart in a giant explosion, during which time new reactions take place that generate elements heavier than iron; this explosion is seen by us as a Supernova (“New Star”) suddenly blazing in the sky, where previously there was just an ordinary star. Such explosions blow into space the heavy elements that had been accumulating in the star’s interior, forming vast filaments of dust around the remnant of the star. It is this material that can later be accumulated, during the formation of second generation stars, to form planetary systems around those stars. Thus the elements of which we are made (the carbon, nitrogen, oxygen and iron nuclei for example) were created in the extreme heat of stellar interiors, and made available for our use by supernova explosions. Without these explosions, we could not exist.

# Rupture…Another Drunken Risibility

How is rupture important to any discourse? Is it the waning of a line of discourse in that the impulsive impact is debilitating over temporality and even spatially or is it a sudden change of the direction altogether hitherto unknown or never before comprehended, a kind of digital break, a sudden change of phase or a sudden phase transition. The latter are called by me ‘ruptures’. But, why do I call it only debilitating to begin with, it could very well be the accelerated impact that could be thought out. a complete dislocation from one discourse to another, bridged by only a kind of conscious memory of the shift. the important factor to be taken into consideration is the determination of the ‘threshold’, where the rupture occurs. This could be cataclysmic as in the case of the extinction of dinosaurs. What is cataclysmic is the eventality of this threshold, a cross over of which is the rupture that is being thought about.

These deliberations are indeed painful. When I was caught up in a paradox in logic (Goodman’s paradox), I could comprehend only to land up in this soup, the thoughts collapse into thinking as if there is a flattening out of the hierarchy between the thought and thinking. Elevation is possible when there is a pre-ordained depression.

For me, suddenly, philosophically, rupture has gained prominence. I am very convinced, if it could be achieved, majority of the philosophical systems in vogue would just annihilate themselves.

I was wondering about the assistance that could come from the field of chaos theory. but then I dismissed it as soon as it had sprung. Chaos Theory is so damn dependent of the sensitivity to initial conditions that it becomes well nigh possible to simulate the results: it becomes deterministic, predictable and thereby, a continuity in what would come about in the future if there is at all a rupture in the sense I mentioned about. This dismisses the notion of a hitherto unthought of break with the past and the present. Knowing the infinitely unknown future based on the finitely known past!!!

Another thing that I thought about was the idea of Clinamen by Lucretius. if I am in a position to link up his idea with the Deleuzean one of the former’s notion of a slight movement in the angle of declination with the latter’s initiation of ‘turbulence’ in the laminar flow, I somehow see this as converging towards the Chaos (read Chaos Theory). This ‘Turbulence’ is still accounted for in that it is still deterministic and predictable. Therefore, a dismissal of Lucretius’ position is validated.