The Locus of Renormalization. Note Quote.

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Since symmetries and the related conservation properties have a major role in physics, it is interesting to consider the paradigmatic case where symmetry changes are at the core of the analysis: critical transitions. In these state transitions, “something is not preserved”. In general, this is expressed by the fact that some symmetries are broken or new ones are obtained after the transition (symmetry changes, corresponding to state changes). At the transition, typically, there is the passage to a new “coherence structure” (a non-trivial scale symmetry); mathematically, this is described by the non-analyticity of the pertinent formal development. Consider the classical para-ferromagnetic transition: the system goes from a disordered state to sudden common orientation of spins, up to the complete ordered state of a unique orientation. Or percolation, often based on the formation of fractal structures, that is the iteration of a statistically invariant motif. Similarly for the formation of a snow flake . . . . In all these circumstances, a “new physical object of observation” is formed. Most of the current analyses deal with transitions at equilibrium; the less studied and more challenging case of far form equilibrium critical transitions may require new mathematical tools, or variants of the powerful renormalization methods. These methods change the pertinent object, yet they are based on symmetries and conservation properties such as energy or other invariants. That is, one obtains a new object, yet not necessarily new observables for the theoretical analysis. Another key mathematical aspect of renormalization is that it analyzes point-wise transitions, that is, mathematically, the physical transition is seen as happening in an isolated mathematical point (isolated with respect to the interval topology, or the topology induced by the usual measurement and the associated metrics).

One can say in full generality that a mathematical frame completely handles the determination of the object it describes as long as no strong enough singularity (i.e. relevant infinity or divergences) shows up to break this very mathematical determination. In classical statistical fields (at criticality) and in quantum field theories this leads to the necessity of using renormalization methods. The point of these methods is that when it is impossible to handle mathematically all the interaction of the system in a direct manner (because they lead to infinite quantities and therefore to no relevant account of the situation), one can still analyze parts of the interactions in a systematic manner, typically within arbitrary scale intervals. This allows us to exhibit a symmetry between partial sets of “interactions”, when the arbitrary scales are taken as a parameter.

In this situation, the intelligibility still has an “upward” flavor since renormalization is based on the stability of the equational determination when one considers a part of the interactions occurring in the system. Now, the “locus of the objectivity” is not in the description of the parts but in the stability of the equational determination when taking more and more interactions into account. This is true for critical phenomena, where the parts, atoms for example, can be objectivized outside the system and have a characteristic scale. In general, though, only scale invariance matters and the contingent choice of a fundamental (atomic) scale is irrelevant. Even worse, in quantum fields theories, the parts are not really separable from the whole (this would mean to separate an electron from the field it generates) and there is no relevant elementary scale which would allow ONE to get rid of the infinities (and again this would be quite arbitrary, since the objectivity needs the inter-scale relationship).

In short, even in physics there are situations where the whole is not the sum of the parts because the parts cannot be summed on (this is not specific to quantum fields and is also relevant for classical fields, in principle). In these situations, the intelligibility is obtained by the scale symmetry which is why fundamental scale choices are arbitrary with respect to this phenomena. This choice of the object of quantitative and objective analysis is at the core of the scientific enterprise: looking only at molecules as the only pertinent observable of life is worse than reductionist, it is against the history of physics and its audacious unifications and invention of new observables, scale invariances and even conceptual frames.

As for criticality in biology, there exists substantial empirical evidence that living organisms undergo critical transitions. These are mostly analyzed as limit situations, either never really reached by an organism or as occasional point-wise transitions. Or also, as researchers nicely claim in specific analysis: a biological system, a cell genetic regulatory networks, brain and brain slices …are “poised at criticality”. In other words, critical state transitions happen continually.

Thus, as for the pertinent observables, the phenotypes, we propose to understand evolutionary trajectories as cascades of critical transitions, thus of symmetry changes. In this perspective, one cannot pre-give, nor formally pre-define, the phase space for the biological dynamics, in contrast to what has been done for the profound mathematical frame for physics. This does not forbid a scientific analysis of life. This may just be given in different terms.

As for evolution, there is no possible equational entailment nor a causal structure of determination derived from such entailment, as in physics. The point is that these are better understood and correlated, since the work of Noether and Weyl in the last century, as symmetries in the intended equations, where they express the underlying invariants and invariant preserving transformations. No theoretical symmetries, no equations, thus no laws and no entailed causes allow the mathematical deduction of biological trajectories in pre-given phase spaces – at least not in the deep and strong sense established by the physico-mathematical theories. Observe that the robust, clear, powerful physico-mathematical sense of entailing law has been permeating all sciences, including societal ones, economics among others. If we are correct, this permeating physico-mathematical sense of entailing law must be given up for unentailed diachronic evolution in biology, in economic evolution, and cultural evolution.

As a fundamental example of symmetry change, observe that mitosis yields different proteome distributions, differences in DNA or DNA expressions, in membranes or organelles: the symmetries are not preserved. In a multi-cellular organism, each mitosis asymmetrically reconstructs a new coherent “Kantian whole”, in the sense of the physics of critical transitions: a new tissue matrix, new collagen structure, new cell-to-cell connections . . . . And we undergo millions of mitosis each minute. More, this is not “noise”: this is variability, which yields diversity, which is at the core of evolution and even of stability of an organism or an ecosystem. Organisms and ecosystems are structurally stable, also because they are Kantian wholes that permanently and non-identically reconstruct themselves: they do it in an always different, thus adaptive, way. They change the coherence structure, thus its symmetries. This reconstruction is thus random, but also not random, as it heavily depends on constraints, such as the proteins types imposed by the DNA, the relative geometric distribution of cells in embryogenesis, interactions in an organism, in a niche, but also on the opposite of constraints, the autonomy of Kantian wholes.

In the interaction with the ecosystem, the evolutionary trajectory of an organism is characterized by the co-constitution of new interfaces, i.e. new functions and organs that are the proper observables for the Darwinian analysis. And the change of a (major) function induces a change in the global Kantian whole as a coherence structure, that is it changes the internal symmetries: the fish with the new bladder will swim differently, its heart-vascular system will relevantly change ….

Organisms transform the ecosystem while transforming themselves and they can stand/do it because they have an internal preserved universe. Its stability is maintained also by slightly, yet constantly changing internal symmetries. The notion of extended criticality in biology focuses on the dynamics of symmetry changes and provides an insight into the permanent, ontogenetic and evolutionary adaptability, as long as these changes are compatible with the co-constituted Kantian whole and the ecosystem. As we said, autonomy is integrated in and regulated by constraints, with an organism itself and of an organism within an ecosystem. Autonomy makes no sense without constraints and constraints apply to an autonomous Kantian whole. So constraints shape autonomy, which in turn modifies constraints, within the margin of viability, i.e. within the limits of the interval of extended criticality. The extended critical transition proper to the biological dynamics does not allow one to prestate the symmetries and the correlated phase space.

Consider, say, a microbial ecosystem in a human. It has some 150 different microbial species in the intestinal tract. Each person’s ecosystem is unique, and tends largely to be restored following antibiotic treatment. Each of these microbes is a Kantian whole, and in ways we do not understand yet, the “community” in the intestines co-creates their worlds together, co-creating the niches by which each and all achieve, with the surrounding human tissue, a task closure that is “always” sustained even if it may change by immigration of new microbial species into the community and extinction of old species in the community. With such community membership turnover, or community assembly, the phase space of the system is undergoing continual and open ended changes. Moreover, given the rate of mutation in microbial populations, it is very likely that these microbial communities are also co-evolving with one another on a rapid time scale. Again, the phase space is continually changing as are the symmetries.

Can one have a complete description of actual and potential biological niches? If so, the description seems to be incompressible, in the sense that any linguistic description may require new names and meanings for the new unprestable functions, where functions and their names make only sense in the newly co-constructed biological and historical (linguistic) environment. Even for existing niches, short descriptions are given from a specific perspective (they are very epistemic), looking at a purpose, say. One finds out a feature in a niche, because you observe that if it goes away the intended organisms dies. In other terms, niches are compared by differences: one may not be able to prove that two niches are identical or equivalent (in supporting life), but one may show that two niches are different. Once more, there are no symmetries organizing over time these spaces and their internal relations. Mathematically, no symmetry (groups) nor (partial-) order (semigroups) organize the phase spaces of phenotypes, in contrast to physical phase spaces.

Finally, here is one of the many logical challenges posed by evolution: the circularity of the definition of niches is more than the circularity in the definitions. The “in the definitions” circularity concerns the quantities (or quantitative distributions) of given observables. Typically, a numerical function defined by recursion or by impredicative tools yields a circularity in the definition and poses no mathematical nor logical problems, in contemporary logic (this is so also for recursive definitions of metabolic cycles in biology). Similarly, a river flow, which shapes its own border, presents technical difficulties for a careful equational description of its dynamics, but no mathematical nor logical impossibility: one has to optimize a highly non linear and large action/reaction system, yielding a dynamically constructed geodetic, the river path, in perfectly known phase spaces (momentum and space or energy and time, say, as pertinent observables and variables).

The circularity “of the definitions” applies, instead, when it is impossible to prestate the phase space, so the very novel interaction (including the “boundary conditions” in the niche and the biological dynamics) co-defines new observables. The circularity then radically differs from the one in the definition, since it is at the meta-theoretical (meta-linguistic) level: which are the observables and variables to put in the equations? It is not just within prestatable yet circular equations within the theory (ordinary recursion and extended non – linear dynamics), but in the ever changing observables, the phenotypes and the biological functions in a circularly co-specified niche. Mathematically and logically, no law entails the evolution of the biosphere.

Financial Forward Rate “Strings” (Didactic 1)

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Imagine that Julie wants to invest $1 for two years. She can devise two possible strategies. The first one is to put the money in a one-year bond at an interest rate r1. At the end of the year, she must take her money and find another one-year bond, with interest rate r1/2 which is the interest rate in one year on a loan maturing in two years. The final payoff of this strategy is simply (1 + r1)(1 + r1/2). The problem is that Julie cannot know for sure what will be the one-period interest rate r1/2 of next year. Thus, she can only estimate a return by guessing the expectation of r1/2.

Instead of making two separate investments of one year each, Julie could invest her money today in a bond that pays off in two years with interest rate r2. The final payoff is then (1 + r2)2. This second strategy is riskless as she knows for sure her return. Now, this strategy can be reinterpreted along the line of the first strategy as follows. It consists in investing for one year at the rate r1 and for the second year at a forward rate f2. The forward rate is like the r1/2 rate, with the essential difference that it is guaranteed : by buying the two-year bond, Julie can “lock in” an interest rate f2 for the second year.

This simple example illustrates that the set of all possible bonds traded on the market is equivalent to the so-called forward rate curve. The forward rate f(t,x) is thus the interest rate that can be contracted at time t for instantaneously riskless borrowing 1 or lending at time t + x. It is thus a function or curve of the time-to-maturity x2, where x plays the role of a “length” variable, that deforms with time t. Its knowledge is completely equivalent to the set of bond prices P(t,x) at time t that expire at time t + x. The shape of the forward rate curve f(t,x) incessantly fluctuates as a function of time t. These fluctuations are due to a combination of factors, including future expectation of the short-term interest rates, liquidity preferences, market segmentation and trading. It is obvious that the forward rate f (t, x+δx) for δx small can not be very different from f (t,x). It is thus tempting to see f(t,x) as a “string” characterized by a kind of tension which prevents too large local deformations that would not be financially acceptable. This superficial analogy is in the follow up of the repetitious intersections between finance and physics, starting with Bachelier who solved the diffusion equation of Brownian motion as a model of stock market price fluctuations five years before Einstein, continuing with the discovery of the relevance of Lévy laws for cotton price fluctuations by Mandelbrot that can be compared with the present interest of such power laws for the description of physical and natural phenomena. The present investigation delves into how to formalize mathematically this analogy between the forward rate curve and a string. We formulate the term structure of interest rates as the solution of a stochastic partial differential equation (SPDE), following the physical analogy of a continuous curve (string) whose shape moves stochastically through time.

The equation of motion of macroscopic physical strings is derived from conservation laws. The fundamental equations of motion of microscopic strings formulated to describe the fundamental particles derive from global symmetry principles and dualities between long-range and short-range descriptions. Are there similar principles that can guide the determination of the equations of motion of the more down-to-earth financial forward rate “strings”?

Suppose that in the middle ages, before Copernicus and Galileo, the Earth really was stationary at the centre of the universe, and only began moving later on. Imagine that during the nineteenth century, when everyone believed classical physics to be true, that it really was true, and quantum phenomena were non-existent. These are not philosophical musings, but an attempt to portray how physics might look if it actually behaved like the financial markets. Indeed, the financial world is such that any insight is almost immediately used to trade for a profit. As the insight spreads among traders, the “universe” changes accordingly. As G. Soros has pointed out, market players are “actors observing their own deeds”. As E. Derman, head of quantitative strategies at Goldman Sachs, puts it, in physics you are playing against God, who does not change his mind very often. In finance, you are playing against Gods creatures, whose feelings are ephemeral, at best unstable, and the news on which they are based keep streaming in. Value clearly derives from human beings, while mass, charge and electromagnetism apparently do not. This has led to suggestions that a fruitful framework to study finance and economy is to use evolutionary models inspired from biology and genetics.

This does not however guide us much for the determination of “fundamental” equa- tions, if any. Here, we propose to use the condition of absence of arbitrage opportunity and show that this leads to strong constraints on the structure of the governing equations. The basic idea is that, if there are arbitrage opportunities (free lunches), they cannot live long or must be quite subtle, otherwise traders would act on them and arbitrage them away. The no-arbitrage condition is an idealization of a self-consistent dynamical state of the market resulting from the incessant actions of the traders (ar- bitragers). It is not the out-of-fashion equilibrium approximation sometimes described but rather embodies a very subtle cooperative organization of the market.

We consider this condition as the fundamental backbone for the theory. The idea to impose this requirement is not new and is in fact the prerequisite of most models developed in the academic finance community. Modigliani and Miller [here and here] have indeed emphasized the critical role played by arbitrage in determining the value of securities. It is sometimes suggested that transaction costs and other market imperfections make irrelevant the no-arbitrage condition. Let us address briefly this question.

Transaction costs in option replication and other hedging activities have been extensively investigated since they (or other market “imperfections”) clearly disturb the risk-neutral argument and set option theory back a few decades. Transaction costs induce, for obvious reasons, dynamic incompleteness, thus preventing valuation as we know it since Black and Scholes. However, the most efficient dynamic hedgers (market makers) incur essentially no transaction costs when owning options. These specialized market makers compete with each other to provide liquidity in option instruments, and maintain inventories in them. They rationally limit their dynamic replication to their residual exposure, not their global exposure. In addition, the fact that they do not hold options until maturity greatly reduces their costs of dynamic hedging. They have an incentive in the acceleration of financial intermediation. Furthermore, as options are rarely replicated until maturity, the expected transaction costs of the short options depend mostly on the dynamics of the order flow in the option markets – not on the direct costs of transacting. For the efficient operators (and those operators only), markets are more dynamically complete than anticipated. This is not true for a second category of traders, those who merely purchase or sell financial instruments that are subjected to dynamic hedging. They, accordingly, neither are equipped for dynamic hedging, nor have the need for it, thanks to the existence of specialized and more efficient market makers. The examination of their transaction costs in the event of their decision to dynamically replicate their options is of no true theoretical contribution. A second important point is that the existence of transaction costs should not be invoked as an excuse for disregarding the no-arbitrage condition, but, rather should be constructively invoked to study its impacts on the models…..

Genesis and Evaluation of Political Philosophy of Thomas Hobbes. Part 2.

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Hobbes recognizes the nature of the ideal of an exact philosophical morality,which is paradoxical and makes it the backbone of his political philosophy. In his moral philosophy also, the antithesis between the virtue and pseudo-virtue forms a constituent part. He also teaches that true virtue and pseudo-virtue differ only in their reason. Like Plato, he also recognizes only political virtues. Hobbes also distrusts rhetoric, in a way, which recalls Plato.

A pleader commonly thinks he ought to say all he can for the benefit of his client, and therefore has need of a faculty to wrest the sense of words from their true meaning, and the faculty of rhetoric to seduce the jury, and sometimes the judge also, and many other arts which I neither have, nor intend to study.

Basing his reason on Platonic approach, he thought that the difference between the analysis of ordinary values and of passions given in Aristotle’s rhetoric on the one hand, and the theory of ethics on the other, not nearly great enough. While in Aristotle’s view the common passionate valuations have a peculiar consistency and universality, Hobbes, by reason of his radical criticism of opinion as such, cannot but deny them this dignity. 

What Hobbes’ political philosophy owes to Platonism is the antithesis between truth and appearance, the fitting and the great, between reason and passion. From the very outset, Hobbes’ conviction was the antithesis between vanity and fear and for him, it was of fundamental importance for morals. But in the beginning, Hobbes understood this antithesis as an antithesis within the domain of the passions. But when he turned to Plato, he began to conceive this antithesis between vanity and fear as the antithesis between passion and reason. However, resolutely Hobbes demands a completely passionless, purely rational political philosophy, he desires, as it were, in the same breath, that the norm to be set by reason should be in accord with the passions. Respect for applicability determines the seeking after the norm from the outset. With this, Hobbes does not merely tacitly adopt Aristotle’s criticism of Plato’s political philosophy but he goes much beyond Aristotle.

Primary reason for Hobbes’ opposition to Plato, is the motive for turning to Euclid as to the ‘resolutive-compositive’ method. In this method, the given object of investigation is first analysed, traced back to its reasons, and then by completely lucid deduction the object is again reconstituted. The axioms, which Hobbes gains by going back from the existing State to its reasons, and from there he deduces the form of the right State; are according to him, the man’s natural selfishness and the fear of death. Hobbes’ political philosophy differs from Plato in that, in the latter, exactness means the undistorted reliability of the standards, while in the former, exactness means unconditional applicability, under all circumstances. Hobbes took the ‘resolutive-compositive’ method over from Galileo. He believes that by this method he can achieve for political philosophy what Galileo achieved for physics. But the adequacy for physics does not guarantee its adequacy for political philosophy. For while the subject for physics is the natural body, the subject of political philosophy is an artificial body, i.e. a whole that has to be made by men from natural wholes. Thus the concern of political philosophy is not so much knowledge of the artificial body as the production of that body. Political philosophy analyses the existing State into its elements only in order that by a better synthesis of those elements the right State may be produced. Political philosophy thus becomes a technique for the regulation of the State. Its task is to alter the unstable balance of the existing State to the stable balance of the right State. The introduction of Galileo’s method into political philosophy from the outset renounces all discussions of the fundamental political problems, i.e. the elimination of the fundamental question as to the aim of the State.

Hobbes doesn’t question the necessity of political philosophy, i.e. he doesn’t ask first, ‘What is virtue?’ and ‘Can it be taught?’ and ‘What is the aim of the State?’, because for him, these questions are answered by tradition, or by common opinion. The aim of the State is for him as a matter of course peace, i.e. peace at any price. The underlying presupposition is that violent death is the first and greatest and supreme evil. After finding this presupposition as a principle when he analysed the existing State, he proceeds to deduce from it the right State; opposed to Plato, whose consideration of the genesis of the State seems superficially akin, but has the character of reflection, of deliberate questioning of what is good and fitting. Convinced of the absolutely typical character of the mathematical method, according to which one proceeds from axioms to self-evident truths/conclusions, Hobbes fails to realize that in the ‘beginning’, in the ‘evident’ presuppositions whether of mathematics or of politics, the task of ‘dialectic’ is hidden. Hobbes considers it superfluous, even dangerous, to take as one’s point of departure what men say about justice and so forth: ‘the names of virtues and vices…can never be true grounds of any ratiocination’. The application of the ‘resolutive-compositive’ method to political philosophy is of doubtful value as it prevented Hobbes from asking the questions as to the standard. He begins his political philosophy with the question as to the nature of the man in the sense of that which falls to all men before education. If the procedure of deducing the right State is to be significant, the principles themselves contain the answer to the question as to the right State, as to the standard. Hobbes characterizes the two principles viz., limitless self-love on the one hand and that of violent death on the other as he principles of the wrong and the principles of the right. But this characterization does not arise from the analysis, for the analysis can only show the principles of the existing State, and cannot, therefore, teach anything about the rightness and wrongness of those principles, and, on the other hand, this characterization is the presupposition of the synthesis, which as a synthesis of the right State cannot arise until it has been established what is the right. This qualification, which follows the analysis and precedes the synthesis, is certainly into the frame of the ‘resolutive-compositive’ method; but it is not to be understood from this method, either in general or even in particular. The justification of the standard, which is the fundamental part of the political philosophy, is hidden by the ‘resolutive-compositive’ method and even made unrecognizable.

What is justified in this way is indeed not a standard, an obligation; but a right, a claim. According to Hobbes, the basis of politics is not the ‘law of nature’, but the ‘right of nature’. This right is the minimum claim, which as such is fundamentally just, and the origin of any other just claim; more exactly, it is unconditionally just because it can be answered for in face of all men in all circumstances. A claim of this kind is only the claim to defend life and limb. Its opposite is the maximum claim, which is fundamentally unjust, for it cannot be answered for in face of any other man. The maximum claim, the claim man makes by nature, i.e. as long as he is not educated by ‘unforeseen mischances’, is the claim to triumph over all other men. This ‘natural’ claim is checked by fear of violent death and becomes man’s rational minimum claim, and thus ‘right of nature’ comes into being, or atleast comes to light. That is to say, the ‘right of nature’ is the first juridical or moral fact, which arises if one, starts from man’s nature i.e. from man’s natural appetite. The ‘law of nature’ belongs to a much later stage of the progress from human nature to the State: ‘natural right’ is dealt with in the first chapter of De Cive, ‘natural law’ in the second and third chapters.

The ‘law of nature’ owes all its dignity simply to the circumstances that it is the necessary consequence of the ‘right of nature’. We may ask the question as to what is the peculiarity of modern political thought in relation to the classical political thought?  While modern thought starts from the rights of the individual, and conceives the State as existing to secure the conditions of his development, Greek thought starts from the right of the State. Modern and classical political philosophy are fundamentally distinguished in that modern political philosophy takes ‘right’ as the starting point, whereas classical political philosophy has ‘law’ as its starting point.

Hobbes marked an epoch not only by subordinating law to right. He was at the same time ‘the first writer to grasp the full importance of the idea of sovereignty…he must take the credit of being the first to see that the idea of sovereignty lies at the very root of the whole theory of the State; and the first to realize the necessity of fixing precisely where it lies, and what are its functions and its limits’. By this also Hobbes stands in contrast to classical political philosophy: ‘Amongst the most notable omissions of Greek philosophy is the absence of any clear attempt to define the nature of sovereignty, to determine its seat, or settle the ultimate sanction on which it rests’. In classical times, the question, ‘who or what shall rule?’ has the antiquity answer running, ‘the law’. Philosophers who could not acquiesce in the Divine origin of the law justify this answer in the following way: the rational should rule over the irrational (the old over the young, the man over the woman, the master over the slave) and therefore law over men. Granting that there are men who by force of reason are undoubtedly superior to others, would those others submit to them merely on this ground, and obey them? Would they recognize their superiority? But doubt does not stop at that. It is denied that any considerable difference in reasonableness exists between men. Because reason is essentially impotent, it is not enough to reply that reason is the origin and the seat of sovereignty. Thus it becomes fundamentally questionable, which of the men who are equal and alike is to rule over the others, and under which conditions and within which limits, they have a claim to rule. Because all men a re equally reasonable, the reason of one or more individuals must arbitrarily be made the standard reason as an artificial substitute for the lacking natural superiority of the reason. Because reason is impotent, the rational ‘law of nature’ also loses its dignity. In its place we have the ‘right of nature’ which is, indeed, according to reason but dictated not by reason but by the fear of death. The break with rationalism is thus the decisive presupposition for the concept of sovereignty as well as for the supplanting of ‘law’ by ‘right’.

Hobbes in his writings conceives sovereign power not as reason but as will. Hobbes expressly turns against the view still predominant in his age that the holder of the sovereign power is in the same relation to the State as the head to the whole man. The holder of the sovereign power is not the ‘head’, that is, the capacity to deliberate and plan, but the ‘soul’, that is, the capacity to command, in the State. The explicit break with rationalism is thus the reason for the antithesis of modern political thought to classical and is characterized thusly: ‘the Greeks believed in the need of education to tune and harmonize social opinions to the spirit and tone of a fixed and fundamental law. The modern belief is the need of a representation to adjust and harmonize a fluid and changing and subordinate law to the movement of a sovereign public opinion or ‘general will’.

The view of classical rationalism, that only reason justifies dominion, found its most radical expression in Plato’s saying that the only necessary and adequate condition for the weal of a State is that the philosophers should be Kings and Kings philosophers. This amounts to stating that the setting up of a perfect commonwealth depends exclusively on ‘internal policy’ and not at all on foreign policy. From here on, Plato’s theory of justice can be summed up thus: there is no happiness for men without justice; justice means attending to one’s own business, bringing oneself into the right disposition with regard to the transcendent unchanging norm, to which the soul is akin, and not meddling into other people’s affairs; and justice in the State is not different from justice in the individual, except that the State is self-sufficient and can thus practice justice; attending to its own business; incomparably more perfectly than can the individual who is not self-sufficient. The citizens of the perfect State, for this very reason to foreigners, happen to be either allies to be esteemed or foes to be feared. Let us take Plato’s example; if the essence of the thing is to be preferred to its external conditions, to the self-realization and self-assertion of that thing against its external conditions, then, for instance, the right constitution of the body, its health, is to be preferred to its return to its health, to its recovery after its loss of health. By this example, Plato makes clear that the good statesman carries out his legislation with an eye to peace, which is to the good internal constitution of the State, and not with an eye to war, that is, to the assertion of the State against external conditions. Hobbes differs from Plato and asserts that the recovery of health is to be preferred to the undisturbed possession of health. While for Plato and to an extent for Aristotle, and in accordance with the primary interest they attach to home policy, the question of the number of inhabitants of the perfect State, that is, the limits set to the State by its inner necessity, is of decisive importance; Hobbes brushes this question aside in these words: ‘The Multitude sufficient to confide in for our security, is not determined by any certain number, but by comparison by the enemy we feare…’ The primacy of foreign policy is not specifically taught by Hobbes, but it is an integral part of all of modern political philosophy. Immanuel Kant in one of his works has a phrase, which runs like: ‘The problem of establishing a perfect civil constitution is dependent on the problem of a lawful external relation between the States and cannot be solved independently of the solution of the latter problem’.

The antithesis between Platonic and Hobbesian political philosophy, reduced to principle, is that the former orientates itself by speech and the latter from the outset refuses to do so. This refusal originally arises from what may be called natural valuations. While Plato goes back to the truth hidden in the natural valuations and thereof seeks to teach nothing new and unheard of, but to recall what is known to all but not understood, Hobbes, rejecting the natural valuations in principle, goes beyond, goes forward to a new a priori political philosophy, which is of the future and freely projected. Measured by Aristotle’s classical explanation of morals, Platonic moral philosophy is as paradoxical as Hobbes’. But whereas the paradoxical nature of Platonic moral philosophy is as irreversible as the  ‘cave’ existence of men bound to the body, Hobbes’ moral philosophy is destined sooner or later to change from paradox to an accepted form of public opinion. The paradoxical nature of Hobbes’ moral philosophy is the paradox of the surprisingly new, unheard of venture. Whereas Plato retraces natural morals and the orientation provided by them to their origin, Hobbes must attempt in sovereignty, and without this orientation, to discover the principles of morals. Hobbes travels the path, which leads to formal ethics and finally to relativist skepticism. The enormous extension of the claims made on political science leads at least to a denial of the very idea of political science and to the replacement of political science by sociology. Plato does not question the virtue character of courage, to which speech bears witness but simply opposes the over-estimation of courage, which underlies the popular opinion. Hobbes, because he renounced all orientation by speech, goes so far to deny the virtue character of courage. And just as disdain of speech finally leads to relativist skepticism, the negation of courage leads to the controversial position of courage, which becomes more and more acute on the way from Rousseau by Hegel to Nietzsche and is completed by the reabsorption of wisdom by courage, in the view that the ideal is not the object of wisdom, but the hazardous venture of the will.

Relinquishing orientation by speech does not mean that Hobbes ‘forgets’ the question of standards, but that he poses this question only as an afterthought, and, therefore, inadequately. Whereas Plato distinguishes between two kinds of reasons, the good and the necessary, Hobbes recognizes only one kind, the necessary. Since as a result of this he is obliged to take into account the inevitable difference between the good and the necessary within the necessary itself, the question of the standard, of the good, becomes for him the question of what is par excellence necessary, and he discovers the retreat from death as the necessary par excellence. For Hobbes, the denial of natural standards was irrefutably evident on the basis of his materialist metaphysics. Thus this metaphysics is the implicit pre-supposition even of his turning to Euclid, provided that the acceptance of the ‘mathematical’ method presupposes the negation of absolute standards. For the question arises; why did Hobbes decide in favour of materialism? On the ground of what primary conviction was materialism so vividly evident for him? The answer can be based on rough indications i.e. Hobbes’ turn to natural science is to be explained by his interest not so much in nature as in man, in self knowledge of man as he really is, i.e. by the interest that characterized him even in his humanist period. His scientific explanation of sense perception is characterized by the fact that it interprets perception of the higher senses by the sense of touch; and the preference for the sense of touch, which this presupposes is already implied in Hobbes’ original view of fundamental significance of the antithesis between vanity and fear. If Hobbes’ natural science is dependent on his ‘humanist’, that is moral, interests and convictions, on the other hand a particular conception of nature is the implicit basis of his views on moral and political philosophy. It is certain that the conception of nature, which is the presupposition of his political philosophy and the conception of nature, which he explains in his scientific writings, has a kinship and which in principle are to be kept separate. It is for these reasons that his scientific investigations could exert a powerful influence on the evolution of his political philosophy. He could not have maintained his thesis that death is the greatest and supreme evil but for the conviction vouched for by his natural science that the soul is not immortal. His criticism of aristocratic virtue and his denial of any gradation in mankind gains certainty only through his conception of nature, according to which there is no order, that is, no gradation in nature. From this standpoint we can understand the difference between Hobbes’ conception of Pride and the traditional conception. ‘Pride’ in the traditional sense means rebellion against the gradation of beings; it presupposes, therefore, the existence and the obligatory character of that gradation. Hobbes’ conception of ‘Pride’, on the other hand, presupposes the denial of natural gradation; this conception is, indeed, nothing other than a means of ‘explaining’, i.e. of denying that gradation: the allegedly natural gradation concerning the faculties of the mind proceeds from a ‘a vain concept of ones own wisdom, which almost all men think they have in a greater degree, than the Vulgar’. The idea of civilization achieves its telling effect solely by reason of the presupposition that the civilization of human nature can go on boundlessly, because what tradition in agreement with common sense had understood as given and immutable human nature is for the main part a mere ‘natural limit’, which may be over passed. Very little is innate in man; most of what is alleged to come to him from the nature is acquired and therefore mutable, as conditions change; the most important peculiarities of man; speech, reason, sociality are not gifts of nature, but the work of his will. This example creates a duality in his political philosophy. The idea of civilization presupposes that man, by virtue of his intelligence, can place himself outside nature, can rebel against nature. The antithesis of nature and human will is hidden by the monist (materialist-deterministic) metaphysic, which Hobbes found himself forced to adopt simply because he saw no other possibility of escaping the ‘Kingdom of darkness’. This signifies that the moral basis of his political philosophy becomes more and more disguised, the farther the evolution of his natural science progresses. In other words, with the progressive evolution of his natural sciences, vanity, which must of necessity be treated from the moral standpoint, is more and more replaced by the striving for power, which is neutral and therefore more amenable to scientific interpretation. But Hobbes took great care not to follow this path as he thought that consistent naturalism would ruin his political philosophy. To compare Spinoza with Hobbes, Spinoza was more naturalistic than Hobbes. Spinoza relinquished the distinction between ‘might’ and ‘right’ and taught the natural right of all passions. Hobbes, on the other hand, by virtue of the basis of his political philosophy asserted the natural right only of the fear of death. On the other hand, if we consider Montesquieu, who carried the naturalistic analysis of the passions to its logical conclusion, came forward with the result that the State of nature cannot be the war of all against all this clearly exemplifies that if inconsistent naturalism is compatible with Hobbes’ political philosophy, the consistent naturalism, which Hobbes displays in his scientific writings cannot be the foundation of his political philosophy. This foundation must be another conception of nature, which although being related to naturalism is by no means identical to it.

Therefore, the foundation of Hobbes’ political philosophy, which is the moral attitude to which it owes its existence, is objectively prior to the mathematical scientific founding and presentation of that philosophy. The mathematical method and the materialistic metaphysics each in their own way contributed to disguise the original motivation to undermine Hobbes’ political philosophy. Hence, Leviathan is by no means an adequate source for an understanding of Hobbes’ moral and political philosophy, although the presuppositions and conclusions dealing with moral attitude are clearly manifest in the Leviathan.

History and Historicity in the Political Philosophy of Thomas Hobbes

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Hobbes’ early moral and political views may be traced back to the Aristotelian tradition. If this is the case, then it can be said that these views are definitely the materials for his political philosophy but not the seeds for his political philosophy. But his later views are in direct contrast to Aristotelianism. If it may be contended that Hobbes’ taking of considerable elements from Aristotle paved the way for a later break with Aristotle, then a sense of fundamental defect with the Aristotelian philosophy was a must for this break. Hobbes later elaborated these modifications and presented them as systematic objections. This deep dissatisfaction with traditional philosophy must have forced Hobbes for turning to history and thus citing his case in his humanist period. His turning to history is revealed in his revolutionary early thought. His turning to history was definitely intentional with philosophical contentions.

According to Hobbes, philosophy and history are fundamentally different. Philosophy lays down precepts for the right behaviour of men, but then again precepts don’t prove their practical aspects efficaciously. History, not philosophy, gives man prudence.

‘…the principal and proper work of history (is) to instruct and enable men, by the knowledge of actions past, to bear themselves prudently in the present and providently towards the future…’ ‘…the nature of history is merely narrative…look how much a man of understanding might have added to his experience, if he had then lived a beholder of their proceedings, and familiar with the men and business of the time: so much almost may he profit now, by attentive reading of the same here written. He may from the narrations draw out lessons to himself’.

History widens men’s experiences by making men capable of applying the precepts in the individual cases. Hobbes takes it for granted that this philosophy rightly lays down the norms for human actions. He asserts that practical wisdom is at least the sine qua non for moral virtue and this wisdom is gained only through experience. The study of history widens the experience from service to the acquisition of wisdom and thus from service to moral education. Aristotle believes in rational precepts having no influence on most men. But according to Aristotle’s view, what is true of most men is not by any means true of free and noble minded characters who love honour; they obey precepts. As Hobbes doubts the effects of precepts altogether, does he not assert the impotence of reason with reference to all men; can we not say that the dicta of impotence of reason was thoroughly established in his mind, before his engagement with natural science?

The question, by which history originally breaks with philosophy, is the question of effectiveness of rational precept. It purely becomes a matter of application of precepts. These precepts were handed down by Aristotelian ethics. Since Aristotle satisfactorily explicated these precepts, the fundamental problem of philosophy was solved; this gave Hobbes the leisure and ample opportunity to give thought to the secondary problem of the application of precepts. In reference to this application the assertion is made that the precepts are not effective in themselves that they are not followed for their own sake, but under all circumstances it may be made plausible by making use of other measures to ensure their being followed. Hobbes of course does not question the necessity and effectiveness of laws. But now the teachings to be drawn from history slip in as it were midway between the precepts of philosophy and the laws.

‘…(history) doth things with more grace and modestie then the civill lawes and ordinances do: because it is more grace for a man to teach and instruct, then to chastise or punish’.

The teaching to be drawn from history has from now on to fulfill the function for noble natures which, according to Aristotle, was the task of philosophical precepts. The teachings of history replace the precepts of philosophy in the education of the aristocracy.

The opposition of philosophical precept and the historical example based on the doubt of the efficacy of the precept is punctuated in the literature of the sixteenth century. It need only mean, we must attribute to a regrettable shortcoming on the part of the majority of men that they do not obey the precepts of philosophy, that they do not love virtue for itself, but for all its reward, which is praise. This doubt also means that the true motive of virtue is honour and glory. It essentially implies aristocratic virtue. As a result of the close connection between history and honour or glory, the more virtue is envisaged as aristocratic virtue, the keener will be the interest in history. Hobbes often quotes Lipsius as an authority for his views on history. Through Lipsius’ political philosophy, Hobbes successfully accomplished turning to history. What is felt as a lack is not so much the scientific writing of history; it is recognized that from all time histories have been written which are adequate for every possible demand; not even directions for the writing of history, but above all methodical readings of the histories already in existence. With a view to the teaching of history by methodically reading it is to be gained for the right ordering of human actions. The only clear knowledge of the application of the norms, which obtain for human actions, which have taken place in the past. History seeks the application and realization of precepts, the conditions and results of that realization. Unlike poetry, whose main objective is to give pleasure, history and philosophy derives its objectivity in seriousness. Hobbes names history and philosophy as the two fundamental branches of human knowledge.

If the main emphasis of history is to instruct and enable men, to bear themselves prudently in the present and providently in the future, undertaking a methodic utilization of history implies that a methodic education for prudence is aimed at. This education of prudence is to be sought by placing the whole available experience of mankind at our disposal, there has to be no room for any elements of chance. To the question, ‘How is one to behave in an individual case?’, one is no longer to receive the Aristotelian answer of how a sensible man would behave, but one receives for the particular case concrete maxims gained from the study of history. In this education, words and actions are important only in reference to aims. It is only through history that the reader is to be taught which kinds of aims are salutary and destructive. The systematic transformation to history, finds its most complete expression in Bacon’s philosophy.

According to Bacon, moral philosophy as the study of virtue and duty has been perfectly worked out in classical philosophy. But he opines that the fundamental shortcoming of ancient philosophy is the limiting factor that imposes itself on the description of nature of good versus the heroical descriptions of virtue, duty and felicity. As Bacon expressly says of a particular desideratum; a doctrine of the vices peculiar to the individual vocations; but as he thinks in all cases, they will seek what men sought to do, but what men really do. Traditional philosophy is blind to these materials; but the real solace comes about in the study of history. So if the neglect of history is surmounted, one of the weightiest reasons for the inadequacy and uselessness of scholasticism is given way to. Bacon makes a plea for history of literature; which he thinks has been neglected and going into this study makes him sure of making men wise.

‘History is natural, civil, ecclesiastical, and literary; whereof the first three I allow as extant, the fourth I note as deficient. For no man hath propounded to himself the general state of learning to be described and represented from age to age…without which the history of the world seemeth to me to be as the statua of Polyphemus with his eyes out; that part being wanting which doth most shew the spirit and life of the person…The use and end of which work I do not so much design for curiosity and satisfaction of those that are the lovers of learning, but chiefly for a more serious and grave purpose; which is this in few words, that it will make learned men wise in the use and administration of learning. For it is not St. Augustine’s not St. Ambrose’s works that will make so wise a divine, as ecclesiastical history, thoroughly read and observed; and the same reason is of learning’.

Bacon’s interest in history is its applicative tendencies. Bacon vehemently advocated the philosophy’s turning to history. But why? is the question? The primary reason for such a turn augments the most important material for philosophy because philosophic intent is shifting from physics and metaphysics to morals and politics.

According to Aristotle’s assertion, this change of interest takes place as soon as man becomes the consideration of being the highest being in the world. If, however, one looks back to Plato, to whom moral and political problems are of incomparably greater importance than to Aristotle, and who yet no less than Aristotle raised his gaze away from man to the eternal order, one must hold that it is not the conviction man’s superiority to all existing creatures but the conviction of the transcendence of good over all being, which is the reason why philosophic investigation begins with the ethical and political problems, with the question of the right life and the right society. This turn is caused not by the enhanced interest in the question of the good and the best form of State; but by the enhanced interest in man. The division of philosophy into natural philosophy and human philosophy is based on the systematic distinction between man and the world, which Bacon makes in express controversy against ancient philosophy.

‘…the works of God…show the omnipotency and the wisdom of the Maker, but not his image: and therefore therein the heathen opinion differeth from the sacred truth; for they supposed the world to be the image of God, and man to be exact or compendious image of the world, but the Scriptures never vouchsafe to attribute to the world that honour, as to be the image of God, but only the work of His hands; neither do they speak of any other image of God, but man…’

When the man is considered as the most excellent work of nature; then man instead of eternal order which transcends man becomes the central theme of philosophy. The ideal of contemplative life when substituted with moral virtues still ends up in a fiasco for explaining of the turn of philosophy to history. It is not the substitution of the contemplative ideal by moral virtue, in particular by the Biblical demands for justice and charity, but the systematic doubt of the efficacy of precept, which is added to this substitution, is the reason why philosophy turns to history. Bacon doubts the efficacy of rational precepts. The ancient philosophers, he says, ‘fortified and entrenched virtue and duty, as much as discourse can do, against popular and corrupt opinions’.

The reason for the turning of philosophy to history is thus the conviction of the impotence of reason, added to the enhanced interest in man. The impotence of reason is not the incapacity to establish or justify norms. It is not the way in which precepts are given to men, whether by reason or by revelation, the difficulty, which leads to the study of history, would still remain. The fact is that man does not obey the transcendent norm, whether it be rational or revealed, which is the reason of the study of history. History is studied to remedy man’s disobedience. In the sixteenth century, the reason why philosophy turned to history is the repression of the morality of obedience. As long as the distinction between philosophic knowledge on the one hand and the applicative techniques on the other hand is retained; there is at least implicitly and in principle a recognition of the pre-eminence of obedience over every other motive for action. Induction from history teaches one to distinguish between aims which justify themselves and lead to success, and aims which come to grief. The receipts to be gained from history bear only on success and failure. According to Bodin, he says in his Works, history is the easiest and the most obscure of sciences and is independent of every other science. Its subject is the study of aims and projects. By the distinction between good and bad aims, it makes possible knowledge of the norms for human actions.

Hobbes’ political philosophy, which from this time was gradually maturing precisely, had the function of replacing history, as history was understood in Bodin’s words. Hobbes’ political philosophy in its fundamental parts aimed at distinguishing between the good and the bad and thus leading to the discovery of the norms. Thus from the time of the formation of the new political philosophy, history sinks back into its philosophic insignificance; with the important difference being; in the new political philosophy, in contrast to the traditional, history is taken up and conserved. From this point of view one can appreciate the fact that Hobbes, who was particularly preoccupied with history up to the time of his return to philosophy, gives less and less thought to history as his political philosophy develops. As late as the Elements of Law, it is emphasized in a special paragraph that

‘belief…in many cases is no less free from doubt, than perfect and manifest knowledge…there be many things which we receive from report of others, of which it is impossible to imagine any cause of doubt: for what can be opposed against the consent of all men, in things they can know, and have no cause to report otherwise than they are, unless a man would say that all the world had conspired to deceive him’.

The more Hobbes learns to distinguish sharply between what is and what should be, the more the ideal character of the Leviathan becomes clear in his mind, the less significance has history for him. As a result, the distinctions between history, which is serious and seeks truth, and poetry, and the superiority of history over poetry, lose their former justification. History is thrust into the background in the measure that the new political philosophy gains clarity. For the new political philosophy fulfils the function, which had to be fulfilled by history, as, long as traditional political philosophy was acknowledged as valid. The necessity of political philosophy is shown because most men do not obey precepts. And the same presupposition, which caused the turn to history, is the basis of Hobbes’ political philosophy: the replacement of the morality of obedience by the morality of prudence.

‘All that is required, both in faith and manners, for man’s salvation, is, I confess, set down in Scriptures as plainly as can be. “Children, obey your parents in all things…Let all men be subject to the higher powers…” are words of the Scripture, which are well enough understood; but neither children, nor the greatest part of men, do understand why it is their duty to do so. They see not that the safety of the commonwealth, and consequently their own, depends upon their doing it. Every man by nature, without discipline, does in all his actions look upon, as far as he can see, the benefit that shall redound to himself from his obedience….the Scripture says one thing, and they think another, weighing the commodities or incommodities of this present life only, which are in their sight, never putting into the scales the good and evil of the life to come, which they see not’.

Bacon’s criticism of the Aristotelian morals that it does not teach the realization of virtues therefore becomes an element also in Hobbes’ criticism of Aristotle. For the turn to history had taken place precisely because traditional philosophy showed no way to the application of norms. This failure is remedied by the new political philosophy, whose boast it is, that it, in contrast to traditional philosophy, teaches an applicable morality. Hobbes allows the validity of the aristocratic virtue, completing it by a morality, which is systematically applicable and which appeals to the greatest part of men. Hobbes acknowledges the binding force of the Ten Commandments and only denies that they are applicable without more detailed interpretation by the secular power. In the same way, Hobbes admits the natural inequality, and only contests that this inequality is of any practical importance. Hobbes also concedes that the Civil Government be ordained as a means to bring us to a spiritual felicity, and thus that all earthly things are means to eternal bliss. But he denies that from this hierarchy of things earthly and things eternal, anything can be deduced as to the relative position of the holder of secular power and the holder of spiritual power. With this, Hobbes lets us see that even if there were an eternal order, he would take into consideration only the actual behaviour of men, and that his whole interest is centered on man, on application, on the use of means.

The shifting of interest from the eternal order to man found its expression in turning of philosophy to history. Hobbes doesn’t have the intention justifying the traditional norms in a way more practicable for application than was the way of traditional philosophy; he altogether denies the applicability of traditional morals; whether of ancient philosophy or of Biblical Christianity.  He not only showed that Aristotle did not show the way to a way of realization of the norms, but also that he did not even rightly define the norms. Hobbes wishes to play the passions one against the other, in order to show the way for the realization of already established norms, he wishes to draw up a political philosophy which will be in harmony with the passions from the outset. And after Hobbes found in the fear of the violent death, a truly applicable principle of political philosophy, it is again in accordance with the interest in the application that he progresses from this foundation to the establishment of the law of nature. The right to defend life, which man has from nature by the reason of the inescapable fear of death, becomes a right to all things and all actions; since a right to the end is invalid without a right to the necessary means. In order to avoid the arbitrariness, the uncertainty of what a wise man would decide under unforeseen circumstances, he rules that each man has a right to all things and all actions, since anyone under some circumstances may consider that anything or action is a necessary means for the defence of his life. The express premise of this finding is the equality of all men. Since there is no natural order, the difference between the wise minority and the unwise majority loses the fundamental importance it had for traditional political philosophy. Hobbes’ political philosophy first pushes history back into its old insignificance for philosophy. To this extent, it is true to say that Hobbes’ political philosophy is unhistorical. To make this judgment cognizant is however, not so much that Hobbes took no interest in history as that he made incorrect assertions as to history being the basis of his political philosophy. Hobbes’ fundamental error was his assumption that man’s primitive condition was the war of everyone against everyone. Hobbes cannot rest content with findings as to the historical origin States, for they give no answer to the only important question, which concerns the right order of society. So in the criticism that Hobbes’ political philosophy is ‘unhistorical’, the only statement that is justified is that Hobbes considered the philosophic grounding of the principles of all judgment on political subjects more fundamental, incomparably more important than the most thoroughly founded historical knowledge.

Hobbes considers the State of nature not as an historical fact, but a necessary construction. It is essential to his political philosophy that it should begin with the description of the State of nature, and that it should let the State emerge from the State of nature.  But he acknowledges that the subject of his political philosophy, is a history, a genesis, and not an order, which is static and perfect. To clarify this point, one has to compare Hobbes’ ‘compositive’ method with Aristotle’s ‘genetic’ method. When Aristotle depicts the genesis of the city as the perfect community out of primitive communities, the understanding of perfect organism is the main presupposition for the understanding of its constituent parts, the more primitive communities. For Hobbes, the imperfection of the primitive condition, or the State of nature, is perceived not by looking to the already, even if only cursorily clarified, idea of the State as the perfect community, but by fully understanding the experience of the State of nature. As for Hobbes the primitive condition is irrational, so for Hegel

‘knowing as it is found at the start, mind in its immediate and primitive stages, is without the essential nature of mind, is sense-conciousness’.

Hobbes has no intention of measuring the imperfect by a standard that transcends it, but as they simply look on, while the imperfect by its own movement annuls itself, tests itself. This is the meaning of Hobbes’ argument that the man who wishes to remain in the State of nature contradicts himself, that the mutual fear that characterizes the State of nature is the motive for abolishing the State of nature. The premise for an immanent testing, which necessarily finds its expression within the framework of a typical history is for Hobbes, the rejection for the morality of obedience. For Hobbes, at all events, history finally becomes superfluous, because for him political philosophy itself becomes a typical history. His political philosophy becomes historical because for him order is not immutable, eternal, in existence from the beginning, but is produced only at the end of a process; because for him order is not independent of human volition, but is borne up by a human volition alone. For this, political philosophy no longer has the function, as it had in classical antiquity, of reminding political life of the eternally immutable prototype of the perfect State, but the peculiarly modern task of delineating for the first time the programme of the essentially future perfect State. The repression of history in favour of philosophy means in reality the repression of the past; of the ancient, which is an image of the eternal; in favour of the future.

If the order of man’s world springs from man’s will alone, there is no philosophical or theological security for that order. Man then can convince himself of his capacity to order his world only by the fact of his ordering activity. Therefore according to Hobbes’ assumptions, one must turn to real history. Thus, the State of nature, which at first was intended as merely typical, again takes on an historical significance; not, indeed, as a condition of absolute lack of order, but as a condition of extremely defective order. The real history has as its function to vouch for the possibility of further progress by perception of progress already made. After that; historically, perhaps even earlier; its function is to free man from the might of the past, from the authority of antiquity, from prejudices. Authority loses its prestige when its historical origin and evolution are traced; as a result of historical criticism man’s limitations show themselves as limits set by himself, and therefore to be over passed. It is by the doubt of the transcendent eternal order by which man’s reason was assumed to be guided and hence by the conviction of the impotence of reason, that first of all the turning of philosophy to history is caused, and then the process of historicizing philosophy itself.

Quantum Field Theory and Evolution of Forward Rates in Quantitative Finance. Note Quote.

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Applications of physics to finance are well known, and the application of quantum mechanics to the theory of option pricing is well known. Hence it is natural to utilize the formalism of quantum field theory to study the evolution of forward rates. Quantum field theory models of the term structure originated with Baaquie. The intuition behind quantum field theory models of the term structure stems from allowing each forward rate maturity to both evolve randomly and be imperfectly correlated with every other maturity. This may also be accomplished by increasing the number of random factors in the original HJM towards infinity. However, the infinite number of factors in a field theory model are linked via a single function that governs the correlation between forward rate maturities. Thus, instead of estimating additional volatility functions in a multifactor HJM framework, one additional parameter is sufficient for a field theory model to instill imperfect correlation between every forward rate maturity. As the correlation between forward rate maturities approaches unity, field theory models reduce to the standard one1 factor HJM model. Therefore, the fundamental difference between finite factor HJM and field theory models is the minimal structure the latter requires to instill imperfect correlation between forward rates. The Heath-Jarrow-Morton framework refers to a class of models that are derived by directly modeling the dynamics of instantaneous forward-rates. The central insight of this framework is to recognize that there is an explicit relationship between the drift and volatility parameters of the forward-rate dynamics in a no-arbitrage world. The familiar short-rate models can be derived in the HJM framework but in general, however, HJM models are non-Markovian. As a result, it is not possible to use the PDE-based computational approach for pricing derivatives. Instead, discrete-time HJM models and Monte-Carlo methods are often used in practice. Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle.

A Lagrangian is introduced to describe the field. The Lagrangian has the advantage over Brownian motion of being able to control fluctuations in the field, hence forward rates, with respect to maturity through the addition of a maturity dependent gradient as detailed in the definition below. The action of the field integrates the Lagrangian over time and when exponentiated and normalized serves as the probability distribution for forward rate curves. The propagator measures the correlation in the field and captures the effect the field at time t and maturity x has on maturity x′ at time t′. In the one factor HJM model, the propagator equals one which allows the quick recovery of one factor HJM results. Previous research has begun with the propagator or “correlation” function for the field instead of deriving this quantity from the Lagrangian. More importantly, the Lagrangian and its associated action generate a path integral that facilitates the solution of contingent claims and hedge parameters. However, previous term structure models have not defined the Lagrangian and are therefore unable to utilize the path integral in their applications. The Feynman path integral, path integral in short, is a fundamental quantity that provides a generating function for forward rate curves. Although crucial for pricing and hedging, the path integral has not appeared in previous term structure models with generalized continuous random processes.

Notation

Let t0 denote the current time and T the set of forward rate maturities with t0 ≤ T . The upper bound on the forward rate maturities is the constant TFR which constrains the forward rate maturities T to lie within the interval [t0, t0 + TFR].

To illustrate the field theory approach, the original finite factor HJM model is derived using field theory principles in appendix A. In the case of a one factor model, the derivation does not involve the propagator as the propagator is identically one when forward rates are perfectly correlated. However, the propagator is non trivial for field theory models as it governs the imperfect correlation between forward rate maturities. Let A(t,x) be a two dimensional field driving the evolution of forward rates f (t, x) through time. Following Baaquie, the Lagrangian of the field is defined as

Definition:

The Lagrangian of the field equals

L[A] = -1/2TFR  {A2(t, x) + 1/μ2(∂A(t,x)∂x)2} —– (1)

Definition is not unique, other Lagrangians exist and would imply different propagators. However, the Lagrangian in the definition is sufficient to explain the contribution of field theory ∂A(t,x)∂x  that controls field fluctuations in the direction of the forward rate maturity. The constant μ measures the strength of the fluctuations in the maturity direction. The Lagrangian in the definition implies the field is continuous, Gaussian, and Markovian. Forward rates involving the field are expressed below where the drift and volatility functions satisfy the usual regularity conditions.

∂f(t,x)/∂t = α (t, x) + σ (t, x)A(t, x) —– (2)

The forward rate process in equation (2) incorporates existing term structure research on Brown- ian sheets, stochastic strings, etc that have been used in previous continuous term structure models. Note that equation (2) is easily generalized to the K factor case by introducing K independent and identical fields Ai(t, x). Forward rates could then be defined as

∂f(t, x)/∂t = α (t, x) + ∑i=1K σi(t, x)Ai(t, x) —– (3)

However, a multifactor HJM model can be reproduced without introducing multiple fields. In fact, under specific correlation functions, the field theory model reduces to a multifactor HJM model without any additional fields to proxy for additional Brownian motions.

Proposition:

Lagrangian of Multifactor HJM

The Lagrangian describing the random process of a K-factor HJM model is given by

L[A] = −1/2 A(t, x)G−1(t, x, x′)A(t, x′) —– (4)

where

∂f(t, x)/∂t = α(t, x) + A(t, x)

and G−1(t, x, x′)A(t, x′) denotes the inverse of the function.

G(t, x, x′) = ∑i=1K σi(t, x) σi(t, x’) —– (5)

The above proposition is an interesting academic exercise to illustrate the parallel between field theory and traditional multifactor HJM models. However, multifactor HJM models have the disadvantages associated with a finite dimensional basis. Therefore, this approach is not pursued in later empirical work. In addition, it is possible for forward rates to be perfectly correlated within a segment of the forward rate curve but imperfectly correlated with forward rates in other segments. For example, one could designate short, medium, and long maturities of the forward rate curve. This situation is not identical to the multifactor HJM model but justifies certain market practices that distinguish between short, medium, and long term durations when hedging. However, more complicated correlation functions would be required; compromising model parsimony and reintroducing the same conceptual problems of finite factor models. Furthermore, there is little economic intuition to justify why the correlation between forward rates should be discontinuous.

Local Gauge Transformations in Locally Gauge Invariant Relativistic Field Theory

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The question arises of whether local space-time symmetries – arbitrary co-ordinate transformations that leave the explicit form of the equations of motion unaffected – also have an active interpretation. As in the case of local gauge symmetry, it has been argued in the literature that the introduction of a force is required to ‘restore’ local symmetry.  In the case of arbitrary co-ordinate transformations, the force invoked is gravity. Once again, we believe that the arguments (though seductive) are wrong, and that it is important to see why. Kosso’s discussion of arbitrary coordinate transformations is analogous to his argument with respect to local gauge transformations. He writes:

Observing this symmetry requires comparing experimental outcomes between two reference frames that are in variable relative motion, frames that are relatively accelerating or rotating….One can, in principle, observe that this sort of transformation has occurred. … just look out of the window and you can see if you are speeding up or turning with respect to some object that defines a coordinate system in the reference frame of the ground…Now do the experiments to see if the invariance is true. Do the same experiments in the original reference frame that is stationary on the ground, and again in the accelerating reference frame of the train, and see if the physics is the same. One can run the same experiments, with mechanical forces or with light and electromagnetic forces, and observe the results, so the invariance should be observable…But when the experiments are done, the invariance is not directly observed. Spurious forces appear in the accelerating system, objects move spontaneously, light bends, and so on. … The physics is different.

In other words, if we place ourselves at rest first in an inertial reference frame, and then in a non-inertial reference frame, our observations will be distinguishable. For example, in the non-inertial reference frame objects that are seemingly force-free will appear to accelerate, and so we will have to introduce extra, ‘spurious’, forces to account for this accelerated motion. The transformation described by Kosso is clearly not a symmetry transformation. Despite that, his claim appears to be that if we move to General Relativity, this transformation becomes a symmetry transformation. In order to assess this claim, let’s begin by considering Kosso’s experiment from the point of view of classical physics.

Suppose that we describe these observations using Newtonian physics and Maxwell’s equations. We would not be surprised that our descriptions differ depending on the choice of coordinate system: arbitrary coordinate transformations are not symmetries of the Newtonian and Maxwell equations of motion as usually expressed. Nevertheless, we are free to re-write Newtonian and Maxwellian physics in generally covariant form. But notice: the arbitrary coordinate transformations now apply not just to the Newtonian particles and the Maxwellian electromagnetic fields, but also to the metric, and this is necessary for general covariance.

Kosso’s example is given in terms of passive transformations – transformations of the coordinate systems in which we re-coordinatise the fields. In the Kosso experiment, however, we re-coordinatise the matter fields without re-coordinatising the metric field. This is not achieved by a mere coordinate transformation in generally covariant classical theory: a passive arbitrary coordinate transformation induces a re-coordinatisation of not only the matter fields but also the metric. The two states described by Kosso are not related by an arbitrary coordinate transformation in generally covariant classical theory. Further, such a coordinate transformation applied to only the matter and electromagnetic fields is not a symmetry of the equations of Newtonian and Maxwellian physics, regardless of whether those equations are written in generally covariant form.

Suppose that we use General Relativity to describe the above observations. Kosso suggests that in General Relativity the observations made in an inertial reference frame will indeed be related by a symmetry transformation to those made in a non-inertial reference frame. He writes:

The invariance can be restored by revising the physics, by adding a specific dynamical principle. This is why the local symmetry is a dynamical symmetry. We can add to the physics a claim about a specific force that restores the invariance. It is a force that exactly compensates for the local transform. In the case of the general theory of relativity the dynamical principle is the principle of equivalence, and the force is gravity. … With gravity included in the physics and with the windows of the train shuttered, there is no way to tell if the transformation, the acceleration, has taken place. That is, there is now no difference in the outcome of experiments between the transformed and untransformed systems. The force pulling objects to the back of the train could just as well be gravity. Thus the physics, all things including gravity considered, is invariant from one locally transformed frame to the next. The symmetry is restored.

This analysis mixes together the equivalence principle with the meaning of invariance under arbitrary coordinate transformations in a way which seems to us to be confused, with the consequence that the account of local symmetry in General Relativity is mistaken.

Einstein’s field equations are covariant under arbitrary smooth coordinate transformations. However, as with generally covariant Newtonian physics, these symmetry transformations are transformations of the matter fields (such as particles and electromagnetic radiation) combined with transformations of the metric. Kosso’s example, as we have already emphasised, re-coordinatises the matter fields without re-coordinatising the metric field. So, the two states described by Kosso are not related by an arbitrary coordinate transformation even in General Relativity. We can put the point vividly by locating ourselves at the origin of the coordinate system: I will always be able to tell whether the train, myself, and its other contents are all freely falling together, or whether there is a relative acceleration of the other contents relative to the train and me (in which case the other contents would appear to be flung around). This is completely independent of what coordinate system I use – my conclusion is the same regardless of whether I use a coordinate system at rest with respect to the train or one that is accelerating arbitrarily. (This coordinate independence is, of course, the symmetry that Kosso sought in the opening quotation above, but his analysis is mistaken.)

What, then, of the equivalence principle? The Kosso transformation leads to a physically and observationally distinct scenario, and the principle of equivalence is not relevant to the difference between those scenarios. What the principle of equivalence tells us is that the effect in the second scenario, where the contents of the train appear to accelerate to the back of the train, may be due to acceleration of the train in the absence of a gravitational field, or due to the presence of a gravitational field in which the contents of the train are in free fall but the train is not. Mere coordinate transformations cannot be used to bring real physical forces in and out of existence.

It is perhaps worthwhile briefly indicating the analogy between this case and the gauge case. Active arbitrary coordinate transformations in General Relativity involve transformations of both the matter fields and the metric, and they are symmetry transformations having no observable consequences. Coordinate transformations applied to the matter fields alone are no more symmetry transformations in General Relativity than they are in Newtonian physics (whether written in generally covariant form or not). Such transformations do have observational consequences. Analogously, local gauge transformations in locally gauge invariant relativistic field theory are transformations of both the particle fields and the gauge fields, and they are symmetry transformations having no observable consequences. Local phase transformations alone (i.e. local gauge transformations of the matter fields alone) are no more symmetries of this theory than they are of the globally phase invariant theory of free particles. Neither an arbitrary coordinate transformation in General Relativity, nor a local gauge transformation in locally gauge invariant relativistic field theory, can bring forces in and out of existence: no generation of gravitational effects, and no changes to the interference pattern.

Roger Penrose and Artificial Intelligence: Revenance from the Archives and the Archaic.

Let us have a look at Penrose and his criticisms of strong AI, and does he come out as a winner. His Emperor’s New Mind: Concerning Computers, Minds, and The Laws of Physics

sets out to deal a death blow to the project of strong AI. Even while showing humility, like in saying,

My point of view is an unconventional among physicists and is consequently one which is unlikely to be adopted, at present, by computer scientists or physiologists,

he is merely stressing on his speculative musings. Penrosian arguments ala Searle, are definitely opinionated, in making assertions like a conscious mind cannot work like a computer. He grants the possibility of artificial machines coming into existence, and even superseding humans (1), but at every moment remains convinced that algorithmic machines are doomed to subservience. Penrose’s arguments proceed through showing that human intelligence cannot be implemented by any Turing machine equivalent computer, and human mind as not algorithmically based that could capture the Turing machine equivalent. He is even sympathetic to Searle’s Chinese Room argument, despite showing some reservations against its conclusions.

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The speculative nature of his arguments question people as devices which compute that a Turing machine cannot, despite physical laws that allow such a construction of a device as a difficult venture. This is where his quantum theory sets in, with U and R (Unitary process and Reduction process respectively) acting on quantum states that help describe a quantum system. His U and R processes and the states they act upon are not only independent of observers, but at the same time real, thus branding him as a realist. What happens is an interpolation that occurs between Unitary Process and Reductive Process, a new procedure that essentially contains a non-algorithmic element takes shape, which effectuates a future that cannot be computable based on the present, even though it could be determined that way. This radically new concept which is applied to space-time is mapped onto the depths of brain’s structure, and for Penrose, the speculative possibility occurs in what he terms the Phenomenon of Brain Plasticity. As he says,

Somewhere within the depths of the brain, as yet unknown cells are to be found of single quantum sensitivity, such that synapses becoming activate or deactivated through the growth of contraction of dendritic spines…could be governed by something like the processes involved in quasi-crystal growth. Thus, not just one of the possible alternative arrangements is tried out, but vast numbers, all superposed in complex linear superposition.

From the above, it is deduced that the impact is only on the conscious mind, whereas the unconscious mind is left to do with algorithmic computationality. Why is this important for Penrose is, since, as a mathematician believes in the mathematical ideas as populating an ideal Platonic world, and which in turn is accessible only via the intellect. And harking back to the non-locality principle within quantum theory, it is clear that true intellect requires consciousness, and the mathematician’s conscious mind has a direct route to truth. In the meanwhile, there is a position in “many-worlds” (2) view that supersedes Penrose’s quantum realist one. This position rejects the Reduction Process in favor of Unitary Process, by terming the former as a mere illusion. Penrose shows his reservations against this view, as for him, a theory of consciousness needs to be in place prior to “many-worlds” view, and before the latter view could be squared with what one actually observes. Penrose is quite amazed at how many AI reviewers and researchers embrace the “many-worlds” hypothesis, and mocks at them, for their reasons being better supportive of validating AI project. In short, Penrose’s criticism of strong AI is based on the project’s assertion that consciousness can emerge by a complex system of algorithms, whereas for the thinker, a great many things humans involve in are intrinsically non-algorithmic in nature. For Penrose, a system can be deterministic without being algorithmic. He even uses the Turing’s halting theorem (3) to demonstrate the possibility of replication of consciousness. In a public lecture in Kolkata on the 4th of January 2011 (4), Penrose had this to say,

There are many things in Physics which are yet unknown. Unless we unravel them, we cannot think of creating real artificial intelligence. It cannot be achieved through the present system of computing which is based on mathematical algorithm. We need to be able to replicate human consciousness, which, I believe, is possible through physics and quantum mechanics. The good news is that recent experiments indicate that it is possible.

There is an apparent shift in Penrosean ideas via what he calls “correct quantum gravity”, which argues for the rational processes of the mind to be completely algorithmic and probably standing a bright chance to be duplicated by a sufficiently complex computing system. As he quoted from the same lecture in Kolkata,

A few years back, scientists at NASA had contemplated sending intelligent robots to space and sought my inputs. Even though we are still unable to create some device with genuine feelings and understanding, the question remains a disturbing one. Is it ethical to leave a machine which has consciousness in some faraway planet or in space? Honestly, we haven’t reached that stage yet. Having said that, I must add it may not be too far away, either. It is certainly a possibility.

Penrose does meet up with some sympathizers for his view, but his fellow-travelers do not tread along with him for a long distance. For example, in an interview with Sander Olson, Vernor Vinge, despite showing some reluctance to Penrose’s position, accepts that physical aspects of mind, or especially the quantum effects have not been studied in greater detail, but these quantum effects would simply be another thing to be learned with artifacts. Vinge does speculate on other paradigms that could be equally utilized for AI research hitting speed, rather than confining oneself to computer departments to bank on their progress. His speculations (5) have some parallel to what Penrose and Searle would hint at, albeit occasionally. Most of the work in AI could benefit, if AI, neural nets are closely connected to biological life. Rather than banking upon modeling and understanding of biological life with computers, if composite systems relying on biological life for guidance, or for providing features we do not understand quite well as yet to be implemented within the hardware, could be fathomed and made a reality, the program of AI would undoubtedly push the pedal to accelerate. There would probably be no disagreeing with what Aaron Saenz, Senior Editor of singularityhub.com said (6),

Artificial General Intelligence is one of the Holy Grails of science because it is almost mythical in its promise: not a system that simply learns, but one that reaches and exceeds our own kind of intelligence. A truly new form of advanced life. There are many brilliant people trying to find it. Each of these AI researchers have their own approach, their own expectations and their own history of failures and a precious few successes. The products you see on the market today are narrow AI-machines that have very limited ability to learn. As Scott Brown said, “today’s I technology is so primitive that much of the cleverness goes towards inventing business models that do not require good algorithms to succeed.” We’re in the infantile stages of AI. If that. Maybe the fetal stages.

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(1) This is quite apocalyptic sounding like the singularity notion of Ray Kurzweil, which is an extremely disruptive, world-altering event that has the potentiality of forever changing the course of human history. The extermination of humanity by violent machines is not impossible, since there would be no sharp distinctions between men and machines due to the existence of cybernetically enhanced humans and uploaded humans.

(2) “Many-worlds” view was first put forward by Hugh Everett in 1957. According to this view, evolution of state vector regarded realistically, is always governed by deterministic Unitary Process, while Reduction Process remains totally absent from such an evolutionary process. The interesting ramifications of this view are putting conscious observers at the center of the considerations, thus proving the basic assumption that quantum states corresponding to distinct conscious experiences have to be orthogonal (Simon 2009). On the technical side, ‘Orthogonal’ according to quantum mechanics is: two eigenstates of a Hermitian operator, ψm and ψn, are orthogonal if they correspond to different eigenvalues. This means, in Dirac notation, that < ψm | ψn > = 0 unless ψm and ψn correspond to the same eigenvalue. This follows from the fact that Schrödinger’s equation is a Sturm–Liouville equation (in Schrödinger’s formulation) or that observables are given by hermitian operators (in Heisenberg’s formulation).

(3) Halting problem is a decisional problem in computability theory, and is stated as: Given a description of a program, decide whether the program finishes running or continues to run, and will thereby run forever. Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. In a way, the halting problem is undecidable over Turing machines.

(4) Penrose, R. AI may soon become reality. Public lecture delivered in Kolkata on the 4th of Jan 2011. <http://timesofindia.indiatimes.com/city/kolkata-/AI-may-soon-become-reality- Penrose/articleshow/7219700.cms>

(5) Olson, S. Interview with Vernor Vinge in Nanotech.Biz <http://www.nanotech.biz/i.php?id=01_16_09&gt;

(6) Saenz, A. Will Vicarious Systems’ Silicon Valley Pedigree Help it Build AI? in singularityhub.com <http://singularityhub.com/2011/02/03/will-vicarious-systems-silicon-valley-pedigree-help-it-build-agi/&gt;

Cantorian Diagonal Slash

What is Cantor’s diagonal slash? This is often considered to be an absurd argument from physics point of view. Why is that so? The argument says that “infinity of reals is uncountable, and infinity of integers is countable”, thus giving us two different quantifiable infinities.

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What happens if we apply Cantor’s diagonal slash to the integers? How can this be done? Think of the infinite binary tree, which starts at ground level as the trunk and splits in two (bifurcates) as we ascend; if we take the left branch then we assign 0 to the first binary digit, if the right branch then 1. At the next level we do the same….0 if we go left, 1 if we go right…. and we ascend the binary tree all the way to infinity. In this way every possible infinite sequence of binary digits is represented somehow or other (mapped out) by a path up the tree; and every path up the tree corresponds to a unique infinite sequence of binary digits….“the number of routes up the tree is equal to 2n (where n is the number of levels), and the number of nodes (branches) in the tree is equal to (2n)-1.”

Now we can arrange all of these paths in an infinite x infinite binary matrix, of form similar to the infinite x infinite matrix of real numbers that Cantor used for his diagonal slash. What happens if we perform Cantor’s diagonal slash on this infinite binary matrix? According to Cantor, we produce a new infinite binary sequence which is NOT contained within the original matrix (nor is it contained on the infinite binary tree). But how can this be? The matrix contains every possible route up the binary tree, there IS no other route not contained in the matrix. What does this show? That Cantor’s diagonal slash argument is meaningless when applied to infinite matrices.