Entangled State Vectors

Let R₁, R₂ be von Neumann algebras on H such that R₁ ⊆ R′₂. Recall that a state ω of R₁₂ is called a normal product state just in case ω is normal, and there are states ω₁ of R₁ and ω₂ of R₂ such that

ω(AB) = ω1(A)ω2(B) —– (1)

∀ A ∈ R₁, B ∈ R₂. Werner, in dealing with the case of B(Cⁿ) ⊗ B(Cⁿ), defined a density operator D to be classically correlated — the term separable is now more commonly used — just in case D can be approximated in trace norm by convex combinations of density operators of form D₁ ⊗ D₂. Although Werner’s definition of nonseparable states directly generalizes the traditional notion of pure entangled states, he showed that a nonseparable mixed state need not violate a Bell inequality; thus, Bell correlation is in general a sufficient, though not necessary condition for a state’s being non-separable. On the other hand, it has since been shown that nonseparable states often possess more subtle forms of nonlocality, which may be indicated by measurements more general than the single ideal measurements which can indicate Bell correlation.

In terms of the linear functional representation of states, Werner’s separable states are those in the norm closed convex hull of the product states of B(Cⁿ) ⊗ B(Cⁿ). However, in case of the more general setup — i.e., R₁ ⊆ R′₂, where R₁, R₂ are arbitrary von Neumann algebras on H — the choice of topology on the normal state space of R₁₂ will yield in general different definitions of separability. Moreover, it has been argued that norm convergence of a sequence of states can never be verified in the laboratory, and as a result, the appropriate notion of physical approximation is given by the (weaker) weak-∗ topology. And the weak-∗ and norm topologies do not generally coincide even on the normal state space.

For the next proposition, then, we will suppose that the separable states of R₁₂ are those normal states in the weak-∗ closed convex hull of the normal product states. Note that β(ω) = 1 if ω is a product state, and since β is a convex function on the state space, β(ω) = 1 if ω is a convex combination of product states. Furthermore, since β is lower semicontinuous in the weak-∗ topology, β(ω) = 1 for any separable state. Conversely, any Bell correlated state must be nonseparable.

We now introduce some notation that will aid us in the proof of our result. For a state ω of the von Neumann algebra R and an operator A ∈ R, define the state ω^A on R by

ω^A(X) ≡ ω(A∗XA)/ω(A∗A) —– (2)

if ω(A∗A) ≠ 0, and let ω^A = ω otherwise. Suppose now that ω(A∗A) ≠ 0 and ω is a convex combination of states:

ω = ∑_i=1ⁿλ_iω_i —– (3)

Then, letting λ^A_i ≡ ω(A∗A)⁻¹ω_i(A∗A)λ_i, ω^A is again a convex combination

ω^A = ∑_i=1ⁿ λ^A_iω^A_i —– (4)

Moreover, it is not difficult to see that the map ω → ω^A is weak-∗ continuous at any point ρ such that ρ(A∗A) ≠ 0. Indeed, let O₁ = N(ρ^A : X₁,…,X_n, δ) be a weak-∗ neighborhood of ρ_A. Then, taking O₂ = N(ρ : A^∗A,A^∗X₁A,…,A^∗X_nA, δ) and ω ∈ O₂, we have

|ρ(A^∗X_iA) − ω(A^∗X_iA)| < δ —– (5)

for i = 1,…,n, and

|ρ(A^∗A) − ω(A^∗A)| < δ —– (6)

By choosing δ < ρ(A^∗A) ≠ 0, we also have ω(A^∗A) ≠ 0, and thus

|ρ^A(X_i) − ω^A(X_i)| < O(δ) ≤ δ —– (7)

for an appropriate choice of δ. That is, ω^A ∈ O₁ forall ω ∈ O₂ and ω → ω^A is weak-∗ continuous at ρ.

Specializing to the case where R₁ ⊆ R′₂, and R₁₂ = {R₁ ∪ R₂}”, it is clear from the above that for any normal product state ω of R₁₂ and for A ∈ R₁, ω^A is again a normal product state. The same is true if ω is a convex combination of normal product states, or the weak-∗ limit of such combinations. Summarizing the results of this discussion in the following lemma:

Lemma: For any separable state ω of R₁₂ and any A ∈ R₁, ω^A is again separable.

Proposition: Let R₁,R₂ be nonabelian von Neumann algebras such that R₁ ⊆ R′₂. If x is cyclic for R₁, then ω_x is nonseparable across R₁₂.

Proof: There is a normal state ρ of R₁₂ such that β(ρ) = √2. But since all normal states are in the (norm) closed convex hull of vector states, and since β is norm continuous and convex, there is a vector v ∈ S such that β(v) > 1. By the continuity of β (on S), there is an open neighborhood O of v in S such that β(y) > 1 ∀ y∈O. Since x is cyclic for R₁,there is an A ∈ R₁ such that Ax ∈ O. Thus, β(Ax) > 1 which entails that ω_Ax = (ω_x)^A is a nonseparable state for R₁₂. This, by the preceding lemma, entails that ω_x is nonseparable.

Note that if R₁ has at least one cyclic vector x ∈ S, then R₁ has a dense set of cyclic vectors in S. Since each of the corresponding vector states is nonseparable across R₁₂, Proposition shows that if R₁ has a cyclic vector, then the (open) set of vectors inducing nonseparable states across R₁₂ is dense in S. On the other hand, since the existence of a cyclic vector for R₁ is not invariant under isomorphisms of R₁₂, Proposition does not entail that if R₁ has a cyclic vector, then there is a norm dense set of nonseparable states in the entire normal state space of R₁₂. Indeed, if we let R₁ = B(C²) ⊗ I, R₂ = I ⊗ B(C²), then any entangled state vector is cyclic for R₁; but, the set of nonseparable states of B(C²) ⊗ B(C²) is not norm dense. However, if in addition to R₁ or R₂ having a cyclic vector, R₁₂ has a separating vector (as is often the case in quantum field theory), then all normal states of R₁₂ are vector states, and it follows that the nonseparable states will be norm dense in the entire normal state space of R₁₂.

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Loop Quantum Gravity and Nature of Reality. Briefer.

To some “loop quantum gravity is an attempt to define a quantization of gravity paying special attention to the conceptual lessons of general relativity”, while to others it does not have to be about the quantization of gravity but should be “at least conceivable that such a theory marries a classical understanding of gravity with a quantum understanding of matter”

The term ‘loop’ comes from the solution written for every line closed on itself on the proposed structure of quanta’s interactions. John Archibald Wheeler was one of the pioneers in constructing a representation of space which had a granular structure on a very small scale. Together with Bryce DeWitt they produced a mathematical formula known as Wheeler-DeWitt equation, “an equation which should determine the probability of one or another curved space”. The starting point was spacetime of general relativity having “loop-like states”. Having a quantum approach to gravity on closed loops, which are threads of the Faraday lines of the quantum field, constitutes a gravitational field which looks like a spiderweb. A solution could be written for every line closed on itself. Moreover, every line determining a solution of the Wheeler-DeWitt equation describes one of the threads of the spiderweb created by Faraday force lines of the quantum field which are the threads with which the space is woven. The physical Hilbert space as the space of all quantum states of the theory solves all the constraints and thus ought to be considered as the physical states. This implies that the physical Hilbert space of Loop Quantum Gravity is not yet known. The larger space of states which satisfy the first two families of constraints is often termed the kinematical Hilbert space. The one constraint that has so far resisted resolution is the Hamiltonian constraint equation with the seemingly simple form Hˆ|ψ⟩ = 0, the Wheeler-DeWitt equation, where Hˆ is the Hamiltonian operator usually interpreted to generate the dynamical evolution and |ψ⟩ is a quantum state in the kinematical Hilbert space. Of course, the Hamiltonian operator Hˆ is a complicated function(al) of the basic operators corresponding to the basic canonical variables. In fact, the very functional form of Hˆ is debated as several inequivalent candidates are on the table. Insofar as the physical Hilbert space has thus not yet been constructed, Loop Quantum Gravity remains incomplete.

Space, then, is defined based on the nodes on this spiderweb, which is called a spin network, and time, which already lost its fundamental status with special and general relativity, vanishes from the picture of the universe altogether.

Loop quantum gravity combines the dynamic spacetime approach of general relativity with quanta nature of gravity fields. Accordingly, space that bends and stretches are made up of very small particles which are called quanta of space. If one had eyes capable zooming into the space and seeing magnetic fields and quanta, then, by observing the space, one would first witness the quantum field, and then would end up seeing quanta which are extremely small and granular.

Conjuncted: Axiomatizing Artificial Intelligence. Note Quote.

Solomonoff’s work was seminal in that he has single-handedly axiomatized AI, discovering the minimal necessary conditions for any machine to attain general intelligence.

Informally, these axioms are:

AI0 AI must have in its possession a universal computer M (Universality). AI1 AI must be able to learn any solution expressed in M’s code (Learning recursive solutions).
AI2 AI must use probabilistic prediction (Bayes’ theorem).
AI3 AI must embody in its learning a principle of induction (Occam’s razor).

While it may be possible to give a more compact characterization, these are ultimately what is necessary for the kind of general learning that Solomonoff induction achieves. ALP can be seen as a complete formalization of Occam’s razor (as well as Epicurus’s principle)  and thus serve as the foundation of universal induction, capable of solving all AI problems of significance. The axioms are important because they allow us to assess whether a system is capable of general intelligence or not.

Obviously, AI1 entails AI0, therefore AI0 is redundant, and can be omitted entirely, however we stated it separately only for historical reasons, as one of the landmarks of early AI research, in retrospect, was the invention of the universal computer, which goes back to Leibniz’s idea of a universal language (characteristica universalis) that can express every statement in science and mathematics, and has found its perfect embodiment in Turing’s research. A related achievement of early AI was the development of LISP, a universal computer based on lambda calculus (which is a functional model of computation) that has shaped much of early AI research.

Minimum Message Length (MML) principle introduced in 1968 is a formalization of induction developed within the framework of classical information theory, which establishes a trade-off between model complexity and fit-to-data by finding the minimal message that encodes both the model and the data. This trade-off is quite similar to the earlier forms of induction that Solomonoff developed, however independently discovered. Dowe points out that Occam’s razor means choosing the simplest single theory when data is equally matched, which MML formalizes perfectly (and is functional otherwise in the case of inequal fits) while Solomonoff induction maintains a mixture of alternative solutions. However, it was Solomonoff who first observed the importance of universality for AI (AI0-AI1). The plurality of probabilistic approaches to induction supports the importance of AI3 (as well as hinting that diversity of solutions may be useful). AI2, however, does not require much explanation. 

Conceptual Jump Size & Solomonoff Induction. Note Quote.

Let M be a reference machine which corresponds to a universal computer with a prefix-free code. In a prefix-free code, no code is a prefix of another. This is also called a self-delimiting code, as most reasonable computer programming languages are. Ray Solomonoff inquired the probability that an output string x is generated by M considering the whole space of possible programs. By giving each program bitstring p an a priori probability of 2^−|p|, we can ensure that the space of programs meets the probability axioms by the extended Kraft inequality. An instantaneous code (prefix code, tree code) with the code word lengths l₁,…,l_N exists if and only if

∑_i=1^N 2^-L_i ≤ 1

In other words, we imagine that we toss a fair coin to generate each bit of a random program. This probability model of programs entails the following probability mass function (p.m.f.) for strings x ∈ {0, 1}∗:

P_M(x) = ∑_M(p)=x* 2^-|p| —– (1)

which is the probability that a random program will output a prefix of x. P_M(x) is called the algorithmic probability of x for it assumes the definition of program-based probability.

Using this probability model of bitstrings, one can make predictions. Intuitively, we can state that it is impossible to imagine intelligence in the absence of any prediction ability: purely random behavior is decisively non-intelligent. Since, P is a universal probability model, it can be used as the basis of universal prediction, and thus intelligence. Perhaps, Solomonoff’s most significant contributions were in the field of AI, as he envisioned a machine that can learn anything from scratch.

His main proposal for machine learning is inductive inference (Part 1, Part 2), for a variety of problems such as sequence prediction, set induction, operator induction and grammar induction. Without much loss of generality, we can discuss sequence prediction on bitstrings. Assume that there is a computable p.m.f. of bitstrings P₁. Given a bitstring x drawn from P₁, we can define the conditional probability of the next bit simply by normalizing. Algorithmically, we would have to approximate (1) by finding short programs that generate x (the shortest of which is the most probable). In more general induction, we run all models in parallel, quantifying fit-to-data, weighed by the algorithmic probability of the model, to find the best models and construct distributions; the common point being determining good models with high a priori probability. Finding the shortest program in general is undecidable, however, Levin search can be used for this purpose. There are two important results about Solomonoff induction that we shall mention here. First, Solomonoff induction converges very rapidly to the real probability distribution. The convergence theorem shows that the expected total square error is related only to the algorithmic complexity of P₁, which is independent from x. The following bound is discussed at length with a concise proof:

E_P [∑_m=1ⁿ (P(a_m+1 = 1|a₁a₂ …a_m) – P₁(a_m+1 = 1|a₁a₂…a_m))²] ≤ -1/2 ln P(P₁) —– (2)

This bound characterizes the divergence of the Algorithmic Probability (ALP) solution from the real probability distribution P₁. P(P₁) is the a priori probability of P₁ p.m.f. according to our universal distribution P_M. On the right hand side of (2), −lnP_M(P₁) is roughly kln2 where k is the Kolmogorov complexity of P₁ (the length of the shortest program that defines it), thus the total expected error is bounded by a constant, which guarantees that the error decreases very rapidly as example size increases. In algorithmic information theory, the Kolmogorov complexity of an object, such as a piece of text, is the length of the shortest computer program  that produces the object as output. It is measure of the computational resources needed to specify the object, and is also known as descriptive complexity, Kolmogorov–Chaitin complexity, algorithmic entropy, or program-size complexity. Secondly, there is an optimal search algorithm to approximate Solomonoff induction, which adopts Levin’s universal search method to solve the problem of universal induction. Universal search procedure time-shares all candidate programs according to their a priori probability with a clever watch-dog policy to avoid the practical impact of the undecidability of the halting problem. The search procedure starts with a time limit t = t₀, in its iteration tries all candidate programs c with a time limit of t.P(c), and while a solution is not found, it doubles the time limit t. The time t(s)/P (s) for a solution program s taking time t(s) is called the Conceptual Jump Size (CJS), and it is easily shown that Levin Search terminates in at most 2.CJS time. To obtain alternative solutions, one may keep running after the first solution is found, as there may be more probable solutions that need more time. The optimal solution is computable only in the limit, which turns out to be a desirable property of Solomonoff induction, as it is complete and uncomputable.

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

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.