Reductionism of Numerical Complexity: A Wittgensteinian Excursion

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Wittgenstein’s criticism of Russell’s logicist foundation of mathematics contained in (Remarks on the Foundation of Mathematics) consists in saying that it is not the formalized version of mathematical deduction which vouches for the validity of the intuitive version but conversely.

If someone tries to shew that mathematics is not logic, what is he trying to shew? He is surely trying to say something like: If tables, chairs, cupboards, etc. are swathed in enough paper, certainly they will look spherical in the end.

He is not trying to shew that it is impossible that, for every mathematical proof, a Russellian proof can be constructed which (somehow) ‘corresponds’ to it, but rather that the acceptance of such a correspondence does not lean on logic.

Taking up Wittgenstein’s criticism, Hao Wang (Computation, Logic, Philosophy) discusses the view that mathematics “is” axiomatic set theory as one of several possible answers to the question “What is mathematics?”. Wang points out that this view is epistemologically worthless, at least as far as the task of understanding the feature of cognition guiding is concerned:

Mathematics is axiomatic set theory. In a definite sense, all mathematics can be derived from axiomatic set theory. [ . . . ] There are several objections to this identification. [ . . . ] This view leaves unexplained why, of all the possible consequences of set theory, we select only those which happen to be our mathematics today, and why certain mathematical concepts are more interesting than others. It does not help to give us an intuitive grasp of mathematics such as that possessed by a powerful mathematician. By burying, e.g., the individuality of natural numbers, it seeks to explain the more basic and the clearer by the more obscure. It is a little analogous to asserting that all physical objects, such as tables, chairs, etc., are spherical if we swathe them with enough stuff.

Reductionism is an age-old project; a close forerunner of its incarnation in set theory was the arithmetization program of the 19th century. It is interesting that one of its prominent representatives, Richard Dedekind (Essays on the Theory of Numbers), exhibited a quite distanced attitude towards a consequent carrying out of the program:

It appears as something self-evident and not new that every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers [ . . . ] But I see nothing meritorious [ . . . ] in actually performing this wearisome circumlocution and insisting on the use and recognition of no other than rational numbers.

Perec wrote a detective novel without using the letter ‘e’ (La disparition, English A void), thus proving not only that such an enormous enterprise is indeed possible but also that formal constraints sometimes have great aesthetic appeal. The translation of mathematical propositions into a poorer linguistic framework can easily be compared with such painful lipogrammatical exercises. In principle all logical connectives can be simulated in a framework exclusively using Sheffer’s stroke, and all cuts (in Gentzen’s sense) can be eliminated; one can do without common language at all in mathematics and formalize everything and so on: in principle, one could leave out a whole lot of things. However, in doing so one would depart from the true way of thinking employed by the mathematician (who really uses “and” and “not” and cuts and who does not reduce many things to formal systems). Obviously, it is the proof theorist as a working mathematician who is interested in things like the reduction to Sheffer’s stroke since they allow for more concise proofs by induction in the analysis of a logical calculus. Hence this proof theorist has much the same motives as a mathematician working on other problems who avoids a completely formalized treatment of these problems since he is not interested in the proof-theoretical aspect.

There might be quite similar reasons for the interest of some set theorists in expressing usual mathematical constructions exclusively with the expressive means of ZF (i.e., in terms of ∈). But beyond this, is there any philosophical interpretation of such a reduction? In the last analysis, mathematicians always transform (and that means: change) their objects of study in order to make them accessible to certain mathematical treatments. If one considers a mathematical concept as a tool, one does not only use it in a way different from the one in which it would be used if it were considered as an object; moreover, in semiotical representation of it, it is given a form which is different in both cases. In this sense, the proof theorist has to “change” the mathematical proof (which is his or her object of study to be treated with mathematical tools). When stating that something is used as object or as tool, we have always to ask: in which situation, or: by whom.

A second observation is that the translation of propositional formulæ in terms of Sheffer’s stroke in general yields quite complicated new formulæ. What is “simple” here is the particularly small number of symbols needed; but neither the semantics becomes clearer (p|q means “not both p and q”; cognitively, this looks more complex than “p and q” and so on), nor are the formulæ you get “short”. What is looked for in this case, hence, is a reduction of numerical complexity, while the primitive basis attained by the reduction cognitively looks less “natural” than the original situation (or, as Peirce expressed it, “the consciousness in the determined cognition is more lively than in the cognition which determines it”); similarly in the case of cut elimination. In contrast to this, many philosophers are convinced that the primitive basis of operating with sets constitutes really a “natural” basis of mathematical thinking, i.e., such operations are seen as the “standard bricks” of which this thinking is actually made – while no one will reasonably claim that expressions of the type p|q play a similar role for propositional logic. And yet: reduction to set theory does not really have the task of “explanation”. It is true, one thus reduces propositions about “complex” objects to propositions about “simple” objects; the propositions themselves, however, thus become in general more complex. Couched in Fregean terms, one can perhaps more easily grasp their denotation (since the denotation of a proposition is its truth value) but not their meaning. A more involved conceptual framework, however, might lead to simpler propositions (and in most cases has actually just been introduced in order to do so). A parallel argument concerns deductions: in its totality, a deduction becomes more complex (and less intelligible) by a decomposition into elementary steps.

Now, it will be subject to discussion whether in the case of some set operations it is admissible at all to claim that they are basic for thinking (which is certainly true in the case of the connectives of propositional logic). It is perfectly possible that the common sense which organizes the acceptance of certain operations as a natural basis relies on something different, not having the character of some eternal laws of thought: it relies on training.

Is it possible to observe that a surface is coloured red and blue; and not to observe that it is red? Imagine a kind of colour adjective were used for things that are half red and half blue: they are said to be ‘bu’. Now might not someone to be trained to observe whether something is bu; and not to observe whether it is also red? Such a man would then only know how to report: “bu” or “not bu”. And from the first report we could draw the conclusion that the thing was partly red.

Categorial Logic – Paracompleteness versus Paraconsistency. Thought of the Day 46.2

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The fact that logic is content-dependent opens a new horizon concerning the relationship of logic to ontology (or objectology). Although the classical concepts of a priori and a posteriori propositions (or judgments) has lately become rather blurred, there is an undeniable fact: it is certain that the far origin of mathematics is based on empirical practical knowledge, but nobody can claim that higher mathematics is empirical.

Thanks to category theory, it is an established fact that some sort of very important logical systems: the classical and the intuitionistic (with all its axiomatically enriched subsystems), can be interpreted through topoi. And these possibility permits to consider topoi, be it in a Noneist or in a Platonist way, as universes, that is, as ontologies or as objectologies. Now, the association of a topos with its correspondent ontology (or objectology) is quite different from the association of theoretical terms with empirical concepts. Within the frame of a physical theory, if a new fact is discovered in the laboratory, it must be explained through logical deduction (with the due initial conditions and some other details). If a logical conclusion is inferred from the fundamental hypotheses, it must be corroborated through empirical observation. And if the corroboration fails, the theory must be readjusted or even rejected.

In the case of categorial logic, the situation has some similarity with the former case; but we must be careful not to be influenced by apparent coincidences. If we add, as an axiom, the tertium non datur to the formalized intuitionistic logic we obtain classical logic. That is, we can formally pass from the one to the other, just by adding or suppressing the tertium. This fact could induce us to think that, just as in physics, if a logical theory, let’s say, intuitionistic logic, cannot include a true proposition, then its axioms must be readjusted, to make it possible to include it among its theorems. But there is a radical difference: in the semantics of intuitionistic logic, and of any logic, the point of departure is not a set of hypothetical propositions that must be corroborated through experiment; it is a set of propositions that are true under some interpretation. This set can be axiomatic or it can consist in rules of inference, but the theorems of the system are not submitted to verification. The derived propositions are just true, and nothing more. The logician surely tries to find new true propositions but, when they are found (through some effective method, that can be intuitive, as it is in Gödel’s theorem) there are only three possible cases: they can be formally derivable, they can be formally underivable, they can be formally neither derivable nor underivable, that is, undecidable. But undecidability does not induce the logician to readjust or to reject the theory. Nobody tries to add axioms or to diminish them. In physics, when we are handling a theory T, and a new describable phenomenon is found that cannot be deduced from the axioms (plus initial or some other conditions), T must be readjusted or even rejected. A classical logician will never think of changing the axioms or rules of inference of classical logic because it is undecidable. And an intuitionist logician would not care at all to add the tertium to the axioms of Heyting’s system because it cannot be derived within it.

The foregoing considerations sufficiently show that in logic and mathematics there is something that, with full right, can be called “a priori“. And although, as we have said, we must acknowledge that the concepts of a priori and a posteriori are not clear-cut, in some cases, we can rightly speak of synthetical a priori knowledge. For instance, the Gödel’s proposition that affirms its own underivabilty is synthetical and a priori. But there are other propositions, for instance, mathematical induction, that can also be considered as synthetical and a priori. And a great deal of mathematical definitions, that are not abbreviations, are synthetical. For instance, the definition of a monoid action is synthetical (and, of course, a priori) because the concept of a monoid does not have among its characterizing traits the concept of an action, and vice versa.

Categorial logic is, the deepest knowledge of logic that has ever been achieved. But its scope does not encompass the whole field of logic. There are other kinds of logic that are also important and, if we intend to know, as much as possible, what logic is and how it is related to mathematics and ontology (or objectology), we must pay attention to them. From a mathematical and a philosophical point of view, the most important logical non-paracomplete systems are the paraconsistent ones. These systems are something like a dual to paracomplete logics. They are employed in inconsistent theories without producing triviality (in this sense also relevant logics are paraconsistent). In intuitionist logic there are interpretations that, with respect to some topoi, include two false contradictory propositions; whereas in paraconsistent systems we can find interpretations in which there are two contradictory true propositions.

There is, though, a difference between paracompleteness and paraconsistency. Insofar as mathematics is concerned, paracomplete systems had to be coined to cope with very deep problems. The paraconsistent ones, on the other hand, although they have been applied with success to mathematical theories, were conceived for purely philosophical and, in some cases, even for political and ideological motivations. The common point of them all was the need to construe a logical system able to cope with contradictions. That means: to have at one’s disposal a deductive method which offered the possibility of deducing consistent conclusions from inconsistent premisses. Of course, the inconsistency of the premisses had to comply with some (although very wide) conditions to avoid triviality. But these conditions made it possible to cope with paradoxes or antinomies with precision and mathematical sense.

But, philosophically, paraconsistent logic has another very important property: it is used in a spontaneous way to formalize the naive set theory, that is, the kind of theory that pre-Zermelian mathematicians had always employed. And it is, no doubt, important to try to develop mathematics within the frame of naive, spontaneous, mathematical thought, without falling into the artificiality of modern set theory. The formalization of the naive way of mathematical thinking, although every formalization is unavoidably artificial, has opened the possibility of coping with dialectical thought.

Conjuncted: Internal Logic. Thought of the Day 46.1

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So, what exactly is an internal logic? The concept of topos is a generalization of the concept of set. In the categorial language of topoi, the universe of sets is just a topos. The consequence of this generalization is that the universe, or better the conglomerate, of topoi is of overwhelming amplitude. In set theory, the logic employed in the derivation of its theorems is classical. For this reason, the propositions about the different properties of sets are two-valued. There can only be true or false propositions. The traditional fundamental principles: identity, contradiction and excluded third, are absolutely valid.

But if the concept of a topos is a generalization of the concept of set, it is obvious that the logic needed to study, by means of deduction, the properties of all non-set-theoretical topoi, cannot be classic. If it were so, all topoi would coincide with the universe of sets. This fact suggests that to deductively study the properties of a topos, a non-classical logic must be used. And this logic cannot be other than the internal logic of the topos. We know, presently, that the internal logic of all topoi is intuitionistic logic as formalized by Heyting (a disciple of Brouwer). It is very interesting to compare the formal system of classical logic with the intuitionistic one. If both systems are axiomatized, the axioms of classical logic encompass the axioms of intuitionistic logic. The latter has all the axioms of the former, except one: the axiom that formally corresponds to the principle of the excluded middle. This difference can be shown in all kinds of equivalent versions of both logics. But, as Mac Lane says, “in the long run, mathematics is essentially axiomatic.” (Mac Lane). And it is remarkable that, just by suppressing an axiom of classical logic, the soundness of the theory (i.e., intuitionistic logic) can be demonstrated only through the existence of a potentially infinite set of truth-values.

We see, then, that the appellation “internal” is due to the fact that the logic by means of which we study the properties of a topos is a logic that functions within the topos, just as classical logic functions within set theory. As a matter of fact, classical logic is the internal logic of the universe of sets.

Another consequence of the fact that the general internal logic of every topos is the intuitionistic one, is that many different axioms can be added to the axioms of intuitionistic logic. This possibility enriches the internal logic of topoi. Through its application it reveals many new and quite unexpected properties of topoi. This enrichment of logic cannot be made in classical logic because, if we add one or more axioms to it, the new system becomes redundant or inconsistent. This does not happen with intuitionistic logic. So, topos theory shows that classical logic, although very powerful concerning the amount of the resulting theorems, is limited in its mathematical applications. It cannot be applied to study the properties of a mathematical system that cannot be reduced to the system of sets. Of course, if we want, we can utilize classical logic to study the properties of a topos. But, then, there are important properties of the topos that cannot be known, they are occult in the interior of the topos. Classical logic remains external to the topos.

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.