Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for (...) theories of valued fields. (shrink)

Recent years have seen an increasing interest in the study of continuous-time computational models. However, not so much has been done with respect to setting up a complexity theoretic framework for such models. The present paper intends to go a step into this direction. We consider problems over the real numbers which we try to relate to Lyapunov theory for dynamical systems: The global minimizers of particular energy functions are supposed to give solutions of the problem. (...) The structure of such energy functions leads to the introduction of problem classes U and NU; for the systems we are considering they parallel the classical complexity classes P and NP. We then introduce a notion of reducibility among problems and show existence of complete problems for NU and for PU, a polynomial hierarchy of continuous-time problems.For previous work on the computational capabilities of continuous-time systems see the surveys by Cris Moore [9] and by Pekka Orponen [10]. Our paper presents a step into the direction of creating a general framework for a complexity theory of continuous-time systems as outlined in [10]. It is closely related to work done by A. Ben-Hur, H. Siegelmann, and S. Fishman [12, 11]. (shrink)

We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. (...) Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f $\epsilon \mathcal{C}[0, 1]$ computes a non-computable subset of $\mathbb{N}$ ; there is a non-total degree between Turing degrees $a _\eqslantless_{\tau}$ b iff b is a PA degree relative to a; $\mathcal{S} \subseteq 2^{\mathbb{N}}$ is a Scott set iff it is the collection of f-computable subsets of $\mathbb{N}$ for some f $\epsilon \mathcal{C}[O, 1]$ of non-total degree; and there are computably incomparable f, g $\epsilon \mathcal{C}[0, 1]$ which compute exactly the same subsets of $\mathbb{N}$ . Proofs draw from classical analysis and constructive analysis as well as from computability theory. (shrink)

Using Type‐2 theory of effectivity, we define computability notions on the spaces of Lebesgue‐integrable functions on the real line that are based on two natural approaches to integrability from measure theory. We show that Fourier transform and convolution on these spaces are computable operators with respect to these representations. By means of the orthonormal basis of Hermite functions in L2, we show the existence of a linear complexity bound for the Fourier transform. (© 2004 (...) WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

These supporting concepts enable us to much more adequately understand the meaning of apoha. First, a sharp distinction is drawn between the real and the conceptual; the real is particular, unique, momentary and the basis of perception, while the conceptual is universal, general, only supposedly objective and the basis of language. Second, the complex nature of negation discloses the kind of negation meant by apoha. Negation by implication is seen as disclosing the necessary relation between simple affirmations and (...) simple negations. It is in this sense that Dignaga asserts that the meaning of words lies in the negation of the opposite. Third, the idea of apoha as the differentiation of concepts is to be distinguished from the Nyāya-Vaiśe $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s}$$ ika notion of exclusion as a materially present quality in objects.Ratnakirti, and Mok $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s}$$ ākaragupta following him, use the idea of negation by implication to explain apoha as the simultaneity of and logical relation between a positive assertion concerning an epistemological object and a negative assertion concerning all things which are not that epistemological object. For example, Ratnakirti in accordance with the theory of apoha explains that the term cow “harbors” the negation of non-cow in the same way that blue is harbored in the term blue lotus. The exclusion of the other, i.e., the negation of the non-cow, is held by Ratnakirti to be an actually apprehended attribute of the cow cognized (as distinct from the cow in itself). He says “... discrimination of non-cow inevitably arises simultaneously with the perception of the cow, for (the negation of non-cow) is the qualifier (of the individual).” This interaction between the positive and the negation of the other, i.e., the relation of negation by implication, as a simultaneous interaction is necessary lest when, e.g., asked to tie up a cow one ties up a horse instead.Mok $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s}$$ ākaragupta denies three possible interpretations of apoha before putting forth his own definition. The first is that when an external object is conceptualized this is done as “the other (the dissimilar) is discriminated from this.”The second is much the same, only it is in terms of the mental image, i.e., the epistemological object rather than the ontological object as in the first. The third is that apoha means the negation of an assertion, i.e., the simple absence of the other. All three of these interpretations are unacceptable to Mok $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s}$$ ākaragupta. He defines apoha as “the affirmation (of a positive entity) qualified by discrimination (of it from all other entities).”Mok $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s}$$ ākaragupta goes on to explain this simultaneity more fully: ... those who stress affirmation think that after we have known the cow, we consequently determine the discrimination of the essence of the cow from that of the non-cow; those who stress the negative function of apoha are of the opinion that we first know the discrimination of the dissimilar thing and then consequently confirm the thing which is discriminated from others, viz., the cow. Thus (both interpretations) are wrong. For at the time of judgment we do not experience an order of comprehension in which (negation or affirmation) occurs first. In fact, it is not the case that one, having understood the affirmation, later confirms the negation by implication, or that one having understood the negation later confirms what is discriminated (from the dissimilar). Therefore, we say that the very understanding of the cow is the same as the understanding of what is discriminated from the dissimilar.We can see from this that for Ratnakirti and Mok $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s}$$ ākaragupta apoha meant the simultaneous affirmation of and negation of the dissimilar from the (epistemological) object. Mok $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s}$$ ākaragupta's concern with the “time of judgment” also shows that the idea of apoha was understood by them as being psychologically descriptive. In the process of identifying an epistemological object as some particular thing, neither the negative nor the positive function have priority but are, rather, complementary one to the other. This simultaneity and complementarity are supported by the examination of mind “at the time of judgment,” i.e., it is a psychological description of the process of identification of an epistemological object as some particular thing.It is this fact of apoha describing the process of identification which, as described above, places apoha in the inference category of sources of truth. Apoha is clearly, therefore, involved with words and concepts rather than being on the perception side where it would have to be a metaphysical concept describing the real or absolute.Putting the idea of apoha into contemporary terms we can say that it is a psychological descriptor for the bipolar relation of concepts. Dignaga and the other Buddhist logicians are then to be understood here as being involved in producing a phenomenological description of linguistic thought processes. More broadly, as in the discussion of perception, they are involved in the phenomenological description of consciousness, a favorite Buddhist undertaking. As such, the work of the Buddhist logicians fits into the larger soteriological intention of all Buddhist thinkers, since enlightenment is understood to follow from insight into the nature of consciousness as productive of suffering.Some authors have criticized the logicians as being overly concerned with the un-Buddhist occupation of arguing. However, when their work is seen as expanding our understanding of the nature and working of consciousness, the logicians are certainly worthy of our respect. (shrink)

We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational (...) class='Hi'>complexity in a BSS-like manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper. (shrink)

The answer to the question 'what is money?' has changed throughout history. During the Gold Standard era, money was seen as gold or silver (the theory known as bullionism). In the early 20th century, the alternative theory known as chartalism proposed that money was a token chosen by the state for payment of taxes. Today, many economists take an agnostic line, and argue that money is best defined in terms of its function, e.g. as a neutral medium of (...) exchange. This paper argues that none of these approaches adequately describes the nature of money, and proposes a new theory, inspired by non-Newtonian physics, which takes into account the dualistic real/virtual properties and complex nature of money. The theory is applied to the example of the emergence of cybercurrencies. (shrink)

We study the computational complexity of finding a line that bisects simultaneously two sets in the two-dimensional plane, called the pancake problem, using the oracle Turing machine model of Ko. We also study the basic problem of bisecting a set at a given direction. Our main results are: (1) The complexity of bisecting a nice (thick) polynomial-time approximable set at a given direction can be characterized by the counting class #P. (2) The complexity of bisecting simultaneously two (...) linearly separable nice (thick) polynomial-time approximable sets can be characterized by the counting class #P. (3) For either of these two problems, without the thickness condition and the linear separability condition (for the two-set case), it is arbitrarily hard to compute the bisector, even if it is unique. (shrink)

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver [20] and Harrington, Kechris and Louveau [6] show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on , the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on . In this article we examine the (...) effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene’s O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is that of the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington establishing for any recursive ordinal α the existence of singletons whose α-jumps are Turing incomparable. (shrink)

There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The (...) decision problem for entanglement, in: R.S. Cohen et al. (Eds.), Potentiality, Entanglement, and Passion-at-a-Distance: Quantum Mechanical Studies fo Abner Shimony, Vol. 2, Kluwer Academic Publishers, Great Britain, 1997, pp. 177–190]. Unlike some others in the literature, these notions apply not only to certain nice sets, but to general sets in Rn and other appropriate spaces. We consider some motivations for these concepts and the logical relations between them. It has been argued that d.m.z. is especially appropriate for physical applications, and on Rn with the standard measure, it is strictly stronger than r.a. [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382]. Here we show that this is the only implication that holds among our three decidabilities in that setting. Under arbitrary measures, even this implication fails. Yet for intervals of non-zero length, and more generally, convex sets of non-zero measure, the three concepts are equivalent. (shrink)

Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every (...) definable function which is meromorphic on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting. (shrink)

The neutrosophic cubic hesitant fuzzy set can efficiently handle the complex information in a decision-making problem because it combines the advantages of the neutrosophic cubic set and the hesitant fuzzy set. The algebraic operations based on Dombi norms and co-norms are more flexible than the usual algebraic operations as they involve an operational parameter. First, this paper establishes Dombi algebraic operational laws, score functions, and similarity measures in neutrosophic cubic hesitant fuzzy sets. Then, we proposed Dombi exponential operational laws (...) in which the exponents are neutrosophic cubic hesitant fuzzy values and bases are positive real numbers. To use neutrosophic cubic hesitant fuzzy sets in decision-making, we are developing Dombi exponential aggregation operators in the current study. In the end, we present applications of exponential aggregation operators in multiattribute decision-making problems. (shrink)

Research on the oxidation of alloys supports the claim that natural scientists can and do use ideal type concepts when confronted with analytical or computational intractability. In opposition to those who collapse ideal types into 'standard' theoretical concepts, I argue ideal types possess a unique structure, function and axiology. In phenomenologically complex situations, scientists use these features to articulate experiment with theory generally and in particular to discover new boundary conditions. This conceptual articulation is achieved using models rather than (...) objective perceptual attributes alone. The analysis supports a claim of local rather than global identities of methodology. (shrink)

Arthur Kaufmann is one of the most prominent figures among the contemporary philosophers of law in German speaking countries. For many years he was a director of the Institute of Philosophy of Law and Computer Sciences for Law at the University in Munich. Presently, he is a retired professor of this university. Rare in the contemporary legal thought, Arthur Kaufmann's philosophy of law is one with the highest ambitions — it aspires to pinpoint the ultimate foundations of law by explicitly (...) proposing an ontology, a general theory of knowledge and concept of a person. Kaufmann's work derives, first of all, from the thinking of Gustav Radburch, his teacher, and then from ideas of Karl Engish and Hans-Georg Gadamer. The philosophy undertakes to pursue the ultimate foundation of law, law which is understood by Kaufmann, first of all, as a "concrete judgement" that is, what is right in a concrete situation. Justice belongs to the essence of law and "unjust law" is contradictio in adiectio. Kaufmann opposes all those theories, which as the only foundation for establishing just law (Recht) adopt legal norms (Gesetz). In Kaufmann's opinion , such theories are powerless in the face of all types of distortions of law rendered by political forces. He suggests that the basic phenomenon which needs to be explained and which cannot be disregarded by a philosopher of law is so-called "legal lawlessness" ("Gestzliches Unrecht"). "Legal lawlessness" which forms a part of life experience for the people of twentieth century totalitarian states. It proved "with the accuracy of scientific experiment" that the reality of law consists of something more than bare conformity with legal norms. The existence of lex corrupta indicates that law contains something "non-dispositive" which requires acknowledgment of both law-maker and judge. Kaufmann, accepting the convergent concept of truth and cognition, assumes that "non-dispositive" content, emerging as the conformity of a number of cognitive acts of different subjects (inter-subjective communicativeness and verifiability), indicates the presence of being in this cognition. The questions "What is law?" and "What are the principles of a just solution?" lead straight to the ontology of law, to the question about the ontological foundations of law. Kaufmann discerns the ontological foundations of law in the specifically understood "nature of things" and, ultimately, in a "person". He proposes a procedural theory of justice, founded on a "person". In my work, I undertake to reconstruct the train of thought which led Kaufmann to the recognition of a "person" as the ontological foundation of law. In the first part, the conception of philosophy adopted by Kaufmann, initial characteristics of law — of reality which is the subject of analysis, as well as, the requirements for proper philosophical explanation of law posed by Kaufmann are introduced. In the second, Kaufmann's reconstruction of the process of the realisation of law is presented. Next, the conception of analogy which Kaufmann uses when explaining law is analyzed. In the fourth part, Kaufmann's conception of ontological foundations of law is discussed. A critical analysis is carried out in which I demonstrate that the theory of the ontological foundation of law proposed by Kaufmann and the concept of a person included in it do not allow a satisfac¬tory explanation of the phenomenon of "legal lawlessness" and lead to a number of difficulties in the philosophical explanation of law. Finally, the perspectives of a proper formulation of the issue of the ontological foundations of law are drafted in the context of the analyzed theory. My interest is centered on the conception of philosophy adopted by Kaufmann, according to which the existence of the reality is inferred on the basis of a certain configuration of the content of consciousness, whereas at the point of departure of philosophy of law, the data to be explained is a certain process, which is, basically, a process of cognition, while the reality appears only as a condition for the possibility of the occurrence of the process. I wish to argue that the difficulties which appear in the explanation of law are a consequence of the assumed fundamental philosophical solu¬tions, which seem to be characteristic, though usually not assumed explicitly, in philoso¬phy and theory of law dominant at present in continental Europe. Thereby, I wish to show the significance of ontological and epistemological solutions to the possibility of a proper formulation of the problems posed by philosophy and theory of law. Kaufmann proclaims himself in favour of a philosophy which poses questions about the ultimate foundations of understanding of the reality. In epistemology, he assumes that answers to the questions "What is reality like?" and ultimately "What is real?" are inferred on the basis of uniformity of a cognitive acts of different subjects. Cognition of the reality is accomplished exclusively through the content of conceptual material. The two fundamental questions posed by philosophy of law are "What is just law?" and "How is the just law enacted?" The latter is a question about the process of achieving a solution to a concrete case. Since, in Kaufmann's opinion, law does not exist apart from the process of its realisation, an answer to the question about the manner of realisation of law is of fundamental significance to answering the question: "What is law?" and to the explanation of the question about the ontological grounding of law, which is, as well, the foundation of justice. The proper solution has to take into account the moment of "non-dispositive" content of law; its positiveness understood as the reality and, at the same time, it has to point to the principles of the historical transformation of the content. Law, in the primary meaning of the word, always pertains, in Kaufmann's opinion, to a concrete case. A legal norm is solely the "possibility" of law and the entirely real law is ipsa res iusta, that which is just in a given situation. Determination of what is just takes place in a certain type of process performed by a judge (or by man confronted with a choice). Kaufmann aims to reconstruct this process. A question about the ontological foundation of law is a question about the ontological foundations of this process. In the analyzed theory it is formulated as a question about the transcendental conditions, necessary for the possibility of the occurrence of the process: how the reality should be thought to make possible the reconstructed process of the realisation of law. Kaufmann rejects the model for finding a concrete solution based on simple subsump¬tion and proposes a model in which concrete law ensues, based on inference by analogy, through the process of "bringing to conformity" that which is normative with that which is factual. Kaufmann distinguishes three levels in the process of the realisation of law. On the highest level, there are the fundamental legal principles, on the second legal norms, on the third — concrete solutions. The fundamental principles of law are general inasmuch as they cannot be "applied" directly to concrete conditions of life, however, they play an important part in establishing norms. A judge encounters a concrete situation and a system of legal norms. A life situation and norms are situated on inherently different levels of factuality and normativeness. In order to acquire a definite law both a norm (system of norms) and a life situation (Lebenssachverhalt) should undergo a kind of "treatment" which would allow a mutual conformity to be brought to them. Legal norms and definite conditions of life come together in the process of analogical inference in which the "factual state" ("Tatbestand") — which represents a norm, and in the "state of things" ("Sachverhalt") — which represents a specific situation are constructed. A "factual state" is a sense interpreted from a norm with respect to specific conditions of life. The "state of things" is a sense constructed on the basis of concrete conditions of life with respect to norms (system of norms). Legal norms and concrete conditions of life meet in one common sense established during the process of realisation of law. Mutatis mutandis the same refers to the process of composition of legal norms: as the acquisition of concrete law consists in a mutual "synchronization" of norms and concrete conditions of life, so acquisition of legal norms consists of bringing to conformity fundamental principles and possible conditions of life. According to Kaufmann, both of these processes are based on inference through analogy. As this inference is the heart of these processes it is simultaneously a foundation finding just law and justice. How does Kaufmann understand such an inference? As the basis for all justice he assumes a specifically interpreted distributive justice grounded on proportionality. Equality of relations is required between life conditions and their normative qualification. Concrete conditions of life are ascribed normative qualification not through simple application of a general norm. More likely, when we look for a solution we go from one concrete normative qualified case to another, through already known "applications" of norms to a new "application". The relation between life conditions and their normative qualifica¬tion has to be proportional to other, earlier or possible (thought of) assignments of that which is factual to that which is normative. Law as a whole does not consist of a set of norms, but only of a unity of relations. Since law is a, based on proportion, relative unity of a norm and conditions of life, in order to explain law in philosophical manner, the question about ontological base of this unity has to be asked. What is it that makes the relation between a norm and conditions of life "non-dispositive"? What is the basis for such an interpretation of a norm and case which makes it possible to bring a norm and conditions of life into mutual "conformity"? This is a question about a third thing (next to norms and conditions of life), with respect to which the relative identity between a norm and conditions of life occurs, about the intermediary between that which is normative and that which is factual and which provides for the process of establishing of norms, as well as, finding solutions. It is the "sense" in which the idea of law or legal norm and conditions of life have to be identical to be brought to mutual "conformity". In Kaufmann's opinion such a sense is nothing else but the "nature of things" which determines the normative qualification of the reality. Since establishment of this "sense" appears to be "non-dispositive" and controlled inter-subjectively (namely, other subjects will reach a similar result) so, in conformity with the convergent concept of truth, the "nature of things" must be assigned a certain ontological status. According to Kaufmann this is a real relation which occurs between being and obligation, between the conditions of life and normative quality. However, it should be underlined that from the point of view of the analyzed system the "nature of things" is a correlate of constructed sense, a result of a construction which is based on the principle of consistent understanding of senses ("non-normative" and "normative") and is not a reality which is transcendent against the arrangement of senses. In Kaufmann's theory, inference from analogy appears to be a process of reshaping the concepts (senses) governed by tendency to understand the contents appearing in relations between that which is factual and that which is normative in a consistent way. The analogical structure of language (concepts) and recognition of being as composed of an essence and existence is an indispensable requirement for the possibility of the realisation of law, based on specifically understood inference from analogy. It is necessary to assume a moment of existence without content which ensures unity of cognition. Existence emerges thus as a condition of the possibility of cognition. According to Kaufmann, the "nature of things" is the heart of inference through analogy and the basis for establishment of finding of law. Inference from the "state of things" to a norm or from a norm to the "state of things" always means inference through the "nature of things". The "nature of things" is the proper medium of objective legal sense sought in every cognition of law. In Kaufmann's view, the question whether the "nature of things" is ultima ratio of interpretation of law or is only a means of supplement gaps in law or whether it is one of the sources of law, is posed wrongly. The "nature of things" serves neither to supplement the gaps nor is it a source of law as, for example, a legal norm may be. It is a certain kind of "catalyst" necessary in every act of making law and solving a concrete case. Owing to "nature of things" it is possible to bring to a mutual conformity the idea of law and possible conditions of life or legal norms and concrete conditions of life. In Kaufmann's conception the "nature of things" is not yet the ultimate basis for understanding the "non-dispositiveness" of law. The relation between obligation and being is determined in the process of the realisation of law. Both the process itself and that which is transformed in this process are given. A question about the ontological bases of "material" contents undergoing "treatment" in the process of the realisation of law and about being which is the basis of regularity of the occurrence of the process arises. Only this will allow an explanation that the result of the process is not optional. Thus, a question about reality to which law refers and about the subject realising the law has to be formed. To this, Kaufmann gives the following answer: that which is missing is man but not "empirical man" but man as a "person". A "person" understood as a set of relations between man and other people and things. A "person" is the intermediary between those things which are different — norm and case are brought to conformity. A "person" is that which is given and permanent in the process of the realisation of law. It determines the content of law, is "subject" of law; this aspect is described by Kaufmann as the "what" of the process of realisation of law. A "person" consists of precisely just these relations which undergo "treatment" in the process. On the other hand, a "person" is "a place" in which the processes of realisation of law occur, it is the "how" of normative discourse, a "person" is that which determines the procedure of the process, being "outside" of it. This aspect of a "person" is connected with the formal moment of law. A "person" being, at the same time, the "how" and the "what" of the process of the realisation of law, is also, to put it differently, a structural unity of relation and that which constitutes this relation (unity of relatio and relata). According to this approach a "person" is neither an object nor a subject. It exists only "in between". It is not substance. Law is the relation between being and obligation. That which is obligatory is connected with that which is general. That which is general does not exist on its own, it is not completely real. Accordingly, a "person" as such is also not real. It is relational, dynamic and historical. A "person" is not a state but an event. In Kaufmann's opinion, such a concept of a "person" helps to avoid the difficulties connected with the fungibility of law in classical legal positivism. A "person" is that which is given, which is not at free disposal and secures the moment of "non-dispositiveness" of law. Kaufmann concludes: "The idea (»nature«) of law is either the idea of a personal man or is nothing". Theory points at the structure of realising law and explains the process of adoption of general legal norms for a concrete situation. The analysis has shown however, that in this theory a satisfactory answer to the question about the ultimate foundations of law is not given. It seems that in the analyzed theory the understanding of human being takes place through understanding of law. What is good for man as a "person", what is just, what a "person" deserves may be determined only against the existing system of law. A "per¬son" adopted as a basis of law is the reality postulated in the analysis of the process of the realisation of law. It is a condition of possibility of this process ( explaining, on one hand, its unity and, on the other hand, the non-dispositive moments stated in this process). A "person" in the discussed theory is entirely defined by the structure of law, it can be nothing more than that which is given in law, what law refers to, what law is about. Being, which is a "person", is constituted by relations between people and objects, the relations which are based on fundamental links between norms and conditions of life established in a process of bringing them to conformity. It has to be assumed that man as a "person" is a subject of law only as far as realising law "treats" given senses according to their current configuration. The system of law is a starting point and it describes in content what man is as a "person". Moreover, being a "person" is the condition for entering legal relations. Consistently, Kaufmann writes that "empirical man" is not the subject of law, man is not "out of nature" a "person". People become "persons" due to the fact that they acknowledge each other as "persons" — acknowledging, at the same time, law. This acknowledgement is a con¬dition of existence, of the possibility of the occurrence of process of realisation of law and of constituting legal relations which ultimately constitute a "person". Kaufmann assumes, that law tends towards a moral aim: it may and must create an external freedom, without which the internal freedom to fulfil moral obligations cannot develop. However, this postulate is not based on the necessary structure of human being. From the point of view of his system, it is nothing more than only a condition for the possibility of the occurrence of the process of the realisation of law — lack of freedom would destroy the "how" of this process. Thus, the postulate to protect the freedom of personal acts has to be interpreted, in accordance with the analyzed theory, as a postulate, the fulfilment of which aims ultimately at the accomplishment of the very same process of realisation of law itself and not the realisation of a given man. Kaufmann considers a "person" to be an element which unites the system of law as a whole. Law is a structure of relations, which are interdependent and inter-contingent. Consequently, a "person" which is to form the ontological basis of law has to be entity consisting of all relations. Being also the "how" of the process of realisation of law, if a "person" is to warrant its unity, it has to be a common source for all procedures. Hence, a single "person" would constitute a subject of law. Man appears to be only a moment of a certain entirety, realisation of which should be an aim of his actions. Law, creating a "person" as an object and subject of law becomes a primary entity. In the analyzed theory, the basis for determination of aims which law sets to man is not the allocation of man-subject to something which improves him but rather, such relation is only just constituted by law. A question appears, why should aims set in law also be the aims of "empirical man"? Why is this "empirical man" to be punished in the name of a "person" understood in such a way? If, however, it is assumed that what is man is determined by a system which is superior to him, then man has to be understood only as a part of a whole and there are no grounds to prohibit istrumental treatment of man and so the road to all aspects of totalitarianism might be opened. A problem of the application of created theory to the reality arises, the reality which the theory pretends to explain. Ultimately in his theory Kaufmann does not give any systemic grounds for a radical questioning of the validity of any legal norms. Every new norm becomes an equal part of system of norms. It is only its interpretation and application to given conditions of life that may be disputable, however, this refers to all norms without exception. Cohesive inter-pretation of norms and applications is necessary and sufficient for the acquisition of just law. New norms have to be interpreted in the light of others, correspondingly, the other norms require reinterpretation in the light of the new ones. Contradiction in interpretation of a norm does not form a basis for questioning norms but may serve only to question the manner of their interpretation (understanding). Therefore, no grounds exist to assume any legal norm as criminal or unjust, and in consequence, to question any consistently realised system based on formally, properly established norms, as "legal lawlessness". As law and a "person" do not exist without the process of realisation of law, the role of legal safety becomes crucial as the condition for the possibility of the occurrence of the process of realisation of law. Denying legal safety would be tantamount to negation of law in general (also of moral law) as negation of safety takes away, at the same time, the basis for occurrence of the process of realisation of law. Moreover, any lack of legal safety would also mean lack of a basis for the existence of man as a "person". Kaufmann's thesis, that civil disobedience is legalized only when it has a chance to lead to success, consistent with his concept of the foundations of law, seems to point directly to conclusions which deny the facts taken under consideration and doubtlessly Kaufmann's own intentions, since it would have to be assumed that accordingly there are no grounds to question a legal system in force based on violence which secures its operation. Force finally seems to determine which one of the mutually irreconcilable normative systems constitute law and which does not. A legitimate position is one which leads to success, it is the weaker system which is negated. If so, then basically violent imposition of law is not an act directed against the law in force but, to the contrary, realisation of law. In the context of the new system the former system of law may be talked about as unjust solely in the sense of being incapable of being consistently united with the new. However, at the base, ultimately, lies force which reaffirms differences and excludes from the process of realisation of law certain norms and their interpretations. Kaufmann was aiming at grounding of that which is "non-dispositive" in a certain given framework of interpretation. Nevertheless, he does not provide foundations for the understanding of phenomena, which he undertakes to explain at a point of departure. Instead of explaining them the theory negates the possibility of their existence. The reality postulated in regard to "non-dispositive" moments of the reconstructed process of acquiring law consist of a specifically understood "person", which appears in Kaufmann's conceptions as a condition of the possibility of the realisation of law. According to this approach understanding of a "person" may be only a function of law. To understand "legal lawlessness" and foundations of justice it is necessary to look for such theory of law in which understanding of man as a "person" and being is not a function of understanding of law (in which a "person" is not only a condition for possibility of reconstructed process of realisation of law; for possibility of cognition processes). It seems necessary to start from theory of being and a "person" based on broader experience than the one assumed by Kaufmann and reconstruct the ontological foundations of the process of realisation of law only in such perspective. Kaufmann points out that that to which law refers is ipsa res iusta a concrete relation of man to other people and things. This relation, in his theory, appears to be basically only just constituted by law (normative senses "applied" to conditions of life). Therefore, understanding the relation between a given man and other people and things which constitute the aim of his actions, that is understanding of good, is enacted against the background of constitution of senses; constitution which is a result of a process aiming towards consistent understanding of particular contents (of nor¬mative and non-normative senses). "Being" is secondary towards constructed senses it is only their correlate. The primary relation consists of relation of a man to law (system of norms), while the secondary relation is one of man to something which is the aim of his action (relation between man and good). Considering such approach it is difficult to envision a satisfying answer to the fundamental question: why does law put concrete man under any obligation to obey it? The source of this problem can be seen in reduction of the base for understanding good to content of obligation formulated in auto-reflection. Such reduction seems to be a consequence of Kaufmann's adoption of "convergent concept of truth" and in con¬sequence his recognition of indirect, essentialistic grasp of reality formulated in concepts as the basic and only foundation of theory of being and of law. In view of such an approach, analogy of law, concepts and being is the condition for the possibility of the process of transformation of senses which aims at consistent interpretation of all law. Existence is postulated with respect to the possibility of unity of experience and cognition. However, also a different approach to understanding of the problem of being and good is possible. In spontaneous cognition being is affirmed, first of all, not as a certain, non-contradictory, determined content, but as something existing. Together with a cer¬tain content (passed indirectly through notions) existence of being is co-given. The basis for unity of being is not formed by the consistence of content, as it is in the case of the theories departing from the analysis of cognition processes, but by an act of existence realising content (essence). Such an approach makes it also possible to go beyond the convergent concept of truth. It is worth mentioning that allocation of an agent to good is realised not only by the content of duty. A statement that something is good is primary with respect to determination of this good in content. The recognised good always bears some content, however, there are no reasons to base the concept of good exclusively on indirect, formulated in concepts cognition. As primary, can be adopted the relation of man to good and not of man to law. Determination in content appears to be only an articulation of aspectual cognition of being, as an object of action. In such a case the basis for relative unity of norm and conditions of life is not the "nature of things" understood as correlate of sense but it is relation to good based on internal constitution of man as potential, not self-sufficient being. It does not mean, that the moments of the process of realisation of law singled out by Kaufmann are not important to determination of what is just. He, quite rightly, points to significant role played by norms in the evaluation of concrete situations, in man's search for closer specification in content of good innate to him. The structure of process of determining law for a concrete situation, to a great degree corresponds to the processes of determining law which take place not only in the legal sciences. Kaufmann's analyses of the process of realisation of law show the complexity of the structure of these processes and point towards important moments allowing a better understanding of law and man. Nevertheless, these analyses cannot be a basis for construction of philosophical theory of law, theory which hopes to point out the ultimate, ontological foundations for understanding law. Kaufmann's results may become fully valid only in a more general perspective including broader experience at the point of departure. (shrink)

The study of how languages differ in their portrayal of motion events has received much attention since Talmy provided the first detailed account of the phenomenon. Interest has extended from real, or factive motion, to imagined or fictive motion, and from there to metaphorical motion, in which experience in one sensory domain is understood in terms of motion. Studies of metaphorical motion have, however, concentrated so far on a limited number of sensory domains, principally vision, and drawn data from (...) a limited number of textual genres. The aim of this study is to identify the extent to which motion expressions are employed to express organoleptic sensory experiences, that is experiences of taste and smell. Drawing on analytical categories put forward by Talmy in his treatment of Satellite- and Verb-framed languages and on treatments based on Cognitive Metaphor Theory, and particularly the concept of ‘embodiment’, we investigate the variety and complexity of organoleptic Paths evidenced in two self-built corpora of craft beer reviews, one in Spanish, the other in English. We also compare the level of complexity evident in these sensory Path types with those of another, closely related, sensory domain: vision. The study finds, in line with related studies, that English has greater variety and complexity of sensory Paths than Spanish. A more unexpected result is the finding that organoleptic Paths are more varied and complex than visual Paths. This finding, then, may encourage us to reconsider Path complexity differences in sensory domains as a multifactorial issue and not simply as a cognitive-linguistic phenomenon. It is proposed here that external factors such as genre, or specifically, its communicative functions and linguistic constraints, coupled with the unique ‘perceptual landscapes’ created by each sensory modality, may transcend perceived physical boundaries of a Figure’s trajectory. (shrink)

Evolutionary developmental systems theory stresses that selection pressures operate on entire developmental systems rather than just genes. This study extends this approach to language evolution, arguing that selection pressure may operate on two quasi-independent timescales. First, children clearly must acquire language successfully and evolution must equip them with the tools to do so. Second, while this is developing, they must also communicate with others in the moment using partially developed knowledge. These pressures may require different solutions, and their combination (...) may underlie the evolution of complex mechanisms for language development and processing. I present two case studies to illustrate how the demands of both real-time communication and language acquisition may be subtly different. The first case study examines infant-directed speech. A recent view is that IDS underwent cultural to statistical learning mechanisms that infants use to acquire the speech categories of their language. However, recent data suggest is it may not have evolved to enhance development, but rather to serve a more real-time communicative function. The second case study examines the argument for seemingly specialized mechanisms for learning word meanings. Both behavioral and computational work suggest that learning may be much slower and served by general-purpose mechanisms like associative learning. Fast-mapping, then, may be a real-time process meant to serve immediate communication, not learning, by augmenting incomplete vocabulary knowledge with constraints from the current context. Together, these studies suggest that evolutionary accounts consider selection pressure arising from both real-time communicative demands and from the need for accurate language development. (shrink)

We show that the ordered field of real numbers with restricted $\mathbb{R}_{\mathscr{H}}$-definable analytic functions admits quantifier elimination if we add a function symbol $^{-1}$ for the function $x\mapsto \frac{1}{x}$ (with $0^{-1}=0$ by convention), where $\mathbb{R}_{\mathscr{H}}$ is the real field augmented by the functions in the family $\mathscr{H}$ of restricted parts (real and imaginary) of holomorphic functions which satisfies certain conditions. Further, with another condition on $\mathscr{H}$ we show that the structure ($\mathbb{R}_{\mathscr{H}}$, constants) is strongly (...) model complete. (shrink)

We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable (...) Fσ set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly 1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry. (shrink)

We may now ask the question: In what historical perspective should we place the work of Richard Goldschmidt? There is no doubt that in the period 1910–1950 Goldschmidt was an important and prolific figure in the history of biology in general, and of genetics in particular. His textbook on physiological genetics, published in 1938, was an amazing compendium of ideas put forward in the previous half-century about how genes influence physiology and development. His earlier studies on the genetic and geographic (...) distribution of the various species and races of Lymantria contributed soundly to an understanding of the inheritance and development of sex, as well as the genetics of speciation. He was a curious mixture of the experimentalist, empiricist, and theorizer, who could grasp both the large and the small at once and wrap them into the same integrated and coherent package.In insisting on trying to relate genetics to development, Goldschmidt was continually forced to speculate and to forsake experimental fact for generalized theory. In refusing to reconcile a physiological and developmental conception with a corpuscular theory of the gene, he forced himself to substitute for a large body of experimentally verified evidence, vague and general speculations which did not lead in any precise direction. L. C. Dunn summarizes succinctly the effect of Goldschmidt's thinking on the development of genetic theory: Although Goldschmidt's ideas had a considerable influence in directing attention to the developmental processes intervening between genes and characters, they did not lead to the establishment of a general theory of development in genetical terms. In fact, at the end of the period, as signalized by Goldschmidt's book of 1938 [Physiological Gentics] doubt remained whether there was a field with a defined problem which could be identified as developmental genetics.Yet the frequency with which Goldschmidt is cited by later writers suggests that there is a more positive side to his contribution than this. The early Mendelians had recognized the genic constitution of organisms and the Drosophila workers had uncovered in detail the chromosomal mechanism of heredity. These great discoveries were of classical character, solidly based on a multitude of well established facts. By themselves, however, they would not have been sufficient to place genetics in the central position within biology which it was to assume. Beyond the “static” analysis of gene transmission an insight was needed into the dynamic role of genes in cellular physiology, biochemistry, and development. Goldschmidt recognized this aspect and outlined in brilliant generalizations a physiological theory of genetics. Necessarily the prophetic nature of this achievement — romantic in the sense of Ostwald's classification of great scientists and their discoveries — transcended its factual basis. It was the audacity of the theoretician unimpeded by the still scanty data which gave a new focus to biological thought.Some have called Goldschmidt an “obstructionist” in the history of genetics—one whose efforts were largely directed toward useless and continual criticism — challenge for the sake of challenge. As this line of argument goes, time wasted answering Goldschmidt's poorly conceived criticisms could have been better used to pursue new lines of thought within the classical gene theory. Sometimes historians are upbraided for studying the work of such “obstructionists,” or even the work of those whose ideas have later proved to be inadequate. This argument misses the point of history and the lessons which it can teach. Perhaps Goldschmidt did take the time of a number of the classical geneticists who tried to answer his objections to the gene theory; and perhaps many of his own ideas about genes were, by today's view, “wrong.” These are always easier judgments to make by hindsight than at the moment. But that is not the point. Goldschmidt had a considerable influence on his contemporaries, and to dismiss him as an obstructionist because his opponents later proved to be more nearly right, distorts history and obscures significant questions which we might well be asking. It might be valuable to know why Goldschmidt felt compelled to be an iconoclast —and how that impulse, psychological in orgin, led him to overlook critical items of evidence which his opponents considered more carefully. It might also be valuable to try and understand how a contemporary of Goldschmidt or. Morgan really viewed the gene theory — what were its strong and weak point? Such questions are not readily answered by studying the work of only those men who were correct by later standards.Answers to some of the above questions can be gleaned from this study of Goldschmidt — though the first, and most psychologically oriented, question is largely untouched here.On the one hand, Goldschmidt forced classical geneticists to reexamine the foundation of their ideas about the nature, inviolability, and even physical dimensions of genes. He also kept alive the very real problems of gene physiology and its relations to development — a relationship that was given less attention by the Drosophila school in their pursuit of the transmission process. On a broader level he emphasized to the biological community a principle that was in danger of being lost in the enthusiasm surrounding the expansion of the Mendelian-chromosome theory. The point was that all theories in science are transient, logical constructs which should not be held as sacred. Time and again, as the history of science has shown, theories become rigid structures which retard new modes of thought. From Goldschmidt's perspective, the gene theory was in danger of performing just that function, for it was preventing workers from viewing the gene in a functional as well as a structural light.On the other hand, examining the work of critics (or even of outright “obstructionists”) gives the historian an opportunity to observe how the more established community of scholars at some point in time reacts to challenges of its cherished ideas. The fact that many workers in his own day (such as Beadle, Luria, Delbruck, Sturtevant, Bridges, and others), as well as most geneticists today, saw Goldschmidt as having been largely “obstructionist” is useful historical (and/or sociological) data. Clearly, as this paper has tried to show, Goldschmidt's ideas were not all obstructionist, and his viewpoint that genes must be considered functionally was one which was clearly valid, premature as it may have been from an experimental and technique point of view. Perhaps Goldschmidt's major fault was that he developed alternative theories which were not easily testable. Nonetheless, his challenge to a fundamental and established idea brought forth a reaction from many members of the genetic community which indicates how unwilling they were to reconsider the formalized theory on which their work was based. While many of Goldschmidt's criticisms — and particularly his way of making them — arose out of his own idiosyncrasies, they were also a product of his times. His view of the nature of theory-construction, and his rejection of old-style reductionism and mechanism, were part of a growing awareness within the world scientific community that explanations based on reducing complex phenomena to ultimate particles or units were clearly inadequate. The success of the Mendelian-chromosome theory had obscured that fact for many geneticists, but to Goldschmidt it was a point that could not be allowed to pass unnoticed. (shrink)

Some contemporary theories about the origin and the nature of language resort to concepts with no bearing on Darwinian evolutionary hypothesis or evo-devo perspective which are both based on the reconstruction of species morphological structure transformation. These theories, which evoke qualitative leap, cultural evolution, structure/function coevolution as esplicative principles for human evolution, in our opinion, result compatible in some points with the most recent Intelligent Design accounts. Attempting to substantiate itself as a scientific theory, the contemporary ID is ready (...) to give up creationist explanation just to impeding Darwin’s fundamental idea according to which it’s possible to explain evolution only through a gradual material modification of structures. For comprehending a complex phenomenon as human language – according to ID – it’s necessary appealing to a second substance, whatever it is. This idea seems to be at the bottom of all those theories which have rejected monistic structural explanation for embracing functional, psychological or cultural accounts. We consider these kinds of explanation real unresolved residuals of ID, residuals nested in the heart of the most accredited scientific theories. (shrink)

The first-order logical theory Th $({\mathbb{N}},x + 1,F(x))$ is proved to be complete for the class ATIME-ALT $(2^{O(n)},O(n))$ when $F(x) = 2^{x}$ , and the same result holds for $F(x) = c^{x}, x^{c} (c \in {\mathbb{N}}, c \ge 2)$ , and F(x) = tower of x powers of two. The difficult part is the upper bound, which is obtained by using a bounded Ehrenfeucht–Fraïssé game.

A theory of approximation to measurable sets and measurable functions based on the concepts of recursion theory and discrete complexity theory is developed. The approximation method uses a model of oracle Turing machines, and so the computational complexity may be defined in a natural way. This complexity measure may be viewed as a formulation of the average-case complexity of real functions—in contrast to the more restrictive worst-case complexity. The relationship (...) between these two complexity measures is further studied and compared with the notion of the distribution-free probabilistic computation. The computational complexity of the Lebesgue integral of polynomial-time approximable functions is studied and related to the question “FP = ♯P?”. (shrink)

Given any collection F of computable functions over the reals, we show that there exists an algorithm that, given any sentence A containing only bounded quantifiers and functions in F, and any positive rational number delta, decides either “A is true”, or “a delta-strengthening of A is false”. Moreover, if F can be computed in complexity class C, then under mild assumptions, this “delta-decision problem” for bounded Sigma k-sentences resides in Sigma k. The results stand in sharp (...) contrast to the well-known undecidability of the general first-order theories with these functions, and serve as a theoretical basis for the use of numerical methods in decision procedures for formulas over the reals. (shrink)

Introduction. The article is focused on analyzing the utopian direction of Russian cosmism and its influence on the Soviet cosmonautics and the development of society in the USSR. This philosophical theory was created in the period that made it possible to incorporate the applied aspects of utopia into scientific and technological progress and thereby embody a number of steps towards the outer space exploration. The authors have developed criteria and parameters for assessing the utopian component of the Russian cosmism (...) theories, which made it possible to bring this construct to a higher level of abstraction and thereby create a working model for conducting such studies in the context of other utopias of models. The purpose of the article is to show the influence of the Russian cosmism utopia on the cosmonautics development in the USSR, develop empirical criteria for evaluating the phenomenon. Achieving the goal required solving the following tasks: 1) considering and analyzing the subject matter of the cosmism utopia; 2) developing parameters for assessing the impact of utopia on the development of the social system; 3) applying the developed parameters to assess the impact of utopian ideas on the development of the Soviet cosmonautics system. Methods. Developing the theoretical model for assessing social utopias, as well as considering and analyzing the cosmism utopia, required the use of structural-functional and systems analysis. The research was conducted within the framework of a synergistic paradigm. Scientific novelty of the research. The article conceptualizes the concept of utopia. It is shown that most of the definitions of utopia as a socio-political ideal focus on the limitations of its existence: utopia cannot be embodied, often has an unscientific character, does not correlate with the real state of the system, i.e. definitions of utopia are often reduced to the negative format. The authors believe that the influence of utopia on society, as a rule, is positive. It is noted that, along with limitations, utopianism has certain unique essential features that qualitatively affect the social projects implementation. Utopia in the systemic understanding acts as a complex of ideas influencing the development of the system, being both internal (since it is created artificially and consciously by the very elements of the system) and an external factor of influence. Unlike Plato’s eidos, the projection of which is reality, utopia is created inductively, but after its creation it again “descends” to the level of reality, since it begins to influence the social model in which it was created. Results. The article discusses the prognostic and modeling functions of the social utopias of Russian cosmists. It has been proved that one of the essential functions of the Russian cosmism utopias is the formation of an ideal type, towards which, in a historical perspective, the real social system begins to strive. It is convincingly demonstrated that utopia acts as a cognitive support and inevitably forms the canvas along which society begins to move, defining the utopian model as an attractor, although such a goal is not always formulated when creating a utopia. This relationship makes it possible to assess the degree of influence of utopian ideas on the formation of reality in each specific case, which, in turn, provides an opportunity to answer the question of how and to what extent the utopian ideal type participates in determining the characteristics and parameters of a real social system. Conclusions. It was found that the social utopia of cosmists as a cognitive concept is an important effective factor influencing the development of the space industry in the USSR. The parameters adopted in the study allow us to describe the measure of its influence as both an internal and an external factor on the development of the society in which it is implemented. The validity of perceiving the utopia of cosmists as a construct with a certain life cycle, the main part of which is the period of functioning, is stated. During this time period, utopian theory can have a significant impact on the actual development of society from various angles. (shrink)

A Liouville function is an analytic function $H : C \rightarrow C$ with a Taylor series $\Sigma_{n=1}^\infty x^n/a_n$ such the $a_n\prime s$ form a "very fast growing" sequence of integers. In this paper we exhibit the complete first-order theory of the complex field expanded with H.

Helmut Reich’s theory of relational and contextual reasoning is a courageous initiative for the resolution of cognitive conflicts between apparently incompatible or incommensurable views. Built upon Piagetian logico-mathematical reasoning, cognitive complexity theory, and dialectical and analogical reasoning, it includes the development of a both/and logic inclusive of binary either/or logic. Reich provides philosophic, theoretical, and even initial empirical support for the development of this form of reasoning along with a heuristic for its application. A valuable step beyond (...) the limits of binary, static, and formal reasoning, it takes relationship, context, and perspectival variations seriously in an explicitly reflective and iterative system. We can and do address conflicts not resolvable by conventional appeals to logic or evidence, including those at epistemic boundaries or produced by belief-commitment differences. Although this form of reasoning has real promise, including stepping beyond complementarity in the religion-science dialogue, it seems better directed to causally explanatory theories than to other forms of rendering meaning. Finally, its coextension requirement may render it problematic where functionally coherent explananda cannot be identified or are themselves produced or constituted by a belief system. (shrink)

Kolmogorov complexity was originally defined for finitely-representable objects. Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitely-representable objects—such as rational numbers—used to approximate them.This idea will be taken further here by extending the definition to continuous functions over real numbers, based on the fact that every continuous real function can be represented as the limit of a sequence of finitely-representable enclosures, such (...) as polynomials with rational coefficients.Based on this definition, we will prove that for any growth rate imaginable, there are real functions whose Kolmogorov complexities have higher growth rates. In fact, using the concept of prevalence, we will prove that ‘almost every’ continuous real function has such a high-growth Kolmogorov complexity. An asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions will be presented as well. (shrink)

Craft work (for example, ceramics and furniture) poses a dilemma for modern thinking about beauty and the arts. Because of its beauty, craft has more than mere functionality, yet because of its functionality it cannot be fine art. What should craftmakers do? Claim their work to be fine art and forget about functionality? Or remain loyal to functionality (you can eat noodles from the finest ceramic bowl) and face extinction competing with machine-made products of industrial design? Apparently, the tendency is (...) to abandon functionality, though Risatti thinks that is a mistake. Craft is not craft without functionality; bowls too large to lift have become the useless art of sculpture. Function is important because it implies a relation to the hands that make and use the work, guaranteeing human scale and meaning. How much fine art can you say that about? The real threat to craft is not assimilation to fine art, however. The effect of industrial design and machine production is to divorce the way a thing looks from how it is made. Never has so little of our artifactual environment actually been made by hand, and it’s a loss. A loss of connection. A thing made by and for the hands has a sense that transcends our differences and when beautifully made is an eloquent reminder of where ingenuity ultimately comes from. (shrink)

Molecular structure is one of the dispositional attributes of the molecule and counted as an example of affordances. This attribute has been systematically exploited through the development of theories and practice of organic chemistry. (Ochiai 2023, pp. 141–149) The question to be addressed in this study is whether we can legitimately claim that this type of attribute is real. To answer the question, we first clarify what is worthy of the word ‘reality’ in scientific arguments. Whitehead claims that a (...) physical substance like electron is a bundle of spatial-temporal experience. (Mesle 2008, p. 36) The contention is that we cognize the causal relationships of the physical world through the experience of events. What we cognize as real is not unchanging matter but various kinds of events we experience. Based on this recognition we examine the nature of affordances. Affordances are the context-relative dispositional attributes of {agent-world} complexes. (Harré 2014, pp. 77–91; Harré and Llored 2018, pp. 167–186) The affordance of a knife becomes actualized through our experience of cutting. In this way affordances are concerned with work or the function of something useful. They are materialized as tools. This suggests that affordances are real, and so is molecular structure. Molecular structure is characterized by a certain function which molecules show in organic synthesis. Other dispositional attributes of molecules that become actualized in the context of organic chemistry are also examined. We refer to the physical significance of wavefunctions as well, which we discussed in the previous study. (Ochiai 2023, pp. 359–367) Creating function, we expand a world we cognize as real. This view of the world we name ‘functional realism.’ Our conceptual scheme is supported by Quine’s holism. (shrink)

In the target article, Christie, Brusse, et al. argue that selected effect functions do not, in general, explain why a trait exists in a population and, therefore, theories of representational content should not rely on selected effect functions. This response focuses on the claim about functions-for-representation. The role of evolutionary functions in a theory of content is to pick out outcomes that have been systematically stabilized by natural selection. Correctness conditions are conditions involved in explaining (...) how that happened. Selected effect functions can play that role in the complex equilibria that Christie, Brusse, et al. identify. Non-equilibrium cases are also discussed. (shrink)

In [1] Todorov has shown by means of axiomatic set theory that there exists a nonstandard function Δ: *ℝn → * ℂ such that for all continuous functions φ: ℝn → ℂ, equation image.Here *ℝ and *ℂ are the set of the nonstandard real numbers and the set of the nonstandard complex numbers, respectively, and *φ: *ℝn → *ℂ is the nonstandard extension of φ In the present note we want to prove an analogous theorem by predicative (...) means only. (shrink)

The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey (...) of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers. (shrink)

1. One theory or many? In 2004 a very interesting and readable article by Lenore Blum, entitled “Computing over the reals: Where Turing meets Newton,” appeared in the Notices of the American Mathematical Society. It explained a basic model of computation over the reals due to Blum, Michael Shub and Steve Smale (1989), subsequently exposited at length in their influential book, Complexity and Real Computation (1997), coauthored with Felipe Cucker. The ‘Turing’ in the title of Blum’s article (...) refers of course to Alan Turing, famous for his explication of the notion of mechanical computation on discrete data such as the integers. The ‘Newton’ there has to do to with the well known numerical method due to Isaac Newton for approximating the zeros of continuous functions under suitable conditions that is taken to be a paradigm of scientific computing. Finally, the meaning of “Turing meets Newton” in the title of Blum’s article has another, more particular aspect: in connection with the problem of controlling errors in the numerical solution of linear equations and inversion of matrices,Turing (1948) defined a notion of condition for the relation of changes in the output of a computation due to small changes in the input that is analogous to Newton’s definition of the derivative. The thrust of Blum’s 2004 article was that the BSS model of computation on the reals is the appropriate foundation for scientific computing in general. By way of response, two years later another very interesting and equally readable article appeared in the Notices, this time by Mark Braverman and Stephen Cook (2006) entitled “Computing over the reals: Foundations for scientific computing,” in which the authors argued that the requisite foundation is provided by a quite different “bit computation” model, that is in fact prima facie incompatible with the BSS model. The bit computation model goes back to ideas due to Stefan Banach and Stanislaw Mazur in the latter part of the 1930s, but the first publication was not made until Mazur (1963).. (shrink)

In this paper, we simulate the estimation of motion through an interframe difference detection function model and investigate the spatial-temporal context information correlation filtering target tracking algorithm, which is complex and computationally intensive. The basic theory of spatiotemporal context information and correlation filtering is studied to construct a fast target tracking method. The different computational schemes are designed for the flow of multiframe target detection from background removal to noise reduction, to single-frame detection, and finally to multiframe detection, respectively. (...) This enables the ground-based telescope to effectively detect spatial targets in dense stellar backgrounds in both modes. The method is validated by simulations and experiments and can meet the requirements of real projects. The interframe bit attitude estimation is optimized by using the beam-parity method to reduce the interframe estimation noise; a global optimization strategy based on the bit attitude map is used in the back end to reduce the system computation amount and make the global bit attitude estimation more accurate; a loop detection based on the word pocket model is added to the system to reduce the cumulative error. (shrink)

This uncommonly fine textbook of the modern theory of functions of a real variable is particularly well-suited for mathematically mature students in the fields of philosophy and foundations of mathematics, philosophy of physics, probability theory, and statistics. Those who wish to achieve first-hand acquaintance with the quantum theory will also need to have a grasp of the material presented in this book. The first chapter presents a capsule survey of topics in abstract set theory (...) and algebra, including a discussion of relations, functions, the axiom of choice, and the well-ordering theorem, and leading up to the construction of the fields of real and complex numbers. The second chapter is an introduction to topology and spaces of topological functions. The approach is set-theoretical. Chapter three deals with the theory of the Lebesgue integral and measure theory. This chapter is quite comprehensive, rigorous and embellished with many exercises for the student. Subsequent sections include the theories of function spaces, Banach spaces, abstract Hilbert spaces, and differentiation. This is clearly one of the more worthwhile texts in its field.—H. P. K. (shrink)

Enlightened experience (i.e. awakened to the truth) is the most valuable one in most religions including Christianity and Buddhism. As well-known cases of such experience are Apocalypse St. Paul and many Grand Zen masters in Zen Buddhism, it is natural for us to believe that the enlighten is for very talented or speciallytrained ones. However, applying the complexity theory on the structure of enlightenment, based on the power law function, selforganized criticality, phase transition, and emergence, it is clear (...) that the full awakening can be experienced by everyone in daily life. Furthermore, such experiences are not necessary to be limited in religious one. Rediscovering the path to full awakening by complexity theory leads to the demythologization of the enlightenment in Korean Zen Buddhism, which emphasizes a hard apprenticeship in temple for long periods for the purpose. (shrink)

Researching the organization and management of international public affairs (IPA) in the then-new multinational enterprises (MNEs) started in the 1960s. At first, IPA studies kept fairly good pace with what was known about the MNEs' nonmarket environments, their structurings, and their processes. After 1980, a disconnect developed in our knowledge of these three interrelated topics. In particular, much of more recent IPA research (a) fails to reflect the true scope, real actors, and organizational location of the IPA function; (b) (...) remains focused on old issues (e.g., the hierarchical relationship between headquarters and subsidiaries); (c) is mired in complex statistical analyses of remote data; (d) uses relevant theories in rather marginal ways; and (e) overlooks the impact on IPA of such important developments as mergers, acquisitions, and divestments. Yet, investigators can stretch their research methodologies and refocus their studies on the organization and management of the two faces of public affairs—offensive and defensive—in the context of new nonmarket threats and opportunities. (shrink)

Which function is computed by a Turing machine will depend on how the symbols it manipulates are interpreted. Further, by invoking bizarre systems of notation it is easy to define Turing machines that compute textbook examples of uncomputable functions, such as the solution to the decision problem for first-order logic. Thus, the distinction between computable and uncomputable functions depends on the system of notation used. This raises the question: which systems of notation are the relevant ones for determining (...) whether a function is computable? These are the acceptable notations. I argue for a use criterion of acceptability: the acceptable notations for a domain of objects D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document} are the notations that we can use for the usual D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document}-activities. Acceptable notations for natural numbers are ones that we can count with. Acceptable notations for logical formulas are the ones that we can use in inference and logical analysis. And so on. This criterion makes computability a relative notion—whether a function is computable depends on which notations are acceptable, which is relative to our interests and abilities. I argue that this is not a problem, however, since there is independent reason for taking the extension of computable function to be relative to contingent facts. Similarly, the fact that the use criterion makes a mathematical distinction depend on practical concerns is also not a problem because it dovetails with similar phenomena in other areas of computability theory, namely the roles of notation in computation over real numbers and of Extended Church’s Thesis in complexity theory. (shrink)

This is a presentation about joint work between Hector Zenil and Jean-Paul Delahaye. Zenil presents Experimental Algorithmic Theory as Algorithmic Information Theory and NKS, put together in a mixer. Algorithmic Complexity Theory defines the algorithmic complexity k(s) as the length of the shortest program that produces s. But since finding this short program is in general an undecidable question, the only way to approach k(s) is to use compression algorithms. He shows how to use the (...) Compress function in Mathematica to give an idea about the compressibility of various sequences. However, the idea of applying a compression algorithm breaks down for very short sequences. This is true not only for the Compress function, but also for any other compression algorithm. Zenil's approach is to construct a metric of algorithmic complexity for short sequences from scratch. He defines the algorithmic probability as the probability that an arbitrary program produces a sequence. The basic idea is to run a whole class of computational devices such as Turing Machines or Cellular Automata, and compute the distributions of the sequences they generate. Zenil presents a comparison of frequency distributions of sequences generated by 2-state 3-color Turing Machines and 2-color radius 1 Cellular Automata. He also compared these distributions to distributions found in data from the real world, and found that not only there is correlation across different systems, but also that the distributions are rather stable, and the difference between the distributions in abstract systems and real-world data can be attributed to noise. In his paper Zenil elaborates on the nature of the noise he has encountered. Zenil conjectures that the correlation distances between different systems decreases with a larger number of steps, and converge in the infinite limit case. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Rhetoric 39.1 (2006) 72-84 [Access article in PDF] "The Limbo of Ethical Simulacra": A Reply to Ron Greene Dana L. Cloud Department of Communication Studies University of Texas, Austin Steve Macek Department of Speech Communication North Central College James Arnt Aune Department of Communication Texas A&M University In two recent articles, "Another Materialist Rhetoric," and "Rhetoric and Capitalism" (1998, 2004), Ronald Walter Greene pays considerable attention to (...) Marxist rhetorical theory, especially the work of Dana L. Cloud (1994, 2001a, 2001b, 2002) and James Arnt Aune (1994, 2001). These essays usefully point up what is at stake in the debate between poststructuralist theory and Marxism: whether an instrumental, class-based, socialist critical and political project is still feasible and necessary in today's world. Rejecting both Aune's Marxist-humanist approach and Cloud's more economic vision of agency, Greene claims that the changing character of global capitalism and recognition of the "constitutive power" of discourse render rhetorical criticism and practical politics on a Marxist basis untenable. Although we appreciate Greene's engagement with Marxism, we contend that the arguments sketched out in his essays are theoretically and politically flawed.First, Greene's criticisms are directed against a crude caricature of Marxist epistemology that obscures the complexity of theories of representation and ideology in the historical materialist tradition. Second, underwriting Greene's entire project are confused, and in some cases demonstrably false, assessments of the nature and historical trajectory of world capitalism and of the current prospects for class-based social movements. Finally, Greene endorses something like a democratic project but offers no normative criteria for such a project beyond the immanence of always-already present communicative agency and the excesses of joy afforded (to some) by the capitalist system. We will examine each of these shortcomings in turn. [End Page 72] Real Lies: The Complexity of Marxist Epistemology Greene and other poststructuralists charge Marxist theorists in the field of rhetorical studies with holding a dubious representational theory of truth.1 In contrast, Greene embraces a critical approach that examines "how rhetorical practices create the conditions of possibility for a governing apparatus to judge and program reality" and that views "what is in the true" as rhetorically produced (1998, 22). At odds with Greene's approach, Marxists claim that, in a range of ways, ideological discourses reflect, distort, and/or negotiate with the objective social realities, interests, and lived experiences of contending classes in capitalist society. Moreover, we believe that—by means of interpretive, historical, and empirical research—Marxist scholars and activists may develop fuller and more accurate theoretical "representations" of such realities and interests and of the various ways they shape our lives and our political struggles.2This is not to say, however, that the process is simple or mechanistic. Marxist critical practice retains the evaluative power of realism without laying claim to absolute or unmediated empirical knowledge; moreover, we do not believe that ideas are mechanistically generated epiphenomena of economic motive. Discourses, as any communication scholar would agree and as Greene points out, do have effects or "effectivities." Yet anyone interested in a normative political project (which Greene wishes to escape, a point to which we shall return) must recognize the possibility of falsehood. However, as Raymond Geuss (1981) argues, falsehood is not only a matter of empirical misrepresentation. In addition to epistemic falsehood, ideological discourses may exhibit both functional falsehood (referring to how even plausible ideas can bolster oppressive power) and genetic falsehood (ideas promulgated by those with ulterior motives, or intentional mystification).Similarly, analytic Marxist Jon Elster (1985) offers a view of ideology that regards discourses as more than simple epiphenomena of economic situations and struggles.3 Elster describes three overlapping forms of ideological production: the interest-based dissemination of distorted ideas, the failure of social imagination associated with widespread support for the capitalist system and U.S. imperialism, and the psychological comfort of palliative discourses such as religion and patriotism.4 Understanding how widespread adherence to common-sense ideas is accomplished is a project central to... (shrink)

An extended analysis compared to observations shows that modern “globalised” world civilisation has passed through the invisible “complexity threshold”, after which usual “spontaneous”, empirically driven kind of development (“invisible hand” etc.) cannot continue any more without major destructive tendencies. A much deeper, non-simplified understanding of real interaction complexity is necessary in order to cope with such globalised world development problems. Here we introduce the universal definition, fundamental origin, and dynamic equations for a major related quantity of (systemic) (...) risk characterising real complex system development tendencies at any level of dynamics. Practically important conclusions are derived, opening further detailed applications in economy, finance and development practice. (shrink)