Lev Lamberov
Address: 620002, Russia, Yekaterinburg, 51 Lenina av.
Ural Federal University, Philosophy department, Ontology and Theory of Knowledge chair (room 332)
Ural Federal University, Philosophy department, Ontology and Theory of Knowledge chair (room 332)
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Papers by Lev Lamberov
[Standard first-order modal logic is a convenient family of formalisms for studying modalities and their relationships to quantification and predication. The standard semantics for such logics is the semantics of possible worlds. However, as regards predication, it is formulated in such a way that it does not allow an adequate analysis of expressions containing comparisons of objects associated with different possible worlds. However, in some cases, it is necessary to refer to the comparison of objects associated with different possible worlds. Cross-world predication (as opposed to intra-world predication) seems to have a number of obvious ontological advantages. In a natural language, statements with cross-world predication are correct and can be intuitively understood by native speakers of a particular language. Such statements involve comparing what is with what it could be. In addition, we are able to compare objects from different possible worlds with each other, even if there are no such objects in the actual world. However, cross-world predication is not limited to comparisons in general and comparisons of two objects in particular. Comparisons of three or more objects from different possible worlds are allowed, and statements are allowed that say that an arbitrary set of objects from different possible worlds satisfies some criteria. The present article considers the features of CPL. It discusses various standard and non-standard formulas used as axioms of quantification in the formulation of the axiomatic calculus of first-order logic and for standard constant-domain modal logics, as well as formulas expressing the semantic features of quantifiers and their relationships with modalities. The second part of the article discusses the features of CPL in the case of intraworld predication. The third part touches upon the features of cross-world predication. The fourth part gives a brief conclusion. In contrast to standard modal logic, the truth (respectively, validity) of formulas in CPL is determined by a larger number of factors, which primarily include VP-functions used to associate variables with possible worlds. CPL uses a variable domain, so it is important to pay attention to how (with respect to which possible world) values are assigned to one or another formula variable. The latter is affected by VP-functions: if a VP-function is not defined for some variable, then this variable is associated with the current world of evaluation using a “grounded” VP-function. In addition, CPL inherits a number of features that some modal logics without cross-world predication have: variable domain and lambda operators. Also, in CPL, the interpretation of predicate symbols, variables, and individual constants is given in the domain of the model, while quantified variables are assigned values from the domain of the world of evaluation (also associated by VP-functions with the world of evaluation). All these features of CPL must be taken into account when formulating calculi, for example, natural, sequential, and axiomatic.]
[One of the most influential positions in contemporary philosophy of mathematics is mathematical structuralism. The number of different variants of this position is high, and several approaches have been proposed to typologise them. However, none of them is exhaustive, and they do not cover all the variety of variants of mathematical structuralism proposed so far. The paper briefly describes the most representative conceptions of mathematical structuralism, and analyzes the available variants of their typology. It is shown that distinguishing between eliminative and non-eliminative mathematical structuralism, as done by C. Parsons, does not take into account the typical attitude of mathematicians (as opposed to philosophers) to structuralism. This insufficiency is overcome in several approaches highlighting methodological (or non-ontological) varieties of mathematical structuralism. A more detailed typology of mathematical structuralism proposed by E. Reck and M. Price and contains C. Parsons typology as its integral part. The paper shows that the structuralism of patterns (non-eliminative structuralism), which is reduced to the ante rem position of S. Shapiro, does not take into account the post rem structuralism. In a nutshell, the paper proposes an extension of E. Reck and M. Price’s typology which takes into account post rem structuralism.]
[One of the main requirements for mathematical proofs is that of rigor. Rigor guarantees that there is a logical connection between the proposition being proved and the assumptions or axioms being accepted. Traditionally, the rigor of a mathematical proof is thought to be ensured by appealing to formalization. However, this understanding does not correspond to mathematical practice. The vast majority of proofs (both in the past and at present) are given informally and accepted by the community of expert mathematicians without any formalization. The article presents an alternative position regarding the rigor of mathematical proof proposed by S. De Toffoli, which is supposed to be consistent with practice. The article is divided into four parts. The first part considers the traditional understanding of proof as a surveyable and formalizable construct used for persuasion; it discusses the structure of proof and the standard understanding of rigor provided by formalization. The second part of the article discusses the practice of referring to “manipulative imagination” with respect to topology and shows the special status of this ability in “filling in” gaps in informal proofs. It is argued that this ability is developed through an appropriate training and involves only such manipulations that are justified from a strictly formal point of view. The third part discusses an alternative to the traditional notion of proof as a shareable, transferable and formalizable construct, components (“steps”) of which can be enumerated. Also, it introduces the notion of simil-proof which is a means of explaining the mathematical practice of accepting incomplete or error-containing proofs. In the fourth part of the article, the final points are formulated concerning the commitments of accepting relativism and anti-realist ontology when explaining the notions of proof and rigor.]
[The paper is devoted to the problem of knowability and sources of mathematical knowledge from the point of view of eliminative structuralism. A similar difficulty can be formulated for in re structuralism. The paper discusses Benacerraf's dilemma, his criticism of set-theoretic reductionist realism, and raises the problem of their compatibility with the general nominalist orientation of elimitative structuralism.]
[The mathematical structuralism of M. Resnik is a possible solution to the problems posed by P. Benacerraf for any adequate philosophy of mathematics. In the paper, the epistemological part of the concept is briefly outlined, and it is shown how, according to M. Resnik, we can obtain the knowledge of mathematical objects by disregarding the perceptual data. The ontological part, according to which consistency is taken as a criterion of existence, is also considered. To understand positions, M. Resnik uses the metaphor of a geometrical point. Thus, positions cannot be compared with one another in case they belong to different structures just as points cannot be individuated in case they do not belong to the same plane. Mathematical structures can be in relations of congruence, occurrence, and definitional equivalence, but there is no identity relation for them since the positions of the structures do not necessarily coincide. In addition, the paper compares M. Resnik’s conception of structural relativity with W.V.O. Quine’s ontological relativity. From the conception of structural relativity naturally follows what can be called the doctrine of three types of absent facts. Each type of absent facts is explained separately, then it is demonstrated that some interpretations of P. Benacerraf’s identification problem are incorrect.]
[The paper serves as an introduction to the issues discussed in the following articles. It raises the problem (challenge) of integration, according to which an adequate solution of a philosophical problem should simultaneously be an answer to both ontological and epistemological questions. This problem is described speculatively and by referring to P. Benacerraf’s dilemma. In addition, the problem is illustrated by comparing classical and intuitionistic mathematics and also through interpretation of the concept of computer proof. The paper demonstrates that adequate philosophy of mathematics must simultaneously take into account the ontological and epistemological aspects of mathematics and mathematical practice.]
[Mathematical structuralism is characterized by the acceptance of the thesis that the subject matter of mathematics is structures. These structures can be understood in various ways. Structures, whatever they are, are “implemented” by specific collections of objects. One of the ways to classify different variants of mathematical structuralism relates to the possibility of constructing a theory of structures and how such a theory should be constructed. In other words, this method concerns whether it is possible to describe any mathematical structures using some unified mathematical theory, and what kind of mathematical theory it is. From the point of view of the set-theoretic version of mathematical structuralism, mathematics is the science of structures, which can be described using the apparatus of set theory. Specific mathematical theories “grasp” various kinds of set-theoretic structures, and the relation of realization that takes place between these structures and mathematical theories is supposed to be described by model-theoretic means. This version of structuralism presupposes a non-structural understanding of set theory, and the axioms of structure theory have a fixed interpretation. This version of set theory assumes that the entire universe of sets has already been built, that all stages of the cumulative hierarchy have already been realized, which conflicts with the very idea that every stage in the cumulative hierarchy is followed by the next stage. Another consequence of understanding the axioms of set theory as truths with respect to a fixed structure, as well as the fact that set theory as a basis receives a special status and is not interpreted in a structural way, is that such an approach does not imply any kind of set-theoretic pluralism. That is, only one set-theoretic structure is considered the only correct description of the universe of sets, while any alternative options proceeding from other axioms are discarded as false. In order to get rid of the conflict between the fact that the axioms of the theory of structures assume the universe of sets is already built, and all stages of the cumulative hierarchy of sets are already realized, and the principle of indefinite extendability, one can turn to modalities. Due to the fact that the axioms of set theory are understood in a modal way, it is possible to explain in a potentialist way the totality of all sets without the assumption that there is some universe of sets fixed once and for all. Accordingly, this is consistent with the principle of indefinite extendability. Set-theoretic pluralism still turns out to be significantly limited. That is, modal set-theoretic structuralism still does not provide a structuralist explanation for set theory.]
[The paper is devoted to the personality and philosophical views of Eino Kaila (1890–1958). The paper contains a brief biography of the Finnish philosopher, describes his relationship with representatives of the Vienna Circle, gives an overview of his philosophical position. The ideas of E. Kaila are closely related to the ideas of the proponents of neopositivism. However, he resolutely rejects the phenomenalism and criticism of realism proposed by M. Schlick and R. Carnap, and also critically evaluates the principle of verification, the possibility of translating any given single sentence of a scientific theory into the language of experience and the possibility of final verification or falsification of scientific theories. Thus, E. Kaila in a sense anticipates the holistic point of view in the spirit of W.V.O. Quine and a number of I. Lakatos ideas. One of the main ideas of the thinker consists in the adoption of structural realism, according to which structures exist before objects, appear in regularities and invariants, and are grasped by researchers by means of abstraction and mathematization. The professional activity of E. Kaila played a significant role in defining the philosophical atmosphere in the Scandinavian countries and served there as the beginning of the development of analytic philosophy, which later became the dominant philosophical tradition.]
эпистемологические проблемы, связанные со слишком большой длиной
доказательства важных математических результатов, а также с большим и
постоянно возрастающим количеством публикаций по математике.
Предполагается, что эти затруднения могут быть разрешены (хотя бы
частично) путем обращения к компьютерным доказательствам. Однако и
компьютерные доказательства оказываются проблематичными с
эпистемологической точки зрения. И относительно доказательств в
обычной (неформальной) математике, и относительно компьютерных
доказательств в равной степени актуальна проблема их обозримости.
Исходя из традиционного понимания доказательства оно обязательно
должно быть обозримым, иначе оно не будет достигать своей основной
цели - формирования убежденности в правильности доказываемого
математического результата. Около 15 лет назад начал развиваться новый
подход к основаниям математики, сочетающий в себе конструктивистские,
структуралистские черты и ряд преимуществ классического подхода к
математике. Этот подход выстраивается на основе гомотопической теории
типов и носит название унивалентных оснований математики. Благодаря
мощному понятию равенства этот подход позволяет значительно сократить
длину формализованных доказательств, что намечает путь к разрешению
возникших эпистемологических затруднений.
[In recent decades, some epistemological issues have become especially
acute in mathematics. These issues are associated with long proofs of
various important mathematical results, as well as with a large and
constantly increasing number of publications in mathematics. It is
assumed that (at least partially) these difficulties can be resolved
by referring to computer proofs. However, computer proofs also turn
out to be problematic from an epistemological point of view. With
regard to both proofs in ordinary (informal) mathematics and computer
proofs, the problem of their surveyability appears to be fundamental.
Based on the traditional concept of proof, it must be surveyable,
otherwise it will not achieve its main goal, the formation of
conviction in the correctness of the mathematical result being proved.
About 15 years ago, a new approach to the foundations of mathematics
began to develop, combining constructivist, structuralist features and
a number of advantages of the classical approach to mathematics. This
approach is built on the basis of homotopy type theory and is called
the univalent foundations of mathematics. Due to its powerful notion
of equality, this approach can significantly reduce the length of
formalized proofs, which outlines a way to resolve the epistemological
difficulties that have arisen.]
[Standard first-order modal logic is a convenient family of formalisms for studying modalities and their relationships to quantification and predication. The standard semantics for such logics is the semantics of possible worlds. However, as regards predication, it is formulated in such a way that it does not allow an adequate analysis of expressions containing comparisons of objects associated with different possible worlds. However, in some cases, it is necessary to refer to the comparison of objects associated with different possible worlds. Cross-world predication (as opposed to intra-world predication) seems to have a number of obvious ontological advantages. In a natural language, statements with cross-world predication are correct and can be intuitively understood by native speakers of a particular language. Such statements involve comparing what is with what it could be. In addition, we are able to compare objects from different possible worlds with each other, even if there are no such objects in the actual world. However, cross-world predication is not limited to comparisons in general and comparisons of two objects in particular. Comparisons of three or more objects from different possible worlds are allowed, and statements are allowed that say that an arbitrary set of objects from different possible worlds satisfies some criteria. The present article considers the features of CPL. It discusses various standard and non-standard formulas used as axioms of quantification in the formulation of the axiomatic calculus of first-order logic and for standard constant-domain modal logics, as well as formulas expressing the semantic features of quantifiers and their relationships with modalities. The second part of the article discusses the features of CPL in the case of intraworld predication. The third part touches upon the features of cross-world predication. The fourth part gives a brief conclusion. In contrast to standard modal logic, the truth (respectively, validity) of formulas in CPL is determined by a larger number of factors, which primarily include VP-functions used to associate variables with possible worlds. CPL uses a variable domain, so it is important to pay attention to how (with respect to which possible world) values are assigned to one or another formula variable. The latter is affected by VP-functions: if a VP-function is not defined for some variable, then this variable is associated with the current world of evaluation using a “grounded” VP-function. In addition, CPL inherits a number of features that some modal logics without cross-world predication have: variable domain and lambda operators. Also, in CPL, the interpretation of predicate symbols, variables, and individual constants is given in the domain of the model, while quantified variables are assigned values from the domain of the world of evaluation (also associated by VP-functions with the world of evaluation). All these features of CPL must be taken into account when formulating calculi, for example, natural, sequential, and axiomatic.]
[One of the most influential positions in contemporary philosophy of mathematics is mathematical structuralism. The number of different variants of this position is high, and several approaches have been proposed to typologise them. However, none of them is exhaustive, and they do not cover all the variety of variants of mathematical structuralism proposed so far. The paper briefly describes the most representative conceptions of mathematical structuralism, and analyzes the available variants of their typology. It is shown that distinguishing between eliminative and non-eliminative mathematical structuralism, as done by C. Parsons, does not take into account the typical attitude of mathematicians (as opposed to philosophers) to structuralism. This insufficiency is overcome in several approaches highlighting methodological (or non-ontological) varieties of mathematical structuralism. A more detailed typology of mathematical structuralism proposed by E. Reck and M. Price and contains C. Parsons typology as its integral part. The paper shows that the structuralism of patterns (non-eliminative structuralism), which is reduced to the ante rem position of S. Shapiro, does not take into account the post rem structuralism. In a nutshell, the paper proposes an extension of E. Reck and M. Price’s typology which takes into account post rem structuralism.]
[One of the main requirements for mathematical proofs is that of rigor. Rigor guarantees that there is a logical connection between the proposition being proved and the assumptions or axioms being accepted. Traditionally, the rigor of a mathematical proof is thought to be ensured by appealing to formalization. However, this understanding does not correspond to mathematical practice. The vast majority of proofs (both in the past and at present) are given informally and accepted by the community of expert mathematicians without any formalization. The article presents an alternative position regarding the rigor of mathematical proof proposed by S. De Toffoli, which is supposed to be consistent with practice. The article is divided into four parts. The first part considers the traditional understanding of proof as a surveyable and formalizable construct used for persuasion; it discusses the structure of proof and the standard understanding of rigor provided by formalization. The second part of the article discusses the practice of referring to “manipulative imagination” with respect to topology and shows the special status of this ability in “filling in” gaps in informal proofs. It is argued that this ability is developed through an appropriate training and involves only such manipulations that are justified from a strictly formal point of view. The third part discusses an alternative to the traditional notion of proof as a shareable, transferable and formalizable construct, components (“steps”) of which can be enumerated. Also, it introduces the notion of simil-proof which is a means of explaining the mathematical practice of accepting incomplete or error-containing proofs. In the fourth part of the article, the final points are formulated concerning the commitments of accepting relativism and anti-realist ontology when explaining the notions of proof and rigor.]
[The paper is devoted to the problem of knowability and sources of mathematical knowledge from the point of view of eliminative structuralism. A similar difficulty can be formulated for in re structuralism. The paper discusses Benacerraf's dilemma, his criticism of set-theoretic reductionist realism, and raises the problem of their compatibility with the general nominalist orientation of elimitative structuralism.]
[The mathematical structuralism of M. Resnik is a possible solution to the problems posed by P. Benacerraf for any adequate philosophy of mathematics. In the paper, the epistemological part of the concept is briefly outlined, and it is shown how, according to M. Resnik, we can obtain the knowledge of mathematical objects by disregarding the perceptual data. The ontological part, according to which consistency is taken as a criterion of existence, is also considered. To understand positions, M. Resnik uses the metaphor of a geometrical point. Thus, positions cannot be compared with one another in case they belong to different structures just as points cannot be individuated in case they do not belong to the same plane. Mathematical structures can be in relations of congruence, occurrence, and definitional equivalence, but there is no identity relation for them since the positions of the structures do not necessarily coincide. In addition, the paper compares M. Resnik’s conception of structural relativity with W.V.O. Quine’s ontological relativity. From the conception of structural relativity naturally follows what can be called the doctrine of three types of absent facts. Each type of absent facts is explained separately, then it is demonstrated that some interpretations of P. Benacerraf’s identification problem are incorrect.]
[The paper serves as an introduction to the issues discussed in the following articles. It raises the problem (challenge) of integration, according to which an adequate solution of a philosophical problem should simultaneously be an answer to both ontological and epistemological questions. This problem is described speculatively and by referring to P. Benacerraf’s dilemma. In addition, the problem is illustrated by comparing classical and intuitionistic mathematics and also through interpretation of the concept of computer proof. The paper demonstrates that adequate philosophy of mathematics must simultaneously take into account the ontological and epistemological aspects of mathematics and mathematical practice.]
[Mathematical structuralism is characterized by the acceptance of the thesis that the subject matter of mathematics is structures. These structures can be understood in various ways. Structures, whatever they are, are “implemented” by specific collections of objects. One of the ways to classify different variants of mathematical structuralism relates to the possibility of constructing a theory of structures and how such a theory should be constructed. In other words, this method concerns whether it is possible to describe any mathematical structures using some unified mathematical theory, and what kind of mathematical theory it is. From the point of view of the set-theoretic version of mathematical structuralism, mathematics is the science of structures, which can be described using the apparatus of set theory. Specific mathematical theories “grasp” various kinds of set-theoretic structures, and the relation of realization that takes place between these structures and mathematical theories is supposed to be described by model-theoretic means. This version of structuralism presupposes a non-structural understanding of set theory, and the axioms of structure theory have a fixed interpretation. This version of set theory assumes that the entire universe of sets has already been built, that all stages of the cumulative hierarchy have already been realized, which conflicts with the very idea that every stage in the cumulative hierarchy is followed by the next stage. Another consequence of understanding the axioms of set theory as truths with respect to a fixed structure, as well as the fact that set theory as a basis receives a special status and is not interpreted in a structural way, is that such an approach does not imply any kind of set-theoretic pluralism. That is, only one set-theoretic structure is considered the only correct description of the universe of sets, while any alternative options proceeding from other axioms are discarded as false. In order to get rid of the conflict between the fact that the axioms of the theory of structures assume the universe of sets is already built, and all stages of the cumulative hierarchy of sets are already realized, and the principle of indefinite extendability, one can turn to modalities. Due to the fact that the axioms of set theory are understood in a modal way, it is possible to explain in a potentialist way the totality of all sets without the assumption that there is some universe of sets fixed once and for all. Accordingly, this is consistent with the principle of indefinite extendability. Set-theoretic pluralism still turns out to be significantly limited. That is, modal set-theoretic structuralism still does not provide a structuralist explanation for set theory.]
[The paper is devoted to the personality and philosophical views of Eino Kaila (1890–1958). The paper contains a brief biography of the Finnish philosopher, describes his relationship with representatives of the Vienna Circle, gives an overview of his philosophical position. The ideas of E. Kaila are closely related to the ideas of the proponents of neopositivism. However, he resolutely rejects the phenomenalism and criticism of realism proposed by M. Schlick and R. Carnap, and also critically evaluates the principle of verification, the possibility of translating any given single sentence of a scientific theory into the language of experience and the possibility of final verification or falsification of scientific theories. Thus, E. Kaila in a sense anticipates the holistic point of view in the spirit of W.V.O. Quine and a number of I. Lakatos ideas. One of the main ideas of the thinker consists in the adoption of structural realism, according to which structures exist before objects, appear in regularities and invariants, and are grasped by researchers by means of abstraction and mathematization. The professional activity of E. Kaila played a significant role in defining the philosophical atmosphere in the Scandinavian countries and served there as the beginning of the development of analytic philosophy, which later became the dominant philosophical tradition.]
эпистемологические проблемы, связанные со слишком большой длиной
доказательства важных математических результатов, а также с большим и
постоянно возрастающим количеством публикаций по математике.
Предполагается, что эти затруднения могут быть разрешены (хотя бы
частично) путем обращения к компьютерным доказательствам. Однако и
компьютерные доказательства оказываются проблематичными с
эпистемологической точки зрения. И относительно доказательств в
обычной (неформальной) математике, и относительно компьютерных
доказательств в равной степени актуальна проблема их обозримости.
Исходя из традиционного понимания доказательства оно обязательно
должно быть обозримым, иначе оно не будет достигать своей основной
цели - формирования убежденности в правильности доказываемого
математического результата. Около 15 лет назад начал развиваться новый
подход к основаниям математики, сочетающий в себе конструктивистские,
структуралистские черты и ряд преимуществ классического подхода к
математике. Этот подход выстраивается на основе гомотопической теории
типов и носит название унивалентных оснований математики. Благодаря
мощному понятию равенства этот подход позволяет значительно сократить
длину формализованных доказательств, что намечает путь к разрешению
возникших эпистемологических затруднений.
[In recent decades, some epistemological issues have become especially
acute in mathematics. These issues are associated with long proofs of
various important mathematical results, as well as with a large and
constantly increasing number of publications in mathematics. It is
assumed that (at least partially) these difficulties can be resolved
by referring to computer proofs. However, computer proofs also turn
out to be problematic from an epistemological point of view. With
regard to both proofs in ordinary (informal) mathematics and computer
proofs, the problem of their surveyability appears to be fundamental.
Based on the traditional concept of proof, it must be surveyable,
otherwise it will not achieve its main goal, the formation of
conviction in the correctness of the mathematical result being proved.
About 15 years ago, a new approach to the foundations of mathematics
began to develop, combining constructivist, structuralist features and
a number of advantages of the classical approach to mathematics. This
approach is built on the basis of homotopy type theory and is called
the univalent foundations of mathematics. Due to its powerful notion
of equality, this approach can significantly reduce the length of
formalized proofs, which outlines a way to resolve the epistemological
difficulties that have arisen.]